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An Empirical Analysis of the Signaling and Screening Models of Litigation

An Empirical Analysis of the Signaling and Screening Models of Litigation Abstract We present an experimental analysis of the signaling and screening models of litigation. In both models, bargaining failure is driven by asymmetric information. The difference between the models lies in the bargaining structure: In the signaling game, the informed party makes the final offer, while in the screening game the uninformed party makes the final offer. We conduct experiments for both models under a common set of parameter values, allowing only the identity of the party making the final offer to change. We find the anomalous behavior to be more common in the signaling game, but the frequency of this behavior diminishes in the later rounds of the experiment. Across both games, in the later rounds of the experiment over 90% of offers are consistent with the theory. Having the right to make the offer raises a player’s expected payoffs, but by much less than is predicted by theory. Dispute rates across the two games are approximately equal. 1. Introduction Asymmetric information is a leading explanation for the existence of bargaining failures which result in costly litigation. We consider a stylized legal bargaining framework in which an informed “plaintiff” knows whether she has either a strong or weak case against an uninformed “defendant.” Within this class of games, when the informed party makes the final offer, it is called a signaling model, and when the uninformed party makes the final offer it is called the screening model. These informational-based models underlie much of the theoretical work on pretrial bargaining. We present signaling and screening experiments in a litigation context that allows for a side-by-side comparison of each model’s performance.1, Our ultimate goal is to gain a better understanding of the factors which lead to disputes, and understanding behavior in these canonical models of the litigation literature is an important step in this process. Under the parameterization we utilize, the theory predicts distributional effects of moving from a screening game to a signaling game and allows for possible efficiency effects. Typically, the identity of the individual making the offer is not viewed as a choice variable, but if this choice has consequences for efficiency, this identity might be specified in a contracting relationship in which disputes are anticipated. For example, this is observed in contracts specifying the rules for the dissolution of partnerships.2 However, the lack of such terms in contracting situations in which litigation is the dispute resolution mechanism suggests small efficiency and distributional effects associated with the identity of the party making the offer. Under the theory, the dispute rate in our signaling game can be anywhere from zero to eleven percentage points lower than the dispute rate in the screening game.3 We find that dispute rates across the two games are approximately equal. Thus, there are no significant efficiency effects associated with the right to make an offer. In addition, while having the right to make the offer is valuable, it is only about 30–55% as valuable as predicted by theory. Thus in our experiment, the distributional consequences of moving from a screening game to a signaling game are much less than what theory predicts. The signaling game appears to place high cognitive demands upon experimental subjects, and the comparison to the screening game gives us insight about the extent to which this is true. We observe more anomalous offers in the signaling game than in the screening game, but their frequency decreases substantially over time in the signaling experiment. After the sixth round, in both games over 90% of the offers are consistent with the theory. On the other hand, anomalous acceptance behavior, which is almost entirely absent in the screening game, persists in the later rounds of the signaling experiment. As our experimental protocol in the two games is identical except for the identity of the party making the offer, we attribute the differences in anomalous behavior to the greater cognitive demands imposed by the signaling game. In our conclusion, we discuss some implications for naturally occurring litigation. 2. Background There is a large literature in law and economics where trials are an equilibrium outcome of games in which there is asymmetric information. In the screening model of Bebchuk (1984), the uninformed party makes an offer to the informed party. In the signaling model of Reinganum Wilde (1986), the informed party makes an offer to the uninformed party. Much of the subsequent literature on pretrial bargaining is built upon these models. Among other things, this literature analyzes how institutions such as fee shifting and contingency fees affect settlement rates.4 Our experiment is closely based on the Bebchuk and Reinganum and Wilde models. There is an extensive experimental law and economics literature.5 With a few exceptions, this work does not analyze the standard models of pretrial bargaining in which disputes result from asymmetric information. The extensive literature on arbitration has focused on how dispute rates are affected by the choice of arbitration procedure, but this is generally done in a setting of symmetric information.6 A variety of issues have been examined experimentally in the civil litigation literature. These include conditional cost shifting under which the shifting of certain trial costs is a function of either the outcome of trial or of the relationship between the trial outcome and offers made in the pretrial negotiation period. (See, e.g., Coursey and Stanley, 1988 and Main and Park, 2002.) Again, in those experiments, the players are symmetrically informed. In work testing of the Priest and Klein’s (1984) selection hypothesis there is two-sided asymmetric information.7 That work does not constitute a direct evaluation of either the screening or the signaling models that we analyze here. Some of our previous work has used the screening model of litigation as a baseline in the context of analyzing other issues. Pecorino and Van Boening (2004, 2015) analyze voluntary disclosure and Pecorino and Van Boening (2014) consider the effects of asymmetric dispute costs on settlement behavior. Here, we also present the screening game as a baseline. The signaling game serves as the treatment, that is, the switch from having the uninformed party make the offer to having the informed party make the offer. To our knowledge, there has been no systematic empirical comparison of these two games in the litigation context. In standard models of litigation (e.g., Reinganum Wilde (1986) and Bebchuk (1984)), there is an ultimatum game embedded in the larger game of pretrial bargaining between the plaintiff and defendant. The “pie” is the joint saving which is achieved when a settlement is reached and the costs of trial avoided. Because fairness concerns are absent in these models, the recipient is willing to accept an offer which equals her expected payoff at trial. However, from the large experimental literature on the ultimatum game we know behaviorally that the recipient of an offer generally demands some share of the joint surplus and that if too little surplus is proposed, then the offer will be rejected.8 Thus, prior experimental work leads to the expectation that more surplus than predicted under the standard theory will be provided in settlement offers and that there will be disputes not predicted by the theory when the parties cannot agree on what constitutes a fair offer. Both of these behavioral elements are present in our previous work (Pecorino and Van Boening, 2004, 2014, and 2015) involving the screening game. There has been a fairly extensive analysis of signaling games in the experimental economics literature. Some of this work has focused on the performance of certain equilibrium refinement concepts. For example, Brandts and Holt (1992) find mixed support for the refinement concept known as the intuitive criterion, which places restrictions on out of equilibrium beliefs.9Banks et al. (1994) test a range of refinement concepts and find mixed evidence that these refinements are predictive of behavior. Cooper et al. (1997) provide evidence against the empirical validity of the intuitive criterion. None of this previous literature has been in a litigation setting.10 In the civil litigation literature, the refinement D1 (Cho and Kreps, 1987) is required to eliminate all but the pure strategy separating equilibrium which has typically been the focus of this theoretical work. The D1 refinement is stronger than the intuitive criterion meaning it is less likely to hold empirically. Our data reject D1, but given the results cited above, this is unsurprising. Accordingly, our analysis does not focus on the validity of D1. We focus instead on a semi-pooling equilibrium (described below), as our results are more consistent with that prediction. We comment on the failure of the D1 refinement concept as appropriate. 3. The Theory The screening model we describe is a simplified version of Bebchuk (1984), while the signaling model is a simplified version of Reinganum Wilde (1986).11 Both the plaintiff and the defendant are risk neutral. In all of our analyses, the probability that the plaintiff prevails at trial is common knowledge, and we furthermore assume that this probability equals 1. In presenting the theory, we will use the parameter values from our experiment to generate the relevant predictions and we will use the same terminology that we later use in the Section 5. Thus, we will refer to the informed plaintiff as player $$A$$ and the uniformed defendant as player $$B$$. At trial, income is transferred from $$B$$ to $$A$$ which is what distinguishes the player roles. The plaintiff is either type $$A_{H}$$ with a strong case or type $$A_{L}$$ with a weak case. If the case proceeds to trial, player $$A$$ receives judgment $$J^{ i}$$, $$i = H, L$$ with $$J^{ H}$$$$>$$$$J^{ L}$$. Player $$A$$ knows her type, but $$B$$ only knows that with probability $$q$$ he faces a type $$A_{H}$$ plaintiff and with probability $$1 -q$$ he faces a type $$A_{L}$$ plaintiff. The court costs for $$A$$ and $$B$$ are, respectively, $$C_{A}$$ and $$C_{B}$$. These costs are incurred only if the case proceeds to trial. Using this simple environment, we first present the screening model and then the signaling model. 3.1. The Screening Game The stages of the screening game are as follows: 0. Nature determines player $$A$$’s type which is $$A_{H}$$ with probability $$q$$ and $$A_{L}$$ with probability 1 – $$q$$. Player $$A$$ knows her type, but $$B$$ knows only the probability $$q$$ that she is $$A_{H}$$. 1. Player $$B$$ makes an offer $$O_{B}$$ to player $$A$$. 2. Player $$A$$ accepts or rejects the offer. If the offer is accepted, the game ends with the player $$A$$ receiving a payoff of $$O_{B}$$ and player $$B$$ receiving a payoff of –$$O_{B}$$ (i.e., $$B$$ incurs a cost equal to $$O_{B})$$. 3. If the offer is rejected, trial occurs. Player $$A$$ receives the payoff $$J^{i}$$ – $$C_{A}$$ and player $$B$$ receives the payoff –($$J^{i} + C_{B})$$, where $$i = H, L$$. Note that our game description implies the use of the American rule under which both parties bear their own costs of trial. In our experiment we use the following parameter values: $$J^{L} = {\$}1.50$$, $$J^{H} = {\$}4.50$$, $$q = 1/3$$, and $$C_{A} = C_{B} = {\$}0.75$$. In the experiment we present all values in pennies so that, for example, $${\$}$$1.50 is written as 150. We follow that convention in what follows. The plaintiff $$A$$ will accept any offer that leaves her at least as well off as the expected outcome at trial. In other words, $$A_{i}$$ will accept any offer such that $$O_{B} \geqslant J^{i} - C_{A}$$. The defendant $$B$$ is free to make any offer he chooses, but the optimal offer will be one of the following:   \begin{align} O^L_B & = J^{L} - C_{A} = 75,\\ \end{align} (1a)  \begin{align} O^H_B & = J^{H} - C_{A} = 375. \end{align} (1b) Player $$B$$ will choose either the low screening offer $$O_B^L$$ that only $$A_{L}$$ will accept or the high pooling offer $$O_B^H$$ that both plaintiff types will accept. He offers $$O_B^L$$ if   \begin{align} (1 - q)O^L_B + q(J^{H} + C_{B}) < O^H_B. \end{align} (2) The left-hand side represents the expected payout from the offer $$O^L_B$$ which is accepted with probability 1 – $$q$$ and rejected with probability $$q $$ (the probability that $$A_{H}$$ is encountered). When the offer is rejected, $$B$$ proceeds to trial and pays $$J^{H} + C_{B}$$. The right-hand side is the $$B$$’s payout from the higher offer, which is accepted by both player $$A$$ types. Screening offers are made in this model when the probability $$q$$ of encountering a high damage plaintiff is sufficiently small. Rearranging Equation (2), while making use of (1), the condition for $$B$$ to make the low offer may be expressed as follows:   \begin{align} 1/3 = q < \frac{J^H - J^L}{(J^H - J^L)+ C_A + C_B} = 0.67. \end{align} (3) Thus, under our parameter values, the condition is met and we have the predictions that a low screening offer $$O^L_B$$ will be made and that trial occurs with probability $$q = 1/3$$. The predicted dispute rates in our screening game are 0% for $$A_{L}$$, 100% for $$A_{H}$$, and 33% overall. The predictions of this game are summarized in Table 1, which is found in Section 3.3. Table 1. Predictions by Game Type Game  Sender: offer  Dispute rate  $$A$$’s payoff  Recipient decision  Screening  $$B{:}\ O_{B}^{L} =75$$  $$A_{L}$$: 0%  $$A_{L}$$: 75  $$A_{L}$$ accepts $$O_{B} \geqslant $$ 75.        $$A_{H}$$: 100%  $$A_{H}$$: 375  $$A_{H}$$ accepts $$O_{B} \geqslant $$ 375.        $$A$$: 33%  $$A$$: 175$$^{\mathrm{a}}$$     Signaling$$^{\mathrm{b}}$$  $$A_{L}{:}\ O_{A}^{L} =225,$$  $$A_{L}$$: $$\underline {0}$$–25%  $$A_{L}$$: $$\underline {225}$$  $$B$$ accepts $$O_{A} \leqslant $$ 225, and     $$\phantom{A_L{:}}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A_{H}$$: 50–$$\underline {67}$$%  $$A_H$$: 375–$$\underline {425}$$$$^{\mathrm{c}}$$  $$\quad$$ rejects $$O_{A}^{S} \in [375-\underline{525}]$$     $$A_{H}{:}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A$$: $$\underline {22}$$–33%  $$A$$: 275–$$\underline {292}$$$$^{\mathrm{a}}$$  $$\quad$$ at rate $$r$$.$$^{\mathrm{d}}$$  Game  Sender: offer  Dispute rate  $$A$$’s payoff  Recipient decision  Screening  $$B{:}\ O_{B}^{L} =75$$  $$A_{L}$$: 0%  $$A_{L}$$: 75  $$A_{L}$$ accepts $$O_{B} \geqslant $$ 75.        $$A_{H}$$: 100%  $$A_{H}$$: 375  $$A_{H}$$ accepts $$O_{B} \geqslant $$ 375.        $$A$$: 33%  $$A$$: 175$$^{\mathrm{a}}$$     Signaling$$^{\mathrm{b}}$$  $$A_{L}{:}\ O_{A}^{L} =225,$$  $$A_{L}$$: $$\underline {0}$$–25%  $$A_{L}$$: $$\underline {225}$$  $$B$$ accepts $$O_{A} \leqslant $$ 225, and     $$\phantom{A_L{:}}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A_{H}$$: 50–$$\underline {67}$$%  $$A_H$$: 375–$$\underline {425}$$$$^{\mathrm{c}}$$  $$\quad$$ rejects $$O_{A}^{S} \in [375-\underline{525}]$$     $$A_{H}{:}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A$$: $$\underline {22}$$–33%  $$A$$: 275–$$\underline {292}$$$$^{\mathrm{a}}$$  $$\quad$$ at rate $$r$$.$$^{\mathrm{d}}$$  $$^{\mathrm{a}}$$Player $$A $$ expected payoff $$=$$ (1 – $$q)(A_{L}$$ payoff) $$+ q(A_{H}$$ payoff) with $$q =$$ 1/3. $$^{\mathrm{b}}$$Semi-pooling equilibrium in which $$A_{L}$$ and $$A_{H}$$ pool on a single value of $$O_{A}^{S}$$. Underlined values are D1 predictions (see text fn. 16). $$^{\mathrm{c}}$$Player $$A_{H} $$ expected payoff $$= r(375) + (1 - r) O_{A}^{S}$$. See text Equation (6) for $$r$$. $$^{\mathrm{d}}$$$$B$$ also rejects any 226 $$\leqslant O_{A} \leqslant $$ 374. Table 1. Predictions by Game Type Game  Sender: offer  Dispute rate  $$A$$’s payoff  Recipient decision  Screening  $$B{:}\ O_{B}^{L} =75$$  $$A_{L}$$: 0%  $$A_{L}$$: 75  $$A_{L}$$ accepts $$O_{B} \geqslant $$ 75.        $$A_{H}$$: 100%  $$A_{H}$$: 375  $$A_{H}$$ accepts $$O_{B} \geqslant $$ 375.        $$A$$: 33%  $$A$$: 175$$^{\mathrm{a}}$$     Signaling$$^{\mathrm{b}}$$  $$A_{L}{:}\ O_{A}^{L} =225,$$  $$A_{L}$$: $$\underline {0}$$–25%  $$A_{L}$$: $$\underline {225}$$  $$B$$ accepts $$O_{A} \leqslant $$ 225, and     $$\phantom{A_L{:}}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A_{H}$$: 50–$$\underline {67}$$%  $$A_H$$: 375–$$\underline {425}$$$$^{\mathrm{c}}$$  $$\quad$$ rejects $$O_{A}^{S} \in [375-\underline{525}]$$     $$A_{H}{:}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A$$: $$\underline {22}$$–33%  $$A$$: 275–$$\underline {292}$$$$^{\mathrm{a}}$$  $$\quad$$ at rate $$r$$.$$^{\mathrm{d}}$$  Game  Sender: offer  Dispute rate  $$A$$’s payoff  Recipient decision  Screening  $$B{:}\ O_{B}^{L} =75$$  $$A_{L}$$: 0%  $$A_{L}$$: 75  $$A_{L}$$ accepts $$O_{B} \geqslant $$ 75.        $$A_{H}$$: 100%  $$A_{H}$$: 375  $$A_{H}$$ accepts $$O_{B} \geqslant $$ 375.        $$A$$: 33%  $$A$$: 175$$^{\mathrm{a}}$$     Signaling$$^{\mathrm{b}}$$  $$A_{L}{:}\ O_{A}^{L} =225,$$  $$A_{L}$$: $$\underline {0}$$–25%  $$A_{L}$$: $$\underline {225}$$  $$B$$ accepts $$O_{A} \leqslant $$ 225, and     $$\phantom{A_L{:}}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A_{H}$$: 50–$$\underline {67}$$%  $$A_H$$: 375–$$\underline {425}$$$$^{\mathrm{c}}$$  $$\quad$$ rejects $$O_{A}^{S} \in [375-\underline{525}]$$     $$A_{H}{:}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A$$: $$\underline {22}$$–33%  $$A$$: 275–$$\underline {292}$$$$^{\mathrm{a}}$$  $$\quad$$ at rate $$r$$.$$^{\mathrm{d}}$$  $$^{\mathrm{a}}$$Player $$A $$ expected payoff $$=$$ (1 – $$q)(A_{L}$$ payoff) $$+ q(A_{H}$$ payoff) with $$q =$$ 1/3. $$^{\mathrm{b}}$$Semi-pooling equilibrium in which $$A_{L}$$ and $$A_{H}$$ pool on a single value of $$O_{A}^{S}$$. Underlined values are D1 predictions (see text fn. 16). $$^{\mathrm{c}}$$Player $$A_{H} $$ expected payoff $$= r(375) + (1 - r) O_{A}^{S}$$. See text Equation (6) for $$r$$. $$^{\mathrm{d}}$$$$B$$ also rejects any 226 $$\leqslant O_{A} \leqslant $$ 374. 3.2. The Signaling Game The stages of the signaling game are similar to those above with 1$$^\prime $$ and 2$$^\prime $$ replacing 1 and 2. 1$$'$$. Player $$A$$ makes an offer $$O_{A}$$ to player $$B$$. 2$$'$$. Player $$B$$ accepts or rejects the offer. If the offer is accepted, the game ends with player $$A$$ receiving payoff $$O_{A}$$ and player $$B$$ receiving payoff –$$O_{A}$$. The defining feature of the signaling game is that the informed player ($$A)$$ makes the offer to the uninformed player ($$B)$$. Multiple equilibria are a problem in signaling games. In this particular game, the refinement concept D1 has been used to eliminate all but a pure strategy separating equilibrium.12 However, because prior experiments strongly suggest that we should not expect to find support for D1 in our data, we will present predictions for semi-pooling equilibria instead.13 It is worth noting that the model predictions under D1 are endpoints of the predictions intervals for semi-pooling equilibria. Thus, the D1 predictions are a special case of the semi-pooling predictions. Before proceeding, it is useful to present the offers by each player $$A$$ type in a full information game in which $$B$$ knows $$A$$’s type. The offers are as follows:   \begin{align} O^L_A & = J^{L} + C_{B} = 225,\\ \end{align} (4a)  \begin{align} O^H_A & = J^{H} + C_{B} = 525. \end{align} (4b) These offers represent $$B$$’s dispute payoff against each type. This reflects the fact that the theory embeds an ultimatum game in which player $$A$$ has all of the bargaining power. The offer in (4a) will be the revealing offer by $$A_{L}$$ in what follows, while the offer in (4b) represents the upper bound on possible semi-pooling offers.14 In the semi-pooling equilibrium all $$A_{H}$$ players offer $$O_{A}^{S} $$, where $$J^{H} - C_{A} \leqslant O_A^S < O_{A}^{H}$$ (or $$375 \leqslant O^S_A < 525$$). If only $$A_{H}$$ made the offer $$O_{A}^{S} $$, player $$B$$ would accept it, but if player $$B$$ always accepts this, $$A_{L}$$ would also offer $$O_{A}^{S} $$. Thus, there is an equilibrium in mixed strategies in which $$A_{L}$$ sometimes bluffs by offering $$O_{A}^{S} $$ and in which player $$B$$ sometimes rejects $$O_{A}^{S}$$. A mixed strategy equilibrium requires the players to be indifferent between the strategies they mix between. Player $$A_{L}$$ uses a mixed strategy in which she makes the revealing offer $$O_{A}^{L}$$ with probability 1–b and bluffs by offering $$O_{A}^{S}$$ with probability $$b$$. The low offer $$O_{A}^{L}$$ is accepted with probability 1. If the higher offer $$O_{A}^{S}$$ is rejected with probability   \begin{align} r = \frac{O_{A}^{S} -J^L - C_B}{O_{A}^{S} - J^L + C_A} = \frac{O_{A}^{S} - 225}{O_{A}^{S} - 75}, \end{align} (5) then $$A_{L}$$ will be indifferent between the two offers. Upon observing $$O_{A}^{S} $$, player $$B$$ uses Bayes’ rule and the equilibrium probability that $$A_{L}$$ bluffs to update his beliefs about $$A$$’s type. When the probability $$A_{L}$$ bluffs is   \begin{align} b = \left(\frac{q}{1-q}\right) \frac{J^H + C_B + O_{A}^{S}}{O_{A}^{S} - J^L - C_B} = (0.5) \frac{525 - O^S_A}{O_{A}^{S} - 225}, \end{align} (6) then player $$B$$ is indifferent between accepting or rejecting $$O_{A}^{S}$$. The semi-pooling offer can be any offer such that $$375 \leqslant O_{A}^{S} \leqslant 525$$. In our experiment offers are restricted to be whole numbers. Thus, under our design there are 150 possible semi-pooling offers. However, in any particular semi-pooling equilibrium, all the $$A_{H}$$ players and all of the bluffing $$A_{L}$$ players will pool on one and only one value of $$O_{A}^{S}$$.15 Using Equation (5), the predicted dispute rate on semi-pooling offers ranges from $$r =$$ 50% if the offer is $$O_{A}^{S} = 375$$ up to $$r = 67\%$$ if the offer is $$O_{A}^{S} = 525$$. The expected payoff to $$A_{H}$$ in a semi-pooling equilibrium can range from 375 to 425. Using (5) and (6), the dispute rate for $$A_{L}$$ can range from 0% to 25% and the overall dispute rate can range from 22% to 33% in a semi-pooling equilibrium.16 The out-of-equilibrium beliefs and actions are as follows: It is a dominant strategy for $$B$$ to accept an offer $$O_{A} < O_{A}^{L} = 225$$ and this offer is accepted with probability 1. Likewise, it is a dominant strategy for $$B$$ to reject an offer $$O_{A} > O_{A}^{H} = 525$$ and this offer is rejected with probability 1. An offer $$> 226$$ which is not equal to $$O_{A}^{S}$$ is believed to be from $$A_{L}$$ and is rejected with probability 1. 