Mind

Ahmed,, Arif

doi: 10.1093/mind/fzz073pmid: N/A

Abstract The paper discusses Ian Wells’s recent argument (Wells 2019) that there is a decision problem in which followers of Evidential Decision Theory end up poorer than followers of Causal Decision Theory despite having the same opportunities for money. It defends Evidential Decision Theory against Wells’s argument, on the following grounds. (i) Wells's has not presented a decision problem in which his main claim is true. (ii) Four possible decision problems can be generated from his central example, in each of which followers of Evidential Decision Theory do at least as well as followers of Causal Decision Theory (but the former typically have better opportunities for money). (iii) There is another case in which followers of Causal Decision Theory have the same opportunities for making money but end up worse than followers of Evidential Decision Theory. 1. CDT, EDT and Newcomb Causal Decision Theory (CDT) says that a rational agent takes whichever of her options does most to cause the satisfaction of her wants, given her beliefs. Evidential Decision Theory (EDT) says that a rational agent takes whichever of his options constitutes the best evidence for the satisfaction of his wants, where what is evidence for what depends on his beliefs. This paper defends EDT against a recent attack due to Ian Wells (Wells 2019). More formally (and simplifying), let the probability function and the value function respectively represent the agent’s credence (degree of belief) in, and desirability for, an arbitrary proposition. Let be a set of propositions describing the agent’s options and let be a set of propositions describing all possible outcomes in as much detail as concerns the agent. Let some counterfactual-like operator ‘’ reflect causal dependence: says that if the agent were to realize option then the outcome would be ⁠. CDT says that a rational option maximizes ⁠, and EDT that it maximizes ⁠, where the - and -scores of options are: The presence of the conditional credence in (2), instead of ⁠, reflects EDT’s evaluation of an option by its status as evidence of outcomes, whereas CDT cares only about its tendency to cause outcomes, whatever other evidence actually realizing it would provide. EDT and CDT sometimes disagree over what is rational. Consider: Newcomb (NP): There is a transparent box and an opaque box. You can take just the opaque box (‘one-box’) or you can take both (‘two-box’); in either case you keep the contents. The transparent box contains $1,000 (⁠⁠). What the opaque box contains depends on a prediction made yesterday by a predictor in whose accuracy you have enormous confidence. You know that if the predictor predicted that you would one-box then the opaque box contains $1,000,000 (⁠⁠). If the predictor predicted that you would two-box then it is empty. (Cf. Nozick 1969, pp. 207-8) Since the prediction is past, you cannot affect what’s in the opaque box. And whatever is in it, the causal effect of taking both boxes is to make you richer than if you had taken only the opaque one. So CDT counts two-boxing as uniquely rational in Newcomb. But given your knowledge, one-boxing is evidence that the predictor predicted this and, therefore, that you will get ⁠. Similarly, two-boxing is evidence that the predictor predicted that and, therefore, that you will get ⁠. So according to EDT, rationality demands one-boxing. 2. ‘Why Ain’cha Rich?’ and the opportunity defence Defenders of EDT sometimes note that one-boxing generates better average returns than two-boxing, at least if (as is invariably assumed) the accuracy of the predictor matches the agent’s confidence in it. For instance, let Ernie and Clara follow EDT and CDT respectively, both being 99% confident that the predictor is accurate. That is, both are 99% confident that the predictor has predicted that he/she one-boxes, given that he/she does, and 99% confident that the predictor has predicted that he/she two-boxes, given that he/she does. If we run (say) a million Newcomb cases on each, we can expect roughly the following: In 990,000 cases, Ernie one-boxes for a return of ⁠; In 10,000 cases, Ernie one-boxes for nothing; In 990,000 cases, Clara two-boxes for ⁠; In 10,000 cases, Clara two-boxes for ⁠. Here the average return to one-boxing is and that to two-boxing is ⁠. Ernie’s average return exceeds Clara’s by a factor of 90; and after a million trials we can be practically certain that the results will approximate these. So almost certainly Ernie is outperforming Clara. A natural explanation is that Ernie is getting better advice: CDT is misidentifying, and EDT is correctly identifying, what rationality demands in NP. This is the ‘Why Ain’cha Rich?’ (‘WAR’) argument against CDT. I’ll refer to arguments of this type, that is, based on comparison of expected actual return, as WAR-type arguments. Defenders of CDT typically concede that Ernie’s return exceeds Clara’s but insist that the comparison is unfair, because Ernie typically enjoys money-making opportunities that Clara lacks. For instance, Lewis writes on behalf of two-boxers that Ernie’s riches were never (or in this case, almost never) available to them: ‘We were never given any choice about whether to have a million. When we made our choices, there were no millions to be had’ (Lewis 1981, p. 377; see also, for example, Wells 2019, pp. 432-3). So we cannot fairly criticize CDT on the basis of Clara’s foreseeably inferior performance in Newcomb. This is hardly news. But Ian Wells’s recent paper (Wells 2019) is news. It criticizes EDT in the light of Ernie’s foreseeably inferior performance in another problem, whose construction and exposition are the burden of his paper. If Wells's is right then there is, contrary to what anyone expected, a WAR-type objection to Evidential Decision Theory. 