Brad Armendt

The problem of the man who met death in Damascus appeared in the infancy of the theory of rational choice known as causal decision theory. A straightforward, unadorned version of causal decision theory is presented here and applied, along with Brian Skyrms’ deliberation dynamics, to Death in Damascus and similar problems. Decision instability is a fascinating topic, but not a source of difficulty for causal decision theory. Andy Egan’s purported counterexample to causal decision theory, Murder Lesion, is considered; a simple response shows how Murder Lesion and similar examples fail to be counterexamples, and clarifies the use of the unadorned theory in problems of decision instability. I compare unadorned causal decision theory to previous treatments by Frank Arntzenius and by Jim Joyce, and recommend a well-founded heuristic that all three accounts can endorse. Whatever course deliberation takes, causal decision theory is consistently a good guide to rational action.

The primary aim of this paper is the presentation of a foundation for causal decision theory. This is worth doing because causal decision theory (CDT) is philosophically the most adequate rational decision theory now available. I will not defend that claim here by elaborate comparison of the theory with all its competitors, but by providing the foundation. This puts the theory on an equal footing with competitors for which foundations have already been given. It turns out that it will also produce a reply to the most serious objections made so far against CDT and against the particular version of CDT I will defend.

One guide to an argument's significance is the number and variety of refutations it attracts. By this measure, the Dutch book argument has considerable importance.2 Of course this measure alone is not a sure guide to locating arguments deserving of our attention—if a decisive refutation has really been given, we are better off pursuing other topics. But the presence of many and varied counterarguments at least suggests that either the refutations are controversial, or that their target admits of more than one interpretation, or both. The main point of this paper is to focus on a way of understanding the Dutch Book argument (DBA) that avoids many of the well-known criticisms, and to consider how it fares against an important criticism that still remains: the objection that the DBA presupposes value-independence of bets.

To what extent do our beliefs, and how strongly we hold them, depend upon how they matter to us, on what we take to be at stake on them? The idea that beliefs are sometimes stake-sensitive (Armendt 2008, 2013) is further explored here, with a focus on whether beliefs may be stake-sensitive and rational. In contexts of extended deliberation about what to do, beliefs and assessments of options interact. In some deliberations, a belief about what you will do may rationally influence your estimate of the value of doing it; deliberation dynamics provides a framework for modeling such interactions. A distinction is drawn between sensitivity to the magnitude of the stakes, and sensitivity to the shape of the stakes. Contexts of extended deliberation are settings in which some beliefs that p rationally depend on the shape of the stakes on p. The dependence is either rational stake-sensitivity or an outcome of rational learning; empirical evidence concerning contexts of deliberation may lead us to model rational beliefs in one way or the other.

Does the strength of a particular belief depend upon the significance we attach to it? Do we move from one context to another, remaining in the same doxastic state concerning p yet holding a stronger belief that p in one context than in the other? For that to be so, a doxastic state must have a certain sort of context-sensitive complexity. So the question is about the nature of belief states, as we understand them, or as we think a theory should model them. I explore the idea and how it relates to work on imprecise probabilities and second-order confidence.

The idea that beliefs may be stake-sensitive is explored. This is the idea that the strength with which a single, persistent belief is held may vary and depend upon what the believer takes to be at stake. The stakes in question are tied to the truth of the belief—not, as in Pascal’s wager and other cases, to the belief’s presence. Categorical beliefs and degrees of belief are considered; both kinds of account typically exclude the idea and treat belief as stake-invariant , though an exception is briefly described. The role of the assumption of stake-invariance in familiar accounts of degrees of belief is also discussed, and morals are drawn concerning finite and countable Dutch book arguments.

What can rational deliberation indicate about belief? Belief clearly influences deliberation. The principle that rational belief is stake-invariant rules out at least one way that deliberation might influence belief. The principle is widely, if implicitly, held in work on the epistemology of categorical belief, and it is built into the model of choice-guiding degrees of belief that comes to us from Ramsey and de Finetti. Criticisms of subjective probabilism include challenges to the assumption of additive values (the package principle) employed by defenses of probabilism. But the value-interaction phenomena often cited in such challenges are excluded by stake-invariance. A comparison with treatments of categorical belief suggests that the appeal to stake-invariance is not ad hoc. Whether or not to model belief as stake-invariant is a question not settled here.

It is widely held that the influence of risk on rational decisions is not entirely explained by the shape of an agent’s utility curve. Buchak (Erkenntnis, 2013, Risk and rationality, Oxford University Press, Oxford, in press) presents an axiomatic decision theory, risk-weighted expected utility theory (REU), in which decision weights are the agent’s subjective probabilities modified by his risk-function r. REU is briefly described, and the global applicability of r is discussed. Rabin’s (Econometrica 68:1281–1292, 2000) calibration theorem strongly suggests that plausible levels of risk aversion cannot be fully explained by concave utility functions; this provides motivation for REU and other theories. But applied to the synchronic preferences of an individual agent, Rabin’s result is not as problematic as it may first appear. Theories that treat outcomes as gains and losses (e.g. prospect theory and cumulative prospect theory) account for risk sensitivity in a way not available to REU. Reference points that mark the difference between gains and losses are subject to framing, many instances of which cannot be regarded as rational. However, rational decision theory may recognize the difference between gains and losses, without endorsing all ways of fixing the point of reference. In any event, REU is a very interesting theory.