3.3. Predictions Table 1 summarizes our predictions. We focus on three sets of predictions and their comparative statics across the two games. The first involves offers. The predicted difference between player $$B$$ screening offers $$O_{B}^{L}$$ and $$A_{L}$$ revealing offers $$O_{A}^{L}$$ is 150 (75 v. 225). In the signaling game $$A_{L}$$ bluffs and $$A_{H}$$ separating offers are predicted to coincide on a unique semi-pooling offer $$O_{A}^{S} $$. The second involves dispute rates and potential efficiency gains. For offers in the range 75–225, the $$A_{L}$$ v. $$B$$ dispute rate is unaffected by the game structure (both are 0%). In the signaling game, the overall $$A_{L}$$ dispute rate could be as high as 25% because of bluffs by $$A_{L}$$. The predicted $$A_{H}$$ v. $$B $$ dispute rate is 33–50 percentage points lower in the semi-pooling equilibrium compared with the screening game. The overall $$A$$ v. $$B $$ dispute rate could range from no difference (both 33%) to 11 percentage points lower in the signaling game (22% v. 33%). The third involves distributional effects on the player $$A $$ payoffs. Moving from the screening game to the signaling game, the $$A_{L}$$ payoff increases by 150, from 75 to 225. The $$A_{H}$$ expected payoff increases by anywhere from 0 to 50. In total, $$A$$’s expected payoff increases in a range from 100 to 117. These increases represent the value to $$A$$ of having the offer, that is, the expected increase in compensation the plaintiff receives when she has the right to make the settlement offer instead of the defendant. While the theoretical predictions in Table 1 provide benchmarks, these need to be viewed in light of the embedded ultimatum game discussed above in Section 2. Here, the embedded pie is the $$C_{A} + C_{B} =$$ 150 joint surplus from settlement which constitutes the bargained over amount (Pecorino and Van Boening, 2010), and we are skeptical that empirically the offering party will be able to extract all of the surplus. For example, in the screening game it may be the case that $$A_{L}$$ needs $$O_{B}$$$$>$$ 75 before she will accept the offer. When presenting the results, we label offers according to intervals consistent with the theoretical predictions, but allowing for a sharing of the joint surplus from settlement. In the screening game the point prediction for player $$B$$’s offer is $$O_{B} = 75$$, but any offer 75 $$<$$$$O_{B}$$$$<$$ 225 is consistent with the screening behavior, as it is theoretically acceptable to $$A_{L}$$ but not $$A_{H}$$, and leaves both $$A_{L}$$ and $$B$$ with nonnegative surplus from settlement. In the screening game we refer to: Screening offer: An offer by player $$B$$ in the interval 75 $$\leqslant O_{B} \leqslant $$ 225. Pooling offer: An offer by player $$B$$ in the interval 375 $$\leqslant O_{B} \leqslant $$ 525. Analogous reasoning applies in the signaling game. In the signaling game, we refer to: Revealing offer: An offer by player $$A_{L}$$ in the interval 75 $$\leqslant O_{A}$$$$\leqslant $$ 225. Bluffing offer: An offer by player $$A_{L}$$ in the interval 375 $$\leqslant O_{A}$$$$\leqslant $$ 525. Separating offer: An offer by player $$A_{H}$$ in the interval 375 $$\leqslant O_{A}$$$$\leqslant $$ 525. Collectively, we refer to subjects’ offers in these intervals as “consistent with theory.” We are also interested in the frequency of anomalous offers in each game, because ex ante we believe that the signaling game places greater cognitive demands on both players than does the screening game. In both games “between” offers are anomalous: Between offer: An offer by either player in the interval 226 $$\leqslant O \leqslant $$ 374. For player $$B$$ in the screening game this offer is too low to be accepted by $$A_{H}$$, but offers $$>$$ 100% of the surplus from settlement to $$A_{L}$$. In the signaling game, these offers should be rejected by $$B$$ at a 100% rate using the following reasoning: For $$A_{H}$$, an offer in the 226–374 range is dominated by an offer of 375 or more, since the latter equals or exceeds the $$A_{H}$$ dispute payoff. Thus, $$B$$ should place 100% weight on the offer being made by $$A_{L}$$ and reject it as it exceeds his dispute payout of 225 v. $$A_{L}$$. A failure to do so violates the “test of dominated strategies”, a refinement which is weaker (and therefore more likely to be empirically valid) than either the intuitive criterion or D1. (See Kreps, 1990, p. 436.) Given this expected response by $$B$$, $$A_{L}$$ should never make an offer in this range. Any offer $$<$$ 75 or $$>$$ 525 is also anomalous in both games, but we observe very few of these offers, especially in the later rounds of the experiment. In contrast, “between” offers are significantly more common. Note that our predictions are fairly robust to the introduction of risk aversion on the part of either party.17 In the screening game, if $$B$$ is very risk averse, it could induce him to make a pooling offer when a screening offer is otherwise predicted. In the experiment, we observe few (~ 5%) pooling offers. In the signaling game, risk aversion affects the precise mixing probabilities that make $$A_{L}$$ indifferent between a revealing offer and a bluff and $$B$$ indifferent between accepting or rejecting $$O_{A}^{S}$$. 4. Experimental Design Table 2 summarizes the ten sessions in our experimental design, five screening game sessions and five signaling game sessions. Subjects were recruited from summer business classes at the University of Alabama. The number of bargaining pairs per session ranges from 5 to 8, while each session lasted 12–14 rounds18 (see table notes for session-specific details). The player in the role of the plaintiff is referred to as player $$A $$ and the player in the role of the defendant is referred to as player $$B$$. Subjects were not informed ahead of time how many rounds there would be. A typical session, inclusive of an instructional period at the beginning and private payment at the end, lasted between one-and-a-half and two hours. Average payoffs were about $${\$}$$31, ranging from $${\$}$$17 to $${\$}$$47. Subjects were not paid a show-up fee; all earnings were from decision-making. Table 2. Experimental Design             No. of negotiations     Game  No. of sessions  Pairs  Rounds  $$n $$  $$B$$ v. $$ A_{L}$$  $$B $$ v. $$A_{H}$$  Average earnings(min, max)  Screening  5$$^{\mathrm{a}}$$  31  65  402  273  129  $${\$}$$31.89 ($${\$}$$21, $${\$}$$46)  Signaling  5$$^{\mathrm{b}}$$  34  60  408  276  132  $${\$}$$30.63 ($${\$}$$17, $${\$}$$47)              No. of negotiations     Game  No. of sessions  Pairs  Rounds  $$n $$  $$B$$ v. $$ A_{L}$$  $$B $$ v. $$A_{H}$$  Average earnings(min, max)  Screening  5$$^{\mathrm{a}}$$  31  65  402  273  129  $${\$}$$31.89 ($${\$}$$21, $${\$}$$46)  Signaling  5$$^{\mathrm{b}}$$  34  60  408  276  132  $${\$}$$30.63 ($${\$}$$17, $${\$}$$47)  $$^{\mathrm{a}}$$Session Scr1: 7 pairs, 12 rounds, $$n =$$ 84 negotiations (49 $$B$$ v. $$A_{L}$$, 35 $$B$$ v. $$A_{H})$$. Scr2: 5 pairs, 13 rounds, $$n =$$ 65 (47, 18). Scr3: 8 pairs, 13 rounds, $$n =$$ 104 (75, 29). Scr4: 5 pairs, 13 rounds, $$n =$$ 65 (41, 24). Scr5: 6 pairs, 14 rounds, $$n =$$ 84 (61, 23). $$^{\mathrm{b}}$$Session Sig1: 6 pairs, 12 rounds, $$n =$$ 72 negotiations (50 $$B$$ v. $$A_{L}$$, 22 $$B$$ v. $$A_{H})$$. Sig2: 8 pairs, 12 rounds, $$n =$$ 96 (61, 35). Sig3: 8 pairs, 12 rounds, $$n =$$ 96 (68, 28). Sig4: 5 pairs, 12 rounds, $$n =$$ 60 (42, 18). Sig5: 7 pairs, 12 rounds, $$n =$$ 84 (55, 29). Table 2. Experimental Design             No. of negotiations     Game  No. of sessions  Pairs  Rounds  $$n $$  $$B$$ v. $$ A_{L}$$  $$B $$ v. $$A_{H}$$  Average earnings(min, max)  Screening  5$$^{\mathrm{a}}$$  31  65  402  273  129  $${\$}$$31.89 ($${\$}$$21, $${\$}$$46)  Signaling  5$$^{\mathrm{b}}$$  34  60  408  276  132  $${\$}$$30.63 ($${\$}$$17, $${\$}$$47)              No. of negotiations     Game  No. of sessions  Pairs  Rounds  $$n $$  $$B$$ v. $$ A_{L}$$  $$B $$ v. $$A_{H}$$  Average earnings(min, max)  Screening  5$$^{\mathrm{a}}$$  31  65  402  273  129  $${\$}$$31.89 ($${\$}$$21, $${\$}$$46)  Signaling  5$$^{\mathrm{b}}$$  34  60  408  276  132  $${\$}$$30.63 ($${\$}$$17, $${\$}$$47)  $$^{\mathrm{a}}$$Session Scr1: 7 pairs, 12 rounds, $$n =$$ 84 negotiations (49 $$B$$ v. $$A_{L}$$, 35 $$B$$ v. $$A_{H})$$. Scr2: 5 pairs, 13 rounds, $$n =$$ 65 (47, 18). Scr3: 8 pairs, 13 rounds, $$n =$$ 104 (75, 29). Scr4: 5 pairs, 13 rounds, $$n =$$ 65 (41, 24). Scr5: 6 pairs, 14 rounds, $$n =$$ 84 (61, 23). $$^{\mathrm{b}}$$Session Sig1: 6 pairs, 12 rounds, $$n =$$ 72 negotiations (50 $$B$$ v. $$A_{L}$$, 22 $$B$$ v. $$A_{H})$$. Sig2: 8 pairs, 12 rounds, $$n =$$ 96 (61, 35). Sig3: 8 pairs, 12 rounds, $$n =$$ 96 (68, 28). Sig4: 5 pairs, 12 rounds, $$n =$$ 60 (42, 18). Sig5: 7 pairs, 12 rounds, $$n =$$ 84 (55, 29). Upon arrival, subjects were randomly assigned to one of two adjacent rooms, one for player $$A$$ and the other for player $$B$$. An experimenter was assigned to each room. All subjects received common instructions, which included step-by-step practice rounds and earnings calculations from the perspective of both player $$A$$ and player $$B$$.19 Subjects were not informed of their role until the end of the instructions and they maintained that same role throughout the session. The experiment did not utilize verbiage like plaintiff, defendant, judgment at trial, court costs, etc.20 Each subject had a private Record Sheet on which to write decisions, and each experimenter had forms (not visible to subjects) on which to record these decisions. The written decision consisted of either an offer (sender) or an accept/reject choice (recipient). After all subjects in a room had made their decisions, the experimenters met at the entrances to the adjacent rooms, silently copied information from one another’s forms, and then returned to the rooms and wrote the results on the respective subject’s Record Sheet.21 Each round, a subject’s feedback was limited to the offer, accept/reject decision and payoff/cost specific to his or her negotiation, that is, there was no dissemination of information on outcomes for other bargaining pairs. Other than the decisions transmitted by experimenters between the two rooms, there was no communication between the $$A $$ and $$B $$ players or among players within a room. In both experiments, $$A$$’s payoff is the sum of her payoffs from all rounds and $$B$$’s payoff is a lump sum minus the sum of his costs from all rounds.22 The lump sum is known in advance by player $$B$$ but is never revealed to player $$A$$. In all sessions the strangers matching protocol is used, with new random and anonymous pairings prior to the start of each and every round. The sequence in a round of the screening game is as follows: 0. A six-sided die is rolled privately for each Player $$A$$, where a roll of 1, 2, 3, or 4 is called outcome $$L$$ and a roll of 5 or 6 is called outcome $$H$$. Only $$A$$ knows the outcome of the die roll. Player $$B$$ knows how the die roll maps into the outcomes. 1. Player $$B$$ decides on an offer to submit to Player $$A$$. This offer may be any whole number between (and including) 0 and 699. Player $$B$$’s offer is then communicated to Player $$A$$. Player $$A$$ is given a few moments to decide whether or not to accept the offer. Player $$A$$’s decision is then communicated to Player $$B$$. 2. If Player $$A$$ accepts Player $$B$$’s offer, then the round is over for that pair. Players $$A$$’s Payoff for the round $$=$$ Player $$B$$’s offer Player $$B$$’s Cost for the round $$=$$ Player $$B$$’s offer. 3. If Player $$A$$ does not accept $$B$$’s offer, both $$A$$ and $$B$$ incur a fee of 75. $$A$$’s payoff and $$B$$’s cost for the round depend on the die roll and the fees: Under outcome L: Player $$A$$’s Payoff for the round $$=$$ 150 – 75 $$=$$ 75 Player $$B$$’s Cost for the round $$=$$ 150 $$+$$ 75 $$=$$ 225 Under outcome H: Player $$A$$’s Payoff for the round $$=$$ 450 – 75 $$=$$ 375 Player $$B$$’s Cost for the round $$=$$ 450 $$+$$ 75 $$=$$ 525. The description of the game above is very similar in language and appearance to that used in the subjects’ instructions. The information in step 3 was displayed in both rooms by the use of overhead projectors. The overheads included the statement that the same information was displayed in both rooms. The parameters and procedures for the signaling game are identical to the screening game, except that player $$A$$ makes the take-or-leave-it offer to player $$B$$. The steps of a round are identical to the screening game except for the following modifications. 1$$'$$. Player $$A$$ decides on an offer to submit to Player $$B$$. This offer may be any whole number between (and including) 0 and 699. Player $$A$$’s offer is then communicated to Player $$B$$. Player $$B$$ is given a few moments to decide whether or not to accept the offer. Player $$B$$’s decision is then communicated to Player $$A$$. 2$$'$$. If Player $$B$$ accepts Player $$A$$’s offer, then the round is over for that pair. Players $$A$$’s Payoff for the round $$=$$ Player $$A$$’s offer Player $$B$$’s Cost for the round $$=$$ Player $$A$$’s offer. 5. Results We first present data on offer behavior, and we then analyze dispute behavior and the value to player $$A$$ of having the offer. The units of analysis are individual offers, accept/reject decisions and payoffs. We also conducted the analysis treating each session as a single observation, and performed Fisher exact randomization tests for the hypothesis tests. The point estimates and statistical $$P$$-values are quite robust to the unit of analysis, so only the offer level results are presented below. In some tables, we also include data separately from round 7 onward (hereafter R7-end) to evaluate any evolution in subjects’ behavior over time (e.g., learning). These rounds represent the second halves of the individual sessions. 5.1. Offer Behavior Table 3 reports offer frequency distributions using the intervals identified in Section 3.3. In the screening game, 87% of the player $$B$$ offers are screening offers 75–225, 92% in R7-end. In the signaling game, 75% of the $$A_{L}$$ offers are consistent with semi-pooling: 63% are revealing offers 75–225 and 12% are bluffs 375–525. In R7-end, 89% are consistent with semi-pooling: 81% are revealing offers and 8% are bluffs. The lower occurrence of bluffs in R7-end is consistent with the high empirical rejection rate on these offers (see Section 5.2). Somewhat surprisingly, 22% of the $$A_{L}$$ offers are anomalous between offers 226 and 374. Their decline to 11% in R7-end is consistent with the prior work demonstrating the importance of learning in signaling games (e.g., Cooper and Kagel (2008)), but a confounding factor is player $$B$$’s willingness to accept some of these anomalous offers (Section 5.2). Ninety percent of the player $$A_{H}$$ signaling game offers are separating offers 375–525, 96% in R7-end. Collectively, 87% of the screening game offers and 80% of the signaling game offers are consistent with the theory, with the percentages rising to 92% for both games in R7-end.23 Table 3. Offer Frequency Distributions          Proportion (number) of offers in specified interval           Offers consistent with theory  Anomalous offers  Game: player  Rounds  $$n$$  75–225  375–525  226–374  $$<$$ 75, $$>$$ 525  Screening: $$B$$  All  402  0.87 (350)  0.05 (21)  0.06 (24)  0.02 (7)$$^{\mathrm{a}}$$     R7-end  216  0.92 (199)  0.05 (11)  0.03 (6)  – (0)  Signaling: $$A_{L}$$  All  276  0.63 (173)  0.12 (34)  0.22 (60)  0.03 (9)$$^{\mathrm{b}}$$     R7-end  130  0.81 (105)  0.08 (11)  0.11 (14)  – (0)  Signaling: $$A_{H}$$  All  132  0.01 (1)$$^{\mathrm{}}$$  0.91 (119)  0.03 (4)  0.06 (8)$$^{\mathrm{b}}$$     R7-end  74  – (0)  0.96 (71)$$^{\mathrm{c}}$$  0.03 (2)  0.01 (1)$$^{\mathrm{b}}$$           Proportion (number) of offers in specified interval           Offers consistent with theory  Anomalous offers  Game: player  Rounds  $$n$$  75–225  375–525  226–374  $$<$$ 75, $$>$$ 525  Screening: $$B$$  All  402  0.87 (350)  0.05 (21)  0.06 (24)  0.02 (7)$$^{\mathrm{a}}$$     R7-end  216  0.92 (199)  0.05 (11)  0.03 (6)  – (0)  Signaling: $$A_{L}$$  All  276  0.63 (173)  0.12 (34)  0.22 (60)  0.03 (9)$$^{\mathrm{b}}$$     R7-end  130  0.81 (105)  0.08 (11)  0.11 (14)  – (0)  Signaling: $$A_{H}$$  All  132  0.01 (1)$$^{\mathrm{}}$$  0.91 (119)  0.03 (4)  0.06 (8)$$^{\mathrm{b}}$$     R7-end  74  – (0)  0.96 (71)$$^{\mathrm{c}}$$  0.03 (2)  0.01 (1)$$^{\mathrm{b}}$$  $$^{\mathrm{a}}$$Six of these offers are $$<$$75 and one is $$>$$525. $$^{\mathrm{b}}$$All of these offers are $$>$$525. Table 3. Offer Frequency Distributions          Proportion (number) of offers in specified interval           Offers consistent with theory  Anomalous offers  Game: player  Rounds  $$n$$  75–225  375–525  226–374  $$<$$ 75, $$>$$ 525  Screening: $$B$$  All  402  0.87 (350)  0.05 (21)  0.06 (24)  0.02 (7)$$^{\mathrm{a}}$$     R7-end  216  0.92 (199)  0.05 (11)  0.03 (6)  – (0)  Signaling: $$A_{L}$$  All  276  0.63 (173)  0.12 (34)  0.22 (60)  0.03 (9)$$^{\mathrm{b}}$$     R7-end  130  0.81 (105)  0.08 (11)  0.11 (14)  – (0)  Signaling: $$A_{H}$$  All  132  0.01 (1)$$^{\mathrm{}}$$  0.91 (119)  0.03 (4)  0.06 (8)$$^{\mathrm{b}}$$     R7-end  74  – (0)  0.96 (71)$$^{\mathrm{c}}$$  0.03 (2)  0.01 (1)$$^{\mathrm{b}}$$           Proportion (number) of offers in specified interval           Offers consistent with theory  Anomalous offers  Game: player  Rounds  $$n$$  75–225  375–525  226–374  $$<$$ 75, $$>$$ 525  Screening: $$B$$  All  402  0.87 (350)  0.05 (21)  0.06 (24)  0.02 (7)$$^{\mathrm{a}}$$     R7-end  216  0.92 (199)  0.05 (11)  0.03 (6)  – (0)  Signaling: $$A_{L}$$  All  276  0.63 (173)  0.12 (34)  0.22 (60)  0.03 (9)$$^{\mathrm{b}}$$     R7-end  130  0.81 (105)  0.08 (11)  0.11 (14)  – (0)  Signaling: $$A_{H}$$  All  132  0.01 (1)$$^{\mathrm{}}$$  0.91 (119)  0.03 (4)  0.06 (8)$$^{\mathrm{b}}$$     R7-end  74  – (0)  0.96 (71)$$^{\mathrm{c}}$$  0.03 (2)  0.01 (1)$$^{\mathrm{b}}$$  $$^{\mathrm{a}}$$Six of these offers are $$<$$75 and one is $$>$$525. $$^{\mathrm{b}}$$All of these offers are $$>$$525. Table 4 provides analysis on offers consistent with the theory.24 As a robustness check, we also estimate a dummy-variable regression with robust standard errors clustered on sessions, so as to control for session-specific variation. Those results are provided in Appendix Table A1, and they are quite close to those shown in Table 4. Limiting the data to R7-end in Table 4 and in Table A1 yields similar results, so they are omitted. Table 4. Offers Consistent with Theory    Player type and offer interval  Offers 75–225  Offers 375–525     $$B$$  $$A_{L}$$  $$A_{H}$$  $$A_{L}$$ v. $$B$$  $$A_{H}$$ v. $$A_{L}$$  Statistic  75–225  75–225  375–525  375–525  H$$_{\mathrm{0}}$$: Diff. $$=$$ 150  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0  $$n$$  350  173  34  119  Est. diff. $$=$$ 92.0  Est. diff. $$=$$ 37.4  Mean  111.6  203.6  417.5  454.9  $$t =$$ 18.2$$^{\mathrm{a,b}}$$  $$t =$$ 4.12$$^{\mathrm{a}}$$  (SE)  (2.02)  (2.46)  (8.42)  (4.23)        Median  100  220  400  450  Est. diff. $$=$$ 120  Est. diff. $$=$$ 50                 $$z =$$ 18.5$$^{\mathrm{a,c}}$$  $$z =$$ 4.18$$^{\mathrm{a,c}}$$     Player type and offer interval  Offers 75–225  Offers 375–525     $$B$$  $$A_{L}$$  $$A_{H}$$  $$A_{L}$$ v. $$B$$  $$A_{H}$$ v. $$A_{L}$$  Statistic  75–225  75–225  375–525  375–525  H$$_{\mathrm{0}}$$: Diff. $$=$$ 150  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0  $$n$$  350  173  34  119  Est. diff. $$=$$ 92.0  Est. diff. $$=$$ 37.4  Mean  111.6  203.6  417.5  454.9  $$t =$$ 18.2$$^{\mathrm{a,b}}$$  $$t =$$ 4.12$$^{\mathrm{a}}$$  (SE)  (2.02)  (2.46)  (8.42)  (4.23)        Median  100  220  400  450  Est. diff. $$=$$ 120  Est. diff. $$=$$ 50                 $$z =$$ 18.5$$^{\mathrm{a,c}}$$  $$z =$$ 4.18$$^{\mathrm{a,c}}$$  $$^{\mathrm{a}}$$Test statistic has $$P =$$ 0.000 for the given H$$_{\mathrm{0}}$$. $$^{\mathrm{b}}$$Difference-in-means test assumes unequal variances (variance ratio $$F$$ test has $$P =$$ .022). $$^{\mathrm{c}}$$Mann–Whitney rank-sum $$z$$-statistic. Table 4. Offers Consistent with Theory    Player type and offer interval  Offers 75–225  Offers 375–525     $$B$$  $$A_{L}$$  $$A_{H}$$  $$A_{L}$$ v. $$B$$  $$A_{H}$$ v. $$A_{L}$$  Statistic  75–225  75–225  375–525  375–525  H$$_{\mathrm{0}}$$: Diff. $$=$$ 150  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0  $$n$$  350  173  34  119  Est. diff. $$=$$ 92.0  Est. diff. $$=$$ 37.4  Mean  111.6  203.6  417.5  454.9  $$t =$$ 18.2$$^{\mathrm{a,b}}$$  $$t =$$ 4.12$$^{\mathrm{a}}$$  (SE)  (2.02)  (2.46)  (8.42)  (4.23)        Median  100  220  400  450  Est. diff. $$=$$ 120  Est. diff. $$=$$ 50                 $$z =$$ 18.5$$^{\mathrm{a,c}}$$  $$z =$$ 4.18$$^{\mathrm{a,c}}$$     Player type and offer interval  Offers 75–225  Offers 375–525     $$B$$  $$A_{L}$$  $$A_{H}$$  $$A_{L}$$ v. $$B$$  $$A_{H}$$ v. $$A_{L}$$  Statistic  75–225  75–225  375–525  375–525  H$$_{\mathrm{0}}$$: Diff. $$=$$ 150  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0  $$n$$  350  173  34  119  Est. diff. $$=$$ 92.0  Est. diff. $$=$$ 37.4  Mean  111.6  203.6  417.5  454.9  $$t =$$ 18.2$$^{\mathrm{a,b}}$$  $$t =$$ 4.12$$^{\mathrm{a}}$$  (SE)  (2.02)  (2.46)  (8.42)  (4.23)        Median  100  220  400  450  Est. diff. $$=$$ 120  Est. diff. $$=$$ 50                 $$z =$$ 18.5$$^{\mathrm{a,c}}$$  $$z =$$ 4.18$$^{\mathrm{a,c}}$$  $$^{\mathrm{a}}$$Test statistic has $$P =$$ 0.000 for the given H$$_{\mathrm{0}}$$. $$^{\mathrm{b}}$$Difference-in-means test assumes unequal variances (variance ratio $$F$$ test has $$P =$$ .022). $$^{\mathrm{c}}$$Mann–Whitney rank-sum $$z$$-statistic. Theory predicts a difference of 150 between the $$B$$ screening and $$A_{L}$$ revealing offers (75 v. 225). The mean offers are 112 and 204, respectively, and differ by 92. Player $$B$$’s mean offer represents 37 or 25% of the 150 surplus from settlement, and $$A_{L}$$’s mean offer represents 21 or 14% of the surplus.25 The difference of 92 is 61% of the predicted amount. The median offers of 100 and 220 represent less surplus for the recipient (17% and 3%, respectively), and they differ by 120 or 80% of the predicted 150. Further analysis of the data reveals that in both games the preponderance of offers are strongly in the direction of their respective point predictions, but a few include sufficient surplus so as to cause some divergence between the mean and median.26 Nonetheless, both the mean and median differences are statistically different from the 150 prediction. Overall, the $$B$$ screening offers and $$A_{L}$$ revealing offers tend to be stingy, but both contain enough surplus so that their difference is about 60–80% of the predicted amount. In the signaling game, bluffing $$A_{L}$$ players and $$A_{H}$$ are predicted to pool on a single value of the semi-pooling offer $$O_{A}^{S} $$. Empirically, this implies that $$A_{L}$$ and $$A_{H}$$ offers 375–525 will be statistically indistinguishable. Despite the fact that a high percentage of the player $$A$$ offers are consistent with semi-pooling, $$A_{L}$$ and $$A_{H}$$ are unable to successfully coordinate on a unique semi-pooling offer. In particular, the typical $$A_{L}$$ bluff (mean 418, median 400) is substantially lower than the typical $$A_{H}$$ separating offer (mean 445, median 450). The tests on the far-right of Table 4 reject the hypothesis of equal $$A_{L}$$ and $$A_{H}$$ central tendencies on offers 375–525. This coordination failure underscores the challenges posed by signaling games, particularly when feedback is restricted to each litigant’s own negotiation experience. Complicating any attempts at coordination is the empirical dispute behavior, to which we now turn. 5.2. Dispute Behavior Table 5 reports dispute rates between players $$A_{L}$$ and $$B$$. In both games, their predicted dispute rate is 0% over the interval 75–225. The observed rates are 17% in the screening game and 10% in the signaling game. These excess disputes typically occur on offers containing relatively small amounts of surplus, and they are clearly one factor driving the positive surplus in $$B$$ screening offers and $$A_{L}$$ revealing offers.27 In both games, the player