1. Unspecific Evidence

Taken literally, probabilism seems psychologically unrealistic. Aside from certain situations (e.g., where a body of evidence is strongly symmetric, or where the precise values of objective probabilities are known), it seems implausible to think that a person's fine-grained doxastic attitude toward a proposition picks out a unique real number. Yet rarely do those who tend to favor probabilism, e.g., Bayesians, insist that probabilism should be understood in such a literal manner. Instead, the claim from the Bayesian who accepts probabilism is often that a rational agent's doxastic state ought to be able to be modeled with a probability distribution. Models often make idealizing assumptions. We know that taken literally it is psychologically unrealistic to assume that an agent's doxastic state singles out a unique probability distribution in every situation. But we also know that taken literally it is physically unrealistic to assume that planets are perfectly spherical.

So while there are some who oppose probabilism on psychological grounds, I think a more forceful opposition comes from those who oppose probabilism on evidential grounds. The objection is not that conforming with the demands of probabilism is psychologically unrealistic, but rather that sometimes conforming with the demands of probabilism is irrational. Isaac Levi opposed probabilism on these grounds.

As does James Joyce.

So multiple authors think that conforming with the demands of probabilism can be irrational, but what exactly are these situations? One illustrative example comes from Elga (Reference Elga2010).Footnote ¹⁰

That an agent is required to form a non-arbitrary, precise credence in Strange Items seems ridiculous. I do not think that this seems ridiculous because it is too difficult to form a non-arbitrary, precise credence. Consider if, instead, the stranger said that there were nine dice in his bag: three six-sided, three eight-sided dice, and three ten-sided. What's your credence that if all the dice are rolled, the numbers will all be less than 3? I doubt that a number instantly popped into your head, but I expect that you would agree that this body of evidence does support a non-arbitrary, precise credence. If I gave you a pen and paper, you'd be able to figure it out. The lack of a pen and paper in Strange Items is not the issue. Rather the issue seems to be the lack of proper evidential support. Your evidence in Strange Items is not specific enough to warrant a precise credence, so forming a precise credence would not respect your evidence. And we should all agree that rational agents respect their evidence.

Note that it is not the case that a body of evidence is either completely specific or completely unspecific. It may be the case that a body of evidence is specific enough to warrant precise credences in some propositions, but not specific enough to warrant precise credences in other propositions. There are also cases where precise conditional credences are warranted but precise unconditional credences are not warranted. Consider the following example from Scott Sturgeon.

Suppose that you reach into the opaque box and grab a ball. First question: what should your conditional credence be that the ball you grabbed is a red-spotted ball given that the ball you grabbed is blue? First answer: ⅓. Second question: what should your unconditional credence be that the ball you grabbed is a red-spotted ball? Second answer: I have no idea. And it seems that I have no idea because the evidence in this case is not specific enough to uniquely determine a precise unconditional credence that I ought to have.

I hope that the considerations of this section have shown that the idea that evidence can be unspecific is intuitively compelling. My aim in this paper was not to argue in favor of unspecific evidence, but rather to show that disputes over the correct theory of rational choice may not be independent of the dispute over unspecific evidence.

2. Unspecific Evidence and Newcomb's Problem

If one takes seriously the idea that evidence can be unspecific, then Newcomb's Problem deserves a second look.

Let P1 be the state where the predictor predicted that the agent would choose One-Box and P2 be the state where the predictor predicted that the agent would choose Two-Box. Newcomb's Problem can be represented by the following decision matrix:

Here's (a version of) the story many of us have heard before. Newcomb's Problem reveals a conflict between Evidential Decision Theory (EDT) and Dominance. EDT says that when a rational agent faces a decision, he selects the option that maximizes expected utility according to the following formula:

$${\rm EEU}( {\rm O} ) = \sum\limits_i {{\rm U}( {{\rm O}\,\& \,{\rm S}_i} ) {\rm Cr}( {{\rm S}_i\vert {\rm O}} ) } $$

where {S_i} is a partition of causally option-independent possible states of the world, U(O & S_i) is the agent's subjective utility in the option-state pair O & S_i, and Cr(S_i|O) is the agent's conditional credence in S_i conditional on O. “EEU(O)” should be read as “the evidential expected utility of O.”

As long as the predictor is slightly better than chance, EDT will say that the rational choice in Newcomb's Problem is One-Box. For example, suppose that the predictor is 51% accurate and the agent knows this. This leads to the following evidential expected utilities:

$${\rm EEU}( {{\rm One}\hbox{-}{\rm Box}} ) = 0.51( m ) + 0.49(\, {\$ 0} ) = \$ 510{\rm , \;}000$$

$${\rm EEU}( {{\rm Two}\hbox{-}{\rm Box}} ) = 0.49( m + k) + 0.51( k ) = \$ 491{\rm , \;}000$$

Given that EEU(One-Box) > EEU(Two-Box), EDT says that one ought to choose One-Box. This recommendation from EDT conflicts with Dominance:

In Newcomb's Problem, Two-Box dominates One-Box. So Dominance says that One-Box is irrational. Thus, EDT and Dominance are in conflict.