3. Newcomb Coin Toss There is a box containing either a cheque for $6,000 (that is, whose value to the agent is ⁠) or an invoice for $4,000 (value ⁠); which it is was settled last night by tossing a fair coin. The agent cannot (causally) affect what is in the box. The decision problem has two stages. In the second stage, the agent chooses between taking the box for free and buying it for $3,000. Either way the agent keeps its contents. In the next room there is a predictor who knows what is in the box and who has predicted the agent’s choice between taking and buying the box. Suppose that the predictor is perfect. (This is an inessential simplification: Wells 2019, pp. 438-9). In the first stage of the problem the agent chooses whether to pay a $2,000 fee. If the agent pays then she moves straight to stage 2. If the agent does not pay then the predictor sends a positive signal (⁠⁠) or a negative signal ⁠), for instance, by turning on or off a light that is visible if and only if the agent does not pay, in accordance with these rules: If the box contains the invoice and she predicts that the agent buys it, the predictor sends a negative signal; If the box contains the invoice and she predicts that the agent takes it, she sends a positive or negative signal depending on the toss of a fair coin; If the box contains the cheque and she predicts that the agent buys it, she sends a positive or a negative signal depending on the toss of a fair coin; If the box contains the cheque and she predicts that the agent takes it, she sends a positive signal. See Figure 1. Figure 1: Open in new tabDownload slide The Newcomb Coin Toss protocol Figure 1: Open in new tabDownload slide The Newcomb Coin Toss protocol Table 1 Decision Problem . Clara’s average payoff . Ernie’s average payoff . Decision Problem . Clara’s average payoff . Ernie’s average payoff . Open in new tab Table 1 Decision Problem . Clara’s average payoff . Ernie’s average payoff . Decision Problem . Clara’s average payoff . Ernie’s average payoff . Open in new tab Table 2 State . . . . . 0 0.5 0 0.25 0 0 0 0.25 0.5 0 0.5 0 0 0.25 0 0 0 0.25 0 0.5 0.5 0 0.5 0 State . . . . . 0 0.5 0 0.25 0 0 0 0.25 0.5 0 0.5 0 0 0.25 0 0 0 0.25 0 0.5 0.5 0 0.5 0 Open in new tab Table 2 State . . . . . 0 0.5 0 0.25 0 0 0 0.25 0.5 0 0.5 0 0 0.25 0 0 0 0.25 0 0.5 0.5 0 0.5 0 State . . . . . 0 0.5 0 0.25 0 0 0 0.25 0.5 0 0.5 0 0 0.25 0 0 0 0.25 0 0.5 0.5 0 0.5 0 Open in new tab Table 3 Summary of findings . Same DP . State-comp. . Same protocol . Same opp. . Winner . Fair? . No No Yes Yes CDT No Yes Yes Yes Yes Tie Yes Yes Yes Yes Yes Tie Yes Yes Yes Yes No EDT No Yes Yes Yes No EDT No Yes Yes Yes No EDT No Yes Yes Yes Yes EDT Yes . Same DP . State-comp. . Same protocol . Same opp. . Winner . Fair? . No No Yes Yes CDT No Yes Yes Yes Yes Tie Yes Yes Yes Yes Yes Tie Yes Yes Yes Yes No EDT No Yes Yes Yes No EDT No Yes Yes Yes No EDT No Yes Yes Yes Yes EDT Yes Open in new tab Table 3 Summary of findings . Same DP . State-comp. . Same protocol . Same opp. . Winner . Fair? . No No Yes Yes CDT No Yes Yes Yes Yes Tie Yes Yes Yes Yes Yes Tie Yes Yes Yes Yes No EDT No Yes Yes Yes No EDT No Yes Yes Yes No EDT No Yes Yes Yes Yes EDT Yes . Same DP . State-comp. . Same protocol . Same opp. . Winner . Fair? . No No Yes Yes CDT No Yes Yes Yes Yes Tie Yes Yes Yes Yes Yes Tie Yes Yes Yes Yes No EDT No Yes Yes Yes No EDT No Yes Yes Yes No EDT No Yes Yes Yes Yes EDT Yes Open in new tab This two-stage problem is called Newcomb Coin Toss (NCT). Wells's claims that it supports a WAR-type objection to EDT. What does Ernie do? Start with stage 2. Suppose he hasn’t paid the fee and sees a positive signal. This excludes the antecedent of (3): it is not the case that: the box contains the invoice and the predictor has predicted that he buys it. So given that he does buy, and that the predictor is accurate, the box contains the cheque. So buying indicates a net profit for sure. But given that he takes the box, he was predicted to take. So given that he takes, either the antecedent of (4) or the antecedent of (6) is true. Given also a positive signal, it is 2 to 1 that it is the antecedent of (6). So given that he takes, it is 2 to 1 that he gains and 1 to 2 that he loses ⁠. Taking the box therefore indicates expected profit ⁠. Since Ernie follows EDT, we conclude that if he hasn’t paid at stage 1 and observes a positive signal at stage 2 then Ernie buys at stage 2. Suppose alternatively that Ernie hasn’t paid at stage 1 and observes a negative signal. This excludes the antecedent of (6): it is not the case that: the box contains the cheque and the predictor has predicted that he takes it. So given that he takes, and that the predictor is accurate, the box contains the invoice. So taking indicates profit for certain. But given that he buys the box, he was predicted to buy. So given that he buys, either the antecedent of (3) or the antecedent of (5) is true; given also a negative signal, it is 2 to 1 that it is the antecedent of (3). So given that he buys, it is 2 to 1 that he loses (⁠ from the invoice and from the purchase) and 1 to 2 that he gains ⁠. Buying therefore indicates an expected profit ⁠. So if Ernie hasn’t paid at stage 1 and observes a negative signal at stage 2, he again buys. So if Ernie doesn’t pay at stage 1, he buys at stage 2, whatever signal he observes. What if Ernie does pay at stage 1? The choice between buying and taking the box is then no indication of its contents. For EDT, buying is therefore gratuitous, and Ernie takes the box, for an expected net profit ⁠. Return to stage 1. Will Ernie