receiving the offer has a demand for surplus from settlement which is not present in the theory. In Table 5, the rejection rate on $$A_{L}$$ bluffs 375–525 is 85%, 73% in R7-end. Below we discuss $$B$$’s rejection behavior on offers 375–525 (recall that $$B$$ does not know $$A$$’s type upon receiving an offer), but here we note that the observed rate is higher than the 50–67% rate predicted on these offers. Perhaps one consequence of this is the decline in the incidence of $$A_{L}$$ bluffs to 8% in R7-end. Table 5. $$A_{L}$$ v. $$B$$ Dispute Rates    Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.16 (273)  0.17 (241)  0.00 (13)  0.00 (15)  1.0 (4)$$^{\mathrm{a}}$$     R7-end  0.14 (149)  0.15 (136)  0.00 (8)  0.00 (5)  –  Signaling  All  0.32 (276)  0.10 (173)  0.85 (34)  0.55 (60)  1.0 (9)$$^{\mathrm{b}}$$     R7-end  0.18 (130)  0.10 (105)  0.73 (11)  0.43 (14)  –     Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.16 (273)  0.17 (241)  0.00 (13)  0.00 (15)  1.0 (4)$$^{\mathrm{a}}$$     R7-end  0.14 (149)  0.15 (136)  0.00 (8)  0.00 (5)  –  Signaling  All  0.32 (276)  0.10 (173)  0.85 (34)  0.55 (60)  1.0 (9)$$^{\mathrm{b}}$$     R7-end  0.18 (130)  0.10 (105)  0.73 (11)  0.43 (14)  –  $$^{\mathrm{a}}$$All four are $$<$$ 75. $$^{\mathrm{b}}$$All nine are $$>$$ 525. Table 5. $$A_{L}$$ v. $$B$$ Dispute Rates    Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.16 (273)  0.17 (241)  0.00 (13)  0.00 (15)  1.0 (4)$$^{\mathrm{a}}$$     R7-end  0.14 (149)  0.15 (136)  0.00 (8)  0.00 (5)  –  Signaling  All  0.32 (276)  0.10 (173)  0.85 (34)  0.55 (60)  1.0 (9)$$^{\mathrm{b}}$$     R7-end  0.18 (130)  0.10 (105)  0.73 (11)  0.43 (14)  –     Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.16 (273)  0.17 (241)  0.00 (13)  0.00 (15)  1.0 (4)$$^{\mathrm{a}}$$     R7-end  0.14 (149)  0.15 (136)  0.00 (8)  0.00 (5)  –  Signaling  All  0.32 (276)  0.10 (173)  0.85 (34)  0.55 (60)  1.0 (9)$$^{\mathrm{b}}$$     R7-end  0.18 (130)  0.10 (105)  0.73 (11)  0.43 (14)  –  $$^{\mathrm{a}}$$All four are $$<$$ 75. $$^{\mathrm{b}}$$All nine are $$>$$ 525. Table 5 shows one striking anomaly that occurs in the signaling game: the 55% player $$B$$ dispute rate on $$A_{L}$$ between offers 226 and 374, 43% in R7-end.28 The predicted dispute rate on these offers is 100%, but clearly not all players $$B$$ are able to engage in the chain of reasoning which would lead them to reject a between offer (see the discussion on the relatively weak “test of dominated strategies” refinement in Section 3.3 above).29 The frequent acceptance of these anomalous offers helps explain why $$A_{L}$$ was slow to abandon them. Table 5 also shows that in the screening game, the infrequent pooling offers 375–525 are always accepted by $$A_{L}$$, as theory predicts. ($$B$$ is not predicted to make a pooling offer, but if he does, $$A_{L}$$ is predicted to accept it.) Table 6 reports dispute rates between players $$A_{H}$$ and $$B$$. In the screening game, the predicted dispute rate is 100%, conditional on $$A_{H}$$ receiving a screening offer from $$B$$. Over the screening interval 75–225, the observed dispute rate is 98%. The observed dispute rate on all offers is 92%, primarily because $$A_{H}$$ readily accepts the occasional pooling offer 375–525 that she sees. In the signaling game, the predicted $$A_{H}$$ dispute rate is 50–67%. The observed dispute rate for $$A_{H}$$ separating offers 375–525 is 73% overall and 63% in R7-end. This latter percentage is within the predicted range, and it is consistent with the typical $$A_{H}$$ separating offer of about 450.30 Player $$A_{H}$$ does make a handful of offers outside of 375–525, but the $$A_{H}$$ dispute rate on all offers (73%, 65% R7-end) is the same as the $$A_{H}$$ dispute rate on separating offers. Table 6. $$A_{H}$$ v. $$B$$ Dispute Rates       Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.92 (129)  0.98 (109)  0.13 (8)  0.89 (9)  1.0 (3)$$^{\mathrm{a}}$$     R7-end  0.96 (67)  0.98 (63)  0.33 (3)  1.0 (1)  – (0)  Signaling  All  0.73 (132)  1.0 (1)  0.73 (119)  0.50 (4)  1.0 (8)$$^{\mathrm{b}}$$     R7-end  0.65 (74)  – (0)  0.63 (71)  1.0 (2)  1.0 (1)$$^{\mathrm{b}}$$        Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.92 (129)  0.98 (109)  0.13 (8)  0.89 (9)  1.0 (3)$$^{\mathrm{a}}$$     R7-end  0.96 (67)  0.98 (63)  0.33 (3)  1.0 (1)  – (0)  Signaling  All  0.73 (132)  1.0 (1)  0.73 (119)  0.50 (4)  1.0 (8)$$^{\mathrm{b}}$$     R7-end  0.65 (74)  – (0)  0.63 (71)  1.0 (2)  1.0 (1)$$^{\mathrm{b}}$$  $$^{\mathrm{a}}$$ All three are $$<$$ 75. $$^{\mathrm{b}}$$ All eight (one in R7-end) are $$>$$ 525. Table 6. $$A_{H}$$ v. $$B$$ Dispute Rates       Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.92 (129)  0.98 (109)  0.13 (8)  0.89 (9)  1.0 (3)$$^{\mathrm{a}}$$     R7-end  0.96 (67)  0.98 (63)  0.33 (3)  1.0 (1)  – (0)  Signaling  All  0.73 (132)  1.0 (1)  0.73 (119)  0.50 (4)  1.0 (8)$$^{\mathrm{b}}$$     R7-end  0.65 (74)  – (0)  0.63 (71)  1.0 (2)  1.0 (1)$$^{\mathrm{b}}$$        Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.92 (129)  0.98 (109)  0.13 (8)  0.89 (9)  1.0 (3)$$^{\mathrm{a}}$$     R7-end  0.96 (67)  0.98 (63)  0.33 (3)  1.0 (1)  – (0)  Signaling  All  0.73 (132)  1.0 (1)  0.73 (119)  0.50 (4)  1.0 (8)$$^{\mathrm{b}}$$     R7-end  0.65 (74)  – (0)  0.63 (71)  1.0 (2)  1.0 (1)$$^{\mathrm{b}}$$  $$^{\mathrm{a}}$$ All three are $$<$$ 75. $$^{\mathrm{b}}$$ All eight (one in R7-end) are $$>$$ 525. In the signaling game, the predicted dispute rate of 50–67% on offers 375–525 is not contingent on $$A$$’s type as this is not observable by $$B$$ upon receiving the offer. Aggregating the $$A_{L}$$ bluffs in Table 5 with the $$A_{H}$$ separating offers in Table 6 yields a dispute rate of 76% (116/153), 65% (53/82) in R7-end. These rejection rates are somewhat higher than expected given the empirical player $$A$$ offers 375–525.31 Player $$B$$’s empirical rejection rates are 68% (78/115) on offers 375–499 and 100% (38/38) on offers 500–525. In R7-end, these rates are 57% (38/67) and 100% (15/15), respectively. Thus $$B$$ has an implicit demand which forces both $$A_{H}$$ and a bluffing $$A_{L}$$ to experiment with offers $$<$$ 500 to create a positive probability of acceptance.32 Table 7 evaluates dispute rate comparative statics across the two games. The table shows the predicted comparative statics (game-specific predictions are shown in table note $$a)$$ and observed differences separately for $$A_{L} $$v. $$B, A_{H} $$v. $$B$$, and $$A$$ v. $$B$$. The predicted $$A_{L}$$ v. $$B$$ comparative static on the dispute rate is 0–25 percentage points higher in the signaling game semi-pooling equilibrium than in the screening game. The observed dispute rate is sixteen percentage points higher, which is consistent with the prediction. However, both of the individual empirical dispute rates exceed their respective point predictions, due primarily to the three reasons discussed above.33 The predicted $$A_{H}$$ v. $$B$$ comparative static is a decline of 33–50 percentage points in the dispute rate in the signaling game relative to the screening game. The observed difference is a 19-percentage point decline in the signaling game, which is in the predicted direction and statistically different from zero, but well below the minimum expected difference. The less than expected decline is due to the higher than expected dispute rate in the signaling game (73%) and the slightly less than expected dispute rate in the screening game (92%). The overall $$A $$vs. $$B$$ dispute rate is predicted to decline by 0–11 percentage points. The observed difference is an increase of five percentage points; this difference is small economically and not statistically different from zero. Overall, we do not observe any major efficiency effects related to which party has the power to make the offer. Table 7. Comparison of Dispute Rates Across Games       Dispute rates for all rounds, all offers  Negotiation  Prediction$$^{\mathrm{a}}$$  Signaling  Screening  Diff.  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0$$^{\mathrm{b}}$$  $$A_{L}$$ v. $$ B$$  0 to $$+$$25  0.32  0.16  0.16  $$z = 4.32$$        ($$n = 276$$)  ($$n = 273$$)     ($$P = .000$$)  $$A_{H}$$ v. $$ B$$  –33 to –50  0.73  0.92  –0.19  $$z^{}=$$ –4.01        ($$n = 132$$)  ($$n = 129$$)     ($$P = .000$$)  $$A $$ v. $$ B$$  0 to –11  0.45  0.40  0.05  $$z =$$ 1.38        ($$n =$$ 408)  ($$n =$$ 402)     ($$P =.168$$)        Dispute rates for all rounds, all offers  Negotiation  Prediction$$^{\mathrm{a}}$$  Signaling  Screening  Diff.  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0$$^{\mathrm{b}}$$  $$A_{L}$$ v. $$ B$$  0 to $$+$$25  0.32  0.16  0.16  $$z = 4.32$$        ($$n = 276$$)  ($$n = 273$$)     ($$P = .000$$)  $$A_{H}$$ v. $$ B$$  –33 to –50  0.73  0.92  –0.19  $$z^{}=$$ –4.01        ($$n = 132$$)  ($$n = 129$$)     ($$P = .000$$)  $$A $$ v. $$ B$$  0 to –11  0.45  0.40  0.05  $$z =$$ 1.38        ($$n =$$ 408)  ($$n =$$ 402)     ($$P =.168$$)  $$^{\mathrm{a}}$$Predictions are percentage point differences in dispute rates across games. From Table 1, predicted dispute rates are $$A_{L} $$ v. $$ B$$: 0–25% v. 0%, $$A_{H} $$ v. $$B$$: 50–67% v. 100%, and $$A $$ v. $$ B$$: 22–33% v. 33%. $$^{\mathrm{b}}$$Difference-in-proportions binomial test. Table 7. Comparison of Dispute Rates Across Games       Dispute rates for all rounds, all offers  Negotiation  Prediction$$^{\mathrm{a}}$$  Signaling  Screening  Diff.  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0$$^{\mathrm{b}}$$  $$A_{L}$$ v. $$ B$$  0 to $$+$$25  0.32  0.16  0.16  $$z = 4.32$$        ($$n = 276$$)  ($$n = 273$$)     ($$P = .000$$)  $$A_{H}$$ v. $$ B$$  –33 to –50  0.73  0.92  –0.19  $$z^{}=$$ –4.01        ($$n = 132$$)  ($$n = 129$$)     ($$P = .000$$)  $$A $$ v. $$ B$$  0 to –11  0.45  0.40  0.05  $$z =$$ 1.38        ($$n =$$ 408)  ($$n =$$ 402)     ($$P =.168$$)        Dispute rates for all rounds, all offers  Negotiation  Prediction$$^{\mathrm{a}}$$  Signaling  Screening  Diff.  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0$$^{\mathrm{b}}$$  $$A_{L}$$ v. $$ B$$  0 to $$+$$25  0.32  0.16  0.16  $$z = 4.32$$        ($$n = 276$$)  ($$n = 273$$)     ($$P = .000$$)  $$A_{H}$$ v. $$ B$$  –33 to –50  0.73  0.92  –0.19  $$z^{}=$$ –4.01        ($$n = 132$$)  ($$n = 129$$)     ($$P = .000$$)  $$A $$ v. $$ B$$  0 to –11  0.45  0.40  0.05  $$z =$$ 1.38        ($$n =$$ 408)  ($$n =$$ 402)     ($$P =.168$$)  $$^{\mathrm{a}}$$Predictions are percentage point differences in dispute rates across games. From Table 1, predicted dispute rates are $$A_{L} $$ v. $$ B$$: 0–25% v. 0%, $$A_{H} $$ v. $$B$$: 50–67% v. 100%, and $$A $$ v. $$ B$$: 22–33% v. 33%. $$^{\mathrm{b}}$$Difference-in-proportions binomial test. 5.3. The Value to $$A$$ of Having the Offer We now consider the value to player $$A$$ of having the offer. Empirically, the value of the property right depends on the difference between the player $$A$$ signaling game offer and the player $$B$$ screening game offer, as well as the dispute behavior of the respective recipient. In Table 8, we use the mean net payoff earned by player $$A $$ to assess the empirical value of the offer to $$A.$$34 For each player type, we conduct two statistical tests, one for equality of means across games, and one for the predicted difference across games (as the predictions are intervals for $$A_{H}$$ and $$A$$, we use the upper end of the respective interval). As a robustness check, we also estimate a dummy-variable regression with robust standard errors clustered on sessions. Those results are similar to those in Table 8; see Appendix Table A2. We also obtain comparable results using R7-end data. Table 8. Player $$A$$ Net Payoffs       Mean $$A$$ payoff (SE)  Difference-in-means tests     Prediction  Signaling  Screening  Diff.  H$$_0$$: Diff. $$= 0$$  H$$_0$$: Diff. $$=$$ Pred.$$^{\mathrm{a}}$$        $$n =$$ 276  $$n =$$ 273     $$t = 6.33$$  $$t = 15.4$$  $$A_{L}$$  $$+150$$  174.0  130.3  43.7  ($$P = .000$$)  ($$P = .000$$)        (4.92)  (4.84)                 $$n = 132$$  $$n = 129$$           $$A_{H}$$  0 to $$+$$50  386.4  371.5  14.9  $$t = 3.31$$  $$t = 7.82$$        (3.13)  (3.22)     ($$P = .001$$)  ($$P = .000$$)        $$n = 402$$  $$n = 408$$           $$A$$  $$+100$$ to $$+$$117  242.7  207.7  35.0  $$t = 3.92^{\mathrm{b}}$$  $$t = 9.18^{\mathrm{b}}$$        (6.03)  (6.60)     $$(P = 0.001)$$  $$P = 0.000$$        Mean $$A$$ payoff (SE)  Difference-in-means tests     Prediction  Signaling  Screening  Diff.  H$$_0$$: Diff. $$= 0$$  H$$_0$$: Diff. $$=$$ Pred.$$^{\mathrm{a}}$$        $$n =$$ 276  $$n =$$ 273     $$t = 6.33$$  $$t = 15.4$$  $$A_{L}$$  $$+150$$  174.0  130.3  43.7  ($$P = .000$$)  ($$P = .000$$)        (4.92)  (4.84)                 $$n = 132$$  $$n = 129$$           $$A_{H}$$  0 to $$+$$50  386.4  371.5  14.9  $$t = 3.31$$  $$t = 7.82$$        (3.13)  (3.22)     ($$P = .001$$)  ($$P = .000$$)        $$n = 402$$  $$n = 408$$           $$A$$  $$+100$$ to $$+$$117  242.7  207.7  35.0  $$t = 3.92^{\mathrm{b}}$$  $$t = 9.18^{\mathrm{b}}$$        (6.03)  (6.60)     $$(P = 0.001)$$  $$P = 0.000$$  $$^{\mathrm{a}}$$For $$A_{H}$$ test, prediction value is $$+$$50. For player A test, prediction value is $$+$$117. $$^{\mathrm{b}}$$$$t$$-statistic calculated assuming unequal variances as variance ratio $$F$$ test has $$P =$$ 0.098. Table 8. Player $$A$$ Net Payoffs       Mean $$A$$ payoff (SE)  Difference-in-means tests     Prediction  Signaling  Screening  Diff.  H$$_0$$: Diff. $$= 0$$  H$$_0$$: Diff. $$=$$ Pred.$$^{\mathrm{a}}$$        $$n =$$ 276  $$n =$$ 273     $$t = 6.33$$  $$t = 15.4$$  $$A_{L}$$  $$+150$$  174.0  130.3  43.7  ($$P = .000$$)  ($$P = .000$$)        (4.92)  (4.84)                 $$n = 132$$  $$n = 129$$           $$A_{H}$$  0 to $$+$$50  386.4  371.5  14.9  $$t = 3.31$$  $$t = 7.82$$        (3.13)  (3.22)     ($$P = .001$$)  ($$P = .000$$)        $$n = 402$$  $$n = 408$$           $$A$$  $$+100$$ to $$+$$117  242.7  207.7  35.0  $$t = 3.92^{\mathrm{b}}$$  $$t = 9.18^{\mathrm{b}}$$        (6.03)  (6.60)     $$(P = 0.001)$$  $$P = 0.000$$        Mean $$A$$ payoff (SE)  Difference-in-means tests     Prediction  Signaling  Screening  Diff.  H$$_0$$: Diff. $$= 0$$  H$$_0$$: Diff. $$=$$ Pred.$$^{\mathrm{a}}$$        $$n =$$ 276  $$n =$$ 273     $$t = 6.33$$  $$t = 15.4$$  $$A_{L}$$  $$+150$$  174.0  130.3  43.7  ($$P = .000$$)  ($$P = .000$$)        (4.92)  (4.84)                 $$n = 132$$  $$n = 129$$           $$A_{H}$$  0 to $$+$$50  386.4  371.5  14.9  $$t = 3.31$$  $$t = 7.82$$        (3.13)  (3.22)     ($$P = .001$$)  ($$P = .000$$)        $$n = 402$$  $$n = 408$$           $$A$$  $$+100$$ to $$+$$117  242.7  207.7  35.0  $$t = 3.92^{\mathrm{b}}$$  $$t = 9.18^{\mathrm{b}}$$        (6.03)  (6.60)     $$(P = 0.001)$$  $$P = 0.000$$  $$^{\mathrm{a}}$$For $$A_{H}$$ test, prediction value is $$+$$50. For player A test, prediction value is $$+$$117. $$^{\mathrm{b}}$$$$t$$-statistic calculated assuming unequal variances as variance ratio $$F$$ test has $$P =$$ 0.098. The mean payoff for $$A_{L}$$ is about 44 higher in the signaling game. This difference is statistically different from zero, but it is only 29% of the predicted 150. For $$A_{H}$$ the mean payoff is about 15 higher in the signaling game, which is consistent with the semi-pooling equilibrium where the predicted increase for $$A_{H}$$ ranges from 0 to 50. We note that our estimated increase is different both economically and statistically from both the bottom (0) and top (50) ends of that range. The overall average increase for $$A$$ is 35, which is well below the predicted increase of 100–117. Thus, the distributional impact of having the offer in this setting is much less than is predicted by the theory. A combination of reasons explains why $$A$$ gains less than predicted from having the offer. One reason is that $$A_{L}$$ bluffs 375–525 are rejected at an 85% rate, which is much higher than the 50–67% prediction. Another reason is that $$B$$ screening offers, $$A_{L}$$ revealing offers and $$A_{H}$$ separating offers cannot extract all of the surplus from settlement. These offers almost always contain positive surplus for the recipient. Moreover, $$B$$ screening offers and $$A_{L}$$ revealing offers are rejected about 10–17% of the time. If we restrict the analysis to $$B$$ screening offers and $$A_{L}$$ revealing offers, we can gain more insight into the importance of these factors. When we do so (table omitted) we find higher estimates for the value of having the offer, but these are still well below the point predictions. For $$A_{L}$$, the mean payoff increase is 82 or 55% of the predicted 150. Thus failed negotiations on offers in the 75–225 range significantly reduce the predicted distributional impact of having the offer. For $$A_{H}$$, restricting offers to the 375–525 range makes little difference as the estimated value of having the offer remains at about 20. With the restricted offers, the overall average increase for player $$A$$ is 53, which is still well below the predicted range 100–117. 6. Conclusion We provide an experimental study of screening and signaling models in a legal bargaining context. The two asymmetric information games are identical, with the exception of whether the uninformed defendant (the screening game) or the informed plaintiff (the signaling game) makes the pretrial offer. Overall, the screening model performs well in aggregating the data. Player $$B$$ (the defendant) makes screening offers nearly 90% of the time, and his median screening offer contains 1/6 of the joint surplus from settlement. Player $$A_{L}$$ (a plaintiff with a weak case) typically accepts the screening offer, although she sometimes rejects offers containing less than 1/3 of the settlement surplus. The dispute rate for player $$A_{H}$$ (a plaintiff with a strong case) is near the predicted 100%. These results are consistent with the previous experiments in this setting. In the screening game, the recipient is the informed party (player $$A)$$ and she has a fairly trivial decision to make in deciding whether or not to accept the offer. The sender (player $$B)$$ has a more difficult decision, but latching onto the idea of a screening offer does not appear to be an overly demanding task. Although we do observe some fairness concerns, they are insufficient to invalidate the qualitative predictions of the model. Using the same parameters as in the screening game, we switch the identity of the player making the offer to create a signaling game. We believe that this problem is cognitively much more challenging for both the sender (now player $$A)$$ and the recipient (now player $$B)$$. As the sender, $$A_{L}$$ has to decide whether she should bluff and what offer would constitute a good bluff, while $$A_{H}$$ has to decide how much she needs to shade her offer to ensure a reasonable chance of acceptance. As the recipient, player $$B$$ has a difficult decision to make when he is faced with a high offer, because he cannot be sure whether the offer is a revealing offer from $$A_{H}$$ or a bluff by $$A_{L}$$. Potentially complicating these decision are concerns for fairness. Given these difficulties, we expect, and indeed observe, less adherence to theory in the signaling game compared to the screening game. Much of the behavior conforms reasonably well with a semi-pooling equilibrium. The player $$B$$ rejection behavior on $$A_{L}$$ revealing offers and on high offers (which may be $$A_{H}$$ separating offers or $$A_{L}$$ bluffs) is roughly in line with the semi-pooling prediction. Aside from the “between” offers, the player $$A$$ behavior is likewise roughly consistent with the theory. As expected, $$A_{L}$$ makes a revealing offer a high percentage of the time, and the median revealing offer (220) is quite close to the theoretical prediction (225). Player $$A_{H}$$ almost always makes an offer which separates her from a revealing $$A_{L}$$ player. Twelve percent of $$A_{L}$$ offers are bluffs that mimic an $$A_{H}$$ player. $$A_{H}$$ and the bluffing $$A_{L}$$ do not converge on a single offer as they would in a semi-pooling equilibrium, but it would have been rather miraculous if they had. There is a continuum of such equilibria, and player feedback was limited to their own bargaining experience. The most notable deviation from theory is the acceptance of “between” offers that should be rejected using fairly straightforward dominance arguments. This violates a weak refinement concept known as the test of dominated strategies. While the incidence of these anomalous offers by player $$A_{L}$$ falls to ~ 10% in the later rounds, player $$B$$ continues to accept ~ 50% of them.35 Our results contrast with the labor market experiment of Kübler et al. (2008), who find greater adherence to the theory in the signaling game relative to the screening game. A comparison across the two games indicates that the ability to make the offer is not nearly as valuable as suggested by theory. The mean increase in payoffs for player $$A_{L}$$ from moving from the screening to the signaling game is 30–55% of the predicted amount. This is in part due to the existence of disputes not predicted by the theory and also because the player making the offer cannot extract all of the surplus predicted by theory. Overall, we do observe some distributional effects from changing the identity of the player who makes the offer, but they are much smaller in practice than they are in theory. While the dispute rates are approximately equal in the two games, the sources of disputes do differ in some important ways. In both games, disputes likely arise because the recipient makes an implicit demand not present in the theory, which the sender is unwilling to meet and/or unable to observe. In our data, this occurs more frequently in the screening game. Offsetting this, however, is the bluffing and unpredicted “between” offers made by $$A_{L}$$ players in the signaling game. The high dispute rate on these offers offsets the low dispute rate on revealing offers, and the net effect is roughly equivalent dispute rates across the two games. Consequently, we observe no major efficiency implications arising from the assignment of which player is allowed to make the offer. In the later rounds of our screening experiment, anomalous behavior essentially disappears. In the later rounds of our signaling game, anomalous offers are significantly reduced, but anomalous acceptance behavior persists. This is consistent with the idea that the signaling game is more cognitively demanding. The reduction in anomalous offers indicates that some learning did occur in the signaling game. In the laboratory, having players reverse roles might facilitate additional learning, as might public dissemination of some information on negotiations. But real-world plaintiffs typically have limited experience in that role and might never be placed in the role of a defendant. Also, many out-of-court settlements include non-disclosure clauses, which limit what can be learned from the bargaining experience of others. This suggests that the behavior of relatively inexperienced litigants is relevant, even if this inexperience may be mitigated by the presence of an attorney.36 The omission of contract clauses specifying which party makes an offer may reflect the absence of strong efficiency or distributional effects associated with the identity of the party making the settlement offer. Clearly, the signaling model deserves more attention. It is one of the two informational-based models of pretrial bargaining, but the theory typically relies on a refinement concept, D1, which is not supported empirically. Thus, it is vital to gain further data on the actual behavior within this game as this may help better inform the theory. This, in turn, may allow for sharper policy recommendations regarding policies such as fee shifting and the role of costly pretrial discovery. Our experimental design has followed the theory quite closely, but future experiments may incorporate additional features which may be important to legal bargaining. This can include greater context (i.e., labeling the players as plaintiff and defendant) and allowing for free form bargaining in the presence of asymmetric information. The outcomes of experiments without structured bargaining can then be compared to the outcomes under the type of structured bargaining typically assumed in the theory. These empirical comparisons can then help better inform the theory of pretrial bargaining, and help us better understand the nature of disputes. We would like to thank J.J. Prescott, Rudy Santore, and two anonymous referees for providing helpful comments on the article, and the Culverhouse College of Commerce and Business Administration, University of Alabama for providing research support for Paul Pecorino on this project. We would also like to thank William B. Hankins for providing research assistance. Appendix Table A1. Offer Regression with Robust Standard Errors Model$$^{\mathrm{a}}$$: Offer$$_{ijs} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L}$$Reveal$$+ \beta_{\mathrm{2}}$$Sig-A$$_{L}$$Bluff$$+ \beta_{\mathrm{3}}$$Sig-A$$_{H}$$Separate$$+\mu_{is} + \varepsilon_{ijs}$$     Estimated coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  111.6  91.9  305.8  343.3        $$\quad$$ (std.err.)  (3.975)  (5.654)  (7.230)  (7.750)  948.60  0.922  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 676$$  Implied mean  111.6$$^{\mathrm{b}}$$  $$203.6^{\mathrm{b}}$$  417.5$$^{\mathrm{b}}$$  454.9$$^{\mathrm{b}}$$        Model$$^{\mathrm{a}}$$: Offer$$_{ijs} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L}$$Reveal$$+ \beta_{\mathrm{2}}$$Sig-A$$_{L}$$Bluff$$+ \beta_{\mathrm{3}}$$Sig-A$$_{H}$$Separate$$+\mu_{is} + \varepsilon_{ijs}$$     Estimated coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  111.6  91.9  305.8  343.3        $$\quad$$ (std.err.)  (3.975)  (5.654)  (7.230)  (7.750)  948.60  0.922  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 676$$  Implied mean  111.6$$^{\mathrm{b}}$$  $$203.6^{\mathrm{b}}$$  417.5$$^{\mathrm{b}}$$  454.9$$^{\mathrm{b}}$$        $$^{\mathrm{a}}$$Subscripts: $$i =$$ offer, $$j =$$ round, and $$s =$$ session. Dummy variables: baseline is screening game sender $$B$$ and offer 75–225, Sig-A$$_{L}$$Reveal$$=$$ 1 if signaling game sender $$A_{L} $$and revealing offer 75–225 ($$=$$ 0 otherwise), Sig-A$$_{L}$$Bluff$$=$$ 1 if signaling game sender $$A_{L}$$ and bluff offer 375–525 ($$=$$ 0 otherwise), Sig-A$$_{H}$$Separate$$=$$ 1 if signaling game sender $$A_{H} $$and separating offer 375–525 ($$=$$ 0 otherwise). Robust standard errors are clustered on sessions. $$^{\mathrm{b}}$$For the Offers 75–225 H$$_{\mathrm{0}}$$: $$A_{L} $$–$$ B =$$ 150 test in Table 4, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 150 has $$F =$$ 105.5 ($$P =$$ 0.000). For the Offers 375–525 H$$_{\mathrm{0}}$$: $$A_{L} = A_{H}$$ test in Table 4, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}} = \beta_{\mathrm{3}}$$ has $$F =$$ 18.8 ($$P =$$ 0.019). Corresponding to the fn. 25 equal surplus difference-in-means test, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{0}}$$ – 75 $$=$$ 225 – ($$\beta_{\mathrm{0}} + \beta_{\mathrm{1}})$$ has $$F = 7.2 (P = 0.025)$$. Table A1. Offer Regression with Robust Standard Errors Model$$^{\mathrm{a}}$$: Offer$$_{ijs} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L}$$Reveal$$+ \beta_{\mathrm{2}}$$Sig-A$$_{L}$$Bluff$$+ \beta_{\mathrm{3}}$$Sig-A$$_{H}$$Separate$$+\mu_{is} + \varepsilon_{ijs}$$     Estimated coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  111.6  91.9  305.8  343.3        $$\quad$$ (std.err.)  (3.975)  (5.654)  (7.230)  (7.750)  948.60  0.922  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 676$$  Implied mean  111.6$$^{\mathrm{b}}$$  $$203.6^{\mathrm{b}}$$  417.5$$^{\mathrm{b}}$$  454.9$$^{\mathrm{b}}$$        Model$$^{\mathrm{a}}$$: Offer$$_{ijs} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L}$$Reveal$$+ \beta_{\mathrm{2}}$$Sig-A$$_{L}$$Bluff$$+ \beta_{\mathrm{3}}$$Sig-A$$_{H}$$Separate$$+\mu_{is} + \varepsilon_{ijs}$$     Estimated coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  111.6  91.9  305.8  343.3        $$\quad$$ (std.err.)  (3.975)  (5.654)  (7.230)  (7.750)  948.60  0.922  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 676$$  Implied mean  111.6$$^{\mathrm{b}}$$  $$203.6^{\mathrm{b}}$$  417.5$$^{\mathrm{b}}$$  454.9$$^{\mathrm{b}}$$        $$^{\mathrm{a}}$$Subscripts: $$i =$$ offer, $$j =$$ round, and $$s =$$ session. Dummy variables: baseline is screening game sender $$B$$ and offer 75–225, Sig-A$$_{L}$$Reveal$$=$$ 1 if signaling game sender $$A_{L} $$and revealing offer 75–225 ($$=$$ 0 otherwise), Sig-A$$_{L}$$Bluff$$=$$ 1 if signaling game sender $$A_{L}$$ and bluff offer 375–525 ($$=$$ 0 otherwise), Sig-A$$_{H}$$Separate$$=$$ 1 if signaling game sender $$A_{H} $$and separating offer 375–525 ($$=$$ 0 otherwise). Robust standard errors are clustered on sessions. $$^{\mathrm{b}}$$For the Offers 75–225 H$$_{\mathrm{0}}$$: $$A_{L} $$–$$ B =$$ 150 test in Table 4, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 150 has $$F =$$ 105.5 ($$P =$$ 0.000). For the Offers 375–525 H$$_{\mathrm{0}}$$: $$A_{L} = A_{H}$$ test in Table 4, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}} = \beta_{\mathrm{3}}$$ has $$F =$$ 18.8 ($$P =$$ 0.019). Corresponding to the fn. 25 equal surplus difference-in-means test, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{0}}$$ – 75 $$=$$ 225 – ($$\beta_{\mathrm{0}} + \beta_{\mathrm{1}})$$ has $$F = 7.2 (P = 0.025)$$. Table A2. Player $$A$$ Net Payoff Regression with Robust Standard Errors Model$$^{\mathrm{a,b}}$$: Player $$A$$ net payoff$$_{\mathrm{ijs}} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L} + \beta_{\mathrm{2}}$$Scr-A$$_{H} + \beta_{\mathrm{3}}$$Sig-A$$_{H} + \mu_{is} + \varepsilon_{ijs}$$     Coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  130.3  43.7  241.3  256.1        $$\quad$$ (std. err.)  (8.955)  (12.628)  (9.632)  (9.069)  568.52  0.705  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.001$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 810$$  Implied mean  130.3  174.0  371.5  386.4        Model$$^{\mathrm{a,b}}$$: Player $$A$$ net payoff$$_{\mathrm{ijs}} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L} + \beta_{\mathrm{2}}$$Scr-A$$_{H} + \beta_{\mathrm{3}}$$Sig-A$$_{H} + \mu_{is} + \varepsilon_{ijs}$$     Coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  130.3  43.7  241.3  256.1        $$\quad$$ (std. err.)  (8.955)  (12.628)  (9.632)  (9.069)  568.52  0.705  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.001$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 810$$  Implied mean  130.3  174.0  371.5  386.4        $$^{\mathrm{a}}$$Subscripts: $$i =$$ offer, $$j =$$ round, and $$s =$$ session. Dummy variables: baseline is screening game player $$A_{L}$$, Sig-A$$_{L} =$$ 1 if signaling game player $$A_{L} (=$$ 0 otherwise), Scr-A$$_{L} =$$ 1 if screening game player $$A_{H} (=$$ 0 otherwise), Sig-A$$_{H} =$$ 1 if signaling game player $$A_{H} (=$$ 0 otherwise). Robust standard errors are clustered on sessions. $$^{\mathrm{b}}$$Using the empirical frequencies, the implied mean player $$A$$ payoffs are screening game $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}}) =$$ 215.3 and signaling game $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{1}}) + p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) =$$ 242.7. From Table 2, $$p(L$$| scr) $$=$$ 0.648, $$p(H$$| scr) $$=$$ 0.352, $$p(L$$| sig) $$=$$ 0.676, and $$p(H$$| sig) $$=$$ 0.324. For the Table 8 H$$_{\mathrm{0}}$$: Diff. $$=$$ 0 tests, $$A_{L}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 0 has $$F =$$ 10.96 ($$P =$$ 0.007), $$A_{H}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}}$$ – $$\beta_{\mathrm{3}} =$$ 0 has $$F =$$ 8.84 ($$P =$$ 0.016), and player $$A$$ test $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) =$$ 0 has $$F =$$ 9.43 ($$P =$$ 0.013). For Table 8 tests H$$_{\mathrm{0}}$$: Diff. $$=$$ Pred., $$A_{L}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 150 has $$F =$$ 70.9 ($$P =$$ 0.000), $$A_{H}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}}$$ – $$\beta_{\mathrm{3}} =$$ 50 has $$F =$$ 49.3 ($$P =$$ 0.000), and player $$A$$ test $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) = 117$$ has $$F = 262.4$$ ($$P = 0.000$$). Table A2. Player $$A$$ Net Payoff Regression with Robust Standard Errors Model$$^{\mathrm{a,b}}$$: Player $$A$$ net payoff$$_{\mathrm{ijs}} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L} + \beta_{\mathrm{2}}$$Scr-A$$_{H} + \beta_{\mathrm{3}}$$Sig-A$$_{H} + \mu_{is} + \varepsilon_{ijs}$$     Coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  130.3  43.7  241.3  256.1        $$\quad$$ (std. err.)  (8.955)  (12.628)  (9.632)  (9.069)  568.52  0.705  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.001$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 810$$  Implied mean  130.3  174.0  371.5  386.4        Model$$^{\mathrm{a,b}}$$: Player $$A$$ net payoff$$_{\mathrm{ijs}} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L} + \beta_{\mathrm{2}}$$Scr-A$$_{H} + \beta_{\mathrm{3}}$$Sig-A$$_{H} + \mu_{is} + \varepsilon_{ijs}$$     Coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  130.3  43.7  241.3  256.1        $$\quad$$ (std. err.)  (8.955)  (12.628)  (9.632)  (9.069)  568.52  0.705  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.001$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 810$$  Implied mean  130.3  174.0  371.5  386.4        $$^{\mathrm{a}}$$Subscripts: $$i =$$ offer, $$j =$$ round, and $$s =$$ session. Dummy variables: baseline is screening game player $$A_{L}$$, Sig-A$$_{L} =$$ 1 if signaling game player $$A_{L} (=$$ 0 otherwise), Scr-A$$_{L} =$$ 1 if screening game player $$A_{H} (=$$ 0 otherwise), Sig-A$$_{H} =$$ 1 if signaling game player $$A_{H} (=$$ 0 otherwise). Robust standard errors are clustered on sessions. $$^{\mathrm{b}}$$Using the empirical frequencies, the implied mean player $$A$$ payoffs are screening game $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}}) =$$ 215.3 and signaling game $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{1}}) + p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) =$$ 242.7. From Table 2, $$p(L$$| scr) $$=$$ 0.648, $$p(H$$| scr) $$=$$ 0.352, $$p(L$$| sig) $$=$$ 0.676, and $$p(H$$| sig) $$=$$ 0.324. For the Table 8 H$$_{\mathrm{0}}$$: Diff. $$=$$ 0 tests, $$A_{L}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 0 has $$F =$$ 10.96 ($$P =$$ 0.007), $$A_{H}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}}$$ – $$\beta_{\mathrm{3}} =$$ 0 has $$F =$$ 8.84 ($$P =$$ 0.016), and player $$A$$ test $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) =$$ 0 has $$F =$$ 9.43 ($$P =$$ 0.013). For Table 8 tests H$$_{\mathrm{0}}$$: Diff. $$=$$ Pred., $$A_{L}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 150 has $$F =$$ 70.9 ($$P =$$ 0.000), $$A_{H}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}}$$ – $$\beta_{\mathrm{3}} =$$ 50 has $$F =$$ 49.3 ($$P =$$ 0.000), and player $$A$$ test $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) = 117$$ has $$F = 262.4$$ ($$P = 0.000$$). Footnotes 1. Despite their widespread use, these two models of litigation have not yet been comparatively analyzed in an experimental setting. In this sense, our article is analogous to Kübler et al. (2008), who present a side-by-side comparison of the signaling and screening versions of the Spence (1973) labor market model. However, we find relatively greater support for our screening model, while they find more support for their signaling model. 2. One common provision is that the individual triggering the dissolution of the partnership makes an offer for the shares of the other partner. The second partner can either sell her shares or buy her partner’s shares at the specified price. Although these contract provisions do not exactly match our litigation setting, asymmetric information over the partner’s valuation is a typical component of such models. See Fleischer and Schneider (2012) and McAfee (1992). 3. This is for the parameter values we use. Note that there is no general result implying that dispute rates are always lower in the signaling game. The relative dispute rate across games will depend on the number of types present in the model as well as on other parameter values. 4. For a review of the relevant literature, see Spier (2007), Daughety and Reinganum (2012), and Wickelgren (2013). 5. See Camerer and Talley (2007) and Croson (2009) for surveys of the literature on experimental law and economics. 6. See, for example, Ashenfelter et al. (1992), Dickinson (2004) and (2005), Deck and Farmer (2007), Deck, Farmer and Zeng (2007), and Birkeland (2013). Pecorino and Van Boening (2001) analyze arbitration with asymmetric information. 7. See, for example, Stanley and Coursey (1990). One treatment of Inglis et al. (2005) also adopts this information structure. 8. This literature is initiated by Güth et al. (1982). For a recent survey, see Güth and Kocher (2014). 9. They find that whether or not the predictions of the intuitive criterion hold depends upon the details of the game. Also, Brandts and Holt (1993) show how the dynamics of learning in these signaling games can lead to different outcomes. The intuitive criterion places restrictions on out of equilibrium beliefs. It requires player B to place zero weight on the probability of action by player A if that action is associated with payoffs which are dominated by the payoffs earned by A in equilibrium. 10. Also see Cooper et al. (1997), Cooper and Kagel (2008), Kawagoe and Takizawa (2009), de Haan et al. (2011), Drouvelis et al. (2012) and Jeitschko and Normann (2012). 11. We present a two-type signaling model. For a complete analysis of such a model, see Daughety (2000). 12. See, for example, Daughety and Reinganum (1993, p. 322) or Spier (2007, p. 276, fn. 26.) 13. D1 restricts out of equilibrium beliefs in the following way: Suppose an A$$_{L}$$ plaintiff would be willing to deviate to a particular out-of-equilibrium offer if it were accepted with a probability of 1/2 or higher, but that an A$$_{H}$$ plaintiff would only switch if the same offer were accepted with a probability of 3/4 or higher. In this case, the A$$_{L}$$ plaintiff is considered to have the greater incentive to make the offer and, under D1, the defendant must put a weight of 1 on the probability that such an offer comes from an A$$_{L}$$ plaintiff. See Cho and Kreps (1987). The refinement D1 is stronger than the intuitive criterion because it requires zero weight be placed on out-of-equilibrium actions which are not necessarily dominated by a player’s equilibrium payoff. However, this strengthening of the intuitive criterion is necessary to rule out pooling and semi-pooling equilibria. This is the reason why this refinement is employed in the theoretical literature. 14. We can rule out a pure strategy pooling equilibrium, because our parameters satisfy $$150 = \textit{C}_{A} + \textit{C}_{B} < (1 - q)(J^{H} - J^{L}) = 200$$. A type $$A_{H}$$ plaintiff will accept no less than J$$^{H}$$–C$$_{A}$$ in a pooling equilibrium. The defendant B will prefer to take all types to trial rather than settling at $$J^{H}-C_{A}$$ in a pure pooling equilibrium. Obviously, B would reject all higher offers as well, if they were part of a pure strategy pooling equilibrium. 15. It is not possible to have the A$$_{H}$$ plaintiffs and bluffing A$$_{L}$$ plaintiffs make more than one offer between 375 and 525 because it is not possible to find rejection probabilities that make both plaintiff types indifferent between both offers while also making the A$$_{L}$$ plaintiffs indifferent between bluffing and not bluffing. 16. Under D1, there is a pure strategy separating equilibrium under which A$$_{L}$$ always demands 225 and A$$_{H}$$ always demands 525. The low offer is always accepted and the high offer is rejected with probability r$$=$$ 67%. The high offer and the rejection rate r are at the upper endpoints of the prediction intervals for the semi-pooling equilibrium. The overall dispute rate under D1 is predicted to be 22% which is the lower endpoint of the semi-pooling prediction interval. Under D1, the disputes rates are 0% for A$$_{L}$$ and 67% for A$$_{H}$$. These are, respectively, the lower and upper limits of the dispute rates possible in the semi-pooling equilibrium. 17. In the screening game, once player A receives an offer, she faces no risk: she simply compares the offer to the fixed outcome at trial and chooses whichever is higher. The trial outcome is A’s minimally acceptable offer, which is fixed and therefore not affected by her risk aversion. Thus, a player B engaged in a sorting offer will make the offer in (1a) regardless of A’s risk preference. Under the theory, A$$_{L}$$ always accepts this offer so B incurs no rejection risk and makes the offer in (1a) independent of his own risk preference as well. In the signaling game, player A$$_{L}$$ incurs no risk by making a revealing offer, as this offer is always accepted. An A$$_{H}$$ player incurs risk with a high offer, but under the theory the next lower offer accepted with a positive probability (the low offer J$$^{L}$$$$+$$C$$_{B})$$ is dominated by her dispute payoff J$$^{H}$$– C$$_{A}$$. Thus, risk aversion will not cause A$$_{H}$$ to lower her offer. 18. Our design had a minimum of 12 rounds per session. Subjects were recruited for a two-hour period, and if time permitted we ran additional rounds. 19. We used randomizers (i.e., a die roll and drawing cards from a deck) to generate A and B decisions in the practice rounds, so as to avoid implicitly suggesting to participants what decisions to make during the experiment. 20. Generally speaking, most law and economics experiments which are a direct test of a theory are conducted in a context-free environment. See the discussion in Landeo (2018). 21. The information exchange took about 15–20 seconds. The entrances to the two rooms were directly adjacent so that when the experimenters silently exchanged information, they could closely monitor subjects. Each room had approximately forty seats, and subjects were dispersed so as to ensure privacy and prohibit communication. 22. The lump sum is $${\$}$$68 in the screening game and $${\$}$$73 in the signaling game. The higher lump sum in the signaling game reflects B’s higher expected cost per round in this game. This kept B’s ex ante earnings opportunities approximately equal across the two games. Since B did not know how many rounds the experiment would last, there was no way for him to compute a lump sum per round and make decisions based on such a number. 23. As expected, there is very little support for refinement D1 in the signaling game: only 5% (7/132) of the player A$$_{H}$$ offers match the D1 prediction of 525. 24. If, for example, we include all A$$_{L}$$ offers, this will include bluffs and between offers which provide a large negative surplus to player B if accepted. When all such offers are considered, the average offer from A$$_{L}$$ to B is 265, which represents a surplus of –40 for B . However, this number has little bearing on how A$$_{L}$$ behaves when she, via her offer, reveals her type and attempts to settle with B . Similarly, the inclusion of player B between or pooling offers has little bearing on how B behaves when he makes a screening offer and attempts to settle with A$$_{L}$$. The other offers we omit (i.e., those $$<$$ 75 or $$>$$ 525) are extremely rare, especially in the R7-end. 25. A$$_{L}$$ revealing offers appear to contain relatively less surplus than do B screening offers. The difference-in-means test H$$_{\mathrm{0}}$$: $$\mu_{\mathrm{B}}$$ – 75 $$=$$ 225 – $$\mu_{\mathrm{AL}}$$ has t$$=$$ 4.76 (P$$<$$ 0.000); also see fn. 26. 26. Eighty percent of the B screening offers are 75–125 (1/3 or less of the surplus), while 8% are 200–225 (5/6 or more of the surplus). For A$$_{L}$$ signaling game revealing offers, 81% are 200–225 (1/6 or less of the surplus) and 3% are 75–100 (5/6 or more of the surplus). 27. In the screening game, rejection rates are 25% on offers 75–99 (less than 1/6 of the surplus), 16% on offers 100–125, and 0% on offers above 125. In the signaling game, rejection rates are 13% for offers 201–225 (less than 1/6 of the surplus), and 3% on offers 200 and below. 28. Again, B does not know A’s type upon receiving the offer. Aggregating the signaling game offers 226–374 from Tables 5 and 6, the player B rejection rate for this interval is 55% (35/64) overall, 50% (8/16) in R7-end. 29. A$$_{L}$$ makes 94% of the between offers, so rejection is empirically justified. This anomalous acceptance behavior is fairly widespread among the player B subjects. Half (17/34) accepted at least one between offer, and one-fifth (7/34) did so twice or more. 30. Using Equation (5) and the Table 4 mean A$$_{H}$$ separating offer of 455, r$$=$$ 60.5% (median 450 yields r$$=$$ 60.0%). 31. Using Equation (5), the Table 4 means and medians on offers 375–525 imply a dispute rate of about 56–61%. 32. This explains why the predictions under the refinement D1 are not borne out. The prediction under D1 is that A$$_{H}$$ makes an offer of 525 which is rejected 67% of the time. Empirically these offers are rejected 100% of the time and as a result, A$$_{H}$$ needs to offer well below the D1 prediction to have a positive probability of acceptance. 33. One, in both games excess A$$_{L}$$ v. B disputes occur on offers 75–225. Two, in the signaling game the dispute rate on A$$_{L}$$ bluffs is higher than expected. Three, A$$_{L}$$ unexpectedly makes between offers 226 and 374 in the signaling game and B’s rejection rate is high on those offers, albeit well below the predicted 100%. 34. Our analysis focuses on the mean rather than the median. Because of the high dispute rates for A$$_{H}$$ players, the median is rather uninformative. When dispute rates are 50% or higher, the median payoff will always be 375 for these players. Thus, the median surplus earned by A$$_{H}$$ players is 0 in both games. 35. Also note that the refinement D1 is rejected as descriptive of players’ belief formation. This is in line with other experimental work which has generally rejected the intuitive criterion, a weaker refinement concept than D1. Here, a primary reason D1 fails is the defendant’s (player B’s) uniform rejection of very high offers, which forces the strong A$$_{H}$$ plaintiff to experiment with offers well below the D1 prediction. 36. 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Published by Oxford University Press on behalf of the American Law and Economics Association. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png American Law and Economics Review Oxford University Press

An Empirical Analysis of the Signaling and Screening Models of Litigation

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Abstract