Enter the hero of the story: CDT. CDT says that when a rational agent faces a decision, they select the option that maximizes expected utility according to the following formulaFootnote ¹⁴:

$${\rm CEU}( {\rm O} ) = \sum\limits_{\rm i} {{\rm U}( {{\rm O}\,\& \,{\rm S}_i} ) {\rm Cr}( {{\rm O} \Rightarrow {\rm S}_i} ) } $$

where Cr(O ⇒ S_i) is the agent's unconditional credence in the subjunctive conditional “if I were to choose O, then S_i would be the case.” “CEU(O)” should be read as “the causal expected utility of O.”

CDT does not conflict with Dominance. When the states of a decision are causally option-independent, an informed, rational agent's unconditional credence in O ⇒ S_i ought to be equal to their unconditional credence in S_i (O has no causal influence on the state so the probability of “S if I were to O” is simply the probability of S). So in Newcomb's Problem, we have the following credences:

$${\rm Cr}( {\rm One}\hbox{-}{\rm Box} \Rightarrow {\rm P}1) = {\rm Cr}( {\rm Two}\hbox{-}{\rm Box} \Rightarrow {\rm P}1) = {\rm Cr}( {{\rm P}1} ) $$

$${\rm Cr}( {\rm One}\hbox{-}{\rm Box} \Rightarrow {\rm P}2) = {\rm Cr}( {\rm Two}\hbox{-}{\rm Box} \Rightarrow {\rm P}2) = {\rm Cr}( {{\rm P}2} ) $$

Given that the agent's choice cannot causally affect the contents of the opaque box, the agent's credence that taking only the opaque box will cause there to be money in the opaque box ought to be equal to their unconditional credence that the money is in the opaque box, and so on. From this, it is easy to show that CDT recommends Two-Box for any probabilistically coherent pair of credences Cr(P1) and Cr(P2):

$${\rm CEU}( {{\rm One}\hbox{-}{\rm Box}} ) = {\rm Cr}( {{\rm P}1} ) ( m ) + {\rm Cr}( {{\rm P}2} ) ( {\$ 0} ) $$

$${\rm CEU}( {{\rm Two}\hbox{-}{\rm Box}} ) = {\rm Cr}( {{\rm P}1} ) ( m + k) + {\rm Cr}( {{\rm P}2} ) ( k ) $$

$$\therefore {\rm CEU}( {{\rm Two}\hbox{-}{\rm Box}} ) = {\rm CEV}( {{\rm One}\hbox{-}{\rm Box}} ) + k( {{\rm Cr}( {{\rm P}1} ) + {\rm Cr}( {{\rm P}2} ) } ) $$

So CDT cannot conflict with Dominance in Newcomb's Problem.

This is where the story typically ends. We are told that no matter what unconditional credences an agent has, so long as they are probabilistically coherent, CDT will tell them they should choose Two-Box. But we are never told what unconditional credences an agent should have. So, I ask you, what unconditional credences should the agent have? The only non-arbitrary unconditional credence would be 0.5, but that does not seem warranted. It is not the case that the agent has equal evidence for and against the money being in the opaque box. The only way I could see getting a non-arbitrary unconditional credence would be to use option-credences. However, as I argue in section 4, option credences are not the proper base for a decision – at least in this kind of context. So it seems plausible to me to say that there are no unconditional credences that the agent should have. In other words, Newcomb's Problem is a case with unspecific evidence. This implies that forming a precise unconditional credence in the contents of the opaque box is not warranted. This is so in spite of the fact that precise conditional credences do seem warranted. Suppose the agent saw a detailed track record of the predictor's success in 10,000 cases. It seems very plausible to think that one is warranted in forming precise conditional credences on the basis of this track record.

If this is right, then the traditional story about Newcomb's Problem is only half-correct. It is true that CDT does not conflict with Dominance, but it is not true that CDT explicitly recommends Two-Box. This is because if an agent does not have precise unconditional credences, then CDT does not recommend anything. One cannot calculate CEU unless one has precise credences, and if one cannot calculate CEU, then one cannot extract a recommendation from CDT. This implies that in one sense of “maximization of expected utility” – the EDT sense – a rational agent is in a position to choose an option that maximizes expected utility; however, in a different sense of “maximization of expected utility” – the CDT sense – a rational agent is not in a position to choose an option that maximizes expected utility.Footnote ¹⁵ I think that this conclusion follows naturally from the thesis that evidence can be unspecific.

