Modeling nonhuman conventions: the behavioral ecology of arbitrary action

Modeling nonhuman conventions: the behavioral ecology of arbitrary action Abstract This paper considers the relevance of so-called “Lewisian conventions” to the study of nonhuman animals. Conventions arise in coordination games with multiple equilibria, and the apparent arbitrariness of conventions occurs when processes outside the game itself determine which of several equilibria is ultimately chosen. Well-understood human conventions, such as driving on the left or right, can be seen as equilibria within a game. We consider possible nonhuman conventions, including traditional group locations, dominance, territoriality, and conventional signaling, that can be similarly described. We argue that conventions have been ignored in the study of animal behavior because they have been misunderstood. Yet, students of animal behavior are well prepared to understand and analyze conventions because the basic tools of game theory are already well established in our field. In addition, we argue that a research program exploring nonhuman conventions could greatly enrich the study of animal behavior. INTRODUCTION North Americans drive on the right side of the road, but the British drive on the left. Clearly, both practices are equally useful ways to avoid collisions. But, an American visiting London would not persist in driving on the right, because the benefit of following the left-driving or right-driving rule depends on a sort of agreement or understanding that all the vehicles on a given road will follow the same rule. Similarly, in English, one might order “fish” for dinner; but in Chinese, would ask for “yu.” As with driving rules, there is no a priori reason to prefer one word over the other as a label for gill-bearing aquatic craniates. Yet, you will not get what you expect if you ask for “yu and chips” in London, or “fish he zha shu tiao” in Beijing. Again, as with the driving example, the benefit of yu versus fish depends on an implicit agreement about meaning between the speaker and the listener. In both cases, individuals solve a coordination problem by selecting a rule or practice from a set of alternative possible practices, and the benefit an individual derives from adopting one practice rather than another depends on the expectation that others will follow a complementary practice. We call practices with these 2 properties conventions. Although human behavior offers many familiar and vivid examples of conventional behavior, we argue that the idea of conventions is equally relevant to nonhuman behavior. For example, on the high sagebrush steppes of the Western United States, male sage grouse famously gather to display to females. The spatial locations of these lekking grounds are commonly described as “traditional” (Wiley 1973; Bradbury et al. 1989). Why would a male display here and not there? Given the apparent uniformity of the surrounding steppe, the choice of one location over others seems at least somewhat arbitrary, and it is not far-fetched to postulate that the value of a specific lekking ground flows from an “agreement” about where males and females should meet, just as the value of “fish” over “yu” depends on an implicit agreement between speaker and listener. As another example, males and females share a common interest in avoiding inbreeding, and it is well known that some animals exhibit a pattern of male-biased dispersal (males leave–females stay) whereas others exhibit the reverse (females leave–males stay). As a mechanism to keep related males and females apart, both options would appear to be equally valuable (Perrin and Mazalov 1999), yet as in the problem of driving on the left or right side of the road, the benefit of one pattern over the other would seem to flow from its widespread acceptance. The ideas developed here are derived from the seminal work of the philosopher Lewis (1969). Lewis’s work has been influential in philosophy and the social sciences, but seems to have had little impact in the study of nonhuman behavior. This paper applies Lewis’s ideas to the analysis of nonhuman conventions. Our goal is to show that conventions are both theoretically and empirically tractable, and that they can be analyzed using tools that are already familiar to students of animal behavior. We argue that a conceptually and empirically rigorous study of behavioral conventions can enlarge and enrich our thinking about nonhuman behavior in several ways. First, the defining property of conventional behavior is the availability of alternative forms of coordination and this suggests an under-appreciated source of variation in animal interactions. Second, scientists tend to find what they are looking for, and we postulate that in the absence of a conceptual framework for the analysis of nonhuman conventions, animal behaviorists have either ignored or over-explained many instances of conventional behavior. Finally, the logic of conventions gives us tools for the analysis of animal communication, because as the yu-versus-fish example suggests, meaning can be viewed as a conventional phenomenon. If meaning is conventional, it follows that dishonesty and exaggeration, 2 frequently studied aspects of communication, are necessarily violations of conventions and can be analyzed as such. THE GAME THEORETICAL STRUCTURE OF CONVENTIONS How can we identify, describe, and analyze conventions? For the first example, we develop a model of spatial convention. The model is greatly simplified but based on the real phenomenon of communal roosting. Consider a hypothetical closed landscape that is occupied by 2 nonbreeding egrets, egret A and egret B. Our egrets forage independently during the day but each night they seek the cover of a roosting tree. Our simplified landscape contains 4 suitable trees that we call R1, R2, R3, and R4. Each of the 4 trees is equally acceptable as an overnight roost. If an egret roosts in one of the 4 trees alone, then it experiences an elevated risk of predation. Let the probability of survival as a singleton be 80%. If, however, the 2 egrets roost in the same tree, both individuals have a higher probability of overnight survival, say 95%. We have constructed a simple matrix game in which the strategies available to our 2 players are the 4 overnight roosting locations (Table 1). Table 1 Game matrix for the 2-player communal roosting game Player B R1 R2 R3 R4 Player A R1 .95 .8 .8 .8 R2 .8 .95 .8 .8 R3 .8 .8 .95 .8 R4 .8 .8 .8 .95 Player B R1 R2 R3 R4 Player A R1 .95 .8 .8 .8 R2 .8 .95 .8 .8 R3 .8 .8 .95 .8 R4 .8 .8 .8 .95 Players can choose one of 4 roosting sites, R1–R4, to spend the night. If the players roost together, their odds of survival are better than if they roost alone. View Large Table 1 Game matrix for the 2-player communal roosting game Player B R1 R2 R3 R4 Player A R1 .95 .8 .8 .8 R2 .8 .95 .8 .8 R3 .8 .8 .95 .8 R4 .8 .8 .8 .95 Player B R1 R2 R3 R4 Player A R1 .95 .8 .8 .8 R2 .8 .95 .8 .8 R3 .8 .8 .95 .8 R4 .8 .8 .8 .95 Players can choose one of 4 roosting sites, R1–R4, to spend the night. If the players roost together, their odds of survival are better than if they roost alone. View Large Consider that egret A controls the rows and egret B controls the columns, and the entries show the overnight survival probabilities for egret A (since the game is completely symmetric, egret B’s survival probabilities are the same). You will see immediately that there are 4 equivalent equilibria: both players at R1, both at R2, both at R3, and both at R4. Formally, this is a coordination game: a game with at least 2 pure Nash equilibria. This conceptual structure offers a mathematical representation of a convention. It demonstrates the potential arbitrariness of conventions because it does not matter which roosting site the 2 egrets choose, and it shows that the benefit of conventions derives from agreement, because it does matter that they choose the same roosting site. Simple as this game is, it illustrates 2 important points. First, students of animal behavior are already familiar with game theoretical analysis, and so they already have a key logical tool they need to analyze conventions. Second, we see that the arbitrariness of conventions flows from the fact that the game has multiple pure equilibria. The benefit-via-agreement property of conventions says nothing more than that the solutions are equilibria in which the benefit derived by one player depends on the actions of the other, which is of course what game theory is all about. Although this model may seem too simple to be important in the study of animal behavior, we argue that its structure can in fact apply to a broad range of biologically significant situations. We give several examples below to illustrate this. Shared actions pay As in our example of roosting egrets, there are many situations in which it pays for a pair of animals or a group of animals to adopt “matching” actions. Staying close together either in a roost, flock, herd, or school is a commonly observed behavioral phenomena, but unless all members of the flock are following similar behavioral rules about moving together, the flock would likely disintegrate. Similarly, it surely pays for males and females to come into breeding condition at roughly the same time or in the same environmental conditions. The reader can probably imagine several other situations where a shared behavior or action pays. If this coordination can be accomplished in more than 1 way, then we have a game with multiple equilibria, and the potential for conventional behavior. Complementary actions pay Of course, coordination can also require that players adopt different, complimentary actions. Imagine a simple landscape with 2 territories that both provide a fixed amount of resources (say 2). Imagine 2 competitors who can independently choose to occupy either territory 1 or territory 2. If both competitors occupy the same territory they both obtain one-half of the resources there, but if one occupies territory 1 and the other occupies territory 2 then they both obtain 100% of their territory’s resources, yielding the game matrix shown in Table 2. Table 2 Territoriality game Player B T1 T2 Player A T1 1 2 T2 2 1 Player B T1 T2 Player A T1 1 2 T2 2 1 Players A and B can each occupy one of 2 territories, T1 and T2, each containing an amount of resources (R). If they both occupy the same territory they must split its resources, but if they choose different territories, they each get the full territory’s worth. This game is conventional because the players benefit from coordinating their behavior so they occupy different resources, but the solution is arbitrary; A can occupy T1 and B can occupy T2, or the reverse can occur, with no difference in payoff to the players. View Large Table 2 Territoriality game Player B T1 T2 Player A T1 1 2 T2 2 1 Player B T1 T2 Player A T1 1 2 T2 2 1 Players A and B can each occupy one of 2 territories, T1 and T2, each containing an amount of resources (R). If they both occupy the same territory they must split its resources, but if they choose different territories, they each get the full territory’s worth. This game is conventional because the players benefit from coordinating their behavior so they occupy different resources, but the solution is arbitrary; A can occupy T1 and B can occupy T2, or the reverse can occur, with no difference in payoff to the players. View Large Clearly, it is beneficial for all concerned to recognize a simple “this is mine, that is yours” convention. Similar situations, in which players adopt complementary strategies, are very common in animal behavior. In the classic producer-scrounger model, a group of foragers maximizes its success by splitting between “producers” that actively search for food and “scroungers” that eat what the producers find (Barnard and Sibly 1981). In groups with sentinel behavior, it pays to coordinate vigilance so that someone is always on the lookout, but no more sentinels are present than necessary (Bednekoff and Woolfenden 2006). And, many animal groups have “leaders” that exert more influence over the group’s behavior than “followers,” a difference in strategy that works to minimize conflict (Johnstone and Manica 2011; Smith et al. 2016). Players need not benefit equally for coordination to be important. An informative example arises from thinking about the convention of dominance. Consider a 2-player game in which 2 individuals can play either a dominant or subordinate strategy. Imagine a single resource of value 100. If player A plays “dominant” and player B plays “subordinate,” A obtains 75 units of the resource, but B obtains only 25. If they both play “dominant,” they engage in a fight that costs both players 40 units each, but they are equally likely to win the resource. If they both play subordinate, they do not fight and one player gains the resource at random, so the expected payoff is 50. Table 3 shows the game matrix. Table 3 Dominant/Subordinate game, depicting 2 players competing over a 100 unit resource; the first element in each cell shows the payoff to player A and second shows the payoff to player B Each player can play dominant or subordinate Player B Dominant Subordinate Player A Dominant 10, 10 75, 25 Subordinate 25, 75 50, 50 Player B Dominant Subordinate Player A Dominant 10, 10 75, 25 Subordinate 25, 75 50, 50 Two dominant players will fight over the resource, incurring a cost. A subordinate player will immediately cede three-fourth of the resource to a dominant player. If both players are subordinate, one will gain the resource at random. An equilibrium exists when one player is dominant and the other subordinate, but it is irrelevant which player picks which role. This can be seen by examining each players’ best response to the other’s strategy. If Player A knows that Player B is dominant, it will receive the highest reward from playing subordinate (since 25 > 10); in contrast, if Player B is subordinate, A should play dominant (since 75 > 50). B’s best responses to A are identical. View Large Table 3 Dominant/Subordinate game, depicting 2 players competing over a 100 unit resource; the first element in each cell shows the payoff to player A and second shows the payoff to player B Each player can play dominant or subordinate Player B Dominant Subordinate Player A Dominant 10, 10 75, 25 Subordinate 25, 75 50, 50 Player B Dominant Subordinate Player A Dominant 10, 10 75, 25 Subordinate 25, 75 50, 50 Two dominant players will fight over the resource, incurring a cost. A subordinate player will immediately cede three-fourth of the resource to a dominant player. If both players are subordinate, one will gain the resource at random. An equilibrium exists when one player is dominant and the other subordinate, but it is irrelevant which player picks which role. This can be seen by examining each players’ best response to the other’s strategy. If Player A knows that Player B is dominant, it will receive the highest reward from playing subordinate (since 25 > 10); in contrast, if Player B is subordinate, A should play dominant (since 75 > 50). B’s best responses to A are identical. View Large We see that there are 2 equilibria, one in which player A is dominant and player B is subordinate and another in which player B is dominant and player A is subordinate. The 2 equilibria are equivalent in the sense that they are mirror images of each other, and a game theoretical analysis offers no reason to favor one over the other. From the perspective of the two players, they are quite different in the obvious sense that player 1 is the “winner” in one equilibrium and the “loser” in another. Notwithstanding this asymmetry, it seems perfectly reasonable to think of the choice of equilibrium 1 versus 2 as a problem in conventions. Some readers will recognize our “dominance” game as a game in the “chicken-snowdrift-hawk/dove” family of games (sometimes called anti-coordination games). These games have a long track record in economic and evolutionary game theory (Maynard-Smith and Price 1973; Selten 1980; Maynard-Smith 1982; Hamblin and Hurd 2007). Conventions of meaning Finally, we consider conventional communication through a behaviorally motivated variant of a signaling game, originally proposed by Lewis (1969) to address the conventionality of human language. The game is played by 2 players, a “signaler” and a “receiver.” We suppose that their environment can be in one of 2 states, say, “good” and “bad.” The signaler knows which state is true, but the receiver does not. The signaler has 2 flags, green and red, which can be waved to signal to the receiver. After observing the signaler’s choice of flag, the receiver must choose an action from one of two possibilities that we will call “accept’ and “reject.” If the receiver chooses “accept” in the good state, both the receiver and signaler receive a payoff; similarly, if the receiver chooses “reject” in the bad state, both players benefit. If the receiver makes any other choice both players get nothing. Therefore, we have a mutualism in which both players benefit when the receiver matches its action to the state, yet only the signaler has knowledge of the state that the receiver must match. Now suppose the signaler’s strategy set consists of 2 options: 1) it can wave the red flag when the environment is good, and wave the green flag when the environment is bad (denoted by Good→Red and Bad→Green); or 2) it can wave the green when the state is good and wave the red flag when the state is bad (denoted by Good→Green and Bad→Red). This is to say that the signaler’s strategy takes the form of a state-to-signal “encoding” rule. The receiver’s strategy takes the form of a signal-to-action “decoding” rule. As before, there are 2 possibilities: 1) Red→Accept and Green→Reject, or 2) Green→Accept and Red→Reject (Table 4). Once the strategies have been formulated the game theoretical analysis is hardly