Decision-Theoretic Paradoxes as Voting Paradoxes Rachael Briggs University of Sydney, Griffith University It is a platitude among decision theorists that agents should choose their actions so as to maximize expected value. But exactly how to define expected value is contentious. Evidential decision theory (henceforth EDT), causal decision theory (henceforth CDT), and a theory proposed by Ralph Wedgwood that I will call benchmark theory (BT) all advise agents to maximize different types of expected value. Consequently, their verdicts sometimes conflict. In certain famous cases of conflict���medical Newcomb problems���CDT and BT seem to get things right, while EDT seems to get things wrong. In other cases of conflict, including some recent examples suggested by Egan 2007, EDT and BT seem to get things right, while CDT seems to get things wrong. In still other cases, EDT and CDT seem to get things right, while BT gets things wrong. It���s no accident, I claim, that all three decision theories are subject to counterexamples. Decision rules can be reinterpreted as voting rules, where the voters are the agent���s possible future selves. The problematic examples have the structure of voting paradoxes. Just as voting paradoxes show that no voting rule can do everything we want, decision-theoretic paradoxes show that no decision rule can do everything we want. Luck- ily, the so-called ���tickle defense��� establishes that EDT, CDT, and BT will do everything we want in a wide range of situations. Most decision situations, I thank Dan Greco, Caspar Hare, Brian Hedden, Vann McGee, and an anonymous reviewer at the Philosophical Review for their helpful comments. Philosophical Review, Vol. 119, No. 1, 2010 DOI 10.1215/00318108-2009-024 �� 2010 by Cornell University 1

R A C H A E L B R I G G S Iargue,areanaloguesofvotingsituationsinwhichthevotersunanimously adopt the same set of preferences. In such situations, all plausible voting rules and all plausible decision rules agree. 1. Background Assumptions The decision theories I���ll discuss all rest on a common conceptual frame- work. All three theories advise agents about how to behave in decisionsitua- tions. An agent���s ultimate goal in a decision situation is to attain the out- come with the highest value. But since agents in decision situations are typically uncertain as to how much value their actions will engender, they typically do not know how to maximize value directly. The best they can do is to maximize expected value. The three theories I will discuss provide different conceptions of expected value. Before I begin with a more detailed discussion of decision situa- tions, abitmorediscussionofvalueis inorder. I will assumethatvalues can be meaningfully represented using real numbers, so that (for example) an outcome with a value of thirty is better than an outcome with a value of twenty and worse than an outcome with a value of fifty, but twice as close (in some meaningful sense) to the former as to the latter. There needn���t be anything special about either the value zero or the size of the units: all ofthedecisiontheoriesIdiscussgivethesameresultsunderpositivelinear transformations. (That is, where a is any positive real number and b is any real number, taking the value of each option, multiplying by a, and adding b will not affect the decision theories��� prescriptions.) I will write ���V = v��� for the proposition that the agent obtains an out- come with value v, and I will refer to propositions of the form ���V = v��� as value-level propositions. I make absolutely no substantial assumptions about thenatureofthegoodmeasuredbythevaluefunction:���value���mayberead hedonically, morally, aesthetically, pragmatically, or in whatever other way suits the reader���s fancy. On to decision situations. I���ll represent each decision situation as a triple A,K, C , where A is a finite set of possible actions that the agent believes are available, K is a finite set of dependency hypotheses about how various value-level propositions depend counterfactually on the possible actions, and C is a credence function representing the agent���s beliefs. I���ll ela- borate on each member of the triple in turn. First, the actions. By convention, the actions in A are pairwise incompatible: if the agent can walk, chew gum, or do both at the same time, then the set of actions will include walking and chewing gum and chewing gum and not walking, but it will not include walking or chewing gum. 2

