Abstract
Andy Egan recently drew attention to a class of decision situations that provide a certain kind of informational feedback, which he claims constitute a counterexample to causal decision theory (CDT). Arntzenius and Wallace have sought to vindicate a form of CDT by describing a dynamic process of deliberation that culminates in a “mixed” decision. I show that, for many of the cases in question, this proposal depends on an incorrect way of calculating expected utilities, and argue that it is therefore unsuccessful. I then tentatively defend an alternative proposal by Joyce, which produces a similar process of dynamic deliberation but for a different reason.
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Notes
Incidentally, if an agent is sure she will perform A then A’s CEU and EEU are the same, since \(Cr(A\,\,\Box{\!\!}\rightarrow o)\) and Cr(o / A) are then both equal to the agent’s unconditional credence in o. Thus, once a decision has been made, we can speak unambiguously of the expected utility of that decision.
Op cit, pp. 290–291. This invocation of Piaf’s maxim to criticise the instability of CDT should not be conflated with his invocation of the principle of weak desire reflection (which he describes (p. 277) as a “version of Piaf’s maxim”) to criticise evidential decision theory. Weak desire reflection states that an agent’s evaluation of an option’s desirability at a given time should be equal her expectation, at that time, of her evaluation of its desirability at any given future time, provided that her evaluations change only as a result of updating her credence function according to new information. Arntzenius shows that EDT does not satisfy weak desire reflection (pp. 278–282) but CDT does (pp. 282–285), which he offers as a demonstration of CDT’s superiority over EDT. His criticism of CDT’s instability is based on an entirely separate application of the notion of regret, that does not involve desire reflection.
Wallace (2010) provides a detailed treatment of the use of mixed decisions in cases somewhat similar to those involving type-2 predictors, though he also applies them—erroneously, as we shall see—to a case involving a type-1 predictor, namely the “psychopath button” case (see below). Both Arntzenius and Wallace prove that there is always a ratifiable mixed decision in these cases.
Arntzenius describes his own proposal as “little more than an exposition” of Skyrms’s work (p. 292), which I have therefore referred to to resolve some minor unclarity in Arntzenius’s paper. Some of the details mentioned here are not explicit in Arntzenius but are explicit in Skyrms. Most importantly, Arntzenius is not explicit about the assumption that the CEU of a mixed act is a weighted average the unconditional CEUs of those of the component pure acts, but Skyrms (op cit, pp. 29–30) asserts this without argument by stating that “A state of indecision, P, carries with it an expected utility, the expectation according to the probability vector \(P=\langle p_{1}\,\ldots \,p_{n}\rangle \) of the expected utilities of the acts \(A_{1}\,\ldots \,A_{n}\)” (Unlike Arntzenius, Skyrms draws a notional distinction between the agent’s state of indecision about how to act, which he formalises as a probability vector, and her credences about how she is going to act. This difference is not important for present purposes). Arntzenius also does not state explicitly what he means by the expected utility of the status quo, but Skyrms (pp. 29–30) makes it clear that the expected utility of what he terms a “state of indecision” is that of the default mixed act of that state.
Wedgwood (2011, §3) pursues this and related objections in more detail.
This can be shown as follows. Updating his credences according to a \({1}/{22}\) credence that he will push gives him credence \({1}/{22}\times 0.95+{21}/{22}\times 0.05={1}/{11}\) credence that he is a psycho; the CEU of pushing is then \(CU(PUSH)={1}/{11}\times (-100)+{10}/{11}\times 10=0\), while that of not pushing is always 0. The mixed act’s CEU is, according to DDT, a weighted average of those of the pure acts, and thus also equal to 0; this set of credences is thus a fixed point in Paul’s deliberation.