necessary (though our list of potential strategies is not exhaustive; see Lewis 1969 for a detailed solution). Table 4 Modified Lewis signaling game; the first element in each cell shows the payoff to the signaler and the second shows the payoff to the receiver Receiver Decoding Strategy I Red -> Accept Green -> Reject Decoding Strategy II Red -> Reject Green -> Accept Signaler Encoding Strategy A: Good→Red Bad→Green 1,1 0,0 Encoding Strategy B: Good→Green Bad→Red 0,0 1,1 Receiver Decoding Strategy I Red -> Accept Green -> Reject Decoding Strategy II Red -> Reject Green -> Accept Signaler Encoding Strategy A: Good→Red Bad→Green 1,1 0,0 Encoding Strategy B: Good→Green Bad→Red 0,0 1,1 Two players, a signaler and receiver, occupy an environment that can be “good” or “bad.” If the receiver plays action “accept” when the environment is good, or “reject” when the environment is bad, both players benefit. A mismatch between the environment and the receiver’s action rewards neither player. The receiver is not informed about the nature of the environment, but the signaler is, and can convey this information by showing the receiver a color (red or green). If the players can adopt a complementary set of rules, the receiver can effectively communicate with the signaler about the environment and the signaler can choose its actions appropriately. View Large Table 4 Modified Lewis signaling game; the first element in each cell shows the payoff to the signaler and the second shows the payoff to the receiver Receiver Decoding Strategy I Red -> Accept Green -> Reject Decoding Strategy II Red -> Reject Green -> Accept Signaler Encoding Strategy A: Good→Red Bad→Green 1,1 0,0 Encoding Strategy B: Good→Green Bad→Red 0,0 1,1 Receiver Decoding Strategy I Red -> Accept Green -> Reject Decoding Strategy II Red -> Reject Green -> Accept Signaler Encoding Strategy A: Good→Red Bad→Green 1,1 0,0 Encoding Strategy B: Good→Green Bad→Red 0,0 1,1 Two players, a signaler and receiver, occupy an environment that can be “good” or “bad.” If the receiver plays action “accept” when the environment is good, or “reject” when the environment is bad, both players benefit. A mismatch between the environment and the receiver’s action rewards neither player. The receiver is not informed about the nature of the environment, but the signaler is, and can convey this information by showing the receiver a color (red or green). If the players can adopt a complementary set of rules, the receiver can effectively communicate with the signaler about the environment and the signaler can choose its actions appropriately. View Large Clearly, there are 2 equivalent equilibria in which the signaler’s state-to-signal rule precisely complements the receiver’s signal-to-action rule. We can, therefore, take “meaning” to be the conventional match of a signaler’s state-signal rule with a receiver’s signal-action rule: signalers and receivers can attain an equilibrium if they agree that green “means” good and red “means” bad, but not if they disagree. Expanding this concept, we could consider “honest” or “reliable” signaling to be behavior that is consistent with these established conventions, and “dishonesty” to occur when one player deviates from the convention. This simple game can be expanded to include additional signaling strategies (such as the uninformative “signal” strategy: Good→Green and Bad→Green), and receiver responses (such as always accepting regardless of the signal observed); however, these strategies lead to lower payoffs than the signaling equilibria. The game can also be played with more states and signals, or with a conflict of interest between signaler and receiver, at which point it becomes substantially more complex. Considerable effort has been devoted to modeling how naive individuals or populations playing this game could develop a stable, maximally informative signaling system (Huttegger 2007; Pawlowitsch 2008; Barrett 2009; Huttegger et al. 2009). Defining conventions We have sketched models of 4 behavioral phenomena: communal roosting, territoriality, dominance and signaling. Although our models are simplistic, collectively they illustrate the potential relevance of conventions to animal behavior, because each problem admits alternative or conventional solutions. The definition of conventions has 2 parts. First, conventions arise in situations where animals obtain mutual benefit from coordinated action. As our simplified examples illustrate, beneficial coordination can arise in many types of behavior, from simple aggregation to communication. Second, conventions arise when this coordination can take different forms, specifically, game theoretically stable forms. In each of our examples, animals can coordinate in 2 or more possible ways: our egrets can jointly occupy site 1 or site 4; our signaler-receiver pair can take green to mean good, or red to mean good, and so on. In the terminology of game theory, we define conventions as the multiple (pure) equilibria of coordination games. To understand what conventions are, it is helpful to consider interactions that are not conventional. An interaction in which no benefit is derived from coordination could not have a conventional solution. For example, if player 1’s best action (say action A) is the same regardless of player 2’s behavior, (in game theoretical terminology we would say that action A dominates all other actions), action A is not a convention even if both players adopt it. Even if no single action dominates all others, a game can have a single (or even no) pure equilibrium. Again, an interaction with a single pure equilibrium would not represent a convention because there is no alternative form for the interaction to take. Conventions in repeated games The importance of conventions hinges on the existence of multiple equilibria in social interactions, and we have given several simple examples to illustrate the range of situations in which multiple equilibria can arise. Yet, we have only scratched the surface. The simple mechanism of repetition can, for example, magnify the number of equilibria in an interaction and dramatically increase the opportunities for conventional behavior. This happens because in repeated games players can adopt more complex strategies based on the outcomes of previous interactions. The number of equilibria in a repeated game can be enormous: if our egrets play the nesting game several nights in a row, “Roost at R1” and “Roost at R2” are still equilibria, but so are “Roost at R1 and R2 on alternating nights,” or “Roost in the same site as yesterday if the other player was also there, otherwise choose a random site.” This intuition, formalized as the “folk theorem,” states that almost any behavior can be a stable equilibrium if the game is repeated for long enough (Fudenberg and Maskin 1986). This suggests that we can expect to find multiple equilibria, and hence conventional behavior, in many repeated interactions (Boyd 2006). In this section, we have argued that nonhuman conventions may be very common, possibly even pervasive. Moreover, they may be relevant to a wide range of conceptually important behavioral phenomena such as the origins of meaning and the existence of social inequality. Yet, they are largely unrecognized and unstudied by behavioral ecologists and students of animal behavior. In the remainder of this paper, we consider 2 further topics. First, we review processes that may bias coordination games toward one equilibrium over another, and second we discuss the empirical study of nonhuman conventions. EQUILIBRIUM SELECTION The examples given above suggest that coordination games with multiple equilibria are a common problem in animal behavior and that they potentially include several significant problems such as the origins of meaning, and the explanation of dominance. With this in mind, we return to our simple “roosting location” problem. For simplicity, the game matrix can be rescaled to that in Table 5. Table 5 The roosting site game, normalized to illustrate the problem of equilibrium selection Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 1 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 1 0 R4 0 0 0 1 View Large Table 5 The roosting site game, normalized to illustrate the problem of equilibrium selection Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 1 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 1 0 R4 0 0 0 1 View Large If our pair of egrets adopts a single consistent roosting location, we would say that they are following a convention. Now that we have established that conventions arise in games with multiple equilibria, the problem becomes one of equilibrium selection. Why should our egrets adopt location R1 and not location R3, or any other option? Although there is a body of economic theory that addresses equilibrium selection the broad answer is that equilibrium selection is governed by 2 things: initial conditions, and dynamics. We consider each of these ideas briefly. To see why initial conditions matter, consider a scenario in which both egrets “start” by playing R2. In this case, we can be reasonably confident that the birds will continue to select R2-R2, since they are already there. If instead we imagine a scenario in which the birds begin a sequence of interactions with one bird in R1 and the other R4, then we have relatively little confidence about which equilibrium they will ultimately find (if any). Dynamics refers to the rules of change that each player follows from one interaction to the next. When we say “rules,” we do not necessarily mean that the process is deterministic. We could expect probabilistic transition rules: if I’m in state R1 and I have experience X in the 9th interaction, the probability that I will be at R1 at time 10 is p1, at R2 is p2, and so on for all 4 possible states. These dynamic rules could depend on properties of the candidate equilibria, the dynamic properties of reinforced learning, and evolutionary dynamics. As one might guess, potential variation in initial conditions and dynamics leaves us with a lot of free parameters, and this makes it hard to generalize about equilibrium selection. One way forward is to recognize different types of equilibria. This is the central lesson of Harsayni and Selten’s Nobel prize winning work in economics (Harsanyi and Selten 1988). Harsayni and Selten recognize 2 types of equilibria that can have a “favored” status in equilibrium selection problems: payoff dominant and risk dominant equilibria. A payoff dominant equilibrium is an equilibrium that provides the highest payoff for both players compared to the other equilibria (technically, it is Pareto superior to all other equilibria). In Table 6, R3 versus R3 is a payoff dominant equilibrium. The selection of a payoff dominant equilibrium makes sense in a world of rational, fully informed players. If our egrets could sit down together and review the payoff matrix, they would surely see that roost site 3 is the equilibrium with the highest payoff and choose to roost there. Of course, this is patently silly in the case of our egrets, but the reader can probably imagine somewhat more realistic situations that approach this rational and well-informed idealization—perhaps our egrets follow some sampling procedure before adopting a roosting site convention that allows them to compare payoffs. Table 6 The roosting site game, modified to illustrate the idea of payoff dominance Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 5 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 5 0 R4 0 0 0 1 The R3 versus R3 equilibrium is payoff dominant View Large Table 6 The roosting site game, modified to illustrate the idea of payoff dominance Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 5 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 5 0 R4 0 0 0 1 The R3 versus R3 equilibrium is payoff dominant View Large The criterion of risk dominance makes more sense when equilibria are selected by some type of incremental process. In nonhuman conventions, this could be trial and error learning, natural selection, or some combination of both. Consider Table 7, though the 4 equilibria are all still perfectly valid and have equivalent payoffs, the R2 versus R2 equilibrium has some advantages. If Player B initially plays all 4 options with equal likelihood, or if Player A has no way to predict what option Player B will choose, then Player A can expect a better outcome from playing R2 than from playing any other strategy, since a play of R2 is guaranteed to net some payoff regardless of Player B’s action. The expected payoff from R2 against a random opponent is 14×14+14×1+14×12+14×110=0.4625 whereas the expected payoff from any other option is only 0.25. This suggests that the equilibrium R2 versus R2 should be easiest to reach via some type of hill-climbing algorithm (like learning or natural selection). Similarly, even after an equilibrium is established, R2 versus R2 has an advantage: if player B occasionally deviates, player A will lose less if they have previously adopted the risk dominant equilibrium. Well-cited simulation studies by Young (1993, 1996) show that when payoff- and risk-dominant equilibria are in conflict, risk-dominance is the best predictor of equilibrium selection when the dynamics of equilibrium selection are “evolutionary” in some broad sense. We refer the reader to the original work of Harsanyi and Selten (Harsanyi and Selten 1988; Harsanyi 1995) for formal definitions and the application of these ideas to more complex situations such as asymmetric games. Table 7 The roosting site game, modified to illustrate the idea of risk dominance Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 .25 1 .5 .1 R3 0 0 1 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 .25 1 .5 .1 R3 0 0 1 0 R4 0 0 0 1 The R2 versus R2 equilibrium is risk dominant. View Large Table 7 The roosting site game, modified to illustrate the idea of risk dominance Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 .25 1 .5 .1 R3 0 0 1 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 .25 1 .5 .1 R3 0 0 1 0 R4 0 0 0 1 The R2 versus R2 equilibrium is risk dominant. View Large Although the “dynamics” discussed here may seem more mathematical than biological, readers should appreciate that “dynamics” includes, or potentially includes, many of the basic processes of behavior and cognition. For example, students of decision-making (e.g. Gigerenzer and Selten 2001) sometimes distinguish between global and bounded rationality. As explained above, globally rational players may be able to select payoff dominant equilibria, whereas players constrained to incrementally change their behavior (via some more bounded process) may be more likely to reach risk-dominant equilibria. Similarly, the learning rules and perceptual mechanisms that constrain animals could have profound effects on the equilibria that are ultimately reached. Imagine, for example, that one of our 4 possible roosting trees is distinctive in some way, (e.g. it is taller than the others). If the sites are otherwise equivalent, this distinctiveness might bias equilibrium selection in favor of the more distinctive, or salient, site (Schelling 1980; Vanderschraaf 1995; Mesterton-Gibbons and Adams 2003). Salience can expedite equilibrium selection if all players recognize the same features as salient and are predisposed to respond to them in the same way; what constitutes salience for a given animal depends on the specifics of its sensory system and its environment. CONVENTIONS AS AN EMPIRICAL RESEARCH PROGRAM The significance of the ideas presented here depends, of course, on their relevance to data, and a central goal of this paper is to explore the value of game theoretical conventions in guiding empirical research. A conventions-based empirical program might proceed in 2 ways. First, we can view conventional behavior as a largely unexplored aspect of nonhuman social behavior that deserves more attention in its own right; and second, we argue that many long-standing problems in social behavior can be re-framed in the light of conventions, and that this can bring a common set to tools to bear on a diverse set of problems. These are clearly not mutually exclusive aspects of the study of nonhuman conventions but rather seem to be the extreme points of a spectrum of empirical possibilities. Is it a convention? Is an observed behavior a convention? This is, perhaps, the most basic empirical question suggested by the conceptual framework presented here. In the case of many naturally occurring types of behavioral coordination, the only honest answer is that we simply do not know. To say that something is a convention means that alternative behavior—specifically an alternative equilibrium—exists. In some cases, we suspect that a convention exists because we observe different groups adopting different actions: driving on the left versus driving on the right; male-biased versus female-biased dispersal; lekking at site A versus lekking at site B. Table 8 lists several cases of candidate nonhuman conventions based on observed differences in behavior. Yet, neither the presence or absence of this sort of behavioral variation can be taken as definitive evidence for or against the existence of conventions. Reconsider the example of left and right driving. It is logically possible, although unlikely in this case, that left-driving is the only possible equilibrium in Britain, whereas right-driving is the only possible equilibrium in the United States; so that despite the existence of the behavioral difference neither is a convention. Consider, next the observation that red traffic lights universally mean stop and, to the best of our knowledge, no alternative encodings of light color are used to mean stop. Does this mean that red-to-mean-stop is not a convention? Possibly, but it also seems possible that alternative color-to-mean-stop encodings are possible and have simply not