pay to avoid the signal? Wells's assumes that Ernie knows at stage 1 that he follows EDT at stage 2. So he knows at stage 1 that if he does not pay now then he buys later, and if he does pay then he takes later. At stage 1 it is 50/50 whether the box contains the cheque or the invoice. So from Ernie’s stage 1 perspective, expected net profit given that he buys for $3,000 at stage 2 is ⁠; and expected net profit given that he takes at stage 2 is ⁠. The stage 1 expected payoff of taking exceeds that of buying by ⁠, so it is worth Ernie’s now paying $2,000 to avoid the signal, so that he takes at stage 2. So when all is said and done, Ernie (a) pays the $2,000 fee; (b) avoids the signal; and (c) takes the box. Half the time there is a cheque for $6,000 in the box. Otherwise there is an invoice for $4,000. Overall Ernie makes a foreseeable average loss of $1,000 per trial. A more straightforward argument shows that Clara does not pay the fee, observes a signal and takes the box. For details see Wells’s Appendix; but the intuition is as follows. At stage 2, whatever signal (if any) she observes, Clara knows that whether she buys or takes makes no difference to the contents of the box. CDT therefore always advises her to take. Since she is sufficiently introspective (as Wells's supposes), she knows at stage 1 that she’ll take at stage 2; so paying a fee at stage 1 is gratuitous. Half the time there is a cheque for $6,000 in the box; otherwise there is an invoice for $4,000. So Clara makes a foreseeable average profit of $1,000 per trial. Define a reflective follower of decision theory X as one who follows X and knows that he/she will follow X in the future. In NCT, reflective followers of CDT (for example, Clara) end up foreseeably richer than reflective followers of EDT (for example, Ernie). And not because they have different opportunities. In half of the trials that either party faces, a cheque is in the box, and in half of them an invoice is in the box, because its contents were settled in advance by a fair coin. So a better explanation for the disparity seems to be that CDT is giving rational advice, and EDT is giving irrational advice. Wells's regards this as the best explanation. He therefore rejects EDT on abductive grounds (Wells 2019, p. 442). 4. Comparison of decision problems The argument that Ernie pays at stage 1 requires that Ernie is reflective: he knows at stage 1 that he follows EDT at stage 2. Reflectiveness motivates the claim that at stage 1 Ernie expects to buy at stage 2 unless he pays at stage 1; and that is his reason for paying at stage 1. Similarly, the argument that Clara does not pay at stage 1 requires that she knows at stage 1 that she follows CDT at stage 2. That is what motivates the claim that at stage 1 she expects to take at stage 2 come what may, the latter being her reason for regarding payment at stage 1 as gratuitous. These facts together explain why the WAR-type argument that is presently at issue applies only to reflective agents. But then they also imply that at stage 1 Ernie and Clara face different decision problems. EDT and CDT concern subjective rationality: they settle what is rational relative to given background credences and utilities. But if Ernie and Clara have different credences then their choices have different backgrounds. Specifically: if Ernie regards some relevant future contingency as causally and evidentially dependent on his present choice, but Clara regards her present choice as having neither kind of bearing on an exactly analogous future contingency, then they are choosing against relevantly different backgrounds. They are facing different decision problems. This is the situation in NCT as regards the future contingency that each party expresses by the sentence ‘I will buy the box at stage 2’. Ernie thinks this contingency will be realized if and only if – and if so, because – he now chooses not to pay the fee. Clara thinks that it won’t be realized whether or not she pays, and also that it wouldn’t be realized whether or not she were to pay.1 They face different decision problems at stage 1. NCT makes Ernie worse off in expectation against his decision problem than Clara against hers. But we can’t infer that there is a decision problem in which EDT is giving irrational and CDT rational advice. That would be like arguing: (i) Ernie’s map of London gives him good directions in London; (ii) Clara’s map of London doesn’t give her good directions in Paris; therefore (iii) there is a city of which Ernie has a better map than does Clara. 5. A fair comparison A fairer comparison of EDT and CDT would contrast Ernie’s and Clara’s fortunes in a single decision problem, that is, where their credences concerning all relevant parameters match. Let us see what happens if we force this structure onto NCT in various ways. 5.1 Common accurate credences at stage 1 One way to do it is to stipulate that their stage 1 credences agree about their choices at stage 2. Let us also suppose that these credences are accurate, that is, that across many trials, Clara (Ernie) buys at stage 2 about as often as her (his) stage 1 credences suggest: so at least one of them diverges from his/her favoured decision theory at stage 2.2 Call this problem In what follows I’ll use Wells’s abbreviations: is the proposition that the agent pays at stage 1; that the agent buys the box at stage 2; and that it contains a cheque. Writing for Ernie and Clara’s (common) initial credence: (7) is true because it defines ⁠; (10) because the coin is believed fair. (8) and (9) are justified as follows. The agent’s choice at stage 1 has no evidential bearing on his/her choice at stage 2 except via its causal influence on the