Decision-Theoretic Paradoxes as Voting Paradoxes By convention, the actions in A are also jointly exhaustive: if the agent can refrain from both walking and chewing gum, then not walking and not chewing gum will count as an action (at least assuming that there is nothing else the agent can choose to do or not do at the same time). I will use the same symbol to denote both an action and the proposition to the effect that the action is performed as far as I know, nothing important hinges on this ambiguity. Next, the dependency hypotheses. These are sets of counterfac- tual conditionals. More precisely, each K ��� K is a set containing, for every A ��� A, exactly one nonbacktracking counterfactual conditional of the form A ��� (V = v) and nothing else.1 This provides a new way to write the value function: I will write V (K &A) for the v such that (A ��� (V = v)) ��� K . I���ll make two substantial assumptions about the nature of depen- dency hypotheses. First, I���ll assume that the actions in A are causally irrel- evant to the dependency hypotheses in K���in other words, that an agent cannot change what depends on his or her actions by acting. Second, I���ll assume that in every possible world, some dependency hypothesis K is true���in other words, that no matter what, for every A ��� A, there is some amount of value v that the agent would enjoy if he or she were to perform A. I will use the same symbol to denote both a given dependency hypothe- sisandapropositiontotheeffectthatallthecounterfactualsinthatdepen- dency hypothesis are true as far as I know, nothing important hinges on this ambiguity. The information encoded in A and K can be written in matrix form. Consider a simple situation in which an agent is at the racetrack, deciding whether to bet fifty dollars on horse 1 or on horse 2. Suppose horse1isrunningatthree-to-oneodds,whilehorse2isrunningatnine-to- one odds. Assuming that the agent cares only about money, and assuming that he or she values money linearly, the available actions and dependency hypotheses can be written as follows (where betting on horse 1 is abbrevi- ated ���A1���, and betting on horse 2 is abbreviated ���A2���). K1 K2 K3 A1 150 ���50 ���50 A2 ���50 450 ���50 1. For the purposes of this essay, I assume that the counterfactual dependency between actions and value-level propositions is completely deterministic. See Lewis 1981 for a discussion of probabilistic counterfactual dependency in causal decision theory. 3

R A C H A E L B R I G G S The content of each dependency hypothesis can be read off from the corresponding column of the matrix. If the cell in column Ki and row A j contains the value v, then (A j ��� (V = v)) ��� Ki . So in the above matrix, K1 is the set { A1 ��� (V = 150), A2 ��� (V = ���50)}���the depen- dency hypothesis that obtains if and only if horse 1 wins. Finally, the credence function C . An agent facing a decision situa- tion does not know what he or she will do and may not know which depen- dency hypothesis is true. However, the agent has partial beliefs about both members of A and members of K. These are represented by the credence function C , which assigns real numbers between zero and one to arbitrary Boolean compounds of elements of A and K. I���ll assume that C is a prob- ability function satisfying the Kolmogorov axioms. The information encoded in C can be written in matrix form. The credence function of the agent in the racetrack situation, for instance, might be described as follows. K1 K2 K3 A1 .1 .05 .35 A2 .1 .05 .35 Each cell in the matrix represents a conjunction, and the entry in a given cell represents the credence that the agent assigns to that con- junction. For instance, since the cell in column K1 and row A1 contains the entry .1, the matrix says that C (K1&A1) = .1. Since C is a probabi- lity function, and since the conjunctions represented by the cells are pair- wise incompatible and jointly exhaustive, the entries in the cells must add up to one. So every decision situation A,K, C can be represented as a pair of matrices: the first representing A and K, and the second representing C . Notice that on this method of representation C (V = v) can be read off the two matrices jointly, though it cannot be read off either matrix individ- ually. To compute C (V = v), just examine the first matrix (representing A and K), identify all the cells containing the entry v, and add up the entries of the corresponding cells in the second matrix. One final note on credence functions: I will find it useful to speak of evidential independence. I will say that propositions A and B are eviden- tially independent (for an agent with a credence function C ) just in case C (A | B) = C (A). Evidential independence is always relative to the agent���s credence function C , but I will leave this relativization implicit. 4

Decision-Theoretic Paradoxes as Voting Paradoxes 2. EDT EDT recommends that agents maximize evidentially expected value, or e-expected value for short. The e-expected value of an action A ��� A is de- fined as follows. VE (A) = v C (V = v | A)v (1) In other words, A���s e-expected value is a weighted average of the values of possibleoutcomesthatmightresultfromchoosing A,wheretheweighting of each value v is determined by the agent���s conditional credence in the proposition V = v given A. VE (A) measures the amount of value the agent wouldexpecttoenjoy,wereheorshetoconditionalizeontheinformation that he or she had performed A. EDT can be summed up in the slogan: ���Whenyouhavethepowertomakethenews,makegoodnews���(seeJeffrey 1983, 82���83). One can give an alternative definition of e-expected value in terms of the dependency hypotheses. VE (A) = K ���K C (K | A)V (K & A) (2) This second definition is equivalent to the first and will later prove useful in explaining the relationship between CDT and EDT.2 2. Proof that the two definitions are equivalent: (2) can be rewritten as follows. VE (A) = v {C (K | A) : V (K & A) = v}v But for each K ��� K such that V (K & A) = v, K & A entails V = v. Furthermore if no K ��� K such that V (K & A) = v is true, then some K ��� K such that V (K & A) = v must be true. So for any v, {C (K | A) : V (K & A) = v} = C (V = v | A). Thus, we can rewrite (2) again as VE (A) = v {C (V = v| A)v : V (K &A) = v}v Eliminating the inner sum, VE (A) = v C (V = v| A)v But that���s just (1). 5