Abstract We present an experimental analysis of the signaling and screening models of litigation. In both models, bargaining failure is driven by asymmetric information. The difference between the models lies in the bargaining structure: In the signaling game, the informed party makes the final offer, while in the screening game the uninformed party makes the final offer. We conduct experiments for both models under a common set of parameter values, allowing only the identity of the party making the final offer to change. We find the anomalous behavior to be more common in the signaling game, but the frequency of this behavior diminishes in the later rounds of the experiment. Across both games, in the later rounds of the experiment over 90% of offers are consistent with the theory. Having the right to make the offer raises a player’s expected payoffs, but by much less than is predicted by theory. Dispute rates across the two games are approximately equal. 1. Introduction Asymmetric information is a leading explanation for the existence of bargaining failures which result in costly litigation. We consider a stylized legal bargaining framework in which an informed “plaintiff” knows whether she has either a strong or weak case against an uninformed “defendant.” Within this class of games, when the informed party makes the final offer, it is called a signaling model, and when the uninformed party makes the final offer it is called the screening model. These informational-based models underlie much of the theoretical work on pretrial bargaining. We present signaling and screening experiments in a litigation context that allows for a side-by-side comparison of each model’s performance.1, Our ultimate goal is to gain a better understanding of the factors which lead to disputes, and understanding behavior in these canonical models of the litigation literature is an important step in this process. Under the parameterization we utilize, the theory predicts distributional effects of moving from a screening game to a signaling game and allows for possible efficiency effects. Typically, the identity of the individual making the offer is not viewed as a choice variable, but if this choice has consequences for efficiency, this identity might be specified in a contracting relationship in which disputes are anticipated. For example, this is observed in contracts specifying the rules for the dissolution of partnerships.2 However, the lack of such terms in contracting situations in which litigation is the dispute resolution mechanism suggests small efficiency and distributional effects associated with the identity of the party making the offer. Under the theory, the dispute rate in our signaling game can be anywhere from zero to eleven percentage points lower than the dispute rate in the screening game.3 We find that dispute rates across the two games are approximately equal. Thus, there are no significant efficiency effects associated with the right to make an offer. In addition, while having the right to make the offer is valuable, it is only about 30–55% as valuable as predicted by theory. Thus in our experiment, the distributional consequences of moving from a screening game to a signaling game are much less than what theory predicts. The signaling game appears to place high cognitive demands upon experimental subjects, and the comparison to the screening game gives us insight about the extent to which this is true. We observe more anomalous offers in the signaling game than in the screening game, but their frequency decreases substantially over time in the signaling experiment. After the sixth round, in both games over 90% of the offers are consistent with the theory. On the other hand, anomalous acceptance behavior, which is almost entirely absent in the screening game, persists in the later rounds of the signaling experiment. As our experimental protocol in the two games is identical except for the identity of the party making the offer, we attribute the differences in anomalous behavior to the greater cognitive demands imposed by the signaling game. In our conclusion, we discuss some implications for naturally occurring litigation. 2. Background There is a large literature in law and economics where trials are an equilibrium outcome of games in which there is asymmetric information. In the screening model of Bebchuk (1984), the uninformed party makes an offer to the informed party. In the signaling model of Reinganum Wilde (1986), the informed party makes an offer to the uninformed party. Much of the subsequent literature on pretrial bargaining is built upon these models. Among other things, this literature analyzes how institutions such as fee shifting and contingency fees affect settlement rates.4 Our experiment is closely based on the Bebchuk and Reinganum and Wilde models. There is an extensive experimental law and economics literature.5 With a few exceptions, this work does not analyze the standard models of pretrial bargaining in which disputes result from asymmetric information. The extensive literature on arbitration has focused on how dispute rates are affected by the choice of arbitration procedure, but this is generally done in a setting of symmetric information.6 A variety of issues have been examined experimentally in the civil litigation literature. These include conditional cost shifting under which the shifting of certain trial costs is a function of either the outcome of trial or of the relationship between the trial outcome and offers made in the pretrial negotiation period. (See, e.g., Coursey and Stanley, 1988 and Main and Park, 2002.) Again, in those experiments, the players are symmetrically informed. In work testing of the Priest and Klein’s (1984) selection hypothesis there is two-sided asymmetric information.7 That work does not constitute a direct evaluation of either the screening or the signaling models that we analyze here. Some of our previous work has used the screening model of litigation as a baseline in the context of analyzing other issues. Pecorino and Van Boening (2004, 2015) analyze voluntary disclosure and Pecorino and Van Boening (2014) consider the effects of asymmetric dispute costs on settlement behavior. Here, we also present the screening game as a baseline. The signaling game serves as the treatment, that is, the switch from having the uninformed party make the offer to having the informed party make the offer. To our knowledge, there has been no systematic empirical comparison of these two games in the litigation context. In standard models of litigation (e.g., Reinganum Wilde (1986) and Bebchuk (1984)), there is an ultimatum game embedded in the larger game of pretrial bargaining between the plaintiff and defendant. The “pie” is the joint saving which is achieved when a settlement is reached and the costs of trial avoided. Because fairness concerns are absent in these models, the recipient is willing to accept an offer which equals her expected payoff at trial. However, from the large experimental literature on the ultimatum game we know behaviorally that the recipient of an offer generally demands some share of the joint surplus and that if too little surplus is proposed, then the offer will be rejected.8 Thus, prior experimental work leads to the expectation that more surplus than predicted under the standard theory will be provided in settlement offers and that there will be disputes not predicted by the theory when the parties cannot agree on what constitutes a fair offer. Both of these behavioral elements are present in our previous work (Pecorino and Van Boening, 2004, 2014, and 2015) involving the screening game. There has been a fairly extensive analysis of signaling games in the experimental economics literature. Some of this work has focused on the performance of certain equilibrium refinement concepts. For example, Brandts and Holt (1992) find mixed support for the refinement concept known as the intuitive criterion, which places restrictions on out of equilibrium beliefs.9Banks et al. (1994) test a range of refinement concepts and find mixed evidence that these refinements are predictive of behavior. Cooper et al. (1997) provide evidence against the empirical validity of the intuitive criterion. None of this previous literature has been in a litigation setting.10 In the civil litigation literature, the refinement D1 (Cho and Kreps, 1987) is required to eliminate all but the pure strategy separating equilibrium which has typically been the focus of this theoretical work. The D1 refinement is stronger than the intuitive criterion meaning it is less likely to hold empirically. Our data reject D1, but given the results cited above, this is unsurprising. Accordingly, our analysis does not focus on the validity of D1. We focus instead on a semi-pooling equilibrium (described below), as our results are more consistent with that prediction. We comment on the failure of the D1 refinement concept as appropriate. 3. The Theory The screening model we describe is a simplified version of Bebchuk (1984), while the signaling model is a simplified version of Reinganum Wilde (1986).11 Both the plaintiff and the defendant are risk neutral. In all of our analyses, the probability that the plaintiff prevails at trial is common knowledge, and we furthermore assume that this probability equals 1. In presenting the theory, we will use the parameter values from our experiment to generate the relevant predictions and we will use the same terminology that we later use in the Section 5. Thus, we will refer to the informed plaintiff as player $$A$$ and the uniformed defendant as player $$B$$. At trial, income is transferred from $$B$$ to $$A$$ which is what distinguishes the player roles. The plaintiff is either type $$A_{H}$$ with a strong case or type $$A_{L}$$ with a weak case. If the case proceeds to trial, player $$A$$ receives judgment $$J^{ i}$$, $$i = H, L$$ with $$J^{ H}$$$$>$$$$J^{ L}$$. Player $$A$$ knows her type, but $$B$$ only knows that with probability $$q$$ he faces a type $$A_{H}$$ plaintiff and with probability $$1 -q$$ he faces a type $$A_{L}$$ plaintiff. The court costs for $$A$$ and $$B$$ are, respectively, $$C_{A}$$ and $$C_{B}$$. These costs are incurred only if the case proceeds to trial. Using this simple environment, we first present the screening model and then the signaling model. 3.1. The Screening Game The stages of the screening game are as follows: 0. Nature determines player $$A$$’s type which is $$A_{H}$$ with probability $$q$$ and $$A_{L}$$ with probability 1 – $$q$$. Player $$A$$ knows her type, but $$B$$ knows only the probability $$q$$ that she is $$A_{H}$$. 1. Player $$B$$ makes an offer $$O_{B}$$ to player $$A$$. 2. Player $$A$$ accepts or rejects the offer. If the offer is accepted, the game ends with the player $$A$$ receiving a payoff of $$O_{B}$$ and player $$B$$ receiving a payoff of –$$O_{B}$$ (i.e., $$B$$ incurs a cost equal to $$O_{B})$$. 3. If the offer is rejected, trial occurs. Player $$A$$ receives the payoff $$J^{i}$$ – $$C_{A}$$ and player $$B$$ receives the payoff –($$J^{i} + C_{B})$$, where $$i = H, L$$. Note that our game description implies the use of the American rule under which both parties bear their own costs of trial. In our experiment we use the following parameter values: $$J^{L} = {\$}1.50$$, $$J^{H} = {\$}4.50$$, $$q = 1/3$$, and $$C_{A} = C_{B} = {\$}0.75$$. In the experiment we present all values in pennies so that, for example, $${\$}$$1.50 is written as 150. We follow that convention in what follows. The plaintiff $$A$$ will accept any offer that leaves her at least as well off as the expected outcome at trial. In other words, $$A_{i}$$ will accept any offer such that $$O_{B} \geqslant J^{i} - C_{A}$$. The defendant $$B$$ is free to make any offer he chooses, but the optimal offer will be one of the following:   \begin{align} O^L_B & = J^{L} - C_{A} = 75,\\ \end{align} (1a)  \begin{align} O^H_B & = J^{H} - C_{A} = 375. \end{align} (1b) Player $$B$$ will choose either the low screening offer $$O_B^L$$ that only $$A_{L}$$ will accept or the high pooling offer $$O_B^H$$ that both plaintiff types will accept. He offers $$O_B^L$$ if   \begin{align} (1 - q)O^L_B + q(J^{H} + C_{B}) < O^H_B. \end{align} (2) The left-hand side represents the expected payout from the offer $$O^L_B$$ which is accepted with probability 1 – $$q$$ and rejected with probability $$q $$ (the probability that $$A_{H}$$ is encountered). When the offer is rejected, $$B$$ proceeds to trial and pays $$J^{H} + C_{B}$$. The right-hand side is the $$B$$’s payout from the higher offer, which is accepted by both player $$A$$ types. Screening offers are made in this model when the probability $$q$$ of encountering a high damage plaintiff is sufficiently small. Rearranging Equation (2), while making use of (1), the condition for $$B$$ to make the low offer may be expressed as follows:   \begin{align} 1/3 = q < \frac{J^H - J^L}{(J^H - J^L)+ C_A + C_B} = 0.67. \end{align} (3) Thus, under our parameter values, the condition is met and we have the predictions that a low screening offer $$O^L_B$$ will be made and that trial occurs with probability $$q = 1/3$$. The predicted dispute rates in our screening game are 0% for $$A_{L}$$, 100% for $$A_{H}$$, and 33% overall. The predictions of this game are summarized in Table 1, which is found in Section 3.3. Table 1. Predictions by Game Type Game  Sender: offer  Dispute rate  $$A$$’s payoff  Recipient decision  Screening  $$B{:}\ O_{B}^{L} =75$$  $$A_{L}$$: 0%  $$A_{L}$$: 75  $$A_{L}$$ accepts $$O_{B} \geqslant $$ 75.        $$A_{H}$$: 100%  $$A_{H}$$: 375  $$A_{H}$$ accepts $$O_{B} \geqslant $$ 375.        $$A$$: 33%  $$A$$: 175$$^{\mathrm{a}}$$     Signaling$$^{\mathrm{b}}$$  $$A_{L}{:}\ O_{A}^{L} =225,$$  $$A_{L}$$: $$\underline {0}$$–25%  $$A_{L}$$: $$\underline {225}$$  $$B$$ accepts $$O_{A} \leqslant $$ 225, and     $$\phantom{A_L{:}}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A_{H}$$: 50–$$\underline {67}$$%  $$A_H$$: 375–$$\underline {425}$$$$^{\mathrm{c}}$$  $$\quad$$ rejects $$O_{A}^{S} \in [375-\underline{525}]$$     $$A_{H}{:}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A$$: $$\underline {22}$$–33%  $$A$$: 275–$$\underline {292}$$$$^{\mathrm{a}}$$  $$\quad$$ at rate $$r$$.$$^{\mathrm{d}}$$  Game  Sender: offer  Dispute rate  $$A$$’s payoff  Recipient decision  Screening  $$B{:}\ O_{B}^{L} =75$$  $$A_{L}$$: 0%  $$A_{L}$$: 75  $$A_{L}$$ accepts $$O_{B} \geqslant $$ 75.        $$A_{H}$$: 100%  $$A_{H}$$: 375  $$A_{H}$$ accepts $$O_{B} \geqslant $$ 375.        $$A$$: 33%  $$A$$: 175$$^{\mathrm{a}}$$     Signaling$$^{\mathrm{b}}$$  $$A_{L}{:}\ O_{A}^{L} =225,$$  $$A_{L}$$: $$\underline {0}$$–25%  $$A_{L}$$: $$\underline {225}$$  $$B$$ accepts $$O_{A} \leqslant $$ 225, and     $$\phantom{A_L{:}}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A_{H}$$: 50–$$\underline {67}$$%  $$A_H$$: 375–$$\underline {425}$$$$^{\mathrm{c}}$$  $$\quad$$ rejects $$O_{A}^{S} \in [375-\underline{525}]$$     $$A_{H}{:}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A$$: $$\underline {22}$$–33%  $$A$$: 275–$$\underline {292}$$$$^{\mathrm{a}}$$  $$\quad$$ at rate $$r$$.$$^{\mathrm{d}}$$  $$^{\mathrm{a}}$$Player $$A $$ expected payoff $$=$$ (1 – $$q)(A_{L}$$ payoff) $$+ q(A_{H}$$ payoff) with $$q =$$ 1/3. $$^{\mathrm{b}}$$Semi-pooling equilibrium in which $$A_{L}$$ and $$A_{H}$$ pool on a single value of $$O_{A}^{S}$$. Underlined values are D1 predictions (see text fn. 16). $$^{\mathrm{c}}$$Player $$A_{H} $$ expected payoff $$= r(375) + (1 - r) O_{A}^{S}$$. See text Equation (6) for $$r$$. $$^{\mathrm{d}}$$$$B$$ also rejects any 226 $$\leqslant O_{A} \leqslant $$ 374. Table 1. Predictions by Game Type Game  Sender: offer  Dispute rate  $$A$$’s payoff  Recipient decision  Screening  $$B{:}\ O_{B}^{L} =75$$  $$A_{L}$$: 0%  $$A_{L}$$: 75  $$A_{L}$$ accepts $$O_{B} \geqslant $$ 75.        $$A_{H}$$: 100%  $$A_{H}$$: 375  $$A_{H}$$ accepts $$O_{B} \geqslant $$ 375.        $$A$$: 33%  $$A$$: 175$$^{\mathrm{a}}$$     Signaling$$^{\mathrm{b}}$$  $$A_{L}{:}\ O_{A}^{L} =225,$$  $$A_{L}$$: $$\underline {0}$$–25%  $$A_{L}$$: $$\underline {225}$$  $$B$$ accepts $$O_{A} \leqslant $$ 225, and     $$\phantom{A_L{:}}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A_{H}$$: 50–$$\underline {67}$$%  $$A_H$$: 375–$$\underline {425}$$$$^{\mathrm{c}}$$  $$\quad$$ rejects $$O_{A}^{S} \in [375-\underline{525}]$$     $$A_{H}{:}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A$$: $$\underline {22}$$–33%  $$A$$: 275–$$\underline {292}$$$$^{\mathrm{a}}$$  $$\quad$$ at rate $$r$$.$$^{\mathrm{d}}$$  Game  Sender: offer  Dispute rate  $$A$$’s payoff  Recipient decision  Screening  $$B{:}\ O_{B}^{L} =75$$  $$A_{L}$$: 0%  $$A_{L}$$: 75  $$A_{L}$$ accepts $$O_{B} \geqslant $$ 75.        $$A_{H}$$: 100%  $$A_{H}$$: 375  $$A_{H}$$ accepts $$O_{B} \geqslant $$ 375.        $$A$$: 33%  $$A$$: 175$$^{\mathrm{a}}$$     Signaling$$^{\mathrm{b}}$$  $$A_{L}{:}\ O_{A}^{L} =225,$$  $$A_{L}$$: $$\underline {0}$$–25%  $$A_{L}$$: $$\underline {225}$$  $$B$$ accepts $$O_{A} \leqslant $$ 225, and     $$\phantom{A_L{:}}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A_{H}$$: 50–$$\underline {67}$$%  $$A_H$$: 375–$$\underline {425}$$$$^{\mathrm{c}}$$  $$\quad$$ rejects $$O_{A}^{S} \in [375-\underline{525}]$$     $$A_{H}{:}\ O_{A}^{S} \in [375-\underline{525}]$$  $$A$$: $$\underline {22}$$–33%  $$A$$: 275–$$\underline {292}$$$$^{\mathrm{a}}$$  $$\quad$$ at rate $$r$$.$$^{\mathrm{d}}$$  $$^{\mathrm{a}}$$Player $$A $$ expected payoff $$=$$ (1 – $$q)(A_{L}$$ payoff) $$+ q(A_{H}$$ payoff) with $$q =$$ 1/3. $$^{\mathrm{b}}$$Semi-pooling equilibrium in which $$A_{L}$$ and $$A_{H}$$ pool on a single value of $$O_{A}^{S}$$. Underlined values are D1 predictions (see text fn. 16). $$^{\mathrm{c}}$$Player $$A_{H} $$ expected payoff $$= r(375) + (1 - r) O_{A}^{S}$$. See text Equation (6) for $$r$$. $$^{\mathrm{d}}$$$$B$$ also rejects any 226 $$\leqslant O_{A} \leqslant $$ 374. 3.2. The Signaling Game The stages of the signaling game are similar to those above with 1$$^\prime $$ and 2$$^\prime $$ replacing 1 and 2. 1$$'$$. Player $$A$$ makes an offer $$O_{A}$$ to player $$B$$. 2$$'$$. Player $$B$$ accepts or rejects the offer. If the offer is accepted, the game ends with player $$A$$ receiving payoff $$O_{A}$$ and player $$B$$ receiving payoff –$$O_{A}$$. The defining feature of the signaling game is that the informed player ($$A)$$ makes the offer to the uninformed player ($$B)$$. Multiple equilibria are a problem in signaling games. In this particular game, the refinement concept D1 has been used to eliminate all but a pure strategy separating equilibrium.12 However, because prior experiments strongly suggest that we should not expect to find support for D1 in our data, we will present predictions for semi-pooling equilibria instead.13 It is worth noting that the model predictions under D1 are endpoints of the predictions intervals for semi-pooling equilibria. Thus, the D1 predictions are a special case of the semi-pooling predictions. Before proceeding, it is useful to present the offers by each player $$A$$ type in a full information game in which $$B$$ knows $$A$$’s type. The offers are as follows:   \begin{align} O^L_A & = J^{L} + C_{B} = 225,\\ \end{align} (4a)  \begin{align} O^H_A & = J^{H} + C_{B} = 525. \end{align} (4b) These offers represent $$B$$’s dispute payoff against each type. This reflects the fact that the theory embeds an ultimatum game in which player $$A$$ has all of the bargaining power. The offer in (4a) will be the revealing offer by $$A_{L}$$ in what follows, while the offer in (4b) represents the upper bound on possible semi-pooling offers.14 In the semi-pooling equilibrium all $$A_{H}$$ players offer $$O_{A}^{S} $$, where $$J^{H} - C_{A} \leqslant O_A^S < O_{A}^{H}$$ (or $$375 \leqslant O^S_A < 525$$). If only $$A_{H}$$ made the offer $$O_{A}^{S} $$, player $$B$$ would accept it, but if player $$B$$ always accepts this, $$A_{L}$$ would also offer $$O_{A}^{S} $$. Thus, there is an equilibrium in mixed strategies in which $$A_{L}$$ sometimes bluffs by offering $$O_{A}^{S} $$ and in which player $$B$$ sometimes rejects $$O_{A}^{S}$$. A mixed strategy equilibrium requires the players to be indifferent between the strategies they mix between. Player $$A_{L}$$ uses a mixed strategy in which she makes the revealing offer $$O_{A}^{L}$$ with probability 1–b and bluffs by offering $$O_{A}^{S}$$ with probability $$b$$. The low offer $$O_{A}^{L}$$ is accepted with probability 1. If the higher offer $$O_{A}^{S}$$ is rejected with probability   \begin{align} r = \frac{O_{A}^{S} -J^L - C_B}{O_{A}^{S} - J^L + C_A} = \frac{O_{A}^{S} - 225}{O_{A}^{S} - 75}, \end{align} (5) then $$A_{L}$$ will be indifferent between the two offers. Upon observing $$O_{A}^{S} $$, player $$B$$ uses Bayes’ rule and the equilibrium probability that $$A_{L}$$ bluffs to update his beliefs about $$A$$’s type. When the probability $$A_{L}$$ bluffs is   \begin{align} b = \left(\frac{q}{1-q}\right) \frac{J^H + C_B + O_{A}^{S}}{O_{A}^{S} - J^L - C_B} = (0.5) \frac{525 - O^S_A}{O_{A}^{S} - 225}, \end{align} (6) then player $$B$$ is indifferent between accepting or rejecting $$O_{A}^{S}$$. The semi-pooling offer can be any offer such that $$375 \leqslant O_{A}^{S} \leqslant 525$$. In our experiment offers are restricted to be whole numbers. Thus, under our design there are 150 possible semi-pooling offers. However, in any particular semi-pooling equilibrium, all the $$A_{H}$$ players and all of the bluffing $$A_{L}$$ players will pool on one and only one value of $$O_{A}^{S}$$.15 Using Equation (5), the predicted dispute rate on semi-pooling offers ranges from $$r =$$ 50% if the offer is $$O_{A}^{S} = 375$$ up to $$r = 67\%$$ if the offer is $$O_{A}^{S} = 525$$. The expected payoff to $$A_{H}$$ in a semi-pooling equilibrium can range from 375 to 425. Using (5) and (6), the dispute rate for $$A_{L}$$ can range from 0% to 25% and the overall dispute rate can range from 22% to 33% in a semi-pooling equilibrium.16 The out-of-equilibrium beliefs and actions are as follows: It is a dominant strategy for $$B$$ to accept an offer $$O_{A} < O_{A}^{L} = 225$$ and this offer is accepted with probability 1. Likewise, it is a dominant strategy for $$B$$ to reject an offer $$O_{A} > O_{A}^{H} = 525$$ and this offer is rejected with probability 1. An offer $$> 226$$ which is not equal to $$O_{A}^{S}$$ is believed to be from $$A_{L}$$ and is rejected with probability 1. 