Louis Marinoff [1990] criticizes Axelrod and Hamilton's [1981] use of the concept of an evolutionarily stable strategy, and claims to find an inconsistency between their theory for repeated Prisoner's Dilemma games and empirical results. Marinoff seeks to resolve the inconsistency by arguing that Axelrod and Hamilton's model is ill conceived: he purports to prove, contra Axelrod and Hamilton, that no evolutionarily stable strategy exists in the repeated Prisoner's Dilemma. But his argument is flawed, and moreover, Marinoff gives no good reason for thinking the claimed discrepancy between thory and empirical results exists.

Defenders of sophisticated evidential decision theory (EDT) have argued (1) that its failure to provide correct recommendations in problems where the agent believes himself asymmetrically fallible in executing his choices is no flaw of the theory, and (2) that causal decision theory gives incorrect recommendations in certain examples unless it is supplemented with an additional metatickle or ratifiability deliberation mechanism. In the first part of this paper, I argue that both positions are incorrect. In the second part of the paper, I show how the agent's preferences involved in standard counterexamples to EDT, such as Newcomb's problem, violate the Jeffrey/Bolker preference axioms, specifically the Impartiality axiom.

Two criticisms of Dutch strategy arguments are discussed: One says that the arguments fail because agents who know the arguments can use that knowledge to avoid Dutch strategy vulnerability, even though they violate the norm in question. The second consists of cases alleged to be counterexamples to the norms that Dutch strategy arguments defend. The principle of Reflection and its Dutch strategy argument are discussed, but most attention is given to the rule of Conditionalization and to Jeffrey's rule for fallible learning. I argue that the first criticism should be rejected, and that the second presents no counterexamples to the rationality of commitment to the rules.

Causal decision theory (CDT) is the best theory of rational choice now available.2 I intend to provide some support for that claim in part I of this paper by responding to two criticisms of CDT. The first criticism says that CDT is superfluous, since it does no better in the problems that matter than does evidential decision theory (EDT) at recommending correct choices. A second criticism says that CDT by itself is flawed: according to this view, there are problems in which CDT makes bad recommendations, unless it is supplemented with an additional deliberation mechanism, either involving metatickles or screening for ratifiable choices.3 I will argue in response to the first criticism that CDT is genuinely better than any EDT of the most sophisticated sort: there are problems where CDT gives better recommendations, and they are problems that do matter.

Causal decision theory is a general theory of rational decision, appropriate for simple or complex decision problems. It is an expected utility theory distinguished by its explicit attention to causal features of decision problems, and by the significance it attaches to those features. When the causal structure of a decision problem is uncomplicated, the recommendations of CDT and other theories generally agree. In more complex cases, however, CDT identifies rational decisions where other theories do not. Several varieties of CDT have been offered; they differ in their ways of representing beliefs about causal influence, but as decision theories they are very similar. Each of them was developed as a subjective expected utility theory, and that approach will be assumed here.

In recent years rational decision theories such as Richard Jeffrey's, which recommend that an agent act so as to maximize his conditional expected utility, have come under attack on the grounds that they are unable to adequately handle certain kinds of decision problems. Because of their general structure, these problems are sometimes known as "causal counterexamples" to the theories; well-known examples are Newcomb's problem, the Prisoner's Dilemma, and Fisher's smoking gene problem. Several versions of "causal decision theory" have been presented by Gibbard and Harper, Brian Skyrms, and David Lewis; these theories are designed to deal with the counterexamples correctly. In this dissertation I show how an axiomatic treatment, including representation and uniqueness theorems, for Brian Skyrms' version of causal decision theory can be obtained from work done by Peter Fishburn on another theory; I partially reinterpret Fishhburn's formal theory and then show how the expected utility rule of Skyrms' theory can be derived from the rule of Fishburn's theory. With the interpretation I give of the formal theory, together with an account of conditional preference which I develop, I reply to criticisms which have been made of Skyrms' theory: It has been suggested that the theory is flawed because it requires the use of "K-partitions" of state descriptions which must be identified by judging the causal independence of the states from the possible actions and the causal relevance of the states to the problem. This is thought to be an undesirable dependence of rational decision theory upon an adequate theory of causal independence and causal relevance. I argue that this criticism is mistaken, and I state conditions which identify adequate K-partitions of state descriptions in terms of the behavior of those descriptions in the agent's prference system. Finally, I discuss the axioms of the formal theory under my interpretation of them, and I conclude that, properly understood, they are reasonable requirements for rational conditional preferences, so that the causal decision theory is well-founded.