been adopted for one reason or another. To hypothesize that red-means-stop is a convention is to suppose that at least one other color-to-stop encoding is possible and potentially stable. We see, therefore, that neither the presence nor the absence of multiple forms of coordinated behavior can be taken as definitive evidence for or against the existence of a convention, and it follows that we need some stronger form of inference to establish or refute the conventionality of an observed form of coordination. Table 8 Selected examples of nonhuman conventions Behavior Alternatives Sources Dispersal Males leave and females stay; females leave and males stay (Greenwood 1980; Perrin and Mazalov 1999) Territory partitioning Arbitrary landscape features used to resolve territory border disputes (Mesterton-Gibbons and Adams 2003) Meeting locations Meet at one of several otherwise equivalent locations (Bradbury et al. 1989; Teng et al. 2012) Leader–follower behavior Group leaders could be determined by several mechanisms (experience, inheritance, personality, differences in motivation) (Jaupart et al. 2003; Smith et al. 2016) Conflict resolution If no economic difference between competitors exists, arbitrary differences could be used to efficiently determine winners and losers (Maynard-Smith and Price 1973; Hargreaves- Heap and Varoufakis 2002) Conventional signaling Signal “meaning” is arbitrary and determined by agreement between signaler and receiver, rather than differences in signal costs. (Hurd and Enquist 2005; Tibbetts 2013) Behavior Alternatives Sources Dispersal Males leave and females stay; females leave and males stay (Greenwood 1980; Perrin and Mazalov 1999) Territory partitioning Arbitrary landscape features used to resolve territory border disputes (Mesterton-Gibbons and Adams 2003) Meeting locations Meet at one of several otherwise equivalent locations (Bradbury et al. 1989; Teng et al. 2012) Leader–follower behavior Group leaders could be determined by several mechanisms (experience, inheritance, personality, differences in motivation) (Jaupart et al. 2003; Smith et al. 2016) Conflict resolution If no economic difference between competitors exists, arbitrary differences could be used to efficiently determine winners and losers (Maynard-Smith and Price 1973; Hargreaves- Heap and Varoufakis 2002) Conventional signaling Signal “meaning” is arbitrary and determined by agreement between signaler and receiver, rather than differences in signal costs. (Hurd and Enquist 2005; Tibbetts 2013) View Large Table 8 Selected examples of nonhuman conventions Behavior Alternatives Sources Dispersal Males leave and females stay; females leave and males stay (Greenwood 1980; Perrin and Mazalov 1999) Territory partitioning Arbitrary landscape features used to resolve territory border disputes (Mesterton-Gibbons and Adams 2003) Meeting locations Meet at one of several otherwise equivalent locations (Bradbury et al. 1989; Teng et al. 2012) Leader–follower behavior Group leaders could be determined by several mechanisms (experience, inheritance, personality, differences in motivation) (Jaupart et al. 2003; Smith et al. 2016) Conflict resolution If no economic difference between competitors exists, arbitrary differences could be used to efficiently determine winners and losers (Maynard-Smith and Price 1973; Hargreaves- Heap and Varoufakis 2002) Conventional signaling Signal “meaning” is arbitrary and determined by agreement between signaler and receiver, rather than differences in signal costs. (Hurd and Enquist 2005; Tibbetts 2013) Behavior Alternatives Sources Dispersal Males leave and females stay; females leave and males stay (Greenwood 1980; Perrin and Mazalov 1999) Territory partitioning Arbitrary landscape features used to resolve territory border disputes (Mesterton-Gibbons and Adams 2003) Meeting locations Meet at one of several otherwise equivalent locations (Bradbury et al. 1989; Teng et al. 2012) Leader–follower behavior Group leaders could be determined by several mechanisms (experience, inheritance, personality, differences in motivation) (Jaupart et al. 2003; Smith et al. 2016) Conflict resolution If no economic difference between competitors exists, arbitrary differences could be used to efficiently determine winners and losers (Maynard-Smith and Price 1973; Hargreaves- Heap and Varoufakis 2002) Conventional signaling Signal “meaning” is arbitrary and determined by agreement between signaler and receiver, rather than differences in signal costs. (Hurd and Enquist 2005; Tibbetts 2013) View Large Conventions in the laboratory Perhaps the simplest way to know how many equilibria exist is to experimentally construct game matrices. The work of Polnaszek and Stephens (Polnaszek and Stephens 2014) illustrates the potential of this approach. These authors sought to study of the economics of signaling games without specific thought to conventions. They paired captive blue jays (Cyanocitta cristata) as signaler and receiver in adjacent enclosures separated by windows. The birds played an experimental game in which the receiver’s binary actions (hopping to the left or to the right) controlled both players’ payoffs. Some of the time hopping on the left rewarded both players with food, whereas at other, probabilistically determined times, hopping on the right was rewarded. The signaler could observe a cue light that indicated the receiver’s best action, so that a cue light on the right meant that food would be delivered if the receiver hopped on the right, and so on. The receiver could not see this cue light, but it could see the signaler through the windows. The experimenters assumed that when the cue light indicated the right, the signaler would hop to the right side of its enclosure, and the receiver would subsequently learn to match this behavior. This is what happened most of the time, but Polnaszek and Stephens observed that some pairs of birds found an alternative “anti-matching” solution (Figure 1). In these pairs, the signaler hopped to the left when the cue light indicated the right, and the receiver learned to take the opposite perch, hopping to the right when the signaler went left. Polnaszek and Stephens seem to have unintentionally constructed a Lewisian signaling game in which there are 2 possible “cue light to signal” encodings that complement 2 possible “signal to action” encodings. Interestingly, most pairs in most situations found the expected “matching” equilibrium, suggesting that the matching equilibrium could be more attractive or salient in some way, but a handful still found the conceptually equivalent “anti-matching” equilibrium. Figure 1 View largeDownload slide Evidence of a spontaneous signaling convention. Results from Polnaszek and Stephens’s (2014) study of experimental signaling. The x-axis shows the proportion of trials in which the signaler hopped to the same side as the cue light (which only the signaler could see), and y-axis shows the proportion of trial in which receivers hopped to the same side as the signaler (matched). We see clusters of points in the top right and bottom left corners. The top right corner corresponds to a convention in which the signaler matches the cue-light and receiver matches the signaler; whereas the bottom left corresponds to a convention in which the signal “anti-matches” the cue light and the receiver adopts a position opposite the signaler. Figure 1 View largeDownload slide Evidence of a spontaneous signaling convention. Results from Polnaszek and Stephens’s (2014) study of experimental signaling. The x-axis shows the proportion of trials in which the signaler hopped to the same side as the cue light (which only the signaler could see), and y-axis shows the proportion of trial in which receivers hopped to the same side as the signaler (matched). We see clusters of points in the top right and bottom left corners. The top right corner corresponds to a convention in which the signaler matches the cue-light and receiver matches the signaler; whereas the bottom left corresponds to a convention in which the signal “anti-matches” the cue light and the receiver adopts a position opposite the signaler. Even though this experimental convention arose coincidentally, it suggests that experimental manipulation of conventional behavior is plausible and potentially productive. In forthcoming work, we have extended the observations of Polnaszek and Stephens in several directions (Heinen and Stephens, in preparation). In one experiment, for example, we compare of the behavior of sender-receiver pairs in nonconventional signaling games (i.e. games with a single equilibrium) to their behavior in conventional signaling games (with multiple experimentally created equilibria). The potential power of experimental conventions is that by directly controlling the structure of the coordination problem, we should be able to ask questions such as how conventions are transmitted, how sensory biases shape equilibrium finding, and so on. Conventions in nature Although laboratory studies of conventions have great potential, we obviously need to identify and study the prevalence of conventions in nature to establish the broader importance of conventions in the lives of nonhuman animals. Although there are many questions, we might ask about natural conventions, the most basic is whether an observed form of coordinated action is conventional. The definition of conventions tells us that to say that coordinated behavior X is a convention is to say than at least one stable alternative form of coordination, say Y, exists. So, a hypothesis about the conventionality of an action is necessarily also a hypothesis about alternatives to the action of interest. We might formulate such a hypothesis because we have observed that some groups coordinate in one whereas others coordinate in some alternative way. However, we might also develop a hypothesis about alternatives from some underlying knowledge of the game theoretical structure of the coordination at hand. As we outlined above, the structure of several common coordination problems (e.g. meeting, territoriality, dominance, signaling) strongly hints at the possibility of alternative equilibrium. Suppose then that we formulate the hypothesis that X (which we take to be a pair of actions, one action per player) is a convention because Y is an alternative stable state. How do we test this claim? The logical answer is via a displacement experiment. Our hypothesis holds that a pair of players engaging in stable equilibrium X would also stably engage in candidate equilibrium Y, if they can be moved to this new equilibrium. Alternatively, if Y is not an alternative equilibrium, a shift to Y will not be stable and the system will (eventually) return to X (or some other equilibrium) after the “displacement.” To see how such an experiment might proceed, we reconsider our egret roosting example, supposing for simplicity that two roosting sites, A and B, are available and our pair of egrets roost at A every evening. Now suppose that we somehow shift things so our egrets start to roost at B rather than A. We might do this by making A unavailable temporarily, or even by physically transporting the birds. If, following this displacement, our egrets return to A, then, we would conclude that roosting at A is not a convention, because this result supports that the hypothesis that roosting at A is the only equilibrium in the game and B is not a stable alternative equilibrium. Although operational details of conducting such an experiment could be daunting, recent work suggests it can be feasible in some situations. Teng et al. (2012) investigated “traditional” communal roosting in harvestmen (Prionostemma sp.). These nocturnal arachnids aggregate in spiny palm trees during the day. Only a fraction of the suitable trees are used, and the same trees are used repeatedly. The problem of finding a roosting site seems potentially conventional, if harvestmen are choosing their roosting sites because they are likely to meet other harvestmen there rather than because of some characteristic of the habitat. Teng et al. tested this possibility by relocating groups of harvestmen to unoccupied trees. They found that the harvestmen continued to utilize these new locations, and that some of these experimentally created roosting sites were still in use up to 8 years after the original displacement. This suggests that the new trees represent an alternative stable equilibrium in the “meeting game.” Importantly, in Teng et al.’s experiment the previous equilibria were still available; the sites were located such that the harvestmen could have easily abandoned the new site for a previous one. This is important because it ensures that the game matrix does in fact have multiple equilibria. Other experiments, not necessarily designed with conventions in mind, have similarly displaced animals from an equilibrium by completely removing that equilibrium from the game. Examples include mating aggregations in Przeqalski gazelles (You et al. 2011), and pairwise duetting in canebrake wrens (Rivera-Cáceres et al. 2016). We do not, of course, intend to suggest that displacement experiments will be easy. There are clearly many subtleties to be addressed in implementing such an experiment, and our ability to manipulate some kinds of conventions will necessarily be limited. Our proposed laboratory studies are similarly limited in the kinds of conventions we can address, and the timescale those conventions develop over. We offer these experiments as starting points, and as evidence that the empirical study of conventions is both feasible and important. We have focused our attention here on experimentally determining whether a given behavior is conventional. Clearly, this is not a trivial question and is a necessary first step in any empirical study of conventions. But, this example is intended to illustrate that the experimental study of conventional behavior is possible, and to serve as a springboard for further research ideas. Once a behavior is identified as conventional, many fascinating—and currently unexplored—avenues of investigation remain. We address some of these potential questions further in the discussion. DISCUSSION Following Lewis’s (1969) groundbreaking book, we have presented a straightforward, logically coherent framework for thinking about nonhuman conventions. Although readers will surely recognize the central role that conventions play in human interactions, we know comparatively little about nonhuman conventions. The foundation of the Lewisian approach to conventions is a stunningly simple definition: conventions are equilibria in coordination games with multiple equilibria. As we have argued, this simple definition includes, or potentially includes, a huge number of animal social interactions from grouping, to dominance to signaling At one level, there is nothing surprising in this view. The simple fact that games can have multiple equilibria is basic to any game theoretical approach to behavior. Thinking about these multiple equilibria as conventions is more a change in perspective than an earth-shattering new claim. Yet, we argue that this change in perspective is important for several reasons. First, this framework helps us clarify several misunderstandings about conventions in the behavioral literature (discussed below). Moreover, it helps us see unexpected parallels between disparate social phenomena such as the choice of roost sites and the meaning of signals. Second, it means that the mathematical tools for the analysis of conventions are already widely understood by students of animal behavior. Third, it reframes our thinking about multiple equilibria. Multiple equilibria exist in several important games, but typically we consider only one of these equilibria to be “interesting,” (e.g. in the iterated prisoner’s dilemma we focus on the cooperative equilibrium and consider the defect-defect equilibrium to be “a problem”). In the conventions perspective, all possible equilibria are broadly interesting. Fourth, as we argued above, the Lewisian perspective gives us the logical framework we need to design experiments and craft observations to explore the properties of conventions empirically. Finally, a conventions-based approach to signaling might bring new insights into the study of meaning, exaggeration, and honesty by recognizing that multiple rules of meaning are possible, and it follows that multiple forms of honesty and exaggeration are also possible. Arbitrariness and equivalence A striking feature of many conventions is their apparent arbitrariness, but it is easy to misunderstand or overemphasize this property of conventions. We can reasonably refer to the choice of equilibria with a multi-equilibrium coordination problem as arbitrary because forces outside the game itself will commonly influence equilibrium selection; yet this arbitrariness is not randomness. Reconsidering the example of “fish” versus “yu” in English and Chinese. We recognize these labels as arbitrary because we can imagine a variant of English in which “yu” replaces “fish” that could be perfectly stable and functional (or, in parallel, variant of Chinese in which “fish” replaces “yu”). Yet, it seems unlikely that Chinese speakers and listeners selected the syllable “yu” through sheer randomness. Surely, the history of the region and the phonetic rules of precursor languages have biased the equilibrium-finding process towards a sound like “yu” rather than a sound like “fish.” “Arbitrariness” does not mean that there is no reason for convention adopters to do things this way instead of that—there are often very important and interesting reasons for reaching one equilibrium instead of another. It does mean, however, that things could have been done another way, and that this alternative way would also be stable. Second, an appealing feature of the framework presented here is that it can explain the simultaneous