Given that the historical development of CDT seems to be an attempt to formulate a decision theory that vindicates two-boxing, this may seem like a bad, or at least awkward, result for the causal decision theorist since it seems that I am implying that CDT is silent on Newcomb's Problem.Footnote ¹⁶ But I think that the causal decision theorist could respond by saying that the development of CDT has really been an attempt to formulate a way of calculating expected utility that does not conflict with Dominance. One could say that CDT, strictly speaking, is a conjunction of Dominance and CEU, or one could endorse a generalization of Dominance, what I call Credal Dominance in section 3.2 – these combinations would give an explicit verdict in Newcomb's Problem. I also discuss some possible imprecise decision principles the causal decision theorist could incorporate into their overall decision-theoretic framework in section 3.4.

What is more troubling for the causal decision theorist is a case like The Frustrater, which many consider to be a counterexample to (orthodox) CDT. However, in the next section I argue that the causal decision theorist can appeal to unspecific evidence in order to evade this alleged counterexample.

3. Unspecific Evidence and The Frustrater

3.1 The Frustrater

Before I discuss Spencer and Wells' case, consider the following simple case:

Envelope and Boxes: You have a choice between an envelope, a red box, and a blue box. The envelope contains $40. The two boxes together contain $100, however, you do not know how the $100 is distributed between the two boxes.

What option do you choose? Perhaps you take the safe choice and select the guaranteed $40, or maybe you take the risk and choose one of the boxes. Both seem reasonable to me. The point of this simple case is not to make an argument about which option is the rational one to choose. Instead, my point is that if one is on board with the possibility of unspecific evidence, then this seems like a case with unspecific evidence. The evidence regarding the contents of the boxes is incomplete and symmetrically ambiguous. So it seems reasonable to me to suggest that it would be irrational for an agent to form precise credences when they face Envelope and Boxes. If this is correct, then Envelope and Boxes is a case where maximization of expected utility does not apply.Footnote ¹⁷ And I think it is due to this fact that all three choices strike me as reasonable choices.

Spencer and Wells' case is a variation of Envelope and Boxes Footnote ¹⁸:

The Frustrater: The setup is the same as in Envelope and Boxes, except now you learn that the distribution of the $100 between the red box and blue box is determined as follows. A reliable predictor, called the Frustrater, predicts whether you choose the envelope (“Envelope”), the red box (“Red”), or the blue box (“Blue”). If the Frustrater predicts Blue, they will put $100 in the red box. If they predict Red, they will put $100 in the blue box. If they predict Envelope, then they will put $50 in the red box and $50 in the blue box. They make their prediction, and they distribute the money accordingly, one week before your choice.

Let PR be the state where the Frustrater predicts Red, PB be the state where the Frustrater predicts Blue, and PE be the state where the Frustrater predicts Envelope. We can represent The Frustrater with the following decision matrix:

Spencer and Wells' argument against CDT based on this case is straightforward:

P1. In The Frustrater, Envelope is the uniquely rational choice.

P2. In The Frustrater, CDT says that Envelope is an irrational choice.

C. Therefore, CDT is false. [P1, P2]

In defense of P1, Spencer and Wells appeal to a “why ain'cha rich?” (WAR) argument. The idea is that if we imagine two agents, the envelope-taker and the box-taker, who face The Frustrater many times, we expect that the envelope-taker will win $40 every time and that the box-taker will win nothing nearly every time. The envelope-taking strategy will likely make an agent wealthier than the box-taking strategy. Given that the assumption for a case like this is that the hypothetical agent only cares about maximizing their monetary wealth, then it seems reasonable to conclude that the envelope-taking strategy is the rational strategy. Hence, Envelope is the rational choice in The Frustrater.

The original WAR argument is an argument for one-boxing in Newcomb's Problem. If we imagine that two agents, the one-boxer and the two-boxer, face Newcomb's Problem many times, we expect that the one-boxer will end up richer than the two-boxer. On the assumption that the predictor's predictions are very accurate, we expect that the one-boxer will win $1,000,000 nearly every time, while we expect that the two-boxer will win $1,000 nearly every time. If the goal is to win the most amount of money possible, how could it be that two-boxing is rational? As Lewis (Reference Lewis1981b) puts it, the one-boxer asks the two-boxer, “why ain'cha rich?.”