3.3. Predictions Table 1 summarizes our predictions. We focus on three sets of predictions and their comparative statics across the two games. The first involves offers. The predicted difference between player $$B$$ screening offers $$O_{B}^{L}$$ and $$A_{L}$$ revealing offers $$O_{A}^{L}$$ is 150 (75 v. 225). In the signaling game $$A_{L}$$ bluffs and $$A_{H}$$ separating offers are predicted to coincide on a unique semi-pooling offer $$O_{A}^{S} $$. The second involves dispute rates and potential efficiency gains. For offers in the range 75–225, the $$A_{L}$$ v. $$B$$ dispute rate is unaffected by the game structure (both are 0%). In the signaling game, the overall $$A_{L}$$ dispute rate could be as high as 25% because of bluffs by $$A_{L}$$. The predicted $$A_{H}$$ v. $$B $$ dispute rate is 33–50 percentage points lower in the semi-pooling equilibrium compared with the screening game. The overall $$A$$ v. $$B $$ dispute rate could range from no difference (both 33%) to 11 percentage points lower in the signaling game (22% v. 33%). The third involves distributional effects on the player $$A $$ payoffs. Moving from the screening game to the signaling game, the $$A_{L}$$ payoff increases by 150, from 75 to 225. The $$A_{H}$$ expected payoff increases by anywhere from 0 to 50. In total, $$A$$’s expected payoff increases in a range from 100 to 117. These increases represent the value to $$A$$ of having the offer, that is, the expected increase in compensation the plaintiff receives when she has the right to make the settlement offer instead of the defendant. While the theoretical predictions in Table 1 provide benchmarks, these need to be viewed in light of the embedded ultimatum game discussed above in Section 2. Here, the embedded pie is the $$C_{A} + C_{B} =$$ 150 joint surplus from settlement which constitutes the bargained over amount (Pecorino and Van Boening, 2010), and we are skeptical that empirically the offering party will be able to extract all of the surplus. For example, in the screening game it may be the case that $$A_{L}$$ needs $$O_{B}$$$$>$$ 75 before she will accept the offer. When presenting the results, we label offers according to intervals consistent with the theoretical predictions, but allowing for a sharing of the joint surplus from settlement. In the screening game the point prediction for player $$B$$’s offer is $$O_{B} = 75$$, but any offer 75 $$<$$$$O_{B}$$$$<$$ 225 is consistent with the screening behavior, as it is theoretically acceptable to $$A_{L}$$ but not $$A_{H}$$, and leaves both $$A_{L}$$ and $$B$$ with nonnegative surplus from settlement. In the screening game we refer to: Screening offer: An offer by player $$B$$ in the interval 75 $$\leqslant O_{B} \leqslant $$ 225. Pooling offer: An offer by player $$B$$ in the interval 375 $$\leqslant O_{B} \leqslant $$ 525. Analogous reasoning applies in the signaling game. In the signaling game, we refer to: Revealing offer: An offer by player $$A_{L}$$ in the interval 75 $$\leqslant O_{A}$$$$\leqslant $$ 225. Bluffing offer: An offer by player $$A_{L}$$ in the interval 375 $$\leqslant O_{A}$$$$\leqslant $$ 525. Separating offer: An offer by player $$A_{H}$$ in the interval 375 $$\leqslant O_{A}$$$$\leqslant $$ 525. Collectively, we refer to subjects’ offers in these intervals as “consistent with theory.” We are also interested in the frequency of anomalous offers in each game, because ex ante we believe that the signaling game places greater cognitive demands on both players than does the screening game. In both games “between” offers are anomalous: Between offer: An offer by either player in the interval 226 $$\leqslant O \leqslant $$ 374. For player $$B$$ in the screening game this offer is too low to be accepted by $$A_{H}$$, but offers $$>$$ 100% of the surplus from settlement to $$A_{L}$$. In the signaling game, these offers should be rejected by $$B$$ at a 100% rate using the following reasoning: For $$A_{H}$$, an offer in the 226–374 range is dominated by an offer of 375 or more, since the latter equals or exceeds the $$A_{H}$$ dispute payoff. Thus, $$B$$ should place 100% weight on the offer being made by $$A_{L}$$ and reject it as it exceeds his dispute payout of 225 v. $$A_{L}$$. A failure to do so violates the “test of dominated strategies”, a refinement which is weaker (and therefore more likely to be empirically valid) than either the intuitive criterion or D1. (See Kreps, 1990, p. 436.) Given this expected response by $$B$$, $$A_{L}$$ should never make an offer in this range. Any offer $$<$$ 75 or $$>$$ 525 is also anomalous in both games, but we observe very few of these offers, especially in the later rounds of the experiment. In contrast, “between” offers are significantly more common. Note that our predictions are fairly robust to the introduction of risk aversion on the part of either party.17 In the screening game, if $$B$$ is very risk averse, it could induce him to make a pooling offer when a screening offer is otherwise predicted. In the experiment, we observe few (~ 5%) pooling offers. In the signaling game, risk aversion affects the precise mixing probabilities that make $$A_{L}$$ indifferent between a revealing offer and a bluff and $$B$$ indifferent between accepting or rejecting $$O_{A}^{S}$$. 4. Experimental Design Table 2 summarizes the ten sessions in our experimental design, five screening game sessions and five signaling game sessions. Subjects were recruited from summer business classes at the University of Alabama. The number of bargaining pairs per session ranges from 5 to 8, while each session lasted 12–14 rounds18 (see table notes for session-specific details). The player in the role of the plaintiff is referred to as player $$A $$ and the player in the role of the defendant is referred to as player $$B$$. Subjects were not informed ahead of time how many rounds there would be. A typical session, inclusive of an instructional period at the beginning and private payment at the end, lasted between one-and-a-half and two hours. Average payoffs were about $${\$}$$31, ranging from $${\$}$$17 to $${\$}$$47. Subjects were not paid a show-up fee; all earnings were from decision-making. Table 2. Experimental Design             No. of negotiations     Game  No. of sessions  Pairs  Rounds  $$n $$  $$B$$ v. $$ A_{L}$$  $$B $$ v. $$A_{H}$$  Average earnings(min, max)  Screening  5$$^{\mathrm{a}}$$  31  65  402  273  129  $${\$}$$31.89 ($${\$}$$21, $${\$}$$46)  Signaling  5$$^{\mathrm{b}}$$  34  60  408  276  132  $${\$}$$30.63 ($${\$}$$17, $${\$}$$47)              No. of negotiations     Game  No. of sessions  Pairs  Rounds  $$n $$  $$B$$ v. $$ A_{L}$$  $$B $$ v. $$A_{H}$$  Average earnings(min, max)  Screening  5$$^{\mathrm{a}}$$  31  65  402  273  129  $${\$}$$31.89 ($${\$}$$21, $${\$}$$46)  Signaling  5$$^{\mathrm{b}}$$  34  60  408  276  132  $${\$}$$30.63 ($${\$}$$17, $${\$}$$47)  $$^{\mathrm{a}}$$Session Scr1: 7 pairs, 12 rounds, $$n =$$ 84 negotiations (49 $$B$$ v. $$A_{L}$$, 35 $$B$$ v. $$A_{H})$$. Scr2: 5 pairs, 13 rounds, $$n =$$ 65 (47, 18). Scr3: 8 pairs, 13 rounds, $$n =$$ 104 (75, 29). Scr4: 5 pairs, 13 rounds, $$n =$$ 65 (41, 24). Scr5: 6 pairs, 14 rounds, $$n =$$ 84 (61, 23). $$^{\mathrm{b}}$$Session Sig1: 6 pairs, 12 rounds, $$n =$$ 72 negotiations (50 $$B$$ v. $$A_{L}$$, 22 $$B$$ v. $$A_{H})$$. Sig2: 8 pairs, 12 rounds, $$n =$$ 96 (61, 35). Sig3: 8 pairs, 12 rounds, $$n =$$ 96 (68, 28). Sig4: 5 pairs, 12 rounds, $$n =$$ 60 (42, 18). Sig5: 7 pairs, 12 rounds, $$n =$$ 84 (55, 29). Table 2. Experimental Design             No. of negotiations     Game  No. of sessions  Pairs  Rounds  $$n $$  $$B$$ v. $$ A_{L}$$  $$B $$ v. $$A_{H}$$  Average earnings(min, max)  Screening  5$$^{\mathrm{a}}$$  31  65  402  273  129  $${\$}$$31.89 ($${\$}$$21, $${\$}$$46)  Signaling  5$$^{\mathrm{b}}$$  34  60  408  276  132  $${\$}$$30.63 ($${\$}$$17, $${\$}$$47)              No. of negotiations     Game  No. of sessions  Pairs  Rounds  $$n $$  $$B$$ v. $$ A_{L}$$  $$B $$ v. $$A_{H}$$  Average earnings(min, max)  Screening  5$$^{\mathrm{a}}$$  31  65  402  273  129  $${\$}$$31.89 ($${\$}$$21, $${\$}$$46)  Signaling  5$$^{\mathrm{b}}$$  34  60  408  276  132  $${\$}$$30.63 ($${\$}$$17, $${\$}$$47)  $$^{\mathrm{a}}$$Session Scr1: 7 pairs, 12 rounds, $$n =$$ 84 negotiations (49 $$B$$ v. $$A_{L}$$, 35 $$B$$ v. $$A_{H})$$. Scr2: 5 pairs, 13 rounds, $$n =$$ 65 (47, 18). Scr3: 8 pairs, 13 rounds, $$n =$$ 104 (75, 29). Scr4: 5 pairs, 13 rounds, $$n =$$ 65 (41, 24). Scr5: 6 pairs, 14 rounds, $$n =$$ 84 (61, 23). $$^{\mathrm{b}}$$Session Sig1: 6 pairs, 12 rounds, $$n =$$ 72 negotiations (50 $$B$$ v. $$A_{L}$$, 22 $$B$$ v. $$A_{H})$$. Sig2: 8 pairs, 12 rounds, $$n =$$ 96 (61, 35). Sig3: 8 pairs, 12 rounds, $$n =$$ 96 (68, 28). Sig4: 5 pairs, 12 rounds, $$n =$$ 60 (42, 18). Sig5: 7 pairs, 12 rounds, $$n =$$ 84 (55, 29). Upon arrival, subjects were randomly assigned to one of two adjacent rooms, one for player $$A$$ and the other for player $$B$$. An experimenter was assigned to each room. All subjects received common instructions, which included step-by-step practice rounds and earnings calculations from the perspective of both player $$A$$ and player $$B$$.19 Subjects were not informed of their role until the end of the instructions and they maintained that same role throughout the session. The experiment did not utilize verbiage like plaintiff, defendant, judgment at trial, court costs, etc.20 Each subject had a private Record Sheet on which to write decisions, and each experimenter had forms (not visible to subjects) on which to record these decisions. The written decision consisted of either an offer (sender) or an accept/reject choice (recipient). After all subjects in a room had made their decisions, the experimenters met at the entrances to the adjacent rooms, silently copied information from one another’s forms, and then returned to the rooms and wrote the results on the respective subject’s Record Sheet.21 Each round, a subject’s feedback was limited to the offer, accept/reject decision and payoff/cost specific to his or her negotiation, that is, there was no dissemination of information on outcomes for other bargaining pairs. Other than the decisions transmitted by experimenters between the two rooms, there was no communication between the $$A $$ and $$B $$ players or among players within a room. In both experiments, $$A$$’s payoff is the sum of her payoffs from all rounds and $$B$$’s payoff is a lump sum minus the sum of his costs from all rounds.22 The lump sum is known in advance by player $$B$$ but is never revealed to player $$A$$. In all sessions the strangers matching protocol is used, with new random and anonymous pairings prior to the start of each and every round. The sequence in a round of the screening game is as follows: 0. A six-sided die is rolled privately for each Player $$A$$, where a roll of 1, 2, 3, or 4 is called outcome $$L$$ and a roll of 5 or 6 is called outcome $$H$$. Only $$A$$ knows the outcome of the die roll. Player $$B$$ knows how the die roll maps into the outcomes. 1. Player $$B$$ decides on an offer to submit to Player $$A$$. This offer may be any whole number between (and including) 0 and 699. Player $$B$$’s offer is then communicated to Player $$A$$. Player $$A$$ is given a few moments to decide whether or not to accept the offer. Player $$A$$’s decision is then communicated to Player $$B$$. 2. If Player $$A$$ accepts Player $$B$$’s offer, then the round is over for that pair. Players $$A$$’s Payoff for the round $$=$$ Player $$B$$’s offer Player $$B$$’s Cost for the round $$=$$ Player $$B$$’s offer. 3. If Player $$A$$ does not accept $$B$$’s offer, both $$A$$ and $$B$$ incur a fee of 75. $$A$$’s payoff and $$B$$’s cost for the round depend on the die roll and the fees: Under outcome L: Player $$A$$’s Payoff for the round $$=$$ 150 – 75 $$=$$ 75 Player $$B$$’s Cost for the round $$=$$ 150 $$+$$ 75 $$=$$ 225 Under outcome H: Player $$A$$’s Payoff for the round $$=$$ 450 – 75 $$=$$ 375 Player $$B$$’s Cost for the round $$=$$ 450 $$+$$ 75 $$=$$ 525. The description of the game above is very similar in language and appearance to that used in the subjects’ instructions. The information in step 3 was displayed in both rooms by the use of overhead projectors. The overheads included the statement that the same information was displayed in both rooms. The parameters and procedures for the signaling game are identical to the screening game, except that player $$A$$ makes the take-or-leave-it offer to player $$B$$. The steps of a round are identical to the screening game except for the following modifications. 1$$'$$. Player $$A$$ decides on an offer to submit to Player $$B$$. This offer may be any whole number between (and including) 0 and 699. Player $$A$$’s offer is then communicated to Player $$B$$. Player $$B$$ is given a few moments to decide whether or not to accept the offer. Player $$B$$’s decision is then communicated to Player $$A$$. 2$$'$$. If Player $$B$$ accepts Player $$A$$’s offer, then the round is over for that pair. Players $$A$$’s Payoff for the round $$=$$ Player $$A$$’s offer Player $$B$$’s Cost for the round $$=$$ Player $$A$$’s offer. 5. Results We first present data on offer behavior, and we then analyze dispute behavior and the value to player $$A$$ of having the offer. The units of analysis are individual offers, accept/reject decisions and payoffs. We also conducted the analysis treating each session as a single observation, and performed Fisher exact randomization tests for the hypothesis tests. The point estimates and statistical $$P$$-values are quite robust to the unit of analysis, so only the offer level results are presented below. In some tables, we also include data separately from round 7 onward (hereafter R7-end) to evaluate any evolution in subjects’ behavior over time (e.g., learning). These rounds represent the second halves of the individual sessions. 5.1. Offer Behavior Table 3 reports offer frequency distributions using the intervals identified in Section 3.3. In the screening game, 87% of the player $$B$$ offers are screening offers 75–225, 92% in R7-end. In the signaling game, 75% of the $$A_{L}$$ offers are consistent with semi-pooling: 63% are revealing offers 75–225 and 12% are bluffs 375–525. In R7-end, 89% are consistent with semi-pooling: 81% are revealing offers and 8% are bluffs. The lower occurrence of bluffs in R7-end is consistent with the high empirical rejection rate on these offers (see Section 5.2). Somewhat surprisingly, 22% of the $$A_{L}$$ offers are anomalous between offers 226 and 374. Their decline to 11% in R7-end is consistent with the prior work demonstrating the importance of learning in signaling games (e.g., Cooper and Kagel (2008)), but a confounding factor is player $$B$$’s willingness to accept some of these anomalous offers (Section 5.2). Ninety percent of the player $$A_{H}$$ signaling game offers are separating offers 375–525, 96% in R7-end. Collectively, 87% of the screening game offers and 80% of the signaling game offers are consistent with the theory, with the percentages rising to 92% for both games in R7-end.23 Table 3. Offer Frequency Distributions          Proportion (number) of offers in specified interval           Offers consistent with theory  Anomalous offers  Game: player  Rounds  $$n$$  75–225  375–525  226–374  $$<$$ 75, $$>$$ 525  Screening: $$B$$  All  402  0.87 (350)  0.05 (21)  0.06 (24)  0.02 (7)$$^{\mathrm{a}}$$     R7-end  216  0.92 (199)  0.05 (11)  0.03 (6)  – (0)  Signaling: $$A_{L}$$  All  276  0.63 (173)  0.12 (34)  0.22 (60)  0.03 (9)$$^{\mathrm{b}}$$     R7-end  130  0.81 (105)  0.08 (11)  0.11 (14)  – (0)  Signaling: $$A_{H}$$  All  132  0.01 (1)$$^{\mathrm{}}$$  0.91 (119)  0.03 (4)  0.06 (8)$$^{\mathrm{b}}$$     R7-end  74  – (0)  0.96 (71)$$^{\mathrm{c}}$$  0.03 (2)  0.01 (1)$$^{\mathrm{b}}$$           Proportion (number) of offers in specified interval           Offers consistent with theory  Anomalous offers  Game: player  Rounds  $$n$$  75–225  375–525  226–374  $$<$$ 75, $$>$$ 525  Screening: $$B$$  All  402  0.87 (350)  0.05 (21)  0.06 (24)  0.02 (7)$$^{\mathrm{a}}$$     R7-end  216  0.92 (199)  0.05 (11)  0.03 (6)  – (0)  Signaling: $$A_{L}$$  All  276  0.63 (173)  0.12 (34)  0.22 (60)  0.03 (9)$$^{\mathrm{b}}$$     R7-end  130  0.81 (105)  0.08 (11)  0.11 (14)  – (0)  Signaling: $$A_{H}$$  All  132  0.01 (1)$$^{\mathrm{}}$$  0.91 (119)  0.03 (4)  0.06 (8)$$^{\mathrm{b}}$$     R7-end  74  – (0)  0.96 (71)$$^{\mathrm{c}}$$  0.03 (2)  0.01 (1)$$^{\mathrm{b}}$$  $$^{\mathrm{a}}$$Six of these offers are $$<$$75 and one is $$>$$525. $$^{\mathrm{b}}$$All of these offers are $$>$$525. Table 3. Offer Frequency Distributions          Proportion (number) of offers in specified interval           Offers consistent with theory  Anomalous offers  Game: player  Rounds  $$n$$  75–225  375–525  226–374  $$<$$ 75, $$>$$ 525  Screening: $$B$$  All  402  0.87 (350)  0.05 (21)  0.06 (24)  0.02 (7)$$^{\mathrm{a}}$$     R7-end  216  0.92 (199)  0.05 (11)  0.03 (6)  – (0)  Signaling: $$A_{L}$$  All  276  0.63 (173)  0.12 (34)  0.22 (60)  0.03 (9)$$^{\mathrm{b}}$$     R7-end  130  0.81 (105)  0.08 (11)  0.11 (14)  – (0)  Signaling: $$A_{H}$$  All  132  0.01 (1)$$^{\mathrm{}}$$  0.91 (119)  0.03 (4)  0.06 (8)$$^{\mathrm{b}}$$     R7-end  74  – (0)  0.96 (71)$$^{\mathrm{c}}$$  0.03 (2)  0.01 (1)$$^{\mathrm{b}}$$           Proportion (number) of offers in specified interval           Offers consistent with theory  Anomalous offers  Game: player  Rounds  $$n$$  75–225  375–525  226–374  $$<$$ 75, $$>$$ 525  Screening: $$B$$  All  402  0.87 (350)  0.05 (21)  0.06 (24)  0.02 (7)$$^{\mathrm{a}}$$     R7-end  216  0.92 (199)  0.05 (11)  0.03 (6)  – (0)  Signaling: $$A_{L}$$  All  276  0.63 (173)  0.12 (34)  0.22 (60)  0.03 (9)$$^{\mathrm{b}}$$     R7-end  130  0.81 (105)  0.08 (11)  0.11 (14)  – (0)  Signaling: $$A_{H}$$  All  132  0.01 (1)$$^{\mathrm{}}$$  0.91 (119)  0.03 (4)  0.06 (8)$$^{\mathrm{b}}$$     R7-end  74  – (0)  0.96 (71)$$^{\mathrm{c}}$$  0.03 (2)  0.01 (1)$$^{\mathrm{b}}$$  $$^{\mathrm{a}}$$Six of these offers are $$<$$75 and one is $$>$$525. $$^{\mathrm{b}}$$All of these offers are $$>$$525. Table 4 provides analysis on offers consistent with the theory.24 As a robustness check, we also estimate a dummy-variable regression with robust standard errors clustered on sessions, so as to control for session-specific variation. Those results are provided in Appendix Table A1, and they are quite close to those shown in Table 4. Limiting the data to R7-end in Table 4 and in Table A1 yields similar results, so they are omitted. Table 4. Offers Consistent with Theory    Player type and offer interval  Offers 75–225  Offers 375–525     $$B$$  $$A_{L}$$  $$A_{H}$$  $$A_{L}$$ v. $$B$$  $$A_{H}$$ v. $$A_{L}$$  Statistic  75–225  75–225  375–525  375–525  H$$_{\mathrm{0}}$$: Diff. $$=$$ 150  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0  $$n$$  350  173  34  119  Est. diff. $$=$$ 92.0  Est. diff. $$=$$ 37.4  Mean  111.6  203.6  417.5  454.9  $$t =$$ 18.2$$^{\mathrm{a,b}}$$  $$t =$$ 4.12$$^{\mathrm{a}}$$  (SE)  (2.02)  (2.46)  (8.42)  (4.23)        Median  100  220  400  450  Est. diff. $$=$$ 120  Est. diff. $$=$$ 50                 $$z =$$ 18.5$$^{\mathrm{a,c}}$$  $$z =$$ 4.18$$^{\mathrm{a,c}}$$     Player type and offer interval  Offers 75–225  Offers 375–525     $$B$$  $$A_{L}$$  $$A_{H}$$  $$A_{L}$$ v. $$B$$  $$A_{H}$$ v. $$A_{L}$$  Statistic  75–225  75–225  375–525  375–525  H$$_{\mathrm{0}}$$: Diff. $$=$$ 150  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0  $$n$$  350  173  34  119  Est. diff. $$=$$ 92.0  Est. diff. $$=$$ 37.4  Mean  111.6  203.6  417.5  454.9  $$t =$$ 18.2$$^{\mathrm{a,b}}$$  $$t =$$ 4.12$$^{\mathrm{a}}$$  (SE)  (2.02)  (2.46)  (8.42)  (4.23)        Median  100  220  400  450  Est. diff. $$=$$ 120  Est. diff. $$=$$ 50                 $$z =$$ 18.5$$^{\mathrm{a,c}}$$  $$z =$$ 4.18$$^{\mathrm{a,c}}$$  $$^{\mathrm{a}}$$Test statistic has $$P =$$ 0.000 for the given H$$_{\mathrm{0}}$$. $$^{\mathrm{b}}$$Difference-in-means test assumes unequal variances (variance ratio $$F$$ test has $$P =$$ .022). $$^{\mathrm{c}}$$Mann–Whitney rank-sum $$z$$-statistic. Table 4. Offers Consistent with Theory    Player type and offer interval  Offers 75–225  Offers 375–525     $$B$$  $$A_{L}$$  $$A_{H}$$  $$A_{L}$$ v. $$B$$  $$A_{H}$$ v. $$A_{L}$$  Statistic  75–225  75–225  375–525  375–525  H$$_{\mathrm{0}}$$: Diff. $$=$$ 150  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0  $$n$$  350  173  34  119  Est. diff. $$=$$ 92.0  Est. diff. $$=$$ 37.4  Mean  111.6  203.6  417.5  454.9  $$t =$$ 18.2$$^{\mathrm{a,b}}$$  $$t =$$ 4.12$$^{\mathrm{a}}$$  (SE)  (2.02)  (2.46)  (8.42)  (4.23)        Median  100  220  400  450  Est. diff. $$=$$ 120  Est. diff. $$=$$ 50                 $$z =$$ 18.5$$^{\mathrm{a,c}}$$  $$z =$$ 4.18$$^{\mathrm{a,c}}$$     Player type and offer interval  Offers 75–225  Offers 375–525     $$B$$  $$A_{L}$$  $$A_{H}$$  $$A_{L}$$ v. $$B$$  $$A_{H}$$ v. $$A_{L}$$  Statistic  75–225  75–225  375–525  375–525  H$$_{\mathrm{0}}$$: Diff. $$=$$ 150  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0  $$n$$  350  173  34  119  Est. diff. $$=$$ 92.0  Est. diff. $$=$$ 37.4  Mean  111.6  203.6  417.5  454.9  $$t =$$ 18.2$$^{\mathrm{a,b}}$$  $$t =$$ 4.12$$^{\mathrm{a}}$$  (SE)  (2.02)  (2.46)  (8.42)  (4.23)        Median  100  220  400  450  Est. diff. $$=$$ 120  Est. diff. $$=$$ 50                 $$z =$$ 18.5$$^{\mathrm{a,c}}$$  $$z =$$ 4.18$$^{\mathrm{a,c}}$$  $$^{\mathrm{a}}$$Test statistic has $$P =$$ 0.000 for the given H$$_{\mathrm{0}}$$. $$^{\mathrm{b}}$$Difference-in-means test assumes unequal variances (variance ratio $$F$$ test has $$P =$$ .022). $$^{\mathrm{c}}$$Mann–Whitney rank-sum $$z$$-statistic. Theory predicts a difference of 150 between the $$B$$ screening and $$A_{L}$$ revealing offers (75 v. 225). The mean offers are 112 and 204, respectively, and differ by 92. Player $$B$$’s mean offer represents 37 or 25% of the 150 surplus from settlement, and $$A_{L}$$’s mean offer represents 21 or 14% of the surplus.25 The difference of 92 is 61% of the predicted amount. The median offers of 100 and 220 represent less surplus for the recipient (17% and 3%, respectively), and they differ by 120 or 80% of the predicted 150. Further analysis of the data reveals that in both games the preponderance of offers are strongly in the direction of their respective point predictions, but a few include sufficient surplus so as to cause some divergence between the mean and median.26 Nonetheless, both the mean and median differences are statistically different from the 150 prediction. Overall, the $$B$$ screening offers and $$A_{L}$$ revealing offers tend to be stingy, but both contain enough surplus so that their difference is about 60–80% of the predicted amount. In the signaling game, bluffing $$A_{L}$$ players and $$A_{H}$$ are predicted to pool on a single value of the semi-pooling offer $$O_{A}^{S} $$. Empirically, this implies that $$A_{L}$$ and $$A_{H}$$ offers 375–525 will be statistically indistinguishable. Despite the fact that a high percentage of the player $$A$$ offers are consistent with semi-pooling, $$A_{L}$$ and $$A_{H}$$ are unable to successfully coordinate on a unique semi-pooling offer. In particular, the typical $$A_{L}$$ bluff (mean 418, median 400) is substantially lower than the typical $$A_{H}$$ separating offer (mean 445, median 450). The tests on the far-right of Table 4 reject the hypothesis of equal $$A_{L}$$ and $$A_{H}$$ central tendencies on offers 375–525. This coordination failure underscores the challenges posed by signaling games, particularly when feedback is restricted to each litigant’s own negotiation experience. Complicating any attempts at coordination is the empirical dispute behavior, to which we now turn. 5.2. Dispute Behavior Table 5 reports dispute rates between players $$A_{L}$$ and $$B$$. In both games, their predicted dispute rate is 0% over the interval 75–225. The observed rates are 17% in the screening game and 10% in the signaling game. These excess disputes typically occur on offers containing relatively small amounts of surplus, and they are clearly one factor driving the positive surplus in $$B$$ screening offers and $$A_{L}$$ revealing offers.27 In both games, the player receiving the offer has a demand for surplus from settlement which is not present in the theory. In Table 5, the rejection rate on $$A_{L}$$ bluffs 375–525 is 85%, 73% in R7-end. Below we discuss $$B$$’s rejection behavior on offers 375–525 (recall that $$B$$ does not know $$A$$’s type upon receiving an offer), but here we note that the observed rate is higher than the 50–67% rate predicted on these offers. Perhaps one consequence of this is the decline in the incidence of $$A_{L}$$ bluffs to 8% in R7-end. Table 5. $$A_{L}$$ v. $$B$$ Dispute Rates    Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.16 (273)  0.17 (241)  0.00 (13)  0.00 (15)  1.0 (4)$$^{\mathrm{a}}$$     R7-end  0.14 (149)  0.15 (136)  0.00 (8)  0.00 (5)  –  Signaling  All  0.32 (276)  0.10 (173)  0.85 (34)  0.55 (60)  1.0 (9)$$^{\mathrm{b}}$$     R7-end  0.18 (130)  0.10 (105)  0.73 (11)  0.43 (14)  –     Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.16 (273)  0.17 (241)  0.00 (13)  0.00 (15)  1.0 (4)$$^{\mathrm{a}}$$     R7-end  0.14 (149)  0.15 (136)  0.00 (8)  0.00 (5)  –  Signaling  All  0.32 (276)  0.10 (173)  0.85 (34)  0.55 (60)  1.0 (9)$$^{\mathrm{b}}$$     R7-end  0.18 (130)  0.10 (105)  0.73 (11)  0.43 (14)  –  $$^{\mathrm{a}}$$All four are $$<$$ 75. $$^{\mathrm{b}}$$All nine are $$>$$ 525. Table 5. $$A_{L}$$ v. $$B$$ Dispute Rates    Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.16 (273)  0.17 (241)  0.00 (13)  0.00 (15)  1.0 (4)$$^{\mathrm{a}}$$     R7-end  0.14 (149)  0.15 (136)  0.00 (8)  0.00 (5)  –  Signaling  All  0.32 (276)  0.10 (173)  0.85 (34)  0.55 (60)  1.0 (9)$$^{\mathrm{b}}$$     R7-end  0.18 (130)  0.10 (105)  0.73 (11)  0.43 (14)  –     Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.16 (273)  0.17 (241)  0.00 (13)  0.00 (15)  1.0 (4)$$^{\mathrm{a}}$$     R7-end  0.14 (149)  0.15 (136)  0.00 (8)  0.00 (5)  –  Signaling  All  0.32 (276)  0.10 (173)  0.85 (34)  0.55 (60)  1.0 (9)$$^{\mathrm{b}}$$     R7-end  0.18 (130)  0.10 (105)  0.73 (11)  0.43 (14)  –  $$^{\mathrm{a}}$$All four are $$<$$ 75. $$^{\mathrm{b}}$$All nine are $$>$$ 525. Table 5 shows one striking anomaly that occurs in the signaling game: the 55% player $$B$$ dispute rate on $$A_{L}$$ between offers 226 and 374, 43% in R7-end.28 The predicted dispute rate on these offers is 100%, but clearly not all players $$B$$ are able to engage in the chain of reasoning which would lead them to reject a between offer (see the discussion on the relatively weak “test of dominated strategies” refinement in Section 3.3 above).29 The frequent acceptance of these anomalous offers helps explain why $$A_{L}$$ was slow to abandon them. Table 5 also shows that in the screening game, the infrequent pooling offers 375–525 are always accepted by $$A_{L}$$, as theory predicts. ($$B$$ is not predicted to make a pooling offer, but if he does, $$A_{L}$$ is predicted to accept it.) Table 6 reports dispute rates between players $$A_{H}$$ and $$B$$. In the screening game, the predicted dispute rate is 100%, conditional on $$A_{H}$$ receiving a screening offer from $$B$$. Over the screening interval 75–225, the observed dispute rate is 98%. The observed dispute rate on all offers is 92%, primarily because $$A_{H}$$ readily accepts the occasional pooling offer 375–525 that she sees. In the signaling game, the predicted $$A_{H}$$ dispute rate is 50–67%. The observed dispute rate for $$A_{H}$$ separating offers 375–525 is 73% overall and 63% in R7-end. This latter percentage is within the predicted range, and it is consistent with the typical $$A_{H}$$ separating offer of about 450.30 Player $$A_{H}$$ does make a handful of offers outside of 375–525, but the $$A_{H}$$ dispute rate on all offers (73%, 65% R7-end) is the same as the $$A_{H}$$ dispute rate on separating offers. Table 6. $$A_{H}$$ v. $$B$$ Dispute Rates       Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.92 (129)  0.98 (109)  0.13 (8)  0.89 (9)  1.0 (3)$$^{\mathrm{a}}$$     R7-end  0.96 (67)  0.98 (63)  0.33 (3)  1.0 (1)  – (0)  Signaling  All  0.73 (132)  1.0 (1)  0.73 (119)  0.50 (4)  1.0 (8)$$^{\mathrm{b}}$$     R7-end  0.65 (74)  – (0)  0.63 (71)  1.0 (2)  1.0 (1)$$^{\mathrm{b}}$$        Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.92 (129)  0.98 (109)  0.13 (8)  0.89 (9)  1.0 (3)$$^{\mathrm{a}}$$     R7-end  0.96 (67)  0.98 (63)  0.33 (3)  1.0 (1)  – (0)  Signaling  All  0.73 (132)  1.0 (1)  0.73 (119)  0.50 (4)  1.0 (8)$$^{\mathrm{b}}$$     R7-end  0.65 (74)  – (0)  0.63 (71)  1.0 (2)  1.0 (1)$$^{\mathrm{b}}$$  $$^{\mathrm{a}}$$ All three are $$<$$ 75. $$^{\mathrm{b}}$$ All eight (one in R7-end) are $$>$$ 525. Table 6. $$A_{H}$$ v. $$B$$ Dispute Rates       Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.92 (129)  0.98 (109)  0.13 (8)  0.89 (9)  1.0 (3)$$^{\mathrm{a}}$$     R7-end  0.96 (67)  0.98 (63)  0.33 (3)  1.0 (1)  – (0)  Signaling  All  0.73 (132)  1.0 (1)  0.73 (119)  0.50 (4)  1.0 (8)$$^{\mathrm{b}}$$     R7-end  0.65 (74)  – (0)  0.63 (71)  1.0 (2)  1.0 (1)$$^{\mathrm{b}}$$        Proportion of disputes (number of offers) in specified interval           Offers consistent with theory  Anomalous offers  Game  Rounds  All offers  75–225  375–525  226–374  $$< 75, > 525$$  Screening  All  0.92 (129)  0.98 (109)  0.13 (8)  0.89 (9)  1.0 (3)$$^{\mathrm{a}}$$     R7-end  0.96 (67)  0.98 (63)  0.33 (3)  1.0 (1)  – (0)  Signaling  All  0.73 (132)  1.0 (1)  0.73 (119)  0.50 (4)  1.0 (8)$$^{\mathrm{b}}$$     R7-end  0.65 (74)  – (0)  0.63 (71)  1.0 (2)  1.0 (1)$$^{\mathrm{b}}$$  $$^{\mathrm{a}}$$ All three are $$<$$ 75. $$^{\mathrm{b}}$$ All eight (one in R7-end) are $$>$$ 525. In the signaling game, the predicted dispute rate of 50–67% on offers 375–525 is not contingent on $$A$$’s type as this is not observable by $$B$$ upon receiving the offer. Aggregating the $$A_{L}$$ bluffs in Table 5 with the $$A_{H}$$ separating offers in Table 6 yields a dispute rate of 76% (116/153), 65% (53/82) in R7-end. These rejection rates are somewhat higher than expected given the empirical player $$A$$ offers 375–525.31 Player $$B$$’s empirical rejection rates are 68% (78/115) on offers 375–499 and 100% (38/38) on offers 500–525. In R7-end, these rates are 57% (38/67) and 100% (15/15), respectively. Thus $$B$$ has an implicit demand which forces both $$A_{H}$$ and a bluffing $$A_{L}$$ to experiment with offers $$<$$ 500 to create a positive probability of acceptance.32 Table 7 evaluates dispute rate comparative statics across the two games. The table shows the predicted comparative statics (game-specific predictions are shown in table note $$a)$$ and observed differences separately for $$A_{L} $$v. $$B, A_{H} $$v. $$B$$, and $$A$$ v. $$B$$. The predicted $$A_{L}$$ v. $$B$$ comparative static on the dispute rate is 0–25 percentage points higher in the signaling game semi-pooling equilibrium than in the screening game. The observed dispute rate is sixteen percentage points higher, which is consistent with the prediction. However, both of the individual empirical dispute rates exceed their respective point predictions, due primarily to the three reasons discussed above.33 The predicted $$A_{H}$$ v. $$B$$ comparative static is a decline of 33–50 percentage points in the dispute rate in the signaling game relative to the screening game. The observed difference is a 19-percentage point decline in the signaling game, which is in the predicted direction and statistically different from zero, but well below the minimum expected difference. The less than expected decline is due to the higher than expected dispute rate in the signaling game (73%) and the slightly less than expected dispute rate in the screening game (92%). The overall $$A $$vs. $$B$$ dispute rate is predicted to decline by 0–11 percentage points. The observed difference is an increase of five percentage points; this difference is small economically and not statistically different from zero. Overall, we do not observe any major efficiency effects related to which party has the power to make the offer. Table 7. Comparison of Dispute Rates Across Games       Dispute rates for all rounds, all offers  Negotiation  Prediction$$^{\mathrm{a}}$$  Signaling  Screening  Diff.  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0$$^{\mathrm{b}}$$  $$A_{L}$$ v. $$ B$$  0 to $$+$$25  0.32  0.16  0.16  $$z = 4.32$$        ($$n = 276$$)  ($$n = 273$$)     ($$P = .000$$)  $$A_{H}$$ v. $$ B$$  –33 to –50  0.73  0.92  –0.19  $$z^{}=$$ –4.01        ($$n = 132$$)  ($$n = 129$$)     ($$P = .000$$)  $$A $$ v. $$ B$$  0 to –11  0.45  0.40  0.05  $$z =$$ 1.38        ($$n =$$ 408)  ($$n =$$ 402)     ($$P =.168$$)        Dispute rates for all rounds, all offers  Negotiation  Prediction$$^{\mathrm{a}}$$  Signaling  Screening  Diff.  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0$$^{\mathrm{b}}$$  $$A_{L}$$ v. $$ B$$  0 to $$+$$25  0.32  0.16  0.16  $$z = 4.32$$        ($$n = 276$$)  ($$n = 273$$)     ($$P = .000$$)  $$A_{H}$$ v. $$ B$$  –33 to –50  0.73  0.92  –0.19  $$z^{}=$$ –4.01        ($$n = 132$$)  ($$n = 129$$)     ($$P = .000$$)  $$A $$ v. $$ B$$  0 to –11  0.45  0.40  0.05  $$z =$$ 1.38        ($$n =$$ 408)  ($$n =$$ 402)     ($$P =.168$$)  $$^{\mathrm{a}}$$Predictions are percentage point differences in dispute rates across games. From Table 1, predicted dispute rates are $$A_{L} $$ v. $$ B$$: 0–25% v. 0%, $$A_{H} $$ v. $$B$$: 50–67% v. 100%, and $$A $$ v. $$ B$$: 22–33% v. 33%. $$^{\mathrm{b}}$$Difference-in-proportions binomial test. Table 7. Comparison of Dispute Rates Across Games       Dispute rates for all rounds, all offers  Negotiation  Prediction$$^{\mathrm{a}}$$  Signaling  Screening  Diff.  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0$$^{\mathrm{b}}$$  $$A_{L}$$ v. $$ B$$  0 to $$+$$25  0.32  0.16  0.16  $$z = 4.32$$        ($$n = 276$$)  ($$n = 273$$)     ($$P = .000$$)  $$A_{H}$$ v. $$ B$$  –33 to –50  0.73  0.92  –0.19  $$z^{}=$$ –4.01        ($$n = 132$$)  ($$n = 129$$)     ($$P = .000$$)  $$A $$ v. $$ B$$  0 to –11  0.45  0.40  0.05  $$z =$$ 1.38        ($$n =$$ 408)  ($$n =$$ 402)     ($$P =.168$$)        Dispute rates for all rounds, all offers  Negotiation  Prediction$$^{\mathrm{a}}$$  Signaling  Screening  Diff.  H$$_{\mathrm{0}}$$: Diff. $$=$$ 0$$^{\mathrm{b}}$$  $$A_{L}$$ v. $$ B$$  0 to $$+$$25  0.32  0.16  0.16  $$z = 4.32$$        ($$n = 276$$)  ($$n = 273$$)     ($$P = .000$$)  $$A_{H}$$ v. $$ B$$  –33 to –50  0.73  0.92  –0.19  $$z^{}=$$ –4.01        ($$n = 132$$)  ($$n = 129$$)     ($$P = .000$$)  $$A $$ v. $$ B$$  0 to –11  0.45  0.40  0.05  $$z =$$ 1.38        ($$n =$$ 408)  ($$n =$$ 402)     ($$P =.168$$)  $$^{\mathrm{a}}$$Predictions are percentage point differences in dispute rates across games. From Table 1, predicted dispute rates are $$A_{L} $$ v. $$ B$$: 0–25% v. 0%, $$A_{H} $$ v. $$B$$: 50–67% v. 100%, and $$A $$ v. $$ B$$: 22–33% v. 33%. $$^{\mathrm{b}}$$Difference-in-proportions binomial test. 5.3. The Value to $$A$$ of Having the Offer We now consider the value to player $$A$$ of having the offer. Empirically, the value of the property right depends on the difference between the player $$A$$ signaling game offer and the player $$B$$ screening game offer, as well as the dispute behavior of the respective recipient. In Table 8, we use the mean net payoff earned by player $$A $$ to assess the empirical value of the offer to $$A.$$34 For each player type, we conduct two statistical tests, one for equality of means across games, and one for the predicted difference across games (as the predictions are intervals for $$A_{H}$$ and $$A$$, we use the upper end of the respective interval). As a robustness check, we also estimate a dummy-variable regression with robust standard errors clustered on sessions. Those results are similar to those in Table 8; see Appendix Table A2. We also obtain comparable results using R7-end data. Table 8. Player $$A$$ Net Payoffs       Mean $$A$$ payoff (SE)  Difference-in-means tests     Prediction  Signaling  Screening  Diff.  H$$_0$$: Diff. $$= 0$$  H$$_0$$: Diff. $$=$$ Pred.$$^{\mathrm{a}}$$        $$n =$$ 276  $$n =$$ 273     $$t = 6.33$$  $$t = 15.4$$  $$A_{L}$$  $$+150$$  174.0  130.3  43.7  ($$P = .000$$)  ($$P = .000$$)        (4.92)  (4.84)                 $$n = 132$$  $$n = 129$$           $$A_{H}$$  0 to $$+$$50  386.4  371.5  14.9  $$t = 3.31$$  $$t = 7.82$$        (3.13)  (3.22)     ($$P = .001$$)  ($$P = .000$$)        $$n = 402$$  $$n = 408$$           $$A$$  $$+100$$ to $$+$$117  242.7  207.7  35.0  $$t = 3.92^{\mathrm{b}}$$  $$t = 9.18^{\mathrm{b}}$$        (6.03)  (6.60)     $$(P = 0.001)$$  $$P = 0.000$$        Mean $$A$$ payoff (SE)  Difference-in-means tests     Prediction  Signaling  Screening  Diff.  H$$_0$$: Diff. $$= 0$$  H$$_0$$: Diff. $$=$$ Pred.$$^{\mathrm{a}}$$        $$n =$$ 276  $$n =$$ 273     $$t = 6.33$$  $$t = 15.4$$  $$A_{L}$$  $$+150$$  174.0  130.3  43.7  ($$P = .000$$)  ($$P = .000$$)        (4.92)  (4.84)                 $$n = 132$$  $$n = 129$$           $$A_{H}$$  0 to $$+$$50  386.4  371.5  14.9  $$t = 3.31$$  $$t = 7.82$$        (3.13)  (3.22)     ($$P = .001$$)  ($$P = .000$$)        $$n = 402$$  $$n = 408$$           $$A$$  $$+100$$ to $$+$$117  242.7  207.7  35.0  $$t = 3.92^{\mathrm{b}}$$  $$t = 9.18^{\mathrm{b}}$$        (6.03)  (6.60)     $$(P = 0.001)$$  $$P = 0.000$$  $$^{\mathrm{a}}$$For $$A_{H}$$ test, prediction value is $$+$$50. For player A test, prediction value is $$+$$117. $$^{\mathrm{b}}$$$$t$$-statistic calculated assuming unequal variances as variance ratio $$F$$ test has $$P =$$ 0.098. Table 8. Player $$A$$ Net Payoffs       Mean $$A$$ payoff (SE)  Difference-in-means tests     Prediction  Signaling  Screening  Diff.  H$$_0$$: Diff. $$= 0$$  H$$_0$$: Diff. $$=$$ Pred.$$^{\mathrm{a}}$$        $$n =$$ 276  $$n =$$ 273     $$t = 6.33$$  $$t = 15.4$$  $$A_{L}$$  $$+150$$  174.0  130.3  43.7  ($$P = .000$$)  ($$P = .000$$)        (4.92)  (4.84)                 $$n = 132$$  $$n = 129$$           $$A_{H}$$  0 to $$+$$50  386.4  371.5  14.9  $$t = 3.31$$  $$t = 7.82$$        (3.13)  (3.22)     ($$P = .001$$)  ($$P = .000$$)        $$n = 402$$  $$n = 408$$           $$A$$  $$+100$$ to $$+$$117  242.7  207.7  35.0  $$t = 3.92^{\mathrm{b}}$$  $$t = 9.18^{\mathrm{b}}$$        (6.03)  (6.60)     $$(P = 0.001)$$  $$P = 0.000$$        Mean $$A$$ payoff (SE)  Difference-in-means tests     Prediction  Signaling  Screening  Diff.  H$$_0$$: Diff. $$= 0$$  H$$_0$$: Diff. $$=$$ Pred.$$^{\mathrm{a}}$$        $$n =$$ 276  $$n =$$ 273     $$t = 6.33$$  $$t = 15.4$$  $$A_{L}$$  $$+150$$  174.0  130.3  43.7  ($$P = .000$$)  ($$P = .000$$)        (4.92)  (4.84)                 $$n = 132$$  $$n = 129$$           $$A_{H}$$  0 to $$+$$50  386.4  371.5  14.9  $$t = 3.31$$  $$t = 7.82$$        (3.13)  (3.22)     ($$P = .001$$)  ($$P = .000$$)        $$n = 402$$  $$n = 408$$           $$A$$  $$+100$$ to $$+$$117  242.7  207.7  35.0  $$t = 3.92^{\mathrm{b}}$$  $$t = 9.18^{\mathrm{b}}$$        (6.03)  (6.60)     $$(P = 0.001)$$  $$P = 0.000$$  $$^{\mathrm{a}}$$For $$A_{H}$$ test, prediction value is $$+$$50. For player A test, prediction value is $$+$$117. $$^{\mathrm{b}}$$$$t$$-statistic calculated assuming unequal variances as variance ratio $$F$$ test has $$P =$$ 0.098. The mean payoff for $$A_{L}$$ is about 44 higher in the signaling game. This difference is statistically different from zero, but it is only 29% of the predicted 150. For $$A_{H}$$ the mean payoff is about 15 higher in the signaling game, which is consistent with the semi-pooling equilibrium where the predicted increase for $$A_{H}$$ ranges from 0 to 50. We note that our estimated increase is different both economically and statistically from both the bottom (0) and top (50) ends of that range. The overall average increase for $$A$$ is 35, which is well below the predicted increase of 100–117. Thus, the distributional impact of having the offer in this setting is much less than is predicted by the theory. A combination of reasons explains why $$A$$ gains less than predicted from having the offer. One reason is that $$A_{L}$$ bluffs 375–525 are rejected at an 85% rate, which is much higher than the 50–67% prediction. Another reason is that $$B$$ screening offers, $$A_{L}$$ revealing offers and $$A_{H}$$ separating offers cannot extract all of the surplus from settlement. These offers almost always contain positive surplus for the recipient. Moreover, $$B$$ screening offers and $$A_{L}$$ revealing offers are rejected about 10–17% of the time. If we restrict the analysis to $$B$$ screening offers and $$A_{L}$$ revealing offers, we can gain more insight into the importance of these factors. When we do so (table omitted) we find higher estimates for the value of having the offer, but these are still well below the point predictions. For $$A_{L}$$, the mean payoff increase is 82 or 55% of the predicted 150. Thus failed negotiations on offers in the 75–225 range significantly reduce the predicted distributional impact of having the offer. For $$A_{H}$$, restricting offers to the 375–525 range makes little difference as the estimated value of having the offer remains at about 20. With the restricted offers, the overall average increase for player $$A$$ is 53, which is still well below the predicted range 100–117. 6. Conclusion We provide an experimental study of screening and signaling models in a legal bargaining context. The two asymmetric information games are identical, with the exception of whether the uninformed defendant (the screening game) or the informed plaintiff (the signaling game) makes the pretrial offer. Overall, the screening model performs well in aggregating the data. Player $$B$$ (the defendant) makes screening offers nearly 90% of the time, and his median screening offer contains 1/6 of the joint surplus from settlement. Player $$A_{L}$$ (a plaintiff with a weak case) typically accepts the screening offer, although she sometimes rejects offers containing less than 1/3 of the settlement surplus. The dispute rate for player $$A_{H}$$ (a plaintiff with a strong case) is near the predicted 100%. These results are consistent with the previous experiments in this setting. In the screening game, the recipient is the informed party (player $$A)$$ and she has a fairly trivial decision to make in deciding whether or not to accept the offer. The sender (player $$B)$$ has a more difficult decision, but latching onto the idea of a screening offer does not appear to be an overly demanding task. Although we do observe some fairness concerns, they are insufficient to invalidate the qualitative predictions of the model. Using the same parameters as in the screening game, we switch the identity of the player making the offer to create a signaling game. We believe that this problem is cognitively much more challenging for both the sender (now player $$A)$$ and the recipient (now player $$B)$$. As the sender, $$A_{L}$$ has to decide whether she should bluff and what offer would constitute a good bluff, while $$A_{H}$$ has to decide how much she needs to shade her offer to ensure a reasonable chance of acceptance. As the recipient, player $$B$$ has a difficult decision to make when he is faced with a high offer, because he cannot be sure whether the offer is a revealing offer from $$A_{H}$$ or a bluff by $$A_{L}$$. Potentially complicating these decision are concerns for fairness. Given these difficulties, we expect, and indeed observe, less adherence to theory in the signaling game compared to the screening game. Much of the behavior conforms reasonably well with a semi-pooling equilibrium. The player $$B$$ rejection behavior on $$A_{L}$$ revealing offers and on high offers (which may be $$A_{H}$$ separating offers or $$A_{L}$$ bluffs) is roughly in line with the semi-pooling prediction. Aside from the “between” offers, the player $$A$$ behavior is likewise roughly consistent with the theory. As expected, $$A_{L}$$ makes a revealing offer a high percentage of the time, and the median revealing offer (220) is quite close to the theoretical prediction (225). Player $$A_{H}$$ almost always makes an offer which separates her from a revealing $$A_{L}$$ player. Twelve percent of $$A_{L}$$ offers are bluffs that mimic an $$A_{H}$$ player. $$A_{H}$$ and the bluffing $$A_{L}$$ do not converge on a single offer as they would in a semi-pooling equilibrium, but it would have been rather miraculous if they had. There is a continuum of such equilibria, and player feedback was limited to their own bargaining experience. The most notable deviation from theory is the acceptance of “between” offers that should be rejected using fairly straightforward dominance arguments. This violates a weak refinement concept known as the test of dominated strategies. While the incidence of these anomalous offers by player $$A_{L}$$ falls to ~ 10% in the later rounds, player $$B$$ continues to accept ~ 50% of them.35 Our results contrast with the labor market experiment of Kübler et al. (2008), who find greater adherence to the theory in the signaling game relative to the screening game. A comparison across the two games indicates that the ability to make the offer is not nearly as valuable as suggested by theory. The mean increase in payoffs for player $$A_{L}$$ from moving from the screening to the signaling game is 30–55% of the predicted amount. This is in part due to the existence of disputes not predicted by the theory and also because the player making the offer cannot extract all of the surplus predicted by theory. Overall, we do observe some distributional effects from changing the identity of the player who makes the offer, but they are much smaller in practice than they are in theory. While the dispute rates are approximately equal in the two games, the sources of disputes do differ in some important ways. In both games, disputes likely arise because the recipient makes an implicit demand not present in the theory, which the sender is unwilling to meet and/or unable to observe. In our data, this occurs more frequently in the screening game. Offsetting this, however, is the bluffing and unpredicted “between” offers made by $$A_{L}$$ players in the signaling game. The high dispute rate on these offers offsets the low dispute rate on revealing offers, and the net effect is roughly equivalent dispute rates across the two games. Consequently, we observe no major efficiency implications arising from the assignment of which player is allowed to make the offer. In the later rounds of our screening experiment, anomalous behavior essentially disappears. In the later rounds of our signaling game, anomalous offers are significantly reduced, but anomalous acceptance behavior persists. This is consistent with the idea that the signaling game is more cognitively demanding. The reduction in anomalous offers indicates that some learning did occur in the signaling game. In the laboratory, having players reverse roles might facilitate additional learning, as might public dissemination of some information on negotiations. But real-world plaintiffs typically have limited experience in that role and might never be placed in the role of a defendant. Also, many out-of-court settlements include non-disclosure clauses, which limit what can be learned from the bargaining experience of others. This suggests that the behavior of relatively inexperienced litigants is relevant, even if this inexperience may be mitigated by the presence of an attorney.36 The omission of contract clauses specifying which party makes an offer may reflect the absence of strong efficiency or distributional effects associated with the identity of the party making the settlement offer. Clearly, the signaling model deserves more attention. It is one of the two informational-based models of pretrial bargaining, but the theory typically relies on a refinement concept, D1, which is not supported empirically. Thus, it is vital to gain further data on the actual behavior within this game as this may help better inform the theory. This, in turn, may allow for sharper policy recommendations regarding policies such as fee shifting and the role of costly pretrial discovery. Our experimental design has followed the theory quite closely, but future experiments may incorporate additional features which may be important to legal bargaining. This can include greater context (i.e., labeling the players as plaintiff and defendant) and allowing for free form bargaining in the presence of asymmetric information. The outcomes of experiments without structured bargaining can then be compared to the outcomes under the type of structured bargaining typically assumed in the theory. These empirical comparisons can then help better inform the theory of pretrial bargaining, and help us better understand the nature of disputes. We would like to thank J.J. Prescott, Rudy Santore, and two anonymous referees for providing helpful comments on the article, and the Culverhouse College of Commerce and Business Administration, University of Alabama for providing research support for Paul Pecorino on this project. We would also like to thank William B. Hankins for providing research assistance. Appendix Table A1. Offer Regression with Robust Standard Errors Model$$^{\mathrm{a}}$$: Offer$$_{ijs} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L}$$Reveal$$+ \beta_{\mathrm{2}}$$Sig-A$$_{L}$$Bluff$$+ \beta_{\mathrm{3}}$$Sig-A$$_{H}$$Separate$$+\mu_{is} + \varepsilon_{ijs}$$     Estimated coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  111.6  91.9  305.8  343.3        $$\quad$$ (std.err.)  (3.975)  (5.654)  (7.230)  (7.750)  948.60  0.922  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 676$$  Implied mean  111.6$$^{\mathrm{b}}$$  $$203.6^{\mathrm{b}}$$  417.5$$^{\mathrm{b}}$$  454.9$$^{\mathrm{b}}$$        Model$$^{\mathrm{a}}$$: Offer$$_{ijs} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L}$$Reveal$$+ \beta_{\mathrm{2}}$$Sig-A$$_{L}$$Bluff$$+ \beta_{\mathrm{3}}$$Sig-A$$_{H}$$Separate$$+\mu_{is} + \varepsilon_{ijs}$$     Estimated coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  111.6  91.9  305.8  343.3        $$\quad$$ (std.err.)  (3.975)  (5.654)  (7.230)  (7.750)  948.60  0.922  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 676$$  Implied mean  111.6$$^{\mathrm{b}}$$  $$203.6^{\mathrm{b}}$$  417.5$$^{\mathrm{b}}$$  454.9$$^{\mathrm{b}}$$        $$^{\mathrm{a}}$$Subscripts: $$i =$$ offer, $$j =$$ round, and $$s =$$ session. Dummy variables: baseline is screening game sender $$B$$ and offer 75–225, Sig-A$$_{L}$$Reveal$$=$$ 1 if signaling game sender $$A_{L} $$and revealing offer 75–225 ($$=$$ 0 otherwise), Sig-A$$_{L}$$Bluff$$=$$ 1 if signaling game sender $$A_{L}$$ and bluff offer 375–525 ($$=$$ 0 otherwise), Sig-A$$_{H}$$Separate$$=$$ 1 if signaling game sender $$A_{H} $$and separating offer 375–525 ($$=$$ 0 otherwise). Robust standard errors are clustered on sessions. $$^{\mathrm{b}}$$For the Offers 75–225 H$$_{\mathrm{0}}$$: $$A_{L} $$–$$ B =$$ 150 test in Table 4, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 150 has $$F =$$ 105.5 ($$P =$$ 0.000). For the Offers 375–525 H$$_{\mathrm{0}}$$: $$A_{L} = A_{H}$$ test in Table 4, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}} = \beta_{\mathrm{3}}$$ has $$F =$$ 18.8 ($$P =$$ 0.019). Corresponding to the fn. 25 equal surplus difference-in-means test, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{0}}$$ – 75 $$=$$ 225 – ($$\beta_{\mathrm{0}} + \beta_{\mathrm{1}})$$ has $$F = 7.2 (P = 0.025)$$. Table A1. Offer Regression with Robust Standard Errors Model$$^{\mathrm{a}}$$: Offer$$_{ijs} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L}$$Reveal$$+ \beta_{\mathrm{2}}$$Sig-A$$_{L}$$Bluff$$+ \beta_{\mathrm{3}}$$Sig-A$$_{H}$$Separate$$+\mu_{is} + \varepsilon_{ijs}$$     Estimated coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  111.6  91.9  305.8  343.3        $$\quad$$ (std.err.)  (3.975)  (5.654)  (7.230)  (7.750)  948.60  0.922  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 676$$  Implied mean  111.6$$^{\mathrm{b}}$$  $$203.6^{\mathrm{b}}$$  417.5$$^{\mathrm{b}}$$  454.9$$^{\mathrm{b}}$$        Model$$^{\mathrm{a}}$$: Offer$$_{ijs} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L}$$Reveal$$+ \beta_{\mathrm{2}}$$Sig-A$$_{L}$$Bluff$$+ \beta_{\mathrm{3}}$$Sig-A$$_{H}$$Separate$$+\mu_{is} + \varepsilon_{ijs}$$     Estimated coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  111.6  91.9  305.8  343.3        $$\quad$$ (std.err.)  (3.975)  (5.654)  (7.230)  (7.750)  948.60  0.922  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 676$$  Implied mean  111.6$$^{\mathrm{b}}$$  $$203.6^{\mathrm{b}}$$  417.5$$^{\mathrm{b}}$$  454.9$$^{\mathrm{b}}$$        $$^{\mathrm{a}}$$Subscripts: $$i =$$ offer, $$j =$$ round, and $$s =$$ session. Dummy variables: baseline is screening game sender $$B$$ and offer 75–225, Sig-A$$_{L}$$Reveal$$=$$ 1 if signaling game sender $$A_{L} $$and revealing offer 75–225 ($$=$$ 0 otherwise), Sig-A$$_{L}$$Bluff$$=$$ 1 if signaling game sender $$A_{L}$$ and bluff offer 375–525 ($$=$$ 0 otherwise), Sig-A$$_{H}$$Separate$$=$$ 1 if signaling game sender $$A_{H} $$and separating offer 375–525 ($$=$$ 0 otherwise). Robust standard errors are clustered on sessions. $$^{\mathrm{b}}$$For the Offers 75–225 H$$_{\mathrm{0}}$$: $$A_{L} $$–$$ B =$$ 150 test in Table 4, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 150 has $$F =$$ 105.5 ($$P =$$ 0.000). For the Offers 375–525 H$$_{\mathrm{0}}$$: $$A_{L} = A_{H}$$ test in Table 4, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}} = \beta_{\mathrm{3}}$$ has $$F =$$ 18.8 ($$P =$$ 0.019). Corresponding to the fn. 25 equal surplus difference-in-means test, H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{0}}$$ – 75 $$=$$ 225 – ($$\beta_{\mathrm{0}} + \beta_{\mathrm{1}})$$ has $$F = 7.2 (P = 0.025)$$. Table A2. Player $$A$$ Net Payoff Regression with Robust Standard Errors Model$$^{\mathrm{a,b}}$$: Player $$A$$ net payoff$$_{\mathrm{ijs}} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L} + \beta_{\mathrm{2}}$$Scr-A$$_{H} + \beta_{\mathrm{3}}$$Sig-A$$_{H} + \mu_{is} + \varepsilon_{ijs}$$     Coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  130.3  43.7  241.3  256.1        $$\quad$$ (std. err.)  (8.955)  (12.628)  (9.632)  (9.069)  568.52  0.705  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.001$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 810$$  Implied mean  130.3  174.0  371.5  386.4        Model$$^{\mathrm{a,b}}$$: Player $$A$$ net payoff$$_{\mathrm{ijs}} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L} + \beta_{\mathrm{2}}$$Scr-A$$_{H} + \beta_{\mathrm{3}}$$Sig-A$$_{H} + \mu_{is} + \varepsilon_{ijs}$$     Coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  130.3  43.7  241.3  256.1        $$\quad$$ (std. err.)  (8.955)  (12.628)  (9.632)  (9.069)  568.52  0.705  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.001$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 810$$  Implied mean  130.3  174.0  371.5  386.4        $$^{\mathrm{a}}$$Subscripts: $$i =$$ offer, $$j =$$ round, and $$s =$$ session. Dummy variables: baseline is screening game player $$A_{L}$$, Sig-A$$_{L} =$$ 1 if signaling game player $$A_{L} (=$$ 0 otherwise), Scr-A$$_{L} =$$ 1 if screening game player $$A_{H} (=$$ 0 otherwise), Sig-A$$_{H} =$$ 1 if signaling game player $$A_{H} (=$$ 0 otherwise). Robust standard errors are clustered on sessions. $$^{\mathrm{b}}$$Using the empirical frequencies, the implied mean player $$A$$ payoffs are screening game $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}}) =$$ 215.3 and signaling game $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{1}}) + p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) =$$ 242.7. From Table 2, $$p(L$$| scr) $$=$$ 0.648, $$p(H$$| scr) $$=$$ 0.352, $$p(L$$| sig) $$=$$ 0.676, and $$p(H$$| sig) $$=$$ 0.324. For the Table 8 H$$_{\mathrm{0}}$$: Diff. $$=$$ 0 tests, $$A_{L}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 0 has $$F =$$ 10.96 ($$P =$$ 0.007), $$A_{H}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}}$$ – $$\beta_{\mathrm{3}} =$$ 0 has $$F =$$ 8.84 ($$P =$$ 0.016), and player $$A$$ test $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) =$$ 0 has $$F =$$ 9.43 ($$P =$$ 0.013). For Table 8 tests H$$_{\mathrm{0}}$$: Diff. $$=$$ Pred., $$A_{L}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 150 has $$F =$$ 70.9 ($$P =$$ 0.000), $$A_{H}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}}$$ – $$\beta_{\mathrm{3}} =$$ 50 has $$F =$$ 49.3 ($$P =$$ 0.000), and player $$A$$ test $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) = 117$$ has $$F = 262.4$$ ($$P = 0.000$$). Table A2. Player $$A$$ Net Payoff Regression with Robust Standard Errors Model$$^{\mathrm{a,b}}$$: Player $$A$$ net payoff$$_{\mathrm{ijs}} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L} + \beta_{\mathrm{2}}$$Scr-A$$_{H} + \beta_{\mathrm{3}}$$Sig-A$$_{H} + \mu_{is} + \varepsilon_{ijs}$$     Coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  130.3  43.7  241.3  256.1        $$\quad$$ (std. err.)  (8.955)  (12.628)  (9.632)  (9.069)  568.52  0.705  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.001$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 810$$  Implied mean  130.3  174.0  371.5  386.4        Model$$^{\mathrm{a,b}}$$: Player $$A$$ net payoff$$_{\mathrm{ijs}} = \beta_{\mathrm{0}} + \beta_{\mathrm{1}}$$Sig-A$$_{L} + \beta_{\mathrm{2}}$$Scr-A$$_{H} + \beta_{\mathrm{3}}$$Sig-A$$_{H} + \mu_{is} + \varepsilon_{ijs}$$     Coefficients  Summary statistics     $$\beta_{\mathrm{0}}$$  $$\beta_{\mathrm{1}}$$  $$\beta_{\mathrm{2}}$$  $$\beta_{\mathrm{3}}$$  $$F$$  $$R^{\mathrm{2}}$$  Estimate  130.3  43.7  241.3  256.1        $$\quad$$ (std. err.)  (8.955)  (12.628)  (9.632)  (9.069)  568.52  0.705  H$$_{\mathrm{0}}$$: $$\beta_{i} = 0$$  $$P = 0.000$$  $$P = 0.001$$  $$P = 0.000$$  $$P = 0.000$$  $$P = 0.000$$  $$n = 810$$  Implied mean  130.3  174.0  371.5  386.4        $$^{\mathrm{a}}$$Subscripts: $$i =$$ offer, $$j =$$ round, and $$s =$$ session. Dummy variables: baseline is screening game player $$A_{L}$$, Sig-A$$_{L} =$$ 1 if signaling game player $$A_{L} (=$$ 0 otherwise), Scr-A$$_{L} =$$ 1 if screening game player $$A_{H} (=$$ 0 otherwise), Sig-A$$_{H} =$$ 1 if signaling game player $$A_{H} (=$$ 0 otherwise). Robust standard errors are clustered on sessions. $$^{\mathrm{b}}$$Using the empirical frequencies, the implied mean player $$A$$ payoffs are screening game $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}}) =$$ 215.3 and signaling game $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{1}}) + p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) =$$ 242.7. From Table 2, $$p(L$$| scr) $$=$$ 0.648, $$p(H$$| scr) $$=$$ 0.352, $$p(L$$| sig) $$=$$ 0.676, and $$p(H$$| sig) $$=$$ 0.324. For the Table 8 H$$_{\mathrm{0}}$$: Diff. $$=$$ 0 tests, $$A_{L}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 0 has $$F =$$ 10.96 ($$P =$$ 0.007), $$A_{H}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}}$$ – $$\beta_{\mathrm{3}} =$$ 0 has $$F =$$ 8.84 ($$P =$$ 0.016), and player $$A$$ test $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) =$$ 0 has $$F =$$ 9.43 ($$P =$$ 0.013). For Table 8 tests H$$_{\mathrm{0}}$$: Diff. $$=$$ Pred., $$A_{L}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{1}} =$$ 150 has $$F =$$ 70.9 ($$P =$$ 0.000), $$A_{H}$$ test H$$_{\mathrm{0}}$$: $$\beta_{\mathrm{2}}$$ – $$\beta_{\mathrm{3}} =$$ 50 has $$F =$$ 49.3 ($$P =$$ 0.000), and player $$A$$ test $$p(L$$| scr)($$\beta_{\mathrm{0}}) + p(H$$| scr)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(L$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{2}})$$ – $$p(H$$| sig)($$\beta_{\mathrm{0}} + \beta_{\mathrm{3}}) = 117$$ has $$F = 262.4$$ ($$P = 0.000$$). Footnotes 1. Despite their widespread use, these two models of litigation have not yet been comparatively analyzed in an experimental setting. In this sense, our article is analogous to Kübler et al. (2008), who present a side-by-side comparison of the signaling and screening versions of the Spence (1973) labor market model. However, we find relatively greater support for our screening model, while they find more support for their signaling model. 2. One common provision is that the individual triggering the dissolution of the partnership makes an offer for the shares of the other partner. The second partner can either sell her shares or buy her partner’s shares at the specified price. Although these contract provisions do not exactly match our litigation setting, asymmetric information over the partner’s valuation is a typical component of such models. See Fleischer and Schneider (2012) and McAfee (1992). 3. This is for the parameter values we use. Note that there is no general result implying that dispute rates are always lower in the signaling game. The relative dispute rate across games will depend on the number of types present in the model as well as on other parameter values. 4. For a review of the relevant literature, see Spier (2007), Daughety and Reinganum (2012), and Wickelgren (2013). 5. See Camerer and Talley (2007) and Croson (2009) for surveys of the literature on experimental law and economics. 6. See, for example, Ashenfelter et al. (1992), Dickinson (2004) and (2005), Deck and Farmer (2007), Deck, Farmer and Zeng (2007), and Birkeland (2013). Pecorino and Van Boening (2001) analyze arbitration with asymmetric information. 7. See, for example, Stanley and Coursey (1990). One treatment of Inglis et al. (2005) also adopts this information structure. 8. This literature is initiated by Güth et al. (1982). For a recent survey, see Güth and Kocher (2014). 9. They find that whether or not the predictions of the intuitive criterion hold depends upon the details of the game. Also, Brandts and Holt (1993) show how the dynamics of learning in these signaling games can lead to different outcomes. The intuitive criterion places restrictions on out of equilibrium beliefs. It requires player B to place zero weight on the probability of action by player A if that action is associated with payoffs which are dominated by the payoffs earned by A in equilibrium. 10. Also see Cooper et al. (1997), Cooper and Kagel (2008), Kawagoe and Takizawa (2009), de Haan et al. (2011), Drouvelis et al. (2012) and Jeitschko and Normann (2012). 11. We present a two-type signaling model. For a complete analysis of such a model, see Daughety (2000). 12. See, for example, Daughety and Reinganum (1993, p. 322) or Spier (2007, p. 276, fn. 26.) 13. D1 restricts out of equilibrium beliefs in the following way: Suppose an A$$_{L}$$ plaintiff would be willing to deviate to a particular out-of-equilibrium offer if it were accepted with a probability of 1/2 or higher, but that an A$$_{H}$$ plaintiff would only switch if the same offer were accepted with a probability of 3/4 or higher. In this case, the A$$_{L}$$ plaintiff is considered to have the greater incentive to make the offer and, under D1, the defendant must put a weight of 1 on the probability that such an offer comes from an A$$_{L}$$ plaintiff. See Cho and Kreps (1987). The refinement D1 is stronger than the intuitive criterion because it requires zero weight be placed on out-of-equilibrium actions which are not necessarily dominated by a player’s equilibrium payoff. However, this strengthening of the intuitive criterion is necessary to rule out pooling and semi-pooling equilibria. This is the reason why this refinement is employed in the theoretical literature. 14. We can rule out a pure strategy pooling equilibrium, because our parameters satisfy $$150 = \textit{C}_{A} + \textit{C}_{B} < (1 - q)(J^{H} - J^{L}) = 200$$. A type $$A_{H}$$ plaintiff will accept no less than J$$^{H}$$–C$$_{A}$$ in a pooling equilibrium. The defendant B will prefer to take all types to trial rather than settling at $$J^{H}-C_{A}$$ in a pure pooling equilibrium. Obviously, B would reject all higher offers as well, if they were part of a pure strategy pooling equilibrium. 15. It is not possible to have the A$$_{H}$$ plaintiffs and bluffing A$$_{L}$$ plaintiffs make more than one offer between 375 and 525 because it is not possible to find rejection probabilities that make both plaintiff types indifferent between both offers while also making the A$$_{L}$$ plaintiffs indifferent between bluffing and not bluffing. 16. Under D1, there is a pure strategy separating equilibrium under which A$$_{L}$$ always demands 225 and A$$_{H}$$ always demands 525. The low offer is always accepted and the high offer is rejected with probability r$$=$$ 67%. The high offer and the rejection rate r are at the upper endpoints of the prediction intervals for the semi-pooling equilibrium. The overall dispute rate under D1 is predicted to be 22% which is the lower endpoint of the semi-pooling prediction interval. Under D1, the disputes rates are 0% for A$$_{L}$$ and 67% for A$$_{H}$$. These are, respectively, the lower and upper limits of the dispute rates possible in the semi-pooling equilibrium. 17. In the screening game, once player A receives an offer, she faces no risk: she simply compares the offer to the fixed outcome at trial and chooses whichever is higher. The trial outcome is A’s minimally acceptable offer, which is fixed and therefore not affected by her risk aversion. Thus, a player B engaged in a sorting offer will make the offer in (1a) regardless of A’s risk preference. Under the theory, A$$_{L}$$ always accepts this offer so B incurs no rejection risk and makes the offer in (1a) independent of his own risk preference as well. In the signaling game, player A$$_{L}$$ incurs no risk by making a revealing offer, as this offer is always accepted. An A$$_{H}$$ player incurs risk with a high offer, but under the theory the next lower offer accepted with a positive probability (the low offer J$$^{L}$$$$+$$C$$_{B})$$ is dominated by her dispute payoff J$$^{H}$$– C$$_{A}$$. Thus, risk aversion will not cause A$$_{H}$$ to lower her offer. 18. Our design had a minimum of 12 rounds per session. Subjects were recruited for a two-hour period, and if time permitted we ran additional rounds. 19. We used randomizers (i.e., a die roll and drawing cards from a deck) to generate A and B decisions in the practice rounds, so as to avoid implicitly suggesting to participants what decisions to make during the experiment. 20. Generally speaking, most law and economics experiments which are a direct test of a theory are conducted in a context-free environment. See the discussion in Landeo (2018). 21. The information exchange took about 15–20 seconds. The entrances to the two rooms were directly adjacent so that when the experimenters silently exchanged information, they could closely monitor subjects. Each room had approximately forty seats, and subjects were dispersed so as to ensure privacy and prohibit communication. 22. The lump sum is $${\$}$$68 in the screening game and $${\$}$$73 in the signaling game. The higher lump sum in the signaling game reflects B’s higher expected cost per round in this game. This kept B’s ex ante earnings opportunities approximately equal across the two games. Since B did not know how many rounds the experiment would last, there was no way for him to compute a lump sum per round and make decisions based on such a number. 23. As expected, there is very little support for refinement D1 in the signaling game: only 5% (7/132) of the player A$$_{H}$$ offers match the D1 prediction of 525. 24. If, for example, we include all A$$_{L}$$ offers, this will include bluffs and between offers which provide a large negative surplus to player B if accepted. When all such offers are considered, the average offer from A$$_{L}$$ to B is 265, which represents a surplus of –40 for B . However, this number has little bearing on how A$$_{L}$$ behaves when she, via her offer, reveals her type and attempts to settle with B . Similarly, the inclusion of player B between or pooling offers has little bearing on how B behaves when he makes a screening offer and attempts to settle with A$$_{L}$$. The other offers we omit (i.e., those $$<$$ 75 or $$>$$ 525) are extremely rare, especially in the R7-end. 25. A$$_{L}$$ revealing offers appear to contain relatively less surplus than do B screening offers. The difference-in-means test H$$_{\mathrm{0}}$$: $$\mu_{\mathrm{B}}$$ – 75 $$=$$ 225 – $$\mu_{\mathrm{AL}}$$ has t$$=$$ 4.76 (P$$<$$ 0.000); also see fn. 26. 26. Eighty percent of the B screening offers are 75–125 (1/3 or less of the surplus), while 8% are 200–225 (5/6 or more of the surplus). For A$$_{L}$$ signaling game revealing offers, 81% are 200–225 (1/6 or less of the surplus) and 3% are 75–100 (5/6 or more of the surplus). 27. In the screening game, rejection rates are 25% on offers 75–99 (less than 1/6 of the surplus), 16% on offers 100–125, and 0% on offers above 125. In the signaling game, rejection rates are 13% for offers 201–225 (less than 1/6 of the surplus), and 3% on offers 200 and below. 28. Again, B does not know A’s type upon receiving the offer. Aggregating the signaling game offers 226–374 from Tables 5 and 6, the player B rejection rate for this interval is 55% (35/64) overall, 50% (8/16) in R7-end. 29. A$$_{L}$$ makes 94% of the between offers, so rejection is empirically justified. This anomalous acceptance behavior is fairly widespread among the player B subjects. Half (17/34) accepted at least one between offer, and one-fifth (7/34) did so twice or more. 30. Using Equation (5) and the Table 4 mean A$$_{H}$$ separating offer of 455, r$$=$$ 60.5% (median 450 yields r$$=$$ 60.0%). 31. Using Equation (5), the Table 4 means and medians on offers 375–525 imply a dispute rate of about 56–61%. 32. This explains why the predictions under the refinement D1 are not borne out. The prediction under D1 is that A$$_{H}$$ makes an offer of 525 which is rejected 67% of the time. Empirically these offers are rejected 100% of the time and as a result, A$$_{H}$$ needs to offer well below the D1 prediction to have a positive probability of acceptance. 33. One, in both games excess A$$_{L}$$ v. B disputes occur on offers 75–225. Two, in the signaling game the dispute rate on A$$_{L}$$ bluffs is higher than expected. Three, A$$_{L}$$ unexpectedly makes between offers 226 and 374 in the signaling game and B’s rejection rate is high on those offers, albeit well below the predicted 100%. 34. Our analysis focuses on the mean rather than the median. Because of the high dispute rates for A$$_{H}$$ players, the median is rather uninformative. When dispute rates are 50% or higher, the median payoff will always be 375 for these players. Thus, the median surplus earned by A$$_{H}$$ players is 0 in both games. 35. Also note that the refinement D1 is rejected as descriptive of players’ belief formation. This is in line with other experimental work which has generally rejected the intuitive criterion, a weaker refinement concept than D1. Here, a primary reason D1 fails is the defendant’s (player B’s) uniform rejection of very high offers, which forces the strong A$$_{H}$$ plaintiff to experiment with offers well below the D1 prediction. 36. 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Published: Apr 1, 2018

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