existence of equivalent solutions to a coordination problem (as in the vivid example of left vs. right driving), but this strict equivalence is not a necessary feature of conventions. The conventional solutions to a coordination game can have different payoffs, and different dynamic properties. Moreover, there is no requirement that the players in a game experience the same consequences at a conventional equilibrium. Many potentially important behavioral conventions, such as problems of dominance or territoriality, are likely to be strongly asymmetric (for example, if owners do better than nonowners). Traditions are not necessarily conventions Several behavioral phenomena, from foraging strategies to burrow use, are commonly described as traditional (Whiten et al. 2005; Muller and Cant 2010; Thornton et al. 2010; Aplin et al. 2015). Some of these putative traditions involve the choice of a physical location: for example, traditional roosting sites (Teng et al. 2012), foraging sites (Fishlock et al. 2016), or lekking grounds (Bradbury et al. 1989). Other traditions involve the choice of alternative actions. In recent experimental studies, for example, investigators offer animals a problem with multiple solutions, and they conclude that a tradition exists if group members use the same method to solve the experimental problem (e.g. Bonnie et al. 2007; Muller and Cant, 2010; Aplin et al., 2015; Prestipino et al., 2016). Some of these experimental traditions are probably also conventions, but not all. Conventions necessarily involve some sort of coordination or behavioral complementarity, whereas a tradition could in principle be a solitary activity. So spatial traditions that involve coordination of some type—e.g. roosting to reduce predation costs, lekking to attract females—could be seen as conventions, whereas the tradition of opening a puzzle box by the pulling the red lever would not necessarily be a convention even if it learned by observing others. So, traditions are not necessarily conventions. Moreover, a convention need not be traditional. A pair of individuals could, at least in principle, solve a coordination problem via a novel convention that we not would call a tradition, because it has never been used by others. Connection to social learning Conventions are clearly a social phenomenon since they require interaction, and many important conventions are clearly learned. It follows that at some level social learning must be involved. Yet, the extant social learning literature focuses on situations in observers learn to perform actions similar to what they have observed (Laland 2004). This “imitative” social learning may apply to some conventions—when individuals coordinate by performing similar actions—but it will not apply to all, because in many conventions players must adopt complementary actions (e.g. signalers and receivers). To best of our knowledge the social learning of these sorts of complementary actions is relatively little explored, and potentially quite important. Conventions in behavioral ecology Although we feel that nonhuman conventions are under-appreciated and understudied, behavioral ecologists have previously considered conventions in several situations, and we will discuss 2 important cases here. We note however, that in most cases these older uses of the term “convention” are not in complete agreement with the coherent Lewisian framework presented here. Bourgeois and anti-bourgeois Perhaps the most famous theoretical analysis of “conventional behavior” in classical behavioral ecology is Maynard Smith’s Hawk-Dove-Bourgeois analysis of animal conflict over resources (Maynard-Smith and Price 1973; Maynard-Smith 1982). The textbook analysis of this classical game shows that there is no pure equilibrium in the Hawk-Dove game. However, when we add a new strategy called bourgeosis, which “respects the convention of ownership,” we find that a single pure bourgeois-versus-bourgeois equilibrium arises. Readers will immediately see that in the standard development of this game, bourgeosis-versus-bourgeosis is the only pure equilibrium of the game, and it follows that in the Lewisian framework it is not a convention, because there is no alternative equilibrium. Mathematically inclined readers will realize (Maynard Smith 1982), that a second equilibrium would exist if we introduced a mirror image strategy in which intruders win and owners lose (which we might call anti-bougeosis); and this would rescue Maynard Smith’s claim that bourgeosis-versus-bourgeosis is a convention. Yet, naturally occurring resource conflicts may not be so simple. Several authors have remarked on the apparent absence of anti-bourgeosis-like strategies in nature, and noted that territory owners and intruders likely differ in several ways that make the problem less symmetrical than it initially appeared (Kokko et al. 2006; Sherratt and Mesterton-Gibbons 2015). If as these authors suggest, anti-bourgeosis strategies are not equilibria of resource conflict games, then it is not clear that we can properly call owner-respecting strategies like bougeosis conventions. However, the broader theory, that asymmetries between individuals can be used conventionally to resolve conflicts, remains relevant. Conventional “Signals” Perhaps unsurprisingly, the area of behavioral ecology in which the idea of conventions has been most influential is signaling and communication. A comprehensive review of this literature is beyond the scope of our discussion. Instead, we seek to point out how current and historical thinking about signaling conventions differs from the Lewisian framework offered here. To begin, we remark that the phrase “conventional signal” is potentially misleading. A signal cannot stand on its own as a convention. A signaling convention consists of a complementary pair of encoding and decoding rules. The choice of a particular encoding rule/decoding rule pair can be a convention if other sets of rules are available. The availability of alternative equilibria is the defining idea in the conventionality (or lack thereof) of a signaling system. Much of the existing literature on signaling conventions focuses on the problem of honesty. Early investigators rejected the idea of signaling conventions because they argued that the arbitrariness of “conventional signals” makes it too easy for signalers to cheat (Maynard-Smith and Harper 1988; Zahavi 1993; Guilford and Dawkins 1995). In the Lewis framework, this argument loses some of its force because we define conventions as the equilibria of coordination games; and this means that cheaters who depart from this equilibrium will pay a cost, simply because this is how equilibria are defined. In a broader analysis, this complaint seems to suppose that conventional signals exist independently of the complementary behavior of receivers, which as we have pointed out above does not make sense in the framework presented here. More recent, and more sophisticated, treatments of signaling conventions have continued to emphasize the problem of honesty. These investigators have recognized 2 key properties of “honest conventions” that distinguish them from the other mechanisms that promote honesty. First, the signals produced in signaling conventions need not have high production costs. Second, cheating—departing from the conventional equilibrium in our approach—is punished by receiver action. These 2 properties, low production costs and receiver-dependent costs have been taken to be key indicators of signaling conventions (Searcy and Nowicki 2005; Tibbetts 2013). Although these properties are perfectly compatible with the framework presented here, they should not be seen as definitive properties of signaling conventions. Signaling conventions exist, in Lewisian terms, when there are multiple signaling equilibria, and there is no direct connection between production costs per se and the number of equilibria in a signaling game. For example, we could imagine a situation in which a male could advertise his quality via a costly call or a costly ornament, the fact that both could be costly to produce does not preclude the possibility that they represent alternative signaling conventions. The idea that receiver action creates a cost for signalers who depart from an equilibrium refers to a specific mechanism for maintaining an equilibrium, and importantly this mechanism can operate regardless of the number of signaling equilibria in the larger game. Its connection to Lewisian signaling conventions is therefore unclear, because the existence of receiver-dependent costs is not sufficient to establish the existence of multiple equilibria which is, of course, the defining property of Lewisian conventions. Equilibrium selection literature As we explained in the body of the paper, the Lewisian approach to conventions leads us to ask questions about equilibrium selection; and as we pointed out Harsanyi and Selten’s (1988) ideas are broadly seen as the foundational ideas in this area. We feel is important to acknowledge the emerging and extensive body of literature on this topic. Evolutionary game theory is a broad area of research, and illustrates that evolutionary processes, population dynamics, and temporal and spatial change can all dramatically influence equilibrium selection (Boyd 1992; Hammerstein and Selten 1994; Hofbauer and Sigmund 2003; Roca et al. 2009). The Lewis model of signaling has also been influential in philosophy and economics. Results from simulations suggest that simple learning or evolutionary mechanisms reliably converge on signaling conventions (Huttegger 2007; Pawlowitsch 2008; Catteeuw and Manderick 2014) but several factors can bias this convergence. The number of possible signals can influence equilibrium selection (Barrett 2009; Huttegger and Zollman 2011), as can the predictability of the underlying state being communicated (Huttegger 2007; Alexander 2014). See Skryms (2010) and Huttegger et al (2014) for a further overview. LIMITATIONS AND FURTHER QUESTIONS Level of analysis We have presented a range of simple coordination games including models of communal roosting, dominance, territoriality, and signaling. It should be clear that none of these models is intended as a comprehensive treatment of any specific behavior phenomenon. Instead, we have sketched a series of simplistic models to show the potential applicability of game theoretical conventions to a wide range of behavioral phenomena. Moreover, we make no claim that these analyses are novel or even unique. Instead, we hope to illustrate the common structure and pervasiveness of coordination games with multiple equilibria. Conventions and Mixed equilibria Many readers will know that mixed equilibria are a common feature of many biologically important games. Lewis’s initial development simply ignored mixed equilibria, and this paper, like most treatments of conventions, has followed Lewis’s lead and focused on pure equilibria. The relationship between mixed equilibria and conventions is currently unclear. One approach is to continue to follow Lewis and ignore them. There is some justification for this. Mixed equilibria are typically inefficient compared to the pure strategy equilibria, and they tend to be unstable in simulated coordination games (Young 1993; Hansen 2006; Hamblin and Hurd 2007). An alternative, but as yet poorly developed, approach is to consider them co-equal with the pure equilibria in coordination games, and hence candidate conventions. Mixed strategies can be important in some situations, including the maintenance of honest signaling (Huttegger and Zollman 2010; Polnaszek and Stephens 2014). Evolved or learned conventions? Humans learn the conventional meanings of words, and Americans are perfectly happy to drive on the left when they find themselves in England. Moreover, the flexibility of learned conventions provides many opportunities for experimental manipulation. So, one might naturally think of conventions as learned and not evolved. This might seem jarring to those behavioral ecologists who think of game theory as an exclusively evolutionary approach. Yet, the distinction between learned and evolved behavior is not clear cut. Maynard Smith, the father of evolutionary game theory, took pains to point out that learning, and more broadly experience, must often be involved in the establishment of evolutionary equilibria (Maynard-Smith 1982; Harley and Smith 1983). In the broadest sense, learning is likely to interact with selection to shape the conventional equilibria we observe in nature. The question poised in our subheading is ultimately nonsensical: conventions—like other behavioral phenotypes—are shaped by experience and evolution. Indeed, behavioral ecologists have an important opportunity to contribute to the study of conventions by exploring this interaction. The Lewisian approach to conventions is agnostic about whether the dynamics establishing an equilibrium are evolutionary, experiential, or some combination. Yet, applying Lewis’s ideas over an evolutionary time scale raises some novel questions. What happens, for example, if equilibria come and go during a population’s selective history? Our definition requires that equilibria be simultaneously available in order to be called conventions. So, we can imagine scenarios in which an equilibrium was a convention at some time in the evolutionary past, but it is not a convention now (because the alternative equilibrium has been lost). Clearly, changes like this could also have powerful effects on the equilibrium selection process, by biasing individuals or population toward one conventional equilibrium over another. FUTURE DIRECTIONS Population and the scale of conventions Most readers will think first of population-level conventions, since many of the clearest examples occur at this scale. However, conventions can also occur between as few as 2 individuals. Human couples engage in many pairwise conventions: I use the bathroom first, you do the dishes while I vacuum, and so on. Pairwise conventions are not well studied, but candidates include duetting birds that use “duet codes,” pair-specific rules for which phrases should follow each other (Hall 2009; Rivera-Cáceres et al. 2016). Other candidates include parental care, conflict resolution, or territorial partitioning. Pairwise conventions provide a good opportunity for experimental analysis, because pairs of animals can be manipulated more easily than entire populations. Conventions at the population scale generate many interesting questions about the development and spread of conventions. One might speculate that it will be easier to find conventional equilibria when pairs of individuals interact repeatedly, but that some partner swapping is needed to facilitate the spread of the equilibrium throughout a population. How does this balance affect the spread of a convention? And, what happens when populations with different conventions encounter each other? Models have shown that the dynamics of social interactions can have a profound effect on equilibrium selection (Goyal and Vega-Redondo 2005; Van Cleve and Lehmann 2013) and on the potential for the co-existence of alternative conventions (Bhaskar and Vega-Redondo 2004). Sensory bias and the psychophysics of equilibrium selection Although we have emphasized the economic determinants of equilibrium selection, the sensory and psychophysical predispositions should not be ignored. Otherwise, economically equivalent equilibria could be quite different perceptually. Some of these biases are probably simple, as when one potential roosting tree is taller than the others, whereas others could be quite subtle. A basic idea in psychophysics is that discriminations, even apparently trivial discriminations, take place in a background of internal and external noise (Gescheider 1997). This noise is likely to affect each player’s strategy selection as well as its perceptions of the actions of others. In theory, many (if not all) pre-existing bias problems can be thought of as effects of initial conditions and dynamics, which are, of course, already part of our thinking about equilibrium selection. The fact that we can include them in our models does not make them any less important, because their effects are likely to be pervasive. FINAL SUMMARY We have presented some basic ideas about “conventions,” taken from the human behavioral sciences and we have outlined how these ideas can be applied to nonhuman behavior. Conventions arise in coordination games with multiple equilibria. Although the study of nonhuman conventions is virtually nonexistent, we argue that conventions can arise in nearly every aspect of animal social interactions from simple aggregation to the establishment of meaning. The ideas presented here provide a framework for the conceptual and empirical study of conventions. We advocate a research program that considers the role of conventions in animal interactions both experimentally and observationally. Students of animal behavior are well prepared to study and understand conventions because the basic conceptual tools of game theory are already well known to behavioral ecologists. This logical framework can help us understand such divergent phenomena as the meanings of signals and the existence of dominance hierarchies, and may reveal previously unappreciated patterns of behavior, learning and evolution. FUNDING This work was supported by the University of Minnesota Doctoral Dissertation Fellowship (to V.K.H). Data accessibility No new data was collected for the preparation of this paper. The authors are grateful to T. Polnaszek and T. Rubi for their contributions to the ideas and data presented here, and to L. Barrett, and T. Sherratt, and 4 anonymous reviewers for helpful comments and suggestions. REFERENCES Alexander JM . 2014 . 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Modeling nonhuman conventions: the behavioral ecology of arbitrary action