latter, that is, by causing him/her at stage 2 to face a bet with a particular credence profile. After all, nobody is predicting what happens at stage 1. But if Clara (say) doesn’t pay at stage 1, this has 50% probability of causing the signal to be visibly positive when she chooses between taking and buying the box when the signal is visibly positive, and 50% probability of causing it to be visibly negative when she makes that choice; and in this way her choices at stage 1 may causally affect her choice. Similarly, her choosing to pay the fee at stage 1 will (certainly) cause the signal to be absent when she chooses at stage 2; and again this may causally affect her choice. But there is no other medium through which her choice at stage 1 generates news about what she does at stage 2. So however she decides at stage 1, the causal bearing of that on her decision at stage 2 exhausts its evidential bearing on the latter. All this goes for Ernie too; so in both cases the conditional probabilities of given (that is, the evidential bearing of on ⁠), and the probabilities of the counterfactuals  (that is, the causal bearing of on ⁠) must match. It follows from (1) and (2) that EDT and CDT give identical advice to any agent facing this problem. More precisely, an easy calculation3 gives these scores for the options at stage 1: EDT and CDT, therefore, each permit paying the fee if and only if ⁠; they permit not paying if and only if ⁠. So they give the same advice at this stage if they are giving advice on the same decision problem. Since both do the same at stage 1 in every trial, and since at stage 2 (by accuracy) they buy with equal frequency (that is, ⁠), Ernie and Clara can expect the same average return. 5.2 Common credences at stage 2 Other decision problems arise from stage 2, depending on (i) whether the agent pays the fee; and (ii) if not then whether the signal is positive or negative. There are three cases. If the agent pays at stage 1 then there is no signal. Call this case ⁠. It is a straight choice between taking and buying a box whose contents are causally and evidentially independent of that choice. EDT and CDT both advise taking, for an expected net loss (a cost of because of the payment at stage 1 + expected revenue of from taking the box). Ernie and Clara can expect to do equally well. The contingency that the agent has not paid at stage 1 divides into two sub-cases, depending on whether the signal is positive or negative. After all, these are two different decision problems: the agent’s background credences depend on the signal, so the subjective inputs to EDT and CDT are different in each case. Consider first the sub-case where the signal is positive: call this ⁠. Here, EDT and CDT make different recommendations. If the predictor is considered perfect then EDT recommends buying the box, but CDT always recommends taking. (See §3.) So Ernie and Clara choose differently. Who does better? If the predictor is perfect, Ernie’s buying was predicted; so by (3) there is a cheque in the box. So Ernie pays for a cheque worth for average profit ⁠. Clara takes when the signal is positive, and this too is predicted; so by (4) and (6) it is 2 to 1 that she finds a cheque and 1 to 2 that she finds an invoice. In effect she takes a free gamble that pays two-thirds of the time and otherwise loses ⁠, for average profit ⁠. Similarly, when the signal is negative: call this ⁠. EDT and CDT disagree – CDT recommends taking, EDT buying. And again, Ernie and Clara can expect different profits. On average Ernie loses and Clara loses ⁠.4 In short, four decision problems are extractable from NCT, depending on what the common background credences are supposed to be, and EDT and CDT disagree over two. The average payoffs are summarized below. In every case Ernie is making at least as much as Clara, and in two he is making more. So none of these decision problems could support a WAR-type argument against EDT. Neither, therefore, does NCT itself.5 6. The ‘same decision problem’ criterion I’m assuming that Ernie and Clara’s comparative performance supports a comparative assessment of EDT and CDT only if they face the same decision problem, that is, the same payoff structure and options with the same credences over corresponding states, future acts and outcomes and the same utilities for outcomes. But why is sameness of decision problem a necessary condition for a fair comparison? There are two kinds of reason, depending on what normative subjective decision theory (SDT) is supposed to be doing.6 (a) We might treat SDT as a guiding theory: it tells you what to do given your options, credences and utilities. Suppose you seek such advice, and can take it from CDT or from EDT. Would it be relevant to compare the track record of people who follow EDT with that of people who follow CDT? Well, what you are getting from those theories is the advice that each one gives to people with your credences and utilities. So it would be relevant to compare the actual track record of people who share your credences and utilities and follow EDT, with the actual track record of people who share your credences and utilities and follow CDT. It would not be relevant to compare, say, the actual track record of EDT followers having your credences, with the actual track record of CDT followers having some other credences. That would be like choosing careers by comparing the average income after ten years of people with your abilities, experience and qualifications in career A with the average income after ten years of people with different abilities, experience and qualifications in career B. So on the guiding conception, comparing Ernie and Clara’s performances can only help anyone if