But the WAR argument for one-boxing in Newcomb's Problem faces serious challenges. First, imagine a case that is similar to Newcomb's Problem but now both boxes are transparent. Box 1 has 1,000 dollars in it and a reliable predictor puts 1 million dollars in box 2 if, and only if, they predict that the agent will take only box 2. Call this Transparent Newcomb. While there is entrenched disagreement regarding the rational choice in Newcomb's Problem, both sides typically agree that taking both boxes is the rational choice in Transparent Newcomb.Footnote ¹⁹ Both CDT and EDT agree with this as well. But imagine that there are one-box fetishists who take only box 2 no matter what. Notice that these one-box fetishists are in a position to run a WAR argument in favor of one-boxing in Transparent Newcomb. Assuming that the predictor's predictions are very accurate, we expect that the one-box fetishists will win $1,000,000 nearly every time, while we expect the ordinary agents who take both boxes will win only $1,000 every time. The reasoning given in favor of one-boxing in Transparent Newcomb seems to be identical to the reasoning given in favor of one-boxing in Newcomb's Problem. So if the WAR argument in favor of one-boxing in Newcomb's Problem is sound, then the WAR argument in favor of one-boxing in Transparent Newcomb is sound. Given that we all reject the rationality of one-boxing in Transparent Newcomb, this ought to make us suspicious of the soundness of the WAR argument in the original Newcomb's Problem.Footnote ²⁰

Second, consider the opportunities that one-boxers and two-boxers had when they made their respective decisions. When the one-boxer makes their decision, they likely have a choice between an option worth $1,000,000 and an option worth $1,001,000. So the worst they could possibly do is win $1,000,000. And they knowingly choose the worst option.Footnote ²¹ Compare this to the two-boxer. When they make their decision they likely have a choice between an option worth $0 and an option worth $1,000. The best they could possibly do is win $1,000. And they knowingly choose the best option. It is hard to insist that a person who won $1,000,000 by knowingly doing the worst that they possibly could is more rational than the person who wins $1,000 by knowingly doing the best that they possibly could. Another way to put the point, is that in order for a WAR argument to be sound, the explanation for a disparity in wealth must refer to an agent's irrationality. It seems that in Newcomb's Problem, a perfectly good explanation for the disparity in wealth is the unequal opportunities that the two agents face. The one-boxer has an opportunity that the two-boxer never has, namely the opportunity to win $1,000,000.Footnote ²²

The WAR argument in support of Envelope in The Frustrater seems to avoid the problems with the WAR argument in support of one-boxing in Newcomb's Problem. First, there is not a large disparity in opportunity. The envelope-taker typically has a choice between an option worth $40, an option worth $50, and an option worth $50, while the box-taker typically has a choice between an option worth $40, an option worth $0, and an option worth $100.Footnote ²³ Second, I do not think it is even possible to imagine a transparent version of The Frustrater, so long as we are assuming that the agents facing The Frustrater are not absolutely insane. If all three boxes were transparent, then any reasonable person will take the box that contains $100 or one of the boxes that contain $50, depending on how the money is distributed. There simply cannot be a reliable predictor in a transparent variation of The Frustrater since the predictor is forced to indicate their prediction by how they distribute the money between the boxes, but, by doing that, any reasonable agent facing the transparent version of The Frustrater will falsify the prediction.Footnote ²⁴ This is in contrast to Transparent Newcomb, where it is quite straightforward to imagine a reliable predictor. If a reasonable person is facing Transparent Newcomb, then they will choose both boxes no matter what. Presumably the predictor knows this, so if they are predicting the actions of a reasonable person, they will leave one box empty. Taking both boxes is the rational choice for the agent and it confirms the prediction, unlike the transparent version of The Frustrater where the rational choice for the agent will always disconfirm the prediction.

So, unlike the WAR argument in support of one-boxing, the WAR argument in support of Envelope seems to be in good standing. And if the WAR argument in support of Envelope is in good standing, then it is reasonable to think that P1 of Spencer and Wells' argument is true.

The reasoning in support of P2 is quite straightforward. Given that possible states of The Frustrater are causally option-independent, the agent's credences in the relevant subjunctive conditionals should be equal to their unconditional credences in the possible states, just like in Newcomb's Problem:

$${\rm Cr}( {\rm Envelope} \Rightarrow {\rm PR}) = {\rm Cr}( {\rm Red} \Rightarrow {\rm PR}) = {\rm Cr}( {\rm Blue} \Rightarrow {\rm PR}) = {\rm Cr}( {{\rm PR}} ) = a$$

$${\rm Cr}( {\rm Envelope} \Rightarrow {\rm PB}) = {\rm Cr}( {\rm Red} \Rightarrow {\rm PB}) = {\rm Cr}( {\rm Blue} \Rightarrow {\rm PB}) = {\rm Cr}( {{\rm PB}} ) = b$$

$${\rm Cr}( {\rm Envelope} \Rightarrow {\rm PE}) = {\rm Cr}( {\rm Red} \Rightarrow {\rm PE}) = {\rm Cr}( {\rm Blue} \Rightarrow {\rm PE}) = {\rm Cr}( {{\rm PE}} ) = c$$

Taking the envelope is a guaranteed $40, so CEU(Envelope) = $40.