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Abstract

Abstract This paper considers the relevance of so-called “Lewisian conventions” to the study of nonhuman animals. Conventions arise in coordination games with multiple equilibria, and the apparent arbitrariness of conventions occurs when processes outside the game itself determine which of several equilibria is ultimately chosen. Well-understood human conventions, such as driving on the left or right, can be seen as equilibria within a game. We consider possible nonhuman conventions, including traditional group locations, dominance, territoriality, and conventional signaling, that can be similarly described. We argue that conventions have been ignored in the study of animal behavior because they have been misunderstood. Yet, students of animal behavior are well prepared to understand and analyze conventions because the basic tools of game theory are already well established in our field. In addition, we argue that a research program exploring nonhuman conventions could greatly enrich the study of animal behavior. INTRODUCTION North Americans drive on the right side of the road, but the British drive on the left. Clearly, both practices are equally useful ways to avoid collisions. But, an American visiting London would not persist in driving on the right, because the benefit of following the left-driving or right-driving rule depends on a sort of agreement or understanding that all the vehicles on a given road will follow the same rule. Similarly, in English, one might order “fish” for dinner; but in Chinese, would ask for “yu.” As with driving rules, there is no a priori reason to prefer one word over the other as a label for gill-bearing aquatic craniates. Yet, you will not get what you expect if you ask for “yu and chips” in London, or “fish he zha shu tiao” in Beijing. Again, as with the driving example, the benefit of yu versus fish depends on an implicit agreement about meaning between the speaker and the listener. In both cases, individuals solve a coordination problem by selecting a rule or practice from a set of alternative possible practices, and the benefit an individual derives from adopting one practice rather than another depends on the expectation that others will follow a complementary practice. We call practices with these 2 properties conventions. Although human behavior offers many familiar and vivid examples of conventional behavior, we argue that the idea of conventions is equally relevant to nonhuman behavior. For example, on the high sagebrush steppes of the Western United States, male sage grouse famously gather to display to females. The spatial locations of these lekking grounds are commonly described as “traditional” (Wiley 1973; Bradbury et al. 1989). Why would a male display here and not there? Given the apparent uniformity of the surrounding steppe, the choice of one location over others seems at least somewhat arbitrary, and it is not far-fetched to postulate that the value of a specific lekking ground flows from an “agreement” about where males and females should meet, just as the value of “fish” over “yu” depends on an implicit agreement between speaker and listener. As another example, males and females share a common interest in avoiding inbreeding, and it is well known that some animals exhibit a pattern of male-biased dispersal (males leave–females stay) whereas others exhibit the reverse (females leave–males stay). As a mechanism to keep related males and females apart, both options would appear to be equally valuable (Perrin and Mazalov 1999), yet as in the problem of driving on the left or right side of the road, the benefit of one pattern over the other would seem to flow from its widespread acceptance. The ideas developed here are derived from the seminal work of the philosopher Lewis (1969). Lewis’s work has been influential in philosophy and the social sciences, but seems to have had little impact in the study of nonhuman behavior. This paper applies Lewis’s ideas to the analysis of nonhuman conventions. Our goal is to show that conventions are both theoretically and empirically tractable, and that they can be analyzed using tools that are already familiar to students of animal behavior. We argue that a conceptually and empirically rigorous study of behavioral conventions can enlarge and enrich our thinking about nonhuman behavior in several ways. First, the defining property of conventional behavior is the availability of alternative forms of coordination and this suggests an under-appreciated source of variation in animal interactions. Second, scientists tend to find what they are looking for, and we postulate that in the absence of a conceptual framework for the analysis of nonhuman conventions, animal behaviorists have either ignored or over-explained many instances of conventional behavior. Finally, the logic of conventions gives us tools for the analysis of animal communication, because as the yu-versus-fish example suggests, meaning can be viewed as a conventional phenomenon. If meaning is conventional, it follows that dishonesty and exaggeration, 2 frequently studied aspects of communication, are necessarily violations of conventions and can be analyzed as such. THE GAME THEORETICAL STRUCTURE OF CONVENTIONS How can we identify, describe, and analyze conventions? For the first example, we develop a model of spatial convention. The model is greatly simplified but based on the real phenomenon of communal roosting. Consider a hypothetical closed landscape that is occupied by 2 nonbreeding egrets, egret A and egret B. Our egrets forage independently during the day but each night they seek the cover of a roosting tree. Our simplified landscape contains 4 suitable trees that we call R1, R2, R3, and R4. Each of the 4 trees is equally acceptable as an overnight roost. If an egret roosts in one of the 4 trees alone, then it experiences an elevated risk of predation. Let the probability of survival as a singleton be 80%. If, however, the 2 egrets roost in the same tree, both individuals have a higher probability of overnight survival, say 95%. We have constructed a simple matrix game in which the strategies available to our 2 players are the 4 overnight roosting locations (Table 1). Table 1 Game matrix for the 2-player communal roosting game Player B R1 R2 R3 R4 Player A R1 .95 .8 .8 .8 R2 .8 .95 .8 .8 R3 .8 .8 .95 .8 R4 .8 .8 .8 .95 Player B R1 R2 R3 R4 Player A R1 .95 .8 .8 .8 R2 .8 .95 .8 .8 R3 .8 .8 .95 .8 R4 .8 .8 .8 .95 Players can choose one of 4 roosting sites, R1–R4, to spend the night. If the players roost together, their odds of survival are better than if they roost alone. View Large Table 1 Game matrix for the 2-player communal roosting game Player B R1 R2 R3 R4 Player A R1 .95 .8 .8 .8 R2 .8 .95 .8 .8 R3 .8 .8 .95 .8 R4 .8 .8 .8 .95 Player B R1 R2 R3 R4 Player A R1 .95 .8 .8 .8 R2 .8 .95 .8 .8 R3 .8 .8 .95 .8 R4 .8 .8 .8 .95 Players can choose one of 4 roosting sites, R1–R4, to spend the night. If the players roost together, their odds of survival are better than if they roost alone. View Large Consider that egret A controls the rows and egret B controls the columns, and the entries show the overnight survival probabilities for egret A (since the game is completely symmetric, egret B’s survival probabilities are the same). You will see immediately that there are 4 equivalent equilibria: both players at R1, both at R2, both at R3, and both at R4. Formally, this is a coordination game: a game with at least 2 pure Nash equilibria. This conceptual structure offers a mathematical representation of a convention. It demonstrates the potential arbitrariness of conventions because it does not matter which roosting site the 2 egrets choose, and it shows that the benefit of conventions derives from agreement, because it does matter that they choose the same roosting site. Simple as this game is, it illustrates 2 important points. First, students of animal behavior are already familiar with game theoretical analysis, and so they already have a key logical tool they need to analyze conventions. Second, we see that the arbitrariness of conventions flows from the fact that the game has multiple pure equilibria. The benefit-via-agreement property of conventions says nothing more than that the solutions are equilibria in which the benefit derived by one player depends on the actions of the other, which is of course what game theory is all about. Although this model may seem too simple to be important in the study of animal behavior, we argue that its structure can in fact apply to a broad range of biologically significant situations. We give several examples below to illustrate this. Shared actions pay As in our example of roosting egrets, there are many situations in which it pays for a pair of animals or a group of animals to adopt “matching” actions. Staying close together either in a roost, flock, herd, or school is a commonly observed behavioral phenomena, but unless all members of the flock are following similar behavioral rules about moving together, the flock would likely disintegrate. Similarly, it surely pays for males and females to come into breeding condition at roughly the same time or in the same environmental conditions. The reader can probably imagine several other situations where a shared behavior or action pays. If this coordination can be accomplished in more than 1 way, then we have a game with multiple equilibria, and the potential for conventional behavior. Complementary actions pay Of course, coordination can also require that players adopt different, complimentary actions. Imagine a simple landscape with 2 territories that both provide a fixed amount of resources (say 2). Imagine 2 competitors who can independently choose to occupy either territory 1 or territory 2. If both competitors occupy the same territory they both obtain one-half of the resources there, but if one occupies territory 1 and the other occupies territory 2 then they both obtain 100% of their territory’s resources, yielding the game matrix shown in Table 2. Table 2 Territoriality game Player B T1 T2 Player A T1 1 2 T2 2 1 Player B T1 T2 Player A T1 1 2 T2 2 1 Players A and B can each occupy one of 2 territories, T1 and T2, each containing an amount of resources (R). If they both occupy the same territory they must split its resources, but if they choose different territories, they each get the full territory’s worth. This game is conventional because the players benefit from coordinating their behavior so they occupy different resources, but the solution is arbitrary; A can occupy T1 and B can occupy T2, or the reverse can occur, with no difference in payoff to the players. View Large Table 2 Territoriality game Player B T1 T2 Player A T1 1 2 T2 2 1 Player B T1 T2 Player A T1 1 2 T2 2 1 Players A and B can each occupy one of 2 territories, T1 and T2, each containing an amount of resources (R). If they both occupy the same territory they must split its resources, but if they choose different territories, they each get the full territory’s worth. This game is conventional because the players benefit from coordinating their behavior so they occupy different resources, but the solution is arbitrary; A can occupy T1 and B can occupy T2, or the reverse can occur, with no difference in payoff to the players. View Large Clearly, it is beneficial for all concerned to recognize a simple “this is mine, that is yours” convention. Similar situations, in which players adopt complementary strategies, are very common in animal behavior. In the classic producer-scrounger model, a group of foragers maximizes its success by splitting between “producers” that actively search for food and “scroungers” that eat what the producers find (Barnard and Sibly 1981). In groups with sentinel behavior, it pays to coordinate vigilance so that someone is always on the lookout, but no more sentinels are present than necessary (Bednekoff and Woolfenden 2006). And, many animal groups have “leaders” that exert more influence over the group’s behavior than “followers,” a difference in strategy that works to minimize conflict (Johnstone and Manica 2011; Smith et al. 2016). Players need not benefit equally for coordination to be important. An informative example arises from thinking about the convention of dominance. Consider a 2-player game in which 2 individuals can play either a dominant or subordinate strategy. Imagine a single resource of value 100. If player A plays “dominant” and player B plays “subordinate,” A obtains 75 units of the resource, but B obtains only 25. If they both play “dominant,” they engage in a fight that costs both players 40 units each, but they are equally likely to win the resource. If they both play subordinate, they do not fight and one player gains the resource at random, so the expected payoff is 50. Table 3 shows the game matrix. Table 3 Dominant/Subordinate game, depicting 2 players competing over a 100 unit resource; the first element in each cell shows the payoff to player A and second shows the payoff to player B Each player can play dominant or subordinate Player B Dominant Subordinate Player A Dominant 10, 10 75, 25 Subordinate 25, 75 50, 50 Player B Dominant Subordinate Player A Dominant 10, 10 75, 25 Subordinate 25, 75 50, 50 Two dominant players will fight over the resource, incurring a cost. A subordinate player will immediately cede three-fourth of the resource to a dominant player. If both players are subordinate, one will gain the resource at random. An equilibrium exists when one player is dominant and the other subordinate, but it is irrelevant which player picks which role. This can be seen by examining each players’ best response to the other’s strategy. If Player A knows that Player B is dominant, it will receive the highest reward from playing subordinate (since 25 > 10); in contrast, if Player B is subordinate, A should play dominant (since 75 > 50). B’s best responses to A are identical. View Large Table 3 Dominant/Subordinate game, depicting 2 players competing over a 100 unit resource; the first element in each cell shows the payoff to player A and second shows the payoff to player B Each player can play dominant or subordinate Player B Dominant Subordinate Player A Dominant 10, 10 75, 25 Subordinate 25, 75 50, 50 Player B Dominant Subordinate Player A Dominant 10, 10 75, 25 Subordinate 25, 75 50, 50 Two dominant players will fight over the resource, incurring a cost. A subordinate player will immediately cede three-fourth of the resource to a dominant player. If both players are subordinate, one will gain the resource at random. An equilibrium exists when one player is dominant and the other subordinate, but it is irrelevant which player picks which role. This can be seen by examining each players’ best response to the other’s strategy. If Player A knows that Player B is dominant, it will receive the highest reward from playing subordinate (since 25 > 10); in contrast, if Player B is subordinate, A should play dominant (since 75 > 50). B’s best responses to A are identical. View Large We see that there are 2 equilibria, one in which player A is dominant and player B is subordinate and another in which player B is dominant and player A is subordinate. The 2 equilibria are equivalent in the sense that they are mirror images of each other, and a game theoretical analysis offers no reason to favor one over the other. From the perspective of the two players, they are quite different in the obvious sense that player 1 is the “winner” in one equilibrium and the “loser” in another. Notwithstanding this asymmetry, it seems perfectly reasonable to think of the choice of equilibrium 1 versus 2 as a problem in conventions. Some readers will recognize our “dominance” game as a game in the “chicken-snowdrift-hawk/dove” family of games (sometimes called anti-coordination games). These games have a long track record in economic and evolutionary game theory (Maynard-Smith and Price 1973; Selten 1980; Maynard-Smith 1982; Hamblin and Hurd 2007). Conventions of meaning Finally, we consider conventional communication through a behaviorally motivated variant of a signaling game, originally proposed by Lewis (1969) to address the conventionality of human language. The game is played by 2 players, a “signaler” and a “receiver.” We suppose that their environment can be in one of 2 states, say, “good” and “bad.” The signaler knows which state is true, but the receiver does not. The signaler has 2 flags, green and red, which can be waved to signal to the receiver. After observing the signaler’s choice of flag, the receiver must choose an action from one of two possibilities that we will call “accept’ and “reject.” If the receiver chooses “accept” in the good state, both the receiver and signaler receive a payoff; similarly, if the receiver chooses “reject” in the bad state, both players benefit. If the receiver makes any other choice both players get nothing. Therefore, we have a mutualism in which both players benefit when the receiver matches its action to the state, yet only the signaler has knowledge of the state that the receiver must match. Now suppose the signaler’s strategy set consists of 2 options: 1) it can wave the red flag when the environment is good, and wave the green flag when the environment is bad (denoted by Good→Red and Bad→Green); or 2) it can wave the green when the state is good and wave the red flag when the state is bad (denoted by Good→Green and Bad→Red). This is to say that the signaler’s strategy takes the form of a state-to-signal “encoding” rule. The receiver’s strategy takes the form of a signal-to-action “decoding” rule. As before, there are 2 possibilities: 1) Red→Accept and Green→Reject, or 2) Green→Accept and Red→Reject (Table 4). Once the strategies have been formulated the game theoretical analysis is hardly necessary (though our list of potential strategies is not exhaustive; see