we hold their background credences and preferences equal to one another (and can only help you if we hold them equal to yours). (b) We might alternatively treat SDT as a classifying theory: not as guiding agents, but rather as an impersonal filter by means of which we as theorists can sort agents’ choices into the subjectively rational and the subjectively irrational.7 But whether a choice counts as subjectively rational must be relative to the agent’s beliefs and desires. That you and I make different choices when we think or want different things can hardly show that one of us is irrational. It is plausible enough that we may compare EDT and CDT by comparing the actual performance of agents that make choices that each counts as rational, given fixed credences and utilities (and maybe also equal opportunities). But without such a fixed background, that EDT and CDT classify different options as rational does not even entail that they disagree. No comparison of actual performance could support one over the other. Arguing that way would be like arguing that someone who judges that Mozart wrote the best eighteenth-century operas is more discerning than anyone who judges that Reger wrote the best nineteenth-century organ music, on the grounds that Mozart’s operas are better than Reger’s organ works. Maybe they are, but this hardly impugns either judgment. For instance, the fact that EDT classifies Ernie as rational to pay at stage 1 of NCT, and CDT classifies Clara as rational not to pay, does not show that those theories disagree about subjective rationality. Indeed, at stage 1 of NCT they agree, in that both theories classify Ernie as rational to pay, and both classify Clara as rational not to pay.8 I therefore believe that whichever kind of theory SDT aims at being, Ernie and Clara’s relative performances bear on the relative merits of EDT and CDT only if both parties face the same options and potential outcomes with the same utilities over the latter and the same credences over all relevant parameters; that this necessary condition rules out Wells’s WAR-type argument from NCT; and that in four successor cases that it doesn’t rule out, Ernie is doing better if either of them is, so there is in any case no prospect of such an argument. 7. Other criteria of fair comparison But setting that argument aside, suppose you regard the ‘same decision problem’ criterion as too strong. Clara’s doing better than Ernie in a different decision problem might still support CDT over EDT on this view, if the decision problems stand in some other, suitable relation. Could another relation do that trick? I’ll consider two candidates. Say that agents facing one-off or sequential decision problems face the same protocol if they face: (P1) the same options at each node; (P2) the same rules determining the state of the world – for instance, the rules governing the behaviour of the predictor in Newcomb or NCT; and (P3) the same rules governing how the states of the world and their choices determine the outcomes, for which they have identical utilities. In this definition, the state specifies all choices that ‘nature’ makes, in so far as these affect the agent’s ultimate path through the decision tree. For instance, if in NCT we let say that the box contains a cheque, and write for a positive signal, a negative signal or no signal, then, in an obvious notation, the six possible states are ⁠, ⁠, ⁠, and ⁠. Proposal: we can base fair comparison of theories of rational choice on the relative performance of their followers in decision problems with a common protocol. This would vindicate Wells's. In NCT, reflective versions of Clara and Ernie start out with the same knowledge about (P1) what their options are at each node; (P2) the rules determining what the box contains and what signal (if any) is sent; and (P3) how their choices and these states jointly determine outcomes. NCT therefore faces Ernie and Clara with a single protocol in which Clara outperforms Ernie. But sameness of protocol is not a fair basis of comparison. There are protocols against which reflectively irrational agents outperform reflectively rational ones. Parfit’s Hitch-hiker: The agent is dying in the desert. A driver comes along who offers the agent a ride into the city, but only if the agent promises to visit an ATM once they arrive and give the driver $1,000. The driver will have no way to enforce this after they arrive, but she does have an extraordinary ability to detect lies with total accuracy. If the driver detects a lie then the agent is left in the desert. (Cf. Parfit 1984, p. 7) It is rational after being dropped off not to pay. But an agent who knows that he won’t pay if he gets a ride, won’t get a ride. Reflectively rational agents die in the desert. By contrast, an agent who knows that he will pay sincerely promises to pay. Reflectively irrational agents escape. But reflectively rational and reflectively irrational agents face the same protocol. Both start out with (P1) the same options at each stage (promise/don’t promise at stage 1; pay/don’t pay at stage 2 if it is reached); (P2) the rules governing the states (the agent gets a ride if and only if he makes a sincere promise); (P3) how states and choices jointly determine the outcomes (left in the desert, a ride for free9, a ride for $1,000). So being reflectively irrational pays better against this protocol. For all that Wells's shows, this could be what happens in NCT. You cannot infer from Ernie’s facing the same protocol as Clara that there is anything irrational about the advice that EDT gives. Another candidate relation is state-comparability.10 Let us say that Clara and Ernie face one-off or sequential decision problems