Here are the causal expected utilities of Red and Blue:

$${\rm CEU}( {{\rm Red}} ) = a( {\$ 0} ) + b( {\$ 100} ) + c( {\$ 50} ) = b( {\$ 100} ) + c( {\$ 50} ) $$

$${\rm CEU}( {{\rm Blue}} ) = a( {\$ 100} ) + b( {\$ 0} ) + c( {\$ 50} ) = a( {\$ 100} ) + c( {\$ 50} ) $$

Given that a + b + c = 1, this means that CEU(Red) + CEU(Blue) = $100. So no matter the values of a, b, and c, either CEU(Red) > CEU(Envelope) or CEU(Blue) > CEU(Envelope). As long as an agent has probabilistically coherent unconditional credences, CDT will never recommend Envelope.

This seems to be good so far, but let us return to the topic of unspecific evidence. If Newcomb's Problem and Envelope and Boxes are cases with unspecific evidence, then it seems reasonable to think that The Frustrater is also a case with unspecific evidence. If The Frustrater is a case with unspecific evidence, then if we are to imagine a rational agent facing The Frustrater, we must imagine that they do not form precise unconditional credences in the possible states. From this, it follows that CEU(Red) and CEU(Blue) will be undefined, thus CDT is inapplicable in The Frustrater. This implies that P2 of Spencer and Wells' argument is false. CDT does not say that Envelope is irrational because CDT does not say anything. Thus, on the assumption that the evidence in The Frustrater is unspecific, Spencer and Wells' argument against CDT is unsound.

3.2 Bad Dominance

It is tempting to try to avoid the conclusion that CDT is silent in The Frustrater by appealing to a kind of dominance reasoning. The argument goes like this. Even if it is granted that an agent cannot rationally assign unconditional credences to the possible states in The Frustrater, it is still the case that for any probabilistically coherent credences that one could assign, CDT has the consequence that it would be irrational for someone with those credences to choose Envelope. Therefore, CDT has the consequence that it is irrational to choose Envelope even for an agent who does not have any unconditional credences.

The above reasoning is not sound. To see why, first consider a version of Dominance where instead of using actual utilities we use expected utilities:

An option O credal dominates an option Q if, and only if, for any probabilistically coherent credence function Cr(.), CEU(O) > CEU(Q). Credal Dominance says that when O credal dominates Q, then it is irrational to choose Q over O.

Credal Dominance represents a natural extension of Dominance to expected utility. From this, one could combine Credal Dominance with CDT to get an explicit verdict from CDT in Newcomb's Problem since Two-Box credal dominates One-Box. Note that this is possible even if the agent does not, in fact, have any precise credences. Credal Dominance quantifies over the probabilistically coherent credence functions the agent could have, not the credence function (or functions) they do have.

Unlike in Newcomb's Problem, there is no dominant option in The Frustrater, so we cannot use Dominance to justify a choice. There is also no credal-dominant option in The Frustrater, so we cannot use Credal Dominance to justify a choice of option. It is not the case that for any probabilistically coherent credence function CEU(Red) > CEU(Envelope) and it is not the case that for any probabilistically coherent credence function CEU(Blue) > CEU(Envelope). Rather, in The Frustrater, for any probabilistically coherent credence function either CEU(Red) > CEU(Envelope) or CEU(Blue) > CEU(Envelope).

One may say that there is still a sense in which Envelope is dominated since CDT never says it is rational, but trying to use this sense of dominance in order to finagle an explicit verdict from CDT would be based on a false principle of rational choice. It would be the extension of the following principle from actual utilities to expected utilities:

Bad Dominance: If there is an option O such that for every state S_i there is another option Q such that U(Q ∧ S_i) > U(O ∧ S_i), then it is irrational to choose O.

Note the scope of the quantifier. It is not that there is a Q such that for every S_i, U(Q ∧ S_i) > U(O ∧ S_i) – that's Dominance – but rather it is that for every state S_i there is a Q such that U(Q ∧ S_i) > U(O ∧ S_i). This change in the scope of the quantifier matters. Dominance is a plausible principle of rational choice, whereas Bad Dominance is an implausible principle of rational choice. Consider the following decision matrix:

According to Bad Dominance, it would be irrational to choose A, but this is wrong. Unless one was very confident in S1 or S2, then A is the rational choice. Thus, Bad Dominance is false. But notice that the structure of the actual utilities in this case parallels the causal expected utilities in The Frustrater. Trying to force the conclusion that CDT explicitly rejects Envelope would not be based on dominance reasoning but on bad dominance reasoning. So while using an extension of Dominance to expected utilities in order to get an explicit verdict from CDT in Newcomb's Problem seems acceptable, trying to do the same in The Frustrater is unacceptable, since it would rely on a false principle of rational choice.

4. Objections

In this section, I respond to two objections to my reasoning in section 3. The first objection says that talk of unspecific evidence is simply irrelevant to decision theory. The second objection says that in The Frustrater an agent can form precise unconditional credences by incorporating option-credences – an agent's unconditional credence that they will choose one of the available options – into their deliberation.

