We begin by showing that every theory of the value of uncertain prospects must have one of three unpalatable properties. _Reckless_ theories recommend giving up a sure thing, no matter how good, for an arbitrarily tiny chance of enormous gain; _timid_ theories permit passing up an arbitrarily large potential gain to prevent a tiny increase in risk; _non-transitive_ theories deny the principle that, if A is better than B and B is better than C, then A must be better than (...) C. Having set up this trilemma, we study its horns. Non-transitivity has been much discussed; we focus on drawing out the costs and benefits of recklessness and timidity when it comes to axiology, decision theory, and normative uncertainty. (shrink)

We prove a representation theorem for preference relations over countably infinite lotteries that satisfy a generalized form of the Independence axiom, without assuming Continuity. The representing space consists of lexicographically ordered transfinite sequences of bounded real numbers. This result is generalized to preference orders on abstract superconvex spaces.

Naive versions of decision theory take probabilities and utilities as primitive and use expected value to give norms on rational decision. However, standard decision theory takes rational preference as primitive and uses it to construct probability and utility. This paper shows how to justify a version of the naive theory, by taking dominance as the most basic normatively required preference relation, and then extending it by various conditions under which agents should be indifferent between acts. The resulting theory can make (...) all the decisions of classical expected utility theory, plus more in cases where expected utilities are infinite or undefined. Although the theory requires similarly strong assumptions to classical expected utility theory, versions of the theory can be developed with slightly weaker assumptions, without having to prove a new representation theorem for the weaker theory. This alternate foundation is particularly useful if probability is prior to preference, as suggested by the recent program to base probabilism on accuracy and alethic considerations rather than pragmatic ones. (shrink)

Games such as the St. Petersburg game present serious problems for decision theory.1 The St. Petersburg game invokes an unbounded utility function to produce an infinite expectation for playing the game. The problem is usually presented as a clash between decision theory and intuition: most people are not prepared to pay a large finite sum to buy into this game, yet this is precisely what decision theory suggests we ought to do. But there is another problem associated with the St. (...) Petersburg game. The problem is that standard decision theory counsels us to be indifferent between any two actions that have infinite expected utility. So, for example, consider the decision problem of whether to play the St. Petersburg game or a game where every payoff is $1 higher. Let’s call this second game the Petrograd game (it’s the same as St. Petersburg but with a bit of twentieth century inflation). Standard decision theory is indifferent between these two options. Indeed, it might be argued that any intuition that the Petrograd game is better than the St. Petersburg game is a result of misguided and na¨ıve intuitions about infinity.2 But this argument against the intuition in question is misguided. The Petrograd game is clearly better than the St. Petersburg game. And what is more, there is no confusion about infinity involved in thinking this. When the series of coin tosses comes to an end (and it comes to an end with probability 1), no matter how many tails precede the first head, the payoff for the Petrograd game is one dollar higher than the St. Petersburg game. Whatever the outcome, you are better off playing the Petrograd game. Infinity has nothing to do with it. Indeed, a straightforward application of dominance reasoning backs up this line of reasoning.3 Standard decision theory. (shrink)

This paper is a response to Paul Bartha’s ‘Making Do Without Expectations’. We provide an assessment of the strengths and limitations of two notable extensions of standard decision theory: relative expectation theory and Paul Bartha’s relative utility theory. These extensions are designed to provide intuitive answers to some well-known problems in decision theory involving gaps in expectations. We argue that both RET and RUT go some way towards providing solutions to the problems in question but neither extension solves all the (...) relevant problems. (shrink)

I reason: (1) For any x, if I knew that A contained x, then the odds are even that B contains either 2x or x/2, so the expected amount in B would be 5x/4. So (2) for all x, if I knew that A contained x, I would have an expected gain in switching to B. So (3) I should switch to B. But this seems clearly wrong, as my information about A and B is symmetrical.

This paper contributes to the mathematical foundations of the model for utility theory developed by Richard Jeffrey in The Logic of Decision [5]. In it I discuss the relationship of Jeffrey's to classical models, state and interpret an existence theorem for numerical utilities and subjective probabilities and restate a theorem on their uniqueness.

The Pasadena game invented by Nover and Hájek raises a number of challenges for decision theory. The basic problem is how the game should be evaluated: it has no expectation and hence no well-defined value. Easwaran has shown that the Pasadena game does have a weak expectation, raising the possibility that we can eliminate the value gap by requiring agents to value gambles at their weak expectations. In this paper, I first prove a negative result: there are gambles like the (...) Pasadena game that do not even have a weak expectation. Hence, problematic value gaps remain even if decision theory is extended to take in weak expectations. There is a further challenge: the existence of a ‘value gap’ in the Pasadena game seems to make decision theory inapplicable in a number of cases where the right choice is obvious. The positive contribution of the paper is a theory of ‘relative utilities’, an extension of decision theory that lets us make comparative judgements even among gambles that have no well-defined value. (shrink)

Orthodox decision theory gives no advice to agents who hold two goods to be incommensurate in value because such agents will have incomplete preferences. According to standard treatments, rationality requires complete preferences, so such agents are irrational. Experience shows, however, that incomplete preferences are ubiquitous in ordinary life. In this paper, we aim to do two things: (1) show that there is a good case for revising decision theory so as to allow it to apply non-vacuously to agents with incomplete (...) preferences, and (2) to identify one substantive criterion that any such non-standard decision theory must obey. Our criterion, Competitiveness, is a weaker version of a dominance principle. Despite its modesty, Competitiveness is incompatible with prospectism, a recently developed decision theory for agents with incomplete preferences. We spend the final part of the paper showing why Competitiveness should be retained, and prospectism rejected. (shrink)

The two envelope paradox can be dissolved by looking closely at the connection between conditional and unconditional expectation and by being careful when summing an infinite series of positive and negative terms. The two envelope paradox is not another St. Petersburg paradox and that one does not need to ban talk of infinite expectation values in order to dissolve it. The article ends by posing a new puzzle to do with infinite expectations.

We pose and resolve several vexing decision theoretic puzzles. Some are variants of existing puzzles, such as 'Trumped' (Arntzenius and McCarthy 1997), 'Rouble trouble' (Arntzenius and Barrett 1999), 'The airtight Dutch book' (