Lewis 1969 for a detailed solution). Table 4 Modified Lewis signaling game; the first element in each cell shows the payoff to the signaler and the second shows the payoff to the receiver Receiver Decoding Strategy I Red -> Accept Green -> Reject Decoding Strategy II Red -> Reject Green -> Accept Signaler Encoding Strategy A: Good→Red Bad→Green 1,1 0,0 Encoding Strategy B: Good→Green Bad→Red 0,0 1,1 Receiver Decoding Strategy I Red -> Accept Green -> Reject Decoding Strategy II Red -> Reject Green -> Accept Signaler Encoding Strategy A: Good→Red Bad→Green 1,1 0,0 Encoding Strategy B: Good→Green Bad→Red 0,0 1,1 Two players, a signaler and receiver, occupy an environment that can be “good” or “bad.” If the receiver plays action “accept” when the environment is good, or “reject” when the environment is bad, both players benefit. A mismatch between the environment and the receiver’s action rewards neither player. The receiver is not informed about the nature of the environment, but the signaler is, and can convey this information by showing the receiver a color (red or green). If the players can adopt a complementary set of rules, the receiver can effectively communicate with the signaler about the environment and the signaler can choose its actions appropriately. View Large Table 4 Modified Lewis signaling game; the first element in each cell shows the payoff to the signaler and the second shows the payoff to the receiver Receiver Decoding Strategy I Red -> Accept Green -> Reject Decoding Strategy II Red -> Reject Green -> Accept Signaler Encoding Strategy A: Good→Red Bad→Green 1,1 0,0 Encoding Strategy B: Good→Green Bad→Red 0,0 1,1 Receiver Decoding Strategy I Red -> Accept Green -> Reject Decoding Strategy II Red -> Reject Green -> Accept Signaler Encoding Strategy A: Good→Red Bad→Green 1,1 0,0 Encoding Strategy B: Good→Green Bad→Red 0,0 1,1 Two players, a signaler and receiver, occupy an environment that can be “good” or “bad.” If the receiver plays action “accept” when the environment is good, or “reject” when the environment is bad, both players benefit. A mismatch between the environment and the receiver’s action rewards neither player. The receiver is not informed about the nature of the environment, but the signaler is, and can convey this information by showing the receiver a color (red or green). If the players can adopt a complementary set of rules, the receiver can effectively communicate with the signaler about the environment and the signaler can choose its actions appropriately. View Large Clearly, there are 2 equivalent equilibria in which the signaler’s state-to-signal rule precisely complements the receiver’s signal-to-action rule. We can, therefore, take “meaning” to be the conventional match of a signaler’s state-signal rule with a receiver’s signal-action rule: signalers and receivers can attain an equilibrium if they agree that green “means” good and red “means” bad, but not if they disagree. Expanding this concept, we could consider “honest” or “reliable” signaling to be behavior that is consistent with these established conventions, and “dishonesty” to occur when one player deviates from the convention. This simple game can be expanded to include additional signaling strategies (such as the uninformative “signal” strategy: Good→Green and Bad→Green), and receiver responses (such as always accepting regardless of the signal observed); however, these strategies lead to lower payoffs than the signaling equilibria. The game can also be played with more states and signals, or with a conflict of interest between signaler and receiver, at which point it becomes substantially more complex. Considerable effort has been devoted to modeling how naive individuals or populations playing this game could develop a stable, maximally informative signaling system (Huttegger 2007; Pawlowitsch 2008; Barrett 2009; Huttegger et al. 2009). Defining conventions We have sketched models of 4 behavioral phenomena: communal roosting, territoriality, dominance and signaling. Although our models are simplistic, collectively they illustrate the potential relevance of conventions to animal behavior, because each problem admits alternative or conventional solutions. The definition of conventions has 2 parts. First, conventions arise in situations where animals obtain mutual benefit from coordinated action. As our simplified examples illustrate, beneficial coordination can arise in many types of behavior, from simple aggregation to communication. Second, conventions arise when this coordination can take different forms, specifically, game theoretically stable forms. In each of our examples, animals can coordinate in 2 or more possible ways: our egrets can jointly occupy site 1 or site 4; our signaler-receiver pair can take green to mean good, or red to mean good, and so on. In the terminology of game theory, we define conventions as the multiple (pure) equilibria of coordination games. To understand what conventions are, it is helpful to consider interactions that are not conventional. An interaction in which no benefit is derived from coordination could not have a conventional solution. For example, if player 1’s best action (say action A) is the same regardless of player 2’s behavior, (in game theoretical terminology we would say that action A dominates all other actions), action A is not a convention even if both players adopt it. Even if no single action dominates all others, a game can have a single (or even no) pure equilibrium. Again, an interaction with a single pure equilibrium would not represent a convention because there is no alternative form for the interaction to take. Conventions in repeated games The importance of conventions hinges on the existence of multiple equilibria in social interactions, and we have given several simple examples to illustrate the range of situations in which multiple equilibria can arise. Yet, we have only scratched the surface. The simple mechanism of repetition can, for example, magnify the number of equilibria in an interaction and dramatically increase the opportunities for conventional behavior. This happens because in repeated games players can adopt more complex strategies based on the outcomes of previous interactions. The number of equilibria in a repeated game can be enormous: if our egrets play the nesting game several nights in a row, “Roost at R1” and “Roost at R2” are still equilibria, but so are “Roost at R1 and R2 on alternating nights,” or “Roost in the same site as yesterday if the other player was also there, otherwise choose a random site.” This intuition, formalized as the “folk theorem,” states that almost any behavior can be a stable equilibrium if the game is repeated for long enough (Fudenberg and Maskin 1986). This suggests that we can expect to find multiple equilibria, and hence conventional behavior, in many repeated interactions (Boyd 2006). In this section, we have argued that nonhuman conventions may be very common, possibly even pervasive. Moreover, they may be relevant to a wide range of conceptually important behavioral phenomena such as the origins of meaning and the existence of social inequality. Yet, they are largely unrecognized and unstudied by behavioral ecologists and students of animal behavior. In the remainder of this paper, we consider 2 further topics. First, we review processes that may bias coordination games toward one equilibrium over another, and second we discuss the empirical study of nonhuman conventions. EQUILIBRIUM SELECTION The examples given above suggest that coordination games with multiple equilibria are a common problem in animal behavior and that they potentially include several significant problems such as the origins of meaning, and the explanation of dominance. With this in mind, we return to our simple “roosting location” problem. For simplicity, the game matrix can be rescaled to that in Table 5. Table 5 The roosting site game, normalized to illustrate the problem of equilibrium selection Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 1 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 1 0 R4 0 0 0 1 View Large Table 5 The roosting site game, normalized to illustrate the problem of equilibrium selection Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 1 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 1 0 R4 0 0 0 1 View Large If our pair of egrets adopts a single consistent roosting location, we would say that they are following a convention. Now that we have established that conventions arise in games with multiple equilibria, the problem becomes one of equilibrium selection. Why should our egrets adopt location R1 and not location R3, or any other option? Although there is a body of economic theory that addresses equilibrium selection the broad answer is that equilibrium selection is governed by 2 things: initial conditions, and dynamics. We consider each of these ideas briefly. To see why initial conditions matter, consider a scenario in which both egrets “start” by playing R2. In this case, we can be reasonably confident that the birds will continue to select R2-R2, since they are already there. If instead we imagine a scenario in which the birds begin a sequence of interactions with one bird in R1 and the other R4, then we have relatively little confidence about which equilibrium they will ultimately find (if any). Dynamics refers to the rules of change that each player follows from one interaction to the next. When we say “rules,” we do not necessarily mean that the process is deterministic. We could expect probabilistic transition rules: if I’m in state R1 and I have experience X in the 9th interaction, the probability that I will be at R1 at time 10 is p1, at R2 is p2, and so on for all 4 possible states. These dynamic rules could depend on properties of the candidate equilibria, the dynamic properties of reinforced learning, and evolutionary dynamics. As one might guess, potential variation in initial conditions and dynamics leaves us with a lot of free parameters, and this makes it hard to generalize about equilibrium selection. One way forward is to recognize different types of equilibria. This is the central lesson of Harsayni and Selten’s Nobel prize winning work in economics (Harsanyi and Selten 1988). Harsayni and Selten recognize 2 types of equilibria that can have a “favored” status in equilibrium selection problems: payoff dominant and risk dominant equilibria. A payoff dominant equilibrium is an equilibrium that provides the highest payoff for both players compared to the other equilibria (technically, it is Pareto superior to all other equilibria). In Table 6, R3 versus R3 is a payoff dominant equilibrium. The selection of a payoff dominant equilibrium makes sense in a world of rational, fully informed players. If our egrets could sit down together and review the payoff matrix, they would surely see that roost site 3 is the equilibrium with the highest payoff and choose to roost there. Of course, this is patently silly in the case of our egrets, but the reader can probably imagine somewhat more realistic situations that approach this rational and well-informed idealization—perhaps our egrets follow some sampling procedure before adopting a roosting site convention that allows them to compare payoffs. Table 6 The roosting site game, modified to illustrate the idea of payoff dominance Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 5 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 5 0 R4 0 0 0 1 The R3 versus R3 equilibrium is payoff dominant View Large Table 6 The roosting site game, modified to illustrate the idea of payoff dominance Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 5 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 0 1 0 0 R3 0 0 5 0 R4 0 0 0 1 The R3 versus R3 equilibrium is payoff dominant View Large The criterion of risk dominance makes more sense when equilibria are selected by some type of incremental process. In nonhuman conventions, this could be trial and error learning, natural selection, or some combination of both. Consider Table 7, though the 4 equilibria are all still perfectly valid and have equivalent payoffs, the R2 versus R2 equilibrium has some advantages. If Player B initially plays all 4 options with equal likelihood, or if Player A has no way to predict what option Player B will choose, then Player A can expect a better outcome from playing R2 than from playing any other strategy, since a play of R2 is guaranteed to net some payoff regardless of Player B’s action. The expected payoff from R2 against a random opponent is 14×14+14×1+14×12+14×110=0.4625 whereas the expected payoff from any other option is only 0.25. This suggests that the equilibrium R2 versus R2 should be easiest to reach via some type of hill-climbing algorithm (like learning or natural selection). Similarly, even after an equilibrium is established, R2 versus R2 has an advantage: if player B occasionally deviates, player A will lose less if they have previously adopted the risk dominant equilibrium. Well-cited simulation studies by Young (1993, 1996) show that when payoff- and risk-dominant equilibria are in conflict, risk-dominance is the best predictor of equilibrium selection when the dynamics of equilibrium selection are “evolutionary” in some broad sense. We refer the reader to the original work of Harsanyi and Selten (Harsanyi and Selten 1988; Harsanyi 1995) for formal definitions and the application of these ideas to more complex situations such as asymmetric games. Table 7 The roosting site game, modified to illustrate the idea of risk dominance Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 .25 1 .5 .1 R3 0 0 1 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 .25 1 .5 .1 R3 0 0 1 0 R4 0 0 0 1 The R2 versus R2 equilibrium is risk dominant. View Large Table 7 The roosting site game, modified to illustrate the idea of risk dominance Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 .25 1 .5 .1 R3 0 0 1 0 R4 0 0 0 1 Player B R1 R2 R3 R4 Player A R1 1 0 0 0 R2 .25 1 .5 .1 R3 0 0 1 0 R4 0 0 0 1 The R2 versus R2 equilibrium is risk dominant. View Large Although the “dynamics” discussed here may seem more mathematical than biological, readers should appreciate that “dynamics” includes, or potentially includes, many of the basic processes of behavior and cognition. For example, students of decision-making (e.g. Gigerenzer and Selten 2001) sometimes distinguish between global and bounded rationality. As explained above, globally rational players may be able to select payoff dominant equilibria, whereas players constrained to incrementally change their behavior (via some more bounded process) may be more likely to reach risk-dominant equilibria. Similarly, the learning rules and perceptual mechanisms that constrain animals could have profound effects on the equilibria that are ultimately reached. Imagine, for example, that one of our 4 possible roosting trees is distinctive in some way, (e.g. it is taller than the others). If the sites are otherwise equivalent, this distinctiveness might bias equilibrium selection in favor of the more distinctive, or salient, site (Schelling 1980; Vanderschraaf 1995; Mesterton-Gibbons and Adams 2003). Salience can expedite equilibrium selection if all players recognize the same features as salient and are predisposed to respond to them in the same way; what constitutes salience for a given animal depends on the specifics of its sensory system and its environment. CONVENTIONS AS AN EMPIRICAL RESEARCH PROGRAM The significance of the ideas presented here depends, of course, on their relevance to data, and a central goal of this paper is to explore the value of game theoretical conventions in guiding empirical research. A conventions-based empirical program might proceed in 2 ways. First, we can view conventional behavior as a largely unexplored aspect of nonhuman social behavior that deserves more attention in its own right; and second, we argue that many long-standing problems in social behavior can be re-framed in the light of conventions, and that this can bring a common set to tools to bear on a diverse set of problems. These are clearly not mutually exclusive aspects of the study of nonhuman conventions but rather seem to be the extreme points of a spectrum of empirical possibilities. Is it a convention? Is an observed behavior a convention? This is, perhaps, the most basic empirical question suggested by the conceptual framework presented here. In the case of many naturally occurring types of behavioral coordination, the only honest answer is that we simply do not know. To say that something is a convention means that alternative behavior—specifically an alternative equilibrium—exists. In some cases, we suspect that a convention exists because we observe different groups adopting different actions: driving on the left versus driving on the right; male-biased versus female-biased dispersal; lekking at site A versus lekking at site B. Table 8 lists several cases of candidate nonhuman conventions based on observed differences in behavior. Yet, neither the presence or absence of this sort of behavioral variation can be taken as definitive evidence for or against the existence of conventions. Reconsider the example of left and right driving. It is logically possible, although unlikely in this case, that left-driving is the only possible equilibrium in Britain, whereas right-driving is the only possible equilibrium in the United States; so that despite the existence of the behavioral difference neither is a convention. Consider, next the observation that red traffic lights universally mean stop and, to the best of our knowledge, no alternative encodings of light color are used to mean stop. Does this mean that red-to-mean-stop is not a convention? Possibly, but it also seems possible that alternative color-to-mean-stop encodings are possible and have simply not been adopted for one reason or another. To hypothesize