that are state-comparable if and only if: (S1) they face the same protocol; and (S2) at each decision node they have the same subjective probability distribution over states, conditional on each option. Again, the state specifies all ‘choices of nature’ that affect the agent’s ultimate path through the decision tree. The proposal is that agents are fairly comparable when facing state-comparable decision problems. This faces no difficulty from Parfit’s Hitch-hiker. After all, reflectively rational and reflectively irrational hitch-hikers are not facing state-comparable decision problems, because they violate (S2). The reflectively rational hitch-hiker is confident that the state of being given a ride – which is all that ‘nature chooses’ – does not obtain given that he promises to pay; whereas the reflectively irrational hitch-hiker is confident that it does, given the corresponding assumption. However, it is unclear that NCT faces Ernie and Clara with state-comparable problems. Consider stage 1. Because she is reflective, Clara knows that if she doesn’t pay at stage 1 then she takes at stage 2. So she is certain that the antecedent of (4) or the antecedent of (6) is true, given that she doesn’t pay; and between these it is 50/50 which it is, since it is (she currently reckons) 50/50 whether the box contains a cheque or an invoice. She is therefore certain that if she doesn’t pay at the outset, the state is not: ⁠, ⁠, or ⁠. She is 50% confident of ⁠, 25% of and 25% of on the same condition. Finally, she has 50% confidence of and 50% of ⁠, given that she does pay at the outset. By contrast, because he is reflective, Ernie knows that if he doesn’t pay at stage 1 then he buys at stage two. So he is certain that the antecedent of (3) or the antecedent of (5) is true, given that he doesn’t pay at stage 1; between these it is 50/50 which. He is therefore certain that if he doesn’t pay at the outset, the state is not: ⁠, or ⁠. He is 50% confident of ⁠, 25% confident of and 25% confident of given this assumption. And he has 50% confidence of and 50% of ⁠, given that he does pay at the outset. The following table summarizes these credences, which I’ll write and ⁠, conditional on paying (⁠⁠) and on not paying (⁠⁠): (S2) fails: Ernie and Clara do not have the same credence distributions over states, conditional on each available option at each decision node, because they do not have the same probability distribution over states conditional on not paying, at the initial node. So NCT does not face them with state-comparable problems.11 The previous section suggested that we can only base a WAR-type argument on Clara and Ernie’s relative performance against the same decision problem. Notwithstanding that, I have in this section considered two alternative criteria: sameness of protocol and state-comparability. The first is implausible. The second, though not implausible, does not apply to NCT. 8. The Randomizing Frustrater WAR-type arguments seem impotent against both CDT and EDT. We saw at the outset that a WAR-type argument against CDT based on Newcomb faces the ‘opportunity’ objection: it unfairly compares the misfortunes of Clara, who could almost never have made more than the $1,000 that she almost always got, with the fortunes of Ernie, who could almost never have made less than the million that he almost always got. I then argued that a WAR-type argument against EDT based on Newcomb Coin Toss demands reflectiveness, but then Ernie and Clara face different decision problems. We could not find a single decision problem in which Clara and Ernie have the same opportunities but different expected returns. Here, though, is a case where the jaws meet. The Randomizing Frustrater (RF): There are two opaque boxes, A and B, and an envelope. The agent can take box A, box B, or the envelope. The envelope contains $40. One of the boxes contains $100. Which one it is depends on the reliable prediction of a Randomizing Frustrater, who seeks to frustrate. If he predicted that the agent takes A, box B contains $100. If he predicted that the agent takes B, box A contains $100. If he predicted that the agent takes the envelope, he put $100 in A or B based on the toss of a fair coin.12 EDT always recommends taking the envelope, this being symptomatic of an impending $40 profit, whereas taking either box indicates zero profit. CDT recommends taking A, or recommends taking B, or permits both; it never permits taking the envelope. For, however the agent distributes credence between the hypothesis that the $100 is in A and the hypothesis that it is in B, the expected profit from taking one of them must exceed $40, since nothing that the agent does causally affects where the money is. If both parties face RF repeatedly, Ernie makes $40 every time and Clara makes nothing almost every time. Clara occasionally makes $100 if the Randomizing Frustrater occasionally slips (either by wrongly predicting Clara’s choice or by predicting it correctly but putting $100 in the wrong box). But if he slips in fewer than 40% of trials involving Clara then Ernie does better on average. Ernie to Clara: if you’re so smart, why ain’cha rich? Can Clara plead that her opportunities were somehow worse than Ernie’s – that ‘when I made my choice, there were no riches to be had’? Evidently not. Clara has the same opportunities as Ernie – whenever either faces the problem, there is $100 in one box and there is $40 in the envelope; and if Clara chooses A and B equally often, then it is even true that the $100 is in A just as often amongst the trials that Clara faces as it is amongst the trials that Ernie faces. Whenever Clara is in this situation she can (she has the opportunity to) make as much as