The first objection goes something like this:

This is a solipsistic conception of decision theory – decisions are just in the head. And I have no qualms with understanding decisions in this manner. What I cannot make sense of is how we are supposed to judge whether things go well or poorly for an individual if the only information is the agent's opinions about the world and nothing else – we cannot judge whether their opinions are true/false, good/bad, rational/irrational. This conflicts with how we actually analyze different decision theories and principles.

Consider Newcomb's Problem. Suppose an agent can be represented with the proper quadruple. When we imagine that they choose One-Box, we imagine them looking inside and seeing $1,000,000. This is what we imagine because we imagine that that is the most likely outcome. But facts about the agent's opinions don't demand that we imagine this, it is facts about the case. We assume that the predictor is, in fact, very reliable. If we are not allowed to assume this and, furthermore, we are not allowed to assume that a high value for the conditional credence Cr(P1|One-Box) is rational, then there is no reason to assume that there is $1,000,000 in the opaque box when the agent chooses One-Box. Mere subjective facts on their own may tell me what the agent will expect to happen, but they do not tell me what will actually happen. Facts about the case tell me what will happen. It is the fact that the predictor is reliable that tells me to imagine that the one-boxer will be richer than the two-boxer in Newcomb's Problem, not the fact that a person thinks that the predictor is reliable.

I emphasize that I am not rejecting the view that says the decision an agent faces is determined by their actual credences, rather I am saying that when we are analyzing what theory of rational choice to accept by imagining cases, we ought to imagine that any hypothetical agent in an imagined case has rational credences. So my claim is not that an agent is rationally required to act against their credences if their credences are not supported by the evidence. Rather, my claim is that if an agent acts on credences that are not supported by the evidence, then we cannot use the fact that things go poorly for them as reason for rejecting the theory of rational choice they follow. It only makes sense to use the fact that things go poorly for an agent against a theory of rational choice – as one tries to do in a WAR argument – under the assumption that the agent's credences are supported by the evidence.

The upshot is this. It may be the case that EDT and CDT are subjective theories of rational choice, but this does not mean that evidence is irrelevant to decision theory. When we are trying to assess the status of a particular theory of rational choice by imagining cases, we have to imagine that the doxastic attitudes of the hypothetical agent fit the evidence of the case. But this brings the considerations of this paper front and center. In a case like The Frustrater, we should only imagine an agent whose credences fit the evidence. If precise unconditional credences do not fit the evidence, then what an agent with precise credences would or would not do should not constrain our theorizing about the correct theory of rational choice because those precise credences would be irrational.

One might respond to this by receding even further and suggesting that decision theory is merely about articulating principles of means-end structural rationality. Just like one could consider probabilism to be a structural constraint on doxastic attitudes, theories of decision are mere structural constraints on practical rationality. This is indeed one conception of decision theory, but it cannot work as a response to my arguments. This is because, on this construal of decision theory, a case like The Frustrater cannot possibly be a counterexample to CDT. If we suppose an agent has precise unconditional credences and perpetually takes boxes, there is nothing structurally incoherent with this behavior. The argument from Spencer and Wells is not that there is structural incoherence in the agent that perpetually takes boxes, but rather that taking one of the boxes in The Frustrater is irrational.

The next objection goes like this. Even if we accept that there can be cases of unspecific evidence (e.g., in Strange Items and Envelope & Boxes), the hypothetical agent in Newcomb's Problem and The Frustrater is in a position to justifiably form precise credences. The agent can use the combination of their confidence in the agent's prediction and their confidence in what option they will choose in order to form precise unconditional credences in the relevant states. For example, if the agent is very confident that they will choose Two-Box in Newcomb's Problem, then they ought to be very confident that the opaque box is empty, and vice versa if they are very confident that they will choose One-Box. If they have precise values for the relevant conditional credences and option-credences, then the values for the relevant unconditional credences simply follow from the probability calculus.

I think that there are serious difficulties with the use of option-credence in the suggested manner, at least for cases like The Frustrater. Using option-credences in the manner suggested for The Frustrater will lead to an instability. And though it is possible to stipulate away this instability, doing so would demand using option-credences in a way that is incoherent and/or practically irrational.

Regarding the point about instability, consider a well-known case discussed by Gibbard and Harper (Reference Gibbard, Harper, Harper, Stalnaker and Pearce1978).

The suggestion I am considering is to use my confidence in death's predictive powers and my confidence in where I will go in order to determine my confidence in where death is. So if I am very confident that I will stay in Damascus, then this means I should be very confident that death is in Damascus. And if I am very confident death is in Damascus, then I should ride to Aleppo. An instability looms.

