Synthese (2012) 187:95���122 DOI 10.1007/s11229-011-0023-5 Reversing 30years of discussion: why causal decision theorists should one-box Wolfgang Spohn Received: 6 September 2008 / Accepted: 18 March 2009 / Published online: 26 October 2011 �� Springer Science+Business Media B.V. 2011 Abstract The paper will show how one may rationalize one-boxing in Newcomb���s problem and drinking the toxin in the Toxin puzzle within the confines of causal deci- sion theory by ascending to so-called reflexive decision models which reflect how actions are caused by decision situations (beliefs, desires, and intentions) represented by ordinary unreflexive decision models. Keywords Causal decision theory �� Evidential decision theory �� Newcomb���s problem �� Toxin puzzle �� Reflexive decision theory 1 Introduction Decision theorists have been causal decision theorists (CDTs) all along, I assume, not only since Savage (1954). That there is a position to take has become clear, though, only when an apparent alternative came up, namely Jeffrey���s (1965/1983) so called evidential decision theory (EDT), and a problem, namely Newcomb���s problem (NP) (cf. Nozick 1969), separating the alternatives. Thus, the distinction between CDT and EDT emerged in the late 70s, most conspicuously with Gibbard and Harper (1978). The present state of discussion is a somewhat acquiesced one, I feel. There is no commonly agreed version of CDT,1 presumably because we well enough under- 1 Cf., e.g., the overview in Joyce (1999, Chap. 5) or the papers collected in Campbell and Sowden (1985) and in Sobel (1994). I dedicate this paper to Karel Lambert, a first version of which I have presented on the conference on occasion of his 75th birthday at UCI in April 2003. And I am indebted to the thoughtful remarks of an anonymous referee. W. Spohn (B) Department of Philosophy, University of Konstanz, 78457 Konstanz, Germany e-mail: Wolfgang.Spohn@uni-konstanz.de 123

96 Synthese (2012) 187:95���122 stand (subjective) probabilities and utilities, but still not well enough causation and its relation to probabilities. However, the impression that some version of CDT is the right one is overwhelming, and so is the intuition thereby supported that two-boxing is the right thing to do in NP. Some uneasiness remains, even among CDTs. Still, the uneasiness has never led to a general and generally acceptable version of decision theory that was able to make one-boxing plausible.2 Hence the acquiescence the topic seems fought out and further fighting useless. The uneasiness has a name, the title of Lewis (1981b): if you���re so smart, ���why ain���cha rich?��� The basic answer is: what can we do, if irrationality is rewarded? And CDTs find this answer acceptable in some way or other. Since 1976 when I developed my own account of CDT and NP3 I, too, was an ardent two-boxer and convinced of that answer. Since a few years, though, the answer sounds self-pitying to me and just wrong this must be poor rationality that complains about the reward for irrationality. In this paper I shall explain how we can keep all the insights of CDT and neverthe- less rationalize one-boxing. The gist of the paper is presented in Sect. 2 it will make clear what my almost shamefully simple plot will be. In a way I am finished then. However, since my account there will be merely graphical, i.e., in terms of graphs, I have to develop the theory behind those graphs at least as far as required. This will be less simple and the bulk of the paper in the remaining Sects. 3���6 the final crucial step of my argument will be presented in a precise way only at the end of Sect. 6. I hope my case regarding NP will be convincing. It will be even more convincing, I believe, regarding the Toxin puzzle (TP) fully understanding the latter will open the eyes about the former. For this reason I shall deal with both cases in parallel. 2 The central idea, in graphical terms In NP you are standing before two boxes, and you may take the opaque box containing an unknown amount of money or you may take both, the opaque and the transparent box that you see to contain 1,000 dollars. The unknown amount of money is either nil or a million, depending on an earlier prediction of some being, the predictor, about what you will do if the prediction is that you take only the opaque box, it contains a million dollar, and if the prediction is that you take both boxes the amount is 0. You know all this, and in particular you know that the predictor is remarkably successful (say, in predicting your actions in other situations and other persons in that situation). What will you do? There is no point in rehearsing all the arguments for one- and for two-boxing. Stan- dard CDT says you should two-box: your action cannot have any influence on the prediction the content of the opaque box is fixed, whatever it is and by two-box- ing you end up with 1,000 dollars more in any case two-boxing strictly dominates one-boxing. The story is represented by the following decision graph: 2 Even Jeffrey changed his mind several times cf. Jeffrey (1965/1983, 1988, 1996). 3 Cf. Spohn (1976/1978, Chap. 3 and Sects. 5.1���5.2). This German account was neglected it is very close to Meek and Glymour (1994). For more detailed comparative remarks see Spohn (2001). 123

