Decision Theory

There is currently much discussion about how decision making should proceed when an agent's degrees of belief are imprecise; represented by a set of probability functions. I show that decision rules recently discussed by Sarah Moss, Susanna Rinard and Rohan Sud all suffer from the same defect: they all struggle to rationalize diachronic ambiguity aversion. Since ambiguity aversion is among the motivations for imprecise credence, this suggests that the search for an adequate imprecise decision rule is not yet over.

I present a new argument for the repugnant conclusion. The core of the argument is a risky, intrapersonal analogue of the mere addition paradox. The argument is important for three reasons. First, some solutions to Parfit’s original puzzle do not obviously generalize to the intrapersonal puzzle in a plausible way. Second, it raises independently important questions about how to make decisions under uncertainty for the sake of people whose existence might depend on what we do. And, third, it suggests various (...) difficulties for leading views about the value of a person’s life compared to her nonexistence. (shrink)

The Precautionary Principle (PP) is an influential principle of risk management. It has been widely introduced into environmental legislation, and it plays an important role in most international environmental agreements. Yet, there is little consensus on precisely how to understand and formulate the principle. In this paper I prove some impossibility results for two plausible formulations of the PP as a decision-rule. These results illustrate the difficulty in making the PP consistent with the acceptance of any trade-offs between catastrophic risks (...) and more ordinary goods. (shrink)

In his classic book “the Foundations of Statistics” Savage developed a formal system of rational decision making. The system is based on (i) a set of possible states of the world, (ii) a set of consequences, (iii) a set of acts, which are functions from states to consequences, and (iv) a preference relation over the acts, which represents the preferences of an idealized rational agent. The goal and the culmination of the enterprise is a representation theorem: Any preference relation that (...) satisfies certain arguably acceptable postulates determines a (finitely additive) probability distribution over the states and a utility assignment to the consequences, such that the preferences among acts are determined by their expected utilities. Additional problematic assumptions are however required in Savage's proofs. First, there is a Boolean algebra of events (sets of states) which determines the richness of the set of acts. The probabilities are assigned to members of this algebra. Savage's proof requires that this be a σ-algebra (i.e., closed under infinite countable unions and intersections), which makes for an extremely rich preference relation. On Savage's view we should not require subjective probabilities to be σ-additive. He therefore finds the insistence on a σ-algebra peculiar and is unhappy with it. But he sees no way of avoiding it. Second, the assignment of utilities requires the constant act assumption: for every consequence there is a constant act, which produces that consequence in every state. This assumption is known to be highly counterintuitive. The present work contains two mathematical results. The first, and the more difficult one, shows that the σ-algebra assumption can be dropped. The second states that, as long as utilities are assigned to finite gambles only, the constant act assumption can be replaced by the more plausible and much weaker assumption that there are at least two non-equivalent constant acts. The second result also employs a novel way of deriving utilities in Savage-style systems -- without appealing to von Neumann-Morgenstern lotteries. The paper discusses the notion of “idealized agent" that underlies Savage's approach, and argues that the simplified system, which is adequate for all the actual purposes for which the system is designed, involves a more realistic notion of an idealized agent. (shrink)

Can an agent deliberating about an action A hold a meaningful credence that she will do A? 'No', say some authors, for 'Deliberation Crowds Out Prediction' (DCOP). Others disagree, but we argue here that such disagreements are often terminological. We explain why DCOP holds in a Ramseyian operationalist model of credence, but show that it is trivial to extend this model so that DCOP fails. We then discuss a model due to Joyce, and show that Joyce's rejection of DCOP rests (...) on terminological choices about terms such as 'intention', 'prediction', and 'belief'. Once these choices are in view, they reveal underlying agreement between Joyce and the DCOP-favouring tradition that descends from Ramsey. Joyce's Evidential Autonomy Thesis (EAT) is effectively DCOP, in different terminological clothing. Both principles rest on the so-called 'transparency' of first-person present-tensed reflection on one's own mental states. (shrink)

Innovative research on decision making under ‘deep uncertainty’ is underway in applied fields such as engineering and operational research, largely outside the view of normative theorists grounded in decision theory. Applied methods and tools for decision support under deep uncertainty go beyond standard decision theory in the attention that they give to the structuring (also called framing) of decisions. Decision structuring is an important part of a broader philosophy of managing uncertainty in decision making, and normative decision theorists can both (...) learn from, and contribute to, the growing deep uncertainty decision support literature. (shrink)

There is a puzzle about the very possibility of foreseen but unintended side effects, and solving this puzzle requires us to revise our basic picture of the structure of practical deliberation. The puzzle is that, while it seems that we can rationally foresee, but not intend, bringing about foreseen side effects, it also seems that we rationally must decide to bring about foreseen side effects and that we intend to do whatever we decide to do. I propose solving this puzzle (...) by rejecting the idea that we intend to do whatever we decide to do. My solution involves taking account of the underappreciated role that qualified intentions play in deliberation. I also argue that this solution fares better than those that instead reject the idea that we rationally must decide to bring about foreseen side effects, for these solutions are committed to rejecting the even more compelling idea that decisions rationally serve as the conclusions of practical deliberation. (shrink)