The same kind of instability arises in The Frustrater. Perhaps one thinks that this kind of instability is problematic in both Death in Damascus and The Frustrater, so sees this as a problem for CDT.Footnote ³³ But if this were all The Frustrater showed, then it would be old news – it would not constitute a new challenge to CDT. But it does constitute a new challenge. Namely, that it is intuitively compelling that Envelope is a rationally permissible choice in The Frustrater, yet CDT can never endorse Envelope. Furthermore, this instability only arises when one uses option-credences in order to determine unconditional credences in the calculation of CEU. So even if it is a problematic kind of instability, there are ways of rectifying it without rejecting CDT.Footnote ³⁴

But suppose the suggestion is to stipulate away the instability. So, in The Frustrater, it is stipulated that the agent has a high credence that they will take the red box and, on that basis, has a high credence that $100 is in the blue box. It is also stipulated that they do not change their credences. From this, it would be argued that CDT would say that the agent ought to take the blue box, since taking the blue box has the highest causal expected utility. But then, it would be argued, we should expect the box to be empty given the reliability of the Frustrater's predictions. Thus, CDT is to blame for the agent's financial woes.

I cannot make sense of this. Borrowing a term from Spencer (Reference Spencer2021), say that an agent embodies a theory of rational choice T if, and only if, they always act according to T and they know that they always act according to T. Another way of putting the argument from Spencer and Wells, then, is that an agent who embodies CDT will be financially impoverished relative to the envelope-taker in The Frustrater. With this in mind, I think it is easier to see the incoherence in what is being suggested.

We are told that the agent chooses Blue because Blue has the highest causal expected utility. We are also told that Blue has the highest causal expected utility because the agent has a high credence that $100 is in the blue box. Their high credence that the $100 is in the blue box is based on their high credence that they will choose Red. But, if they embody CDT, then they know that they will choose Blue given that CEU(Blue) > CEU(Red). So we are told to imagine an agent that knows that they will choose Blue because they have a high credence that they will choose Red. This seems incoherent to me.

Whether option-credences should be incorporated into one's deliberative process in general is itself a controversial topic. Isaac Levi famously coined the phrase “deliberation crowds out prediction.” It may look as though I am appealing to Levi's thesis – call it DCOP – in order to defend this last point. Perhaps I am, but different authors seem to understand DCOP in different ways. If I am appealing to DCOP here, then I think I am only appealing to a very weak version of DCOP.

In Hájek's (Reference Hájek2016) critique of DCOP, he defined it as follows: while deliberating about what you'll do, you cannot rationally have credences for what you'll do. This definition is unfortunate because it is ambiguous. What exactly is being claimed in this definition depends on how one understands “cannot” and “rationally.” Many of Hájek's criticisms of DCOP seem to be predicated on understanding “you cannot rationally have credences for what you'll do” as “option-credences are epistemically irrational.” In other words, you cannot have option-credences on pain of epistemic irrationality. This is a possible formulation of DCOP, but it is not the only possible formulation of DCOP. I expect that for many, DCOP is not supposed to be an epistemic constraint on what credences would be epistemically rational or irrational, but rather it is supposed to capture a conceptual point about what it is to be a deliberating agent. The idea is that when an agent takes the deliberative stance, they bracket option-credences. This does not mean that one cannot have option-credences, or that it would be irrational to have them, but rather that they should be bracketed when one is deliberating about what one should do.Footnote ³⁵

While there is a dispute regarding the nature and status of DCOP, I do not think that my objection to the use of option-credences in The Frustrater is based on Hájek's definition of DCOP nor is it predicated on the conceptual understanding of DCOP.Footnote ³⁶ Instead, my objection is that the particular use of option-credences required to establish precise unconditional credences in The Frustrater is practically irrational. Joyce (Reference Joyce2002) points out that there are three worries one could have about incorporating option-credences into practical deliberation.Footnote ³⁷

I think that both Worry-2 and Worry-3 are related to the conceptual understanding of DCOP that I mentioned, however, my worry is Worry-1. What I cannot make sense of is an agent who uses the fact that they have a high credence that they will choose Red as justification for choosing Blue. Here is what Joyce has to say about Worry-1.

I think Joyce is right on this point. Such reasoning is fallacious. And notice that accepting that such reasoning is fallacious does not mean that one must maintain that option-credences are epistemically irrational, nor does it mean that one must maintain that using option-credences is always practically irrational. Rather it is to say that using option-credences in a particular way is practically irrational. But to get precise unconditional credences from option-credences in The Frustrater would require using option-credences in precisely this way.

There are cases where reasoning from “I am likely to A” to “I should A” seems innocuous. Consider a friendly (and very reliable) predictor who put $100 in a black box and $0 in a silver box if, and only if, they predicted you would choose the black box, and $0 in a black box and $100 in a silver box if, and only if, they predicted you would choose the silver box.Footnote ³⁹ Suppose you have a slight compulsion toward shiny things, so have a high credence that you will take the silver box. Consequently, you have a high credence that the friendly predictor puts $100 in the silver box. Here it seems fine, or at least benign, to reason from “I am likely to choose the silver box” to “I should choose the silver box.”

