The Gibbard-Harper Collapse Lemma for Counterfactual Decision Theory∗

There is a problem for the debate between causal decision theory, formulated in terms of counterfactuals, and its traditional rival, evidential decision theory: an agent’s credences in counterfactuals concerning their own acts collapse into evidential probabilities on those acts once diachronic conditionalization on the act is taken into account. Given assumptions that both classical CDTers [6] and their critics (prominently, [4]) accept, it therefore follows that three things cannot be distinct: (i) the probability of a state, given an act; (ii) the probability that if the act were performed, the state would result; and (iii) the probability one would have in that same counterfactual, if one learned the act was actually performed. According to both Evidential and Causal decision theory, a choiceworthy act maximizes expected utility. Evidential decision theory (henceforth ‘EDT’) recommends that one calculate expected utilities using act-conditionalized probabilities on states. Causal decision theory (‘CDT’) rejects this strategy, with the rationale that some acts—say, seeking out a therapist— do not cause the states—say, being anxious—with which they are correlated. Allan Gibbard and William Harper’s “Counterfactuals and Two Kinds of Expected Utility” [6] provides a classic statement of the causal approach, using counterfactual conditionals to express causal relationships that are uniquely of interest to a CDTer. Many in the recent literature, however, hold that tides have turned against CDT. One important factor is a 2007 paper by Andy Egan, which presents several counterexamples to the theory.1 On Egan’s view, classical causal decision theorists—in particular, those who appeal to the counterfactual formulation of the theory—adhere to the motto “do whatever has the best expected outcome, holding fixed [one’s] initial views about the likely causal structure of the world” (96). However, Egan claims, there are situations where agents should not hold such views fixed. In these cases, agents should anticipate their future causal views and leverage those instead, thus taking into account what they will learn by performing the act in question. The force of Egan’s point is now widely discussed and widely accepted.2 In this paper, I focus on an unappreciated formal result, pointed out by Gibbard & Harper [6] in the third section of their classic paper. There, they prove that if an agent is probabilistically coherent, and the semantics for counterfactuals obeys the principle of the Conditional Excluded Middle [24], then the probability of a counterfactual, given its antecedent, collapses to the probability of the consequent, given the antecedent. It follows that “Future-directed Causal Decision Theory” recommends an act just in case classical Evidential Decision Theory does. This complicates the dialectic. By Gibbard & Harper’s result, the work of putting causal information into a simple EDT system is exactly undone by adding a norm of learning. I present a stronger version of the collapse lemma, which applies to Lewis [19]’s weakening of Gibbard & Harper’s axioms. I conclude with some thoughts on how to assess the argument from anticipation in light of these results. ∗Warm thanks to Andy Egan, Alex Kocurek, Reuben Stern, two anonymous reviewers for the Amsterdam Colloquium, and the audience at the 2015 Formal Epistemology Workshop for discussion and feedback. 1 These examples were anticipated by Reed Richter in the 1980s: see Richter [21]. 2See, for example, Arntzenius [1], Briggs [2] and Greaves [7]. For a prominent dissent, see Joyce [13]. The Gibbard-Harper Collapse Lemma Melissa Fusco 1 EDT vs. CDT (in brief) Both EDT and CDT begin with the idea that the value of each of a set of available acts (call them the ai’s) is calculated by identifying a set of states which fix one’s welfare (a set of states sk) and then multiplying the utility of each state-act conjunction by one’s subjective probability, or credence, that that state obtains. This relationship is set out by the expected utility equation in Savage’s Foundations of Statistics,3 which applies the generic statistical notion of expectation to the agent’s welfare function V al(·). For an arbitrary act a,