- $$ \begin{aligned} Cr(Push\, \& \,Psycho)&= {} Cr(Psycho/Push)Cr(Push)\\ &= {} 0.95\times {1}/{22}\\ &= {} {19}/{440}\\ Cr(Push\, \& \,\lnot Psycho) &= {} Cr(\lnot Psycho/Push)Cr(Push)\\ &= {} 0.05\times {1}/{22}\\ &= {} {1}/{440}\\ CU(Push)&= {} -100Cr(Psycho)+10Cr(\lnot Psycho)\\ &= {} -100[Cr(Psycho/Push)Cr(Push)+Cr(Psycho/\lnot Push)Cr(\lnot Push)]\\ &\quad+\,10[Cr(\lnot Psycho/Push)Cr(Push)+Cr(\lnot Psycho/\lnot Push)Cr(\lnot Push)]\\ &= {} -100(0.95\times {1}/{22}+0.05\times {21}/{22})+10(0.05\times {1}/{22}+0.95\times {21}/{22}) \end{aligned}$$
Wallace also considers Death in Damascus, but it is not clear whether he is regarding it as a type-1 or type-2 case. He writes that “[a]ssuming that Death is a very good predictor, the optimal strategy is to choose at random. (If Death’s powers go beyond prediction into actual prophecy, I’m less sure if the analysis applies).” The mention of “actual prophecy” seems to refer to the possibility of death being a type-1 predictor; if so, he is correct that his analysis does not then apply.
This is subject to the proviso that, in standard probability theory, probabilities conditional on an event with probability zero are not well-defined. This proviso is not relevant here, since probabilities conditional on an action with probability zero are irrelevant for expected utility calculations.
Although strong centring follows from counterfactual excluded middle and modus ponens, it might be rejected by somebody who rejects conditional excluded middle, in which case this argument would not go through. Since the formulation of CDT we have been using presupposes counterfactual excluded middle, an alternative formulation would be needed, such as that provided by Lewis (1981). I will show here that Lewis’s formulation also implies the equivalence, argued for in the text, between an act’s conditional CEU, given it is performed, and the expected utility simpliciter of the agent’s conditional credence function, given the act is performed.
Lewis formulates the CEU of an act A as \( CEU(A)=\sum _{k\in K}Cr(k)V(A\, \& \,k)\), where \(V(\cdot )\) denotes evidential expected utility the elements of K are propositions specifying aspects of the state of the world, insofar as they are relevant to the outcome of the decision situation, that the agent cannot causally influence. Since we are considering the case in which the agent is sure she will perform A, this reduces to \(CEU(A/A\,)=\sum _{k\in K}Cr(k)V(k)\). Expanding this by substituting in the definition of evidential expected utility, we have \(CEU(A)=\sum _{k\in K}Cr(k)\sum _{o\in O}Cr(o/k)U(o)\). But rearranging and simplifying this gives \(\sum _{o\in \mathcal {O}}Cr(o)U(o)\), the agent’s expected utility simpliciter, which is thus equivalent to A’s conditional CEU on Lewis’s formulation as well as on the formulation I am using.
This space of epistemically possible worlds must, of course, be sharply distinguished from the modal space of metaphysically possible worlds. Indeed, each epistemically possible world comes with its own modal universe. For example, it is epistemically possible that Hesperus is not Phosphorus; in epistemically possible worlds at which this is true, it is metaphysically neecessary, and so each such world comes with a modal universe consisting entirely of worlds at which Hesperus is not Phosphorus.
I am not claiming that this is the “correct” answer. There are many possible answers, and singling one out as “correct” would require an analysis of the causal structure of the situation that is beyond the scope of the present discussion.
One might still be a lexical ratificationist, thinking that ratifiable decisions are preferable to unratifiable ones with the same CEU; but Egan (pp. 111–112) provides a convincing counterexample to this claim.
Here I am once more setting aside a worry about the dynamics of mixed decisions: namely that of how, once a mixed decision has been resolved into a pure act, the agent is to go through with that act. I think the worry I raise here, about how to go through with an unratifiable pure decision that does not result from a mixed decision but is itself, qua pure decision, the outcome of rational deliberation, is harder to set aside.
References
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Acknowledgments
I am indebted to Frank Arntzenius, John Broome, Harjit Bhogal, Harvey Lederman, and two anonymous reviewers, for their invaluable comments on various versions of this article. This work was supported by the Arts and Humanities Research Council.
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Plommer, B. A New Problem with Mixed Decisions, Or: You’ll Regret Reading This Article, But You Still Should. Erkenn 81, 349–373 (2016). https://doi.org/10.1007/s10670-015-9743-0
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DOI: https://doi.org/10.1007/s10670-015-9743-0