But I maintain that it is not. Suppose instead that the friendly predictor will only put $10 in the silver box if they predicted you would choose the silver box. Furthermore, while your compulsion makes you think it likely that you will choose the silver box, whenever you do choose shiny things it makes you slightly sad. So, while you have a high credence that you will choose the silver box, you know you could prefer to choose the silver box (setting aside the possible contents of the boxes). Admitting that “I am likely to choose the silver box” can be a proper justification for one's choice results in saying that choosing the silver box could be the rational choice in this situation. The high credence that you will choose the silver box leads to a high credence that there is $10 in the silver box, so, for a high enough credence, CEU(silver box) > CEU(black box). But you know that the black box's potential value is greater, you know that you would prefer to choose the black box, and you know that your choice does not causally influence the contents of either box. This seems to me to illustrate the problem with allowing option-credences to play a justificatory role in determining rational choice, regardless of whether the hypothetical predictor is friendly or frustrating.

One possible response to this is to say that, in a case like The Frustrater, it is not the fact that I have a high credence that I will choose Red that is justifying my choice of Blue, but rather it is the fact that I have a high credence that the money is in Blue. So while Worry-1 is a legitimate worry, it is not something one should be worried about in The Frustrater.Footnote ⁴⁰ This is essentially Joyce's view, so while I have invoked some of what he has said in defending my view, he ultimately would disagree with me on this point. Joyce acknowledges that one can present CDT without the use of option-credences, as I have, however, he prefers an account of CDT that does make use of option-credences (see Joyce Reference Joyce1999; Reference Joyce2002; Reference Joyce2012). Joyce suggests that his formulation of CDT cannot fall prey to Worry-1, because the values of option-credences are not inputted directly into the calculation of CEU (Joyce Reference Joyce2002: 80). So Joyce would say that one can and should use option-credences in The Frustrater. Due to the unratifiability of Red or Blue in The Frustrater, the agent that embodies CDT ought to be 50/50 regarding what option they will choose. So Cr(Red) = Cr(Blue) = 0.5. Thus, the agent that embodies CDT will be indifferent between Red and Blue. The idea is that, at deliberative equilibrium, it is the agent's credences in where the money is, not the credences in what they will do, that justifies their indifference.

I still find this suggestion to be problematic. Consider that if the agent were to lose their option-credences, then they would also lose their unconditional credences in the states. This seems to me to indicate that the values of the unconditional credences depend on the values of the option-credences. If this is correct, then the values of CEU(Red) and CEU(Blue) ultimately depend on the values of the option-credences. And given that the justification for choosing Red or Blue is based on the values of CEU(Red) and CEU(Blue) which are themselves based on the agent's option-credences, then it is still a case of Worry-1. If the values that are directly inputted into the calculation of CEU are themselves directly dependent on the values of one's option credences, how is this not still reasoning from the form “I am likely (unlikely) to A, so I should A” but with one extra step added?

I admit that I do not take the above considerations to be decisive, but I hope I have made the case for my way of conceptualizing how a rational, deliberative agent approaches decisions. What I think divides philosophers on this issue is, in part, what stage of the deliberative process one focuses on. I think the kind of perspective that I am advocating for looks bizarre to my opponent, in part, because they are focusing on the endpoint of deliberation. Even in my aforementioned case of an asymmetric friendly predictor, it seems as though my position is that picking the silver box is irrational even if you are overwhelmingly confident that the silver box has $10 and the black box has $0. This is, of course, counterintuitive. However, I think that the opposing view is problematic, in part, because I am focusing on the starting point of deliberation. It seems strange to let one's confidence that one will do A affect the reasonableness of doing A, instead of assessing the reasonableness of A independently.Footnote ⁴¹ So, in the asymmetric friendly predictor, even if you start with the thought you will probably take the silver box, if you assess the options independently of their option-credences, you will conclude that taking the black box is more reasonable. When you make this conclusion, this will presumably affect your option-credences, shifting from a high credence that you will take the silver box to a high credence that you will take the black box. At this point both positions would be in agreement that taking the black box is the rational choice.

Summing up. In section 3, I suggested that if an agent's credences are irrational, then a theory of rational choice cannot be blamed for choices based on these irrational credences. It is a case of garbage in, garbage out. The suggestion I have been considering is that unconditional credences in states in The Frustrater could be formed in a rational manner by using option-credences. However, using one's option-credences in the manner suggested leads to instability in The Frustrater, and stipulating the instability away would force an appeal to the fallacious reasoning “I am likely (unlikely) to A, so I should A.” In the same way that a theory of rational choice cannot be blamed for choices based on irrational credences, a theory of rational choice cannot be blamed for choices based on fallacious reasoning. So it is another case of garbage in, garbage out.