that red-means-stop is a convention is to suppose that at least one other color-to-stop encoding is possible and potentially stable. We see, therefore, that neither the presence nor the absence of multiple forms of coordinated behavior can be taken as definitive evidence for or against the existence of a convention, and it follows that we need some stronger form of inference to establish or refute the conventionality of an observed form of coordination. Table 8 Selected examples of nonhuman conventions Behavior Alternatives Sources Dispersal Males leave and females stay; females leave and males stay (Greenwood 1980; Perrin and Mazalov 1999) Territory partitioning Arbitrary landscape features used to resolve territory border disputes (Mesterton-Gibbons and Adams 2003) Meeting locations Meet at one of several otherwise equivalent locations (Bradbury et al. 1989; Teng et al. 2012) Leader–follower behavior Group leaders could be determined by several mechanisms (experience, inheritance, personality, differences in motivation) (Jaupart et al. 2003; Smith et al. 2016) Conflict resolution If no economic difference between competitors exists, arbitrary differences could be used to efficiently determine winners and losers (Maynard-Smith and Price 1973; Hargreaves- Heap and Varoufakis 2002) Conventional signaling Signal “meaning” is arbitrary and determined by agreement between signaler and receiver, rather than differences in signal costs. (Hurd and Enquist 2005; Tibbetts 2013) Behavior Alternatives Sources Dispersal Males leave and females stay; females leave and males stay (Greenwood 1980; Perrin and Mazalov 1999) Territory partitioning Arbitrary landscape features used to resolve territory border disputes (Mesterton-Gibbons and Adams 2003) Meeting locations Meet at one of several otherwise equivalent locations (Bradbury et al. 1989; Teng et al. 2012) Leader–follower behavior Group leaders could be determined by several mechanisms (experience, inheritance, personality, differences in motivation) (Jaupart et al. 2003; Smith et al. 2016) Conflict resolution If no economic difference between competitors exists, arbitrary differences could be used to efficiently determine winners and losers (Maynard-Smith and Price 1973; Hargreaves- Heap and Varoufakis 2002) Conventional signaling Signal “meaning” is arbitrary and determined by agreement between signaler and receiver, rather than differences in signal costs. (Hurd and Enquist 2005; Tibbetts 2013) View Large Table 8 Selected examples of nonhuman conventions Behavior Alternatives Sources Dispersal Males leave and females stay; females leave and males stay (Greenwood 1980; Perrin and Mazalov 1999) Territory partitioning Arbitrary landscape features used to resolve territory border disputes (Mesterton-Gibbons and Adams 2003) Meeting locations Meet at one of several otherwise equivalent locations (Bradbury et al. 1989; Teng et al. 2012) Leader–follower behavior Group leaders could be determined by several mechanisms (experience, inheritance, personality, differences in motivation) (Jaupart et al. 2003; Smith et al. 2016) Conflict resolution If no economic difference between competitors exists, arbitrary differences could be used to efficiently determine winners and losers (Maynard-Smith and Price 1973; Hargreaves- Heap and Varoufakis 2002) Conventional signaling Signal “meaning” is arbitrary and determined by agreement between signaler and receiver, rather than differences in signal costs. (Hurd and Enquist 2005; Tibbetts 2013) Behavior Alternatives Sources Dispersal Males leave and females stay; females leave and males stay (Greenwood 1980; Perrin and Mazalov 1999) Territory partitioning Arbitrary landscape features used to resolve territory border disputes (Mesterton-Gibbons and Adams 2003) Meeting locations Meet at one of several otherwise equivalent locations (Bradbury et al. 1989; Teng et al. 2012) Leader–follower behavior Group leaders could be determined by several mechanisms (experience, inheritance, personality, differences in motivation) (Jaupart et al. 2003; Smith et al. 2016) Conflict resolution If no economic difference between competitors exists, arbitrary differences could be used to efficiently determine winners and losers (Maynard-Smith and Price 1973; Hargreaves- Heap and Varoufakis 2002) Conventional signaling Signal “meaning” is arbitrary and determined by agreement between signaler and receiver, rather than differences in signal costs. (Hurd and Enquist 2005; Tibbetts 2013) View Large Conventions in the laboratory Perhaps the simplest way to know how many equilibria exist is to experimentally construct game matrices. The work of Polnaszek and Stephens (Polnaszek and Stephens 2014) illustrates the potential of this approach. These authors sought to study of the economics of signaling games without specific thought to conventions. They paired captive blue jays (Cyanocitta cristata) as signaler and receiver in adjacent enclosures separated by windows. The birds played an experimental game in which the receiver’s binary actions (hopping to the left or to the right) controlled both players’ payoffs. Some of the time hopping on the left rewarded both players with food, whereas at other, probabilistically determined times, hopping on the right was rewarded. The signaler could observe a cue light that indicated the receiver’s best action, so that a cue light on the right meant that food would be delivered if the receiver hopped on the right, and so on. The receiver could not see this cue light, but it could see the signaler through the windows. The experimenters assumed that when the cue light indicated the right, the signaler would hop to the right side of its enclosure, and the receiver would subsequently learn to match this behavior. This is what happened most of the time, but Polnaszek and Stephens observed that some pairs of birds found an alternative “anti-matching” solution (Figure 1). In these pairs, the signaler hopped to the left when the cue light indicated the right, and the receiver learned to take the opposite perch, hopping to the right when the signaler went left. Polnaszek and Stephens seem to have unintentionally constructed a Lewisian signaling game in which there are 2 possible “cue light to signal” encodings that complement 2 possible “signal to action” encodings. Interestingly, most pairs in most situations found the expected “matching” equilibrium, suggesting that the matching equilibrium could be more attractive or salient in some way, but a handful still found the conceptually equivalent “anti-matching” equilibrium. Figure 1 View largeDownload slide Evidence of a spontaneous signaling convention. Results from Polnaszek and Stephens’s (2014) study of experimental signaling. The x-axis shows the proportion of trials in which the signaler hopped to the same side as the cue light (which only the signaler could see), and y-axis shows the proportion of trial in which receivers hopped to the same side as the signaler (matched). We see clusters of points in the top right and bottom left corners. The top right corner corresponds to a convention in which the signaler matches the cue-light and receiver matches the signaler; whereas the bottom left corresponds to a convention in which the signal “anti-matches” the cue light and the receiver adopts a position opposite the signaler. Figure 1 View largeDownload slide Evidence of a spontaneous signaling convention. Results from Polnaszek and Stephens’s (2014) study of experimental signaling. The x-axis shows the proportion of trials in which the signaler hopped to the same side as the cue light (which only the signaler could see), and y-axis shows the proportion of trial in which receivers hopped to the same side as the signaler (matched). We see clusters of points in the top right and bottom left corners. The top right corner corresponds to a convention in which the signaler matches the cue-light and receiver matches the signaler; whereas the bottom left corresponds to a convention in which the signal “anti-matches” the cue light and the receiver adopts a position opposite the signaler. Even though this experimental convention arose coincidentally, it suggests that experimental manipulation of conventional behavior is plausible and potentially productive. In forthcoming work, we have extended the observations of Polnaszek and Stephens in several directions (Heinen and Stephens, in preparation). In one experiment, for example, we compare of the behavior of sender-receiver pairs in nonconventional signaling games (i.e. games with a single equilibrium) to their behavior in conventional signaling games (with multiple experimentally created equilibria). The potential power of experimental conventions is that by directly controlling the structure of the coordination problem, we should be able to ask questions such as how conventions are transmitted, how sensory biases shape equilibrium finding, and so on. Conventions in nature Although laboratory studies of conventions have great potential, we obviously need to identify and study the prevalence of conventions in nature to establish the broader importance of conventions in the lives of nonhuman animals. Although there are many questions, we might ask about natural conventions, the most basic is whether an observed form of coordinated action is conventional. The definition of conventions tells us that to say that coordinated behavior X is a convention is to say than at least one stable alternative form of coordination, say Y, exists. So, a hypothesis about the conventionality of an action is necessarily also a hypothesis about alternatives to the action of interest. We might formulate such a hypothesis because we have observed that some groups coordinate in one whereas others coordinate in some alternative way. However, we might also develop a hypothesis about alternatives from some underlying knowledge of the game theoretical structure of the coordination at hand. As we outlined above, the structure of several common coordination problems (e.g. meeting, territoriality, dominance, signaling) strongly hints at the possibility of alternative equilibrium. Suppose then that we formulate the hypothesis that X (which we take to be a pair of actions, one action per player) is a convention because Y is an alternative stable state. How do we test this claim? The logical answer is via a displacement experiment. Our hypothesis holds that a pair of players engaging in stable equilibrium X would also stably engage in candidate equilibrium Y, if they can be moved to this new equilibrium. Alternatively, if Y is not an alternative equilibrium, a shift to Y will not be stable and the system will (eventually) return to X (or some other equilibrium) after the “displacement.” To see how such an experiment might proceed, we reconsider our egret roosting example, supposing for simplicity that two roosting sites, A and B, are available and our pair of egrets roost at A every evening. Now suppose that we somehow shift things so our egrets start to roost at B rather than A. We might do this by making A unavailable temporarily, or even by physically transporting the birds. If, following this displacement, our egrets return to A, then, we would conclude that roosting at A is not a convention, because this result supports that the hypothesis that roosting at A is the only equilibrium in the game and B is not a stable alternative equilibrium. Although operational details of conducting such an experiment could be daunting, recent work suggests it can be feasible in some situations. Teng et al. (2012) investigated “traditional” communal roosting in harvestmen (Prionostemma sp.). These nocturnal arachnids aggregate in spiny palm trees during the day. Only a fraction of the suitable trees are used, and the same trees are used repeatedly. The problem of finding a roosting site seems potentially conventional, if harvestmen are choosing their roosting sites because they are likely to meet other harvestmen there rather than because of some characteristic of the habitat. Teng et al. tested this possibility by relocating groups of harvestmen to unoccupied trees. They found that the harvestmen continued to utilize these new locations, and that some of these experimentally created roosting sites were still in use up to 8 years after the original displacement. This suggests that the new trees represent an alternative stable equilibrium in the “meeting game.” Importantly, in Teng et al.’s experiment the previous equilibria were still available; the sites were located such that the harvestmen could have easily abandoned the new site for a previous one. This is important because it ensures that the game matrix does in fact have multiple equilibria. Other experiments, not necessarily designed with conventions in mind, have similarly displaced animals from an equilibrium by completely removing that equilibrium from the game. Examples include mating aggregations in Przeqalski gazelles (You et al. 2011), and pairwise duetting in canebrake wrens (Rivera-Cáceres et al. 2016). We do not, of course, intend to suggest that displacement experiments will be easy. There are clearly many subtleties to be addressed in implementing such an experiment, and our ability to manipulate some kinds of conventions will necessarily be limited. Our proposed laboratory studies are similarly limited in the kinds of conventions we can address, and the timescale those conventions develop over. We offer these experiments as starting points, and as evidence that the empirical study of conventions is both feasible and important. We have focused our attention here on experimentally determining whether a given behavior is conventional. Clearly, this is not a trivial question and is a necessary first step in any empirical study of conventions. But, this example is intended to illustrate that the experimental study of conventional behavior is possible, and to serve as a springboard for further research ideas. Once a behavior is identified as conventional, many fascinating—and currently unexplored—avenues of investigation remain. We address some of these potential questions further in the discussion. DISCUSSION Following Lewis’s (1969) groundbreaking book, we have presented a straightforward, logically coherent framework for thinking about nonhuman conventions. Although readers will surely recognize the central role that conventions play in human interactions, we know comparatively little about nonhuman conventions. The foundation of the Lewisian approach to conventions is a stunningly simple definition: conventions are equilibria in coordination games with multiple equilibria. As we have argued, this simple definition includes, or potentially includes, a huge number of animal social interactions from grouping, to dominance to signaling At one level, there is nothing surprising in this view. The simple fact that games can have multiple equilibria is basic to any game theoretical approach to behavior. Thinking about these multiple equilibria as conventions is more a change in perspective than an earth-shattering new claim. Yet, we argue that this change in perspective is important for several reasons. First, this framework helps us clarify several misunderstandings about conventions in the behavioral literature (discussed below). Moreover, it helps us see unexpected parallels between disparate social phenomena such as the choice of roost sites and the meaning of signals. Second, it means that the mathematical tools for the analysis of conventions are already widely understood by students of animal behavior. Third, it reframes our thinking about multiple equilibria. Multiple equilibria exist in several important games, but typically we consider only one of these equilibria to be “interesting,” (e.g. in the iterated prisoner’s dilemma we focus on the cooperative equilibrium and consider the defect-defect equilibrium to be “a problem”). In the conventions perspective, all possible equilibria are broadly interesting. Fourth, as we argued above, the Lewisian perspective gives us the logical framework we need to design experiments and craft observations to explore the properties of conventions empirically. Finally, a conventions-based approach to signaling might bring new insights into the study of meaning, exaggeration, and honesty by recognizing that multiple rules of meaning are possible, and it follows that multiple forms of honesty and exaggeration are also possible. Arbitrariness and equivalence A striking feature of many conventions is their apparent arbitrariness, but it is easy to misunderstand or overemphasize this property of conventions. We can reasonably refer to the choice of equilibria with a multi-equilibrium coordination problem as arbitrary because forces outside the game itself will commonly influence equilibrium selection; yet this arbitrariness is not randomness. Reconsidering the example of “fish” versus “yu” in English and Chinese. We recognize these labels as arbitrary because we can imagine a variant of English in which “yu” replaces “fish” that could be perfectly stable and functional (or, in parallel, variant of Chinese in which “fish” replaces “yu”). Yet, it seems unlikely that Chinese speakers and listeners selected the syllable “yu” through sheer randomness. Surely, the history of the region and the phonetic rules of precursor languages have biased the equilibrium-finding process towards a sound like “yu” rather than a sound like “fish.” “Arbitrariness” does not mean that there is no reason for convention adopters to do things this way instead of that—there are often very important and interesting reasons for reaching one equilibrium instead of another. It does mean, however, that things could have been done another way, and that this alternative way would also be stable. Second, an appealing feature of the framework presented here is that it can explain the simultaneous existence of equivalent solutions to a coordination