Synthese (2012) 187:95���122 97 (NP1) B M P Here, time always moves from bottom to top, squares represent action nodes, circles represent chance nodes or occurrence nodes, as I shall rather say (since they need not be chancy unlike the action nodes they are at most under indirect control of the agent and objects of the beliefs of the agent), and the arrows have a causal interpretation. Nodes represent variables here, B is the action variable of one- or two-boxing, P describes the prediction, and M is the monetary outcome. The standard interpretation of X ��� Y is that Y (directly) causally depends on X, but we shall have to consider this more closely later on. Given this interpretation, (NP1) accurately represents the temporal and causal relations of NP from the point of view of the agent. ���Decision graph��� does not seem to be an established term, although its meaning springs to one���s eyes. I shall sketch the theory of decision graphs in Sect. 4 they are precisely what Pearl (2000, p. 23) calls mutilated graphs. Right now, two remarks might suffice. First, decision graphs should not be confused with the familiar decision trees. In a decision tree the nodes represent events or states of affairs, and a branch represents an entire possible course of events. Decision trees are temporarily ordered, but their edges have no causal meaning. By contrast, a decision graph is causally structured, and its nodes represent variables. Of course, it is straightforward to construct the associated tree from the graph. Second,decisiongraphsshouldnotbeconfusedwithinfluencediagrams(cf. Howard and Matheson 1981). The latter contain also informational arrows that end at an action node and start from all those variables about which the agent is informed at that action node.However,thiskindofinformationflowisnottoberepresentedindecisiongraphs it will be taken into account only in what I shall call reflexive decision graphs later on. The theory of decision graphs makes the obvious assumption that in the situation given by (NP1) the causal independence of P from B entails its probabilistic inde- pendence from B. Thus, the dominant action, two-boxing, is also the one maximizing (conditional) expected utility, and hence the one recommended by CDT. Note, by the way, that the temporal relation between B and P is inessential all we need to con- clude in two-boxing is the causal independence of P from B. By adding that P realizes before B, we only make dramatically clear that B cannot have a causal influence on P. What about the remarkable success of the predictor that suggests that given you one-box it is very likely that she will have predicted that you will one-box, and likewise for two-boxing? How do they enter the picture? They don���t. CDTs do not deny them, but they take great pains to explain that they are not the ones to be used in practical deliberation calculating expected utilities and they diverge in how exactly to conceive of the subjective probabilities to be used instead. I shall return to this issue in a bit more detail at the end of Sect. 4. Now suppose, just suppose, that the following decision graph would adequately represent NP: 123

98 Synthese (2012) 187:95���122 (NP2) B M P Then, I take it, it would be beyond dispute that the only reasonable thing to do is one-boxing. Nobody has ever doubted this, if NP were a case of backwards causation, as this graph seems to suggest. It is only that we have explicitly excluded backwards causation in NP! Well, I exclude it, too anything else would be absurd. Still, I shall defend the claim that (NP2) is an adequate representation of NP. Obviously, this can be so only if the arrows do not quite mean what they were so far told to mean. We shall see what their causal significance exactly is. For the moment I am happy with the conditional conclusion: if (NP2) should be adequate, one-boxing would indeed be rational. In order to see what I may be up to with (NP2), let us look at the TP invented by Kavka (1983) it is more suitable for making my point. The story is as follows: At this evening you are approached by a weird scientist. She requests you to form the intention to drink a glass of toxin tomorrow noon. If you drink it, you will feel awful for a few hours, but then you will recover without any after-effects. If and only if you have formed the intention by midnight, you will be rewarded with 10,000 dollars. Whether you have formed the intention can be verified by a cerebroscope the scientist has developed. The reward only depends on your intention or rather the verdict of the cerebroscope what you actually do tomorrow noon is of no further relevance. Of course, you think that 10,000 dollars by far outweigh a few hours of sickness. But how do you get them? It is clear in advance that once you stand before the glass of toxin you have no incentive whatsoever to drink it by then the cerebroscope has made its verdict, whatever it is. Hence, it seems you cannot honestly form that intention. You may pretend as well as you can but this is no way to deceive the cerebroscope. Let us again represent the situation by a decision graph. Note that a decision graph only contains action and occurrence nodes, and the only action and occurrence vari- ables involved in the toxin story are these: (TP1) D F C R D is the action variable of drinking the toxin or not, F tells how you feel tomorrow afternoon, C is the variable for the cerebroscope reading, and R says whether or not you get the reward. The causal influences in the story run only in the way indicated by the arrows. 123

Synthese (2012) 187:95���122 99 Given this representation and given again that the causal independence of C from D implies its probabilistic independence, it is clear that at D only not drinking the toxin maximizes (conditional) expected utility and hence that it is difficult or impossible to have the contrary intention. One will object that (TP1) forgets about the most important variable, the intention to be formed. Yes, certainly. I shall undo this neglect in a moment. The neglect is due to the fact that a decision graph contains all the action and occurrence nodes to be considered in the represented situation for making a decision or forming an intention but it does not contain the intention itself as a separate variable. However, let me first make the same move as in the case of NP. Suppose, just suppose, that the decision graph adequately representing TP would be this: (TP2) D F C R Then, again, drinking the toxin would obviously and uncontestedly be the rational action maximizing conditional expected utility. The only mystery, again, is the arrow from D to C, since I have not yet explained how to avoid the absurd and unwanted interpretation of this arrow as backwards causation. The mystery dissolves when we undo the neglect already observed and explicitly introduce the intention as a separate variable. That is, we shall now distinguish a third kind of node, intentional nodes or rather decision nodes or variables that realize in entire decision situations, mental complexes of beliefs and desires, focusing or con- cluding in a decision or intention to act in a certain way I shall represent such decision nodes by triangles. Most of the literature is quite sloppy at this point and refer to the square action nodes also as choice or decision nodes. Thus the present distinction, that is crucial for this paper, is blurred right from the beginning and cannot be reimported into the picture. So, how should we represent the causal situation of the toxin story with these richer means? This is not entirely clear. One idea is that, willy-nilly, you take the final decision only tomorrow noon when you stand before the glass of toxin. This yields the follow- ing causal diagram or reflexive decision graph, as I shall call it for reasons to become clear soon: (TP1*) D D* F C R 123