Ernie actually makes, by taking the envelope. She knows how to do this, and nothing stops her from doing it. (Her hand is not forced; she isn’t in chains; she hasn’t been brainwashed.) Nothing stops her from making at least $40 every time. Yet she typically ends up with nothing, and on average makes nearly nothing. Are Clara and Ernie facing different decision problems? Again, clearly not. RF (like NP) involves just one decision, so no relevant difference between them arises from any difference in beliefs about what each will do after that decision. And there need be no other subjective difference: we can suppose, for example, that both parties have positive linear utility for money, that both start out 50/50 as to whether the $100 is in A or B, and that both are 99% certain that the prediction is accurate.13 In fact, RF appears to satisfy all candidate relations between decision problems that could support a fair comparison. Ernie and Clara face the same decision problem; they face the same protocol; they face state-comparable decision problems; and they have equal opportunities. The following table adds these facts to our other findings about the suitability of NCT, NP and other protocols as bases for a WAR-type argument: If two people ever face fairly comparable decision problems then Ernie and Clara do so against RF. And Ernie does foreseeably better. 9. Conclusion Those who take WAR-type arguments seriously have tended to imagine them as directed against Causal not Evidential Decision Theory. Wells’s ingenious argument appears to reverse this perspective. I’ve argued that ultimately this reversal turns out to be illusory: the example fails to identify a decision problem, or suitably related decision problems, where followers of EDT do worse than followers of CDT. Moreover, RF is a decision problem where followers of CDT do worse than followers of EDT despite equal opportunities. The overall effect is to vindicate the orthodoxy concerning WAR. Telling us why they are not rich is an obligation on Causal Decision Theorists that remains both serious and undischarged.14 Footnotes 1 I assert the independence ‘indicatively’ and ‘subjunctively’ to emphasize that Clara’s credences consider her stage 2 choice both evidentially and causally independent of her stage 1 choice. 2 What motivates accuracy is that without it, it is hard to see how a WAR-type argument could get started. A difference between Ernie’s and Clara’s long-run performance when one or both is seriously misinformed about their own future choices is both unsurprising and unlikely to reveal much about their relative rationality. 3 Here are the details. The possible prizes, the contingencies that could lead to them and their conditional probabilities, are: What might happen . Prize . . . Pay predictor, buy box, invoice in box: Don’t pay, buy, invoice: Pay, take box, invoice: Don’t pay, take, invoice: Pay, buy, cheque in box: Don’t pay, buy, cheque: Pay, take, cheque: Don’t pay, take, cheque: What might happen . Prize . . . Pay predictor, buy box, invoice in box: Don’t pay, buy, invoice: Pay, take box, invoice: Don’t pay, take, invoice: Pay, buy, cheque in box: Don’t pay, buy, cheque: Pay, take, cheque: Don’t pay, take, cheque: What might happen . Prize . . . Pay predictor, buy box, invoice in box: Don’t pay, buy, invoice: Pay, take box, invoice: Don’t pay, take, invoice: Pay, buy, cheque in box: Don’t pay, buy, cheque: Pay, take, cheque: Don’t pay, take, cheque: What might happen . Prize . . . Pay predictor, buy box, invoice in box: Don’t pay, buy, invoice: Pay, take box, invoice: Don’t pay, take, invoice: Pay, buy, cheque in box: Don’t pay, buy, cheque: Pay, take, cheque: Don’t pay, take, cheque: The headings in columns 3-4 assert and for every prize ⁠. The justification is that is determined by two things: what the agent does, and whether the box contains a cheque, that is, the distribution of truth-values over ⁠, and These are evidentially independent at stage 1 and known causally independent throughout. By evidential independence, we have ⁠, where may be either or ⁠. Since a fair coin settles ⁠, ⁠. Similarly, because and are causally independent, ⁠, whence the requisite identity follows by (8) and (9). Applying (1) and (2) to the table gives and ⁠. 4 What about a third sub-case, call it ⁠, that begins after the agent declines to pay but before he/she observes a signal? Here, Clara outperforms Ernie, because Clara always takes and Ernie always buys. But they face different problems, because (assuming accuracy) Clara starts out 75% confident of a positive signal, and Ernie 75% confident of a negative signal. is structurally identical to Arntzenius’s ‘Yankees / Red Sox’ case (2008, pp. 289-90), for discussion of which see Ahmed and Price (2012). 5 But neither does or support a WAR-type argument against CDT – at least, not if the ‘opportunity defence’ is any good. It is true, for instance, that in ⁠, Ernie outperforms Clara (⁠ expected profit as against ⁠). But Clara has worse opportunities in the sense in which two-boxers in Newcomb typically have worse opportunities. Ernie is repeatedly facing a box that always contains a cheque, and (from the CDT perspective) always fluffs the opportunity to make the most of this; whereas Clara’s box has a cheque in it only two times out of three, but always does as well as she could have. (Of course Ernie will deny that he has fluffed anything: all that is true, he will say, is that if he had done otherwise then he would have done better; but for supporters of EDT the lesson of Newcomb’s Problem is just that there are cases where counterfactual considerations of this sort are irrelevant to practical rationality.) 