Although expected utility theory has proven a fruitful and elegant theory in the finite realm, attempts to generalize it to infinite values have resulted in many paradoxes. In this paper, we argue that the use of John Conway's surreal numbers shall provide a firm mathematical foundation for transfinite decision theory. To that end, we prove a surreal representation theorem and show that our surreal decision theory respects dominance reasoning even in the case of infinite values. We then bring our theory (...) to bear on one of the more venerable decision problems in the literature: Pascal's Wager. Analyzing the wager showcases our theory's virtues and advantages. To that end, we analyze two objections against the wager: Mixed Strategies and Many Gods. After formulating the two objections in the framework of surreal utilities and probabilities, our theory correctly predicts that (1) the pure Pascalian strategy beats all mixed strategies, and (2) what one should do in a Pascalian decision problem depends on what one's credence function is like. Our analysis therefore suggests that although Pascal's Wager is mathematically coherent, it does not deliver what it purports to, a rationally compelling argument that people should lead a religious life regardless of how confident they are in theism and its alternatives. (shrink)

There is a long-standing disagreement in the philosophy of probability and Bayesian decision theory about whether an agent can hold a meaningful credence about an upcoming action, while she deliberates about what to do. Can she believe that it is, say, 70% probable that she will do A, while she chooses whether to do A? No, say some philosophers, for Deliberation Crowds Out Prediction (DCOP), but others disagree. In this paper, we propose a valid core for DCOP, and identify terminological (...) causes for some of the apparent disputes. (shrink)

The problem of inferring probability comparisons between events from an initial set of comparisons arises in several contexts, ranging from decision theory to artificial intelligence to formal semantics. In this paper, we treat the problem as follows: beginning with a binary relation ≥ on events that does not preclude a probabilistic interpretation, in the sense that ≥ has extensions that are probabilistically representable, we characterize the extension ≥+ of ≥ that is exactly the intersection of all probabilistically representable extensions of (...) ≥. This extension ≥+ gives us all the additional comparisons that we are entitled to infer from ≥, based on the assumption that there is some probability measure of which ≥ gives us partial qualitative information. We pay special attention to the problem of extending an order on states to an order on events. In addition to the probabilistic interpretation, this problem has a more general interpretation involving measurement of any additive quantity: e.g., given comparisons between the weights of individual objects, what comparisons between the weights of groups of objects can we infer? (shrink)

This paper defines and defends time-slice epistemology, according to which there are no essentially diachronic norms of rationality. First I motivate and distinguish two notions of time-slice epistemology. Then I defend time-slice theories of action under indeterminacy, i.e. theories about how you should act when the outcome of your decision depends on some indeterminate claim. I raise objections to a theory of action under indeterminacy recently defended by Robbie Williams, and I propose some alternative theories in its place. Throughout this (...) discussion, I defend a more general moral about action under indeterminacy, namely that time-slice theories are supported by strong analogies with ethical theories. In particular, our understanding of agents torn between interpretations of a decision situation should be guided by our theories of agents torn between incommensurable values. (shrink)

Decision theory aims to provide mathematical analysis of which choice one should rationally make in a given situation. Our current decision theory norms have been very successful, however, several problems have proven vexing for standard decision theory. In this paper, I show that these problems all share a similar structure and identify a class of problems which decision theory overvalues. I demonstrate that agents who follow current standard decision theory can be exploited and have their preferences reordered if offered decision (...) problems of this class. I show that preference reordering is a serious problem, which motivates my search for a decision theory which is immune to exploitation. I find Dr. Nick Smith’s theory of Rationally Negligible Probabilities cannot be exploited in this way and discuss why agents should adopt it. (shrink)

Decision theory elucidates, in more ways than one, the, concept of rational behavior under imperfect knowledge of the consequences. On the other hand, the generally accepted concept of rationality refers to the end-means relation. This relation is not translatable into the language of decision theory. Consequently, the latter's claim to have defined in a general way rationality of behavior appears not to be valid.

Various proposals have been made for solving The Two Envelope Problem. But even though the problem itself is easily stated and quite simple, the proposedsolutions have not been. Some involve calculus, some involve considerations about infinite values, and some are complicated in other ways. Moreover, there is not yet any one solution that is widely accepted as correct. In addition to being notable for its simplicity and its lack of a generally agreed-upon solution, The Two Envelope Problem is also notable (...) because it demonstrates that something that has been taken to be a fundamental principle of decision theory is false. The main purpose of this paper is to propose and defend a simple solution to The Two Envelope Problem. But I also want to make a start on the project of figuring out how the relevant fundamental principle of decision theory should be revised. (shrink)

Some philosophers have recently argued that decision-makers ought to take normative uncertainty into account in their decisionmaking. These philosophers argue that, just as it is plausible that we should maximize expected value under empirical uncertainty, it is plausible that we should maximize expected choice-worthiness under normative uncertainty. However, such an approach faces two serious problems: how to deal with merely ordinal theories, which do not give sense to the idea of magnitudes of choice-worthiness; and how, even when theories do give (...) sense to magnitudes of choice-worthiness, to compare magnitudes of choice-worthiness across different theories. Some critics have suggested that these problems are fatal to the project of developing a normative account of decision-making under normative uncertainty. The primary purpose of this article is to show that this is not the case. To this end, I develop an analogy between decision-making under normative uncertainty and the problem of social choice, and then argue that the Borda Rule provides the best way of making decisions in the face of merely ordinal theories and intertheoretic incomparability. (shrink)

Gilboa and Schmeidler provide a paradigm for modelling decision making under uncertainty. Unlike the classical theory of expected utility maximization, case-based decision theory does not assume that decision makers know the possible 'states of the world' or the outcomes, let alone the decision matrix attaching outcomes to act-state pairs. Case-based decision theory suggests that people make decisions by analogies to past cases: they tend to choose acts that performed well in the past in similar situations, and to avoid acts that (...) performed poorly. It is an alternative to expected utility theory when both states of the world and probabilities are neither given in the problem nor can be easily constructed. The authors describe the general theory and its relationship to planning, repeated choice problems, inductive inference, and learning; they highlight its mathematical and philosophical foundations and compare it with expected utility theory as well as with rule-based systems. (shrink)

We commonly face circumstances in which the cumulative negative effects of repeatedly acting in a certain way over time will be significant, although the negative effects of any one such act, taken on its own, are insubstantial. Warren Quinn's puzzle of the self-torturer presents an especially clear example of this type of predicament. This paper considers three different approaches to understanding the rational response to such situations. The first focuses on the conditions under which it is rational to revise one's (...) prior intentions. The second raises the possibility of a fundamental disconnect between the rational assessment of an extended pattern of choices and the assessment of the individual choices that make up that pattern. I show that neither adequately addresses the underlying issues. I propose a third approach, according to which the rational assessment of the “self-torturer's” choices is guided, not by any plan or intention the he has actually adopted, but by the plan or plans it would have been reasonable for him to adopt from the outset. The larger significance of this conclusion is brought out through the identification of conditions under which one's past choices can non-derivatively constrain the rational response to one's present circumstances. (shrink)

How should we make choices where the values we subscribe to give conflicting recommendations? I will be defending a reduction of decision making under conflict to decision making under indeterminacy, in the spirit of Broome. To defend this, I set out and endorse the basic features of decision making under conflict that Ruth Chang identifies. I show that we find exactly those features in cases of decision making under indeterminacy not involving conflicting values. Further, my theory of decision making under (...) indeterminacy predicts and explains these features. Particular attention will be paid to the aspect that Chang emphasizes as the decisive problem for the Broomean reduction: the resolutional residue, or angst, that decision making under conflict evinces. (shrink)

First published in 1982, Ellery Eells' original work on rational decision making had extensive implications for probability theorists, economists, statisticians and psychologists concerned with decision making and the employment of Bayesian principles. His analysis of the philosophical and psychological significance of Bayesian decision theories, causal decision theories and Newcomb's paradox continues to be influential in philosophy of science. His book is now revived for a new generation of readers and presented in a fresh twenty-first-century series livery, including a specially commissioned (...) preface written by Brian Skyrms, illuminating its continuing importance and relevance to philosophical enquiry. (shrink)

It is commonly assumed that preferences are determinate; that is, that an agent who has a preference knows that she has the preference in question and is disposed to act upon it. This paper argues the dubiousness of that assumption. An account of indeterminate preferences in terms of self-predicting subjective probabilities is given, and a decision rule for choices involving indeterminate preferences is proposed. Wolfgang Spohn’s and Isaac Levi ’s arguments against self-predicting probabilities are also considered, in light of Wlodek (...) Rabinowicz’s recent criticism. (shrink)

The Intergovernmental Panel on Climate Change has developed a novel framework for assessing and communicating uncertainty in the findings published in their periodic assessment reports. But how should these uncertainty assessments inform decisions? We take a formal decision-making perspective to investigate how scientific input formulated in the IPCC’s novel framework might inform decisions in a principled way through a normative decision model.

This is a collection of Isaac Levi's philosophical papers. Over the period represented by the work here, Professor Levi has developed an interrelated set of views, in the tradition of Peirce and Dewey, on epistemology and the philosophy of science and social science. This focus has been on the problem of induction and the growth of knowledge, the foundations of probability and the theory of rational decision-making. His most important essays in these areas are assembled here, with an introduction setting (...) out their main themes and connections. As a whole the volume presents a coherent, elaborated position which will be of great interest to a range of philosophers, decision theorists, welfare and social choice theorists and cognitive scientists. (shrink)

This study uses the television show Cash Cab as a natural experiment to investigate gender differences in decision making under uncertainty. As expected, men are much more likely to accept the end-of-game gamble than are women, but men and women appear to weigh performance variables differently when relying on subjective probabilities. At best men base their risky decisions on general aspects of their previous “good” play (not all of which is relevant at the time the decision is made) and at (...) worst fail to condition their risky decisions on any of the relevant information available to them. In sharp contrast, women appear to consider all of the information available to them, including previous “poor” play as well as their most recent confident “good” play, which, by design, is likely the most relevant information to consider. (shrink)