problem (as in the vivid example of left vs. right driving), but this strict equivalence is not a necessary feature of conventions. The conventional solutions to a coordination game can have different payoffs, and different dynamic properties. Moreover, there is no requirement that the players in a game experience the same consequences at a conventional equilibrium. Many potentially important behavioral conventions, such as problems of dominance or territoriality, are likely to be strongly asymmetric (for example, if owners do better than nonowners). Traditions are not necessarily conventions Several behavioral phenomena, from foraging strategies to burrow use, are commonly described as traditional (Whiten et al. 2005; Muller and Cant 2010; Thornton et al. 2010; Aplin et al. 2015). Some of these putative traditions involve the choice of a physical location: for example, traditional roosting sites (Teng et al. 2012), foraging sites (Fishlock et al. 2016), or lekking grounds (Bradbury et al. 1989). Other traditions involve the choice of alternative actions. In recent experimental studies, for example, investigators offer animals a problem with multiple solutions, and they conclude that a tradition exists if group members use the same method to solve the experimental problem (e.g. Bonnie et al. 2007; Muller and Cant, 2010; Aplin et al., 2015; Prestipino et al., 2016). Some of these experimental traditions are probably also conventions, but not all. Conventions necessarily involve some sort of coordination or behavioral complementarity, whereas a tradition could in principle be a solitary activity. So spatial traditions that involve coordination of some type—e.g. roosting to reduce predation costs, lekking to attract females—could be seen as conventions, whereas the tradition of opening a puzzle box by the pulling the red lever would not necessarily be a convention even if it learned by observing others. So, traditions are not necessarily conventions. Moreover, a convention need not be traditional. A pair of individuals could, at least in principle, solve a coordination problem via a novel convention that we not would call a tradition, because it has never been used by others. Connection to social learning Conventions are clearly a social phenomenon since they require interaction, and many important conventions are clearly learned. It follows that at some level social learning must be involved. Yet, the extant social learning literature focuses on situations in observers learn to perform actions similar to what they have observed (Laland 2004). This “imitative” social learning may apply to some conventions—when individuals coordinate by performing similar actions—but it will not apply to all, because in many conventions players must adopt complementary actions (e.g. signalers and receivers). To best of our knowledge the social learning of these sorts of complementary actions is relatively little explored, and potentially quite important. Conventions in behavioral ecology Although we feel that nonhuman conventions are under-appreciated and understudied, behavioral ecologists have previously considered conventions in several situations, and we will discuss 2 important cases here. We note however, that in most cases these older uses of the term “convention” are not in complete agreement with the coherent Lewisian framework presented here. Bourgeois and anti-bourgeois Perhaps the most famous theoretical analysis of “conventional behavior” in classical behavioral ecology is Maynard Smith’s Hawk-Dove-Bourgeois analysis of animal conflict over resources (Maynard-Smith and Price 1973; Maynard-Smith 1982). The textbook analysis of this classical game shows that there is no pure equilibrium in the Hawk-Dove game. However, when we add a new strategy called bourgeosis, which “respects the convention of ownership,” we find that a single pure bourgeois-versus-bourgeois equilibrium arises. Readers will immediately see that in the standard development of this game, bourgeosis-versus-bourgeosis is the only pure equilibrium of the game, and it follows that in the Lewisian framework it is not a convention, because there is no alternative equilibrium. Mathematically inclined readers will realize (Maynard Smith 1982), that a second equilibrium would exist if we introduced a mirror image strategy in which intruders win and owners lose (which we might call anti-bougeosis); and this would rescue Maynard Smith’s claim that bourgeosis-versus-bourgeosis is a convention. Yet, naturally occurring resource conflicts may not be so simple. Several authors have remarked on the apparent absence of anti-bourgeosis-like strategies in nature, and noted that territory owners and intruders likely differ in several ways that make the problem less symmetrical than it initially appeared (Kokko et al. 2006; Sherratt and Mesterton-Gibbons 2015). If as these authors suggest, anti-bourgeosis strategies are not equilibria of resource conflict games, then it is not clear that we can properly call owner-respecting strategies like bougeosis conventions. However, the broader theory, that asymmetries between individuals can be used conventionally to resolve conflicts, remains relevant. Conventional “Signals” Perhaps unsurprisingly, the area of behavioral ecology in which the idea of conventions has been most influential is signaling and communication. A comprehensive review of this literature is beyond the scope of our discussion. Instead, we seek to point out how current and historical thinking about signaling conventions differs from the Lewisian framework offered here. To begin, we remark that the phrase “conventional signal” is potentially misleading. A signal cannot stand on its own as a convention. A signaling convention consists of a complementary pair of encoding and decoding rules. The choice of a particular encoding rule/decoding rule pair can be a convention if other sets of rules are available. The availability of alternative equilibria is the defining idea in the conventionality (or lack thereof) of a signaling system. Much of the existing literature on signaling conventions focuses on the problem of honesty. Early investigators rejected the idea of signaling conventions because they argued that the arbitrariness of “conventional signals” makes it too easy for signalers to cheat (Maynard-Smith and Harper 1988; Zahavi 1993; Guilford and Dawkins 1995). In the Lewis framework, this argument loses some of its force because we define conventions as the equilibria of coordination games; and this means that cheaters who depart from this equilibrium will pay a cost, simply because this is how equilibria are defined. In a broader analysis, this complaint seems to suppose that conventional signals exist independently of the complementary behavior of receivers, which as we have pointed out above does not make sense in the framework presented here. More recent, and more sophisticated, treatments of signaling conventions have continued to emphasize the problem of honesty. These investigators have recognized 2 key properties of “honest conventions” that distinguish them from the other mechanisms that promote honesty. First, the signals produced in signaling conventions need not have high production costs. Second, cheating—departing from the conventional equilibrium in our approach—is punished by receiver action. These 2 properties, low production costs and receiver-dependent costs have been taken to be key indicators of signaling conventions (Searcy and Nowicki 2005; Tibbetts 2013). Although these properties are perfectly compatible with the framework presented here, they should not be seen as definitive properties of signaling conventions. Signaling conventions exist, in Lewisian terms, when there are multiple signaling equilibria, and there is no direct connection between production costs per se and the number of equilibria in a signaling game. For example, we could imagine a situation in which a male could advertise his quality via a costly call or a costly ornament, the fact that both could be costly to produce does not preclude the possibility that they represent alternative signaling conventions. The idea that receiver action creates a cost for signalers who depart from an equilibrium refers to a specific mechanism for maintaining an equilibrium, and importantly this mechanism can operate regardless of the number of signaling equilibria in the larger game. Its connection to Lewisian signaling conventions is therefore unclear, because the existence of receiver-dependent costs is not sufficient to establish the existence of multiple equilibria which is, of course, the defining property of Lewisian conventions. Equilibrium selection literature As we explained in the body of the paper, the Lewisian approach to conventions leads us to ask questions about equilibrium selection; and as we pointed out Harsanyi and Selten’s (1988) ideas are broadly seen as the foundational ideas in this area. We feel is important to acknowledge the emerging and extensive body of literature on this topic. Evolutionary game theory is a broad area of research, and illustrates that evolutionary processes, population dynamics, and temporal and spatial change can all dramatically influence equilibrium selection (Boyd 1992; Hammerstein and Selten 1994; Hofbauer and Sigmund 2003; Roca et al. 2009). The Lewis model of signaling has also been influential in philosophy and economics. Results from simulations suggest that simple learning or evolutionary mechanisms reliably converge on signaling conventions (Huttegger 2007; Pawlowitsch 2008; Catteeuw and Manderick 2014) but several factors can bias this convergence. The number of possible signals can influence equilibrium selection (Barrett 2009; Huttegger and Zollman 2011), as can the predictability of the underlying state being communicated (Huttegger 2007; Alexander 2014). See Skryms (2010) and Huttegger et al (2014) for a further overview. LIMITATIONS AND FURTHER QUESTIONS Level of analysis We have presented a range of simple coordination games including models of communal roosting, dominance, territoriality, and signaling. It should be clear that none of these models is intended as a comprehensive treatment of any specific behavior phenomenon. Instead, we have sketched a series of simplistic models to show the potential applicability of game theoretical conventions to a wide range of behavioral phenomena. Moreover, we make no claim that these analyses are novel or even unique. Instead, we hope to illustrate the common structure and pervasiveness of coordination games with multiple equilibria. Conventions and Mixed equilibria Many readers will know that mixed equilibria are a common feature of many biologically important games. Lewis’s initial development simply ignored mixed equilibria, and this paper, like most treatments of conventions, has followed Lewis’s lead and focused on pure equilibria. The relationship between mixed equilibria and conventions is currently unclear. One approach is to continue to follow Lewis and ignore them. There is some justification for this. Mixed equilibria are typically inefficient compared to the pure strategy equilibria, and they tend to be unstable in simulated coordination games (Young 1993; Hansen 2006; Hamblin and Hurd 2007). An alternative, but as yet poorly developed, approach is to consider them co-equal with the pure equilibria in coordination games, and hence candidate conventions. Mixed strategies can be important in some situations, including the maintenance of honest signaling (Huttegger and Zollman 2010; Polnaszek and Stephens 2014). Evolved or learned conventions? Humans learn the conventional meanings of words, and Americans are perfectly happy to drive on the left when they find themselves in England. Moreover, the flexibility of learned conventions provides many opportunities for experimental manipulation. So, one might naturally think of conventions as learned and not evolved. This might seem jarring to those behavioral ecologists who think of game theory as an exclusively evolutionary approach. Yet, the distinction between learned and evolved behavior is not clear cut. Maynard Smith, the father of evolutionary game theory, took pains to point out that learning, and more broadly experience, must often be involved in the establishment of evolutionary equilibria (Maynard-Smith 1982; Harley and Smith 1983). In the broadest sense, learning is likely to interact with selection to shape the conventional equilibria we observe in nature. The question poised in our subheading is ultimately nonsensical: conventions—like other behavioral phenotypes—are shaped by experience and evolution. Indeed, behavioral ecologists have an important opportunity to contribute to the study of conventions by exploring this interaction. The Lewisian approach to conventions is agnostic about whether the dynamics establishing an equilibrium are evolutionary, experiential, or some combination. Yet, applying Lewis’s ideas over an evolutionary time scale raises some novel questions. What happens, for example, if equilibria come and go during a population’s selective history? Our definition requires that equilibria be simultaneously available in order to be called conventions. So, we can imagine scenarios in which an equilibrium was a convention at some time in the evolutionary past, but it is not a convention now (because the alternative equilibrium has been lost). Clearly, changes like this could also have powerful effects on the equilibrium selection process, by biasing individuals or population toward one conventional equilibrium over another. FUTURE DIRECTIONS Population and the scale of conventions Most readers will think first of population-level conventions, since many of the clearest examples occur at this scale. However, conventions can also occur between as few as 2 individuals. Human couples engage in many pairwise conventions: I use the bathroom first, you do the dishes while I vacuum, and so on. Pairwise conventions are not well studied, but candidates include duetting birds that use “duet codes,” pair-specific rules for which phrases should follow each other (Hall 2009; Rivera-Cáceres et al. 2016). Other candidates include parental care, conflict resolution, or territorial partitioning. Pairwise conventions provide a good opportunity for experimental analysis, because pairs of animals can be manipulated more easily than entire populations. Conventions at the population scale generate many interesting questions about the development and spread of conventions. One might speculate that it will be easier to find conventional equilibria when pairs of individuals interact repeatedly, but that some partner swapping is needed to facilitate the spread of the equilibrium throughout a population. How does this balance affect the spread of a convention? And, what happens when populations with different conventions encounter each other? Models have shown that the dynamics of social interactions can have a profound effect on equilibrium selection (Goyal and Vega-Redondo 2005; Van Cleve and Lehmann 2013) and on the potential for the co-existence of alternative conventions (Bhaskar and Vega-Redondo 2004). Sensory bias and the psychophysics of equilibrium selection Although we have emphasized the economic determinants of equilibrium selection, the sensory and psychophysical predispositions should not be ignored. Otherwise, economically equivalent equilibria could be quite different perceptually. Some of these biases are probably simple, as when one potential roosting tree is taller than the others, whereas others could be quite subtle. A basic idea in psychophysics is that discriminations, even apparently trivial discriminations, take place in a background of internal and external noise (Gescheider 1997). This noise is likely to affect each player’s strategy selection as well as its perceptions of the actions of others. In theory, many (if not all) pre-existing bias problems can be thought of as effects of initial conditions and dynamics, which are, of course, already part of our thinking about equilibrium selection. The fact that we can include them in our models does not make them any less important, because their effects are likely to be pervasive. FINAL SUMMARY We have presented some basic ideas about “conventions,” taken from the human behavioral sciences and we have outlined how these ideas can be applied to nonhuman behavior. Conventions arise in coordination games with multiple equilibria. Although the study of nonhuman conventions is virtually nonexistent, we argue that conventions can arise in nearly every aspect of animal social interactions from simple aggregation to the establishment of meaning. The ideas presented here provide a framework for the conceptual and empirical study of conventions. We advocate a research program that considers the role of conventions in animal interactions both experimentally and observationally. Students of animal behavior are well prepared to study and understand conventions because the basic conceptual tools of game theory are already well known to behavioral ecologists. This logical framework can help us understand such divergent phenomena as the meanings of signals and the existence of dominance hierarchies, and may reveal previously unappreciated patterns of behavior, learning and evolution. FUNDING This work was supported by the University of Minnesota Doctoral Dissertation Fellowship (to V.K.H). Data accessibility No new data was collected for the preparation of this paper. The authors are grateful to T. Polnaszek and T. Rubi for their contributions to the ideas and data presented here, and to L. Barrett, and T. Sherratt, and 4 anonymous reviewers for helpful comments and suggestions. REFERENCES Alexander JM . 2014 . 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Behavioral EcologyOxford University Press

Published: Feb 20, 2018

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