6 The criterion is too strong in requiring a match over all credences and utilities. It must surely be fair to compare Ernie’s and Clara’s relative performance when their credences and utilities differ only over matters that are irrelevant to what either decision theory prescribes, for instance, if (in NCT) their credences differ only over yesterday’s weather, or if their utilities differ only over sums of money that neither has any prospect of winning. So strictly speaking the ‘same decision problem’ criterion should be this: a comparison of relative average performance is fair only when Ernie and Clara match over all matters that make a difference to the options prescribed by the decision theories that each follows. Thus weakening the criterion does not affect its relevance to NCT, because there, each agent’s belief about what she will do at stage 2 given that she observes a signal is relevant to what both CDT and EDT prescribe at stage 1. So I’ll take it for granted in what follows. (Thanks to a referee.) 7 The guiding / classifying distinction parallels a familiar one between internalist conceptions of justification (Feldman and Conee 2001, p. 2). According to accessibilism, whatever justifies your beliefs must be capable of guiding you towards them by virtue of being consciously available. According to mentalism, your being in mental state M can suffice for this or that belief of yours to count as justified, whether or not M is consciously accessible. Bales (unpublished) section 3.1 discusses the analogy. 8 This is why: (reflective) Ernie’s credences at stage 1 satisfy (8) and (9) with and ⁠; so by (11) and (12) EDT and CDT both classify paying as his rational choice. Clara’s credences at stage 1 satisfy (8) and (9) with ⁠; so EDT and CDT both classify not paying as her rational choice. 9 This looks impossible, because the driver always forestalls this possibility if she foresees that giving a ride would lead to it. But in the problem as stated, the driver gives a ride to any agent who mistakenly believes that he will pay; perhaps also to any agent who has no great confidence about whether she would pay (depending on what counts as a lie). Still, you might worry that the outcome remains epistemically impossible for the reflectively rational agent, who is certain that he won’t pay. A simple fix: stipulate that the driver detects lies with only near-total accuracy. All three outcomes are then epistemically possible for reflectively rational and irrational agents alike. And reflectively irrational hitch-hikers still outperform reflectively rational ones. 10 Many thanks to a referee for raising this possibility. 11 Of course, at the outset Ernie and Clara do have the same credence distributions over states given each sequence of options. For instance, both have 0.5 credence in given non-payment at stage 1 and taking at stage 2. But identity of credence over states given option-sequences cannot suffice for fair comparison, because it also applies to reflectively rational and reflectively irrational agents in Parfit’s Hitch-hiker. Both, for instance, have the same (high) credence in getting a ride conditional on promising to pay and (later) paying. 12 This case modifies one of Spencer and Wells’s (2019, p. 34) in which, if the Frustrater predicted that the agent would take the envelope, he put $50 in both A and B. On that version we cannot say that Ernie and Clara have the same opportunities, because Ernie typically only has the opportunity to make $50 whereas Clara almost always has the opportunity to make $100. Inferring that EDT is superior to CDT then requires this additional premise: if Ernie does better than Clara when Clara can do better than Ernie (even though these outcomes are associated with different options in each case), then Ernie is following a superior decision theory. A defender of CDT may reject this. However, in the present, modified version, Ernie and Clara have matching opportunities, in the sense that the same amounts of money are available to each. Moreover, if Clara chooses A and B equally often then over many trials their opportunities are as closely related as they are in NCT itself: in half of the trials that either party faces, there is $100 in box A, and in half of them there is $100 in box B, and in all of them there is $40 in the envelope. Thanks to a referee for helping me to see this. 13 One might object that the argument as stated violates the accuracy condition. Across many trials, the frequencies of states, outcomes, and so on, should match Clara’s credences. (See §5.1.) But if Clara chooses A (say) and has 50% credence that the money is in that box, then in a fair comparison we should require that the money is in box A on 50% of occasions on which she chooses box A; but then her average return in RF is $50, which exceeds Ernie’s. However, accuracy should also apply to Clara’s credence in the accuracy of the predictor, and in her credence that she takes box A. Now if the predictor’s accuracy is known to be and Clara follows CDT then her credence that she takes either box can only stabilize at 0.5. Accuracy when applied quite generally therefore demands that across many trials she chooses A as often as B for an average return per trial of ⁠, which is less than Ernie’s if ⁠. 14 I am grateful to Jack Spencer for discussion and to two referees for this journal for their extensive and very helpful comments on an earlier draft. I am also grateful to the Leverhulme Trust, which supported me via a Research Fellowship (REF-2018-231) during some of the writing of this paper, and to the Effective Altruism Foundation, which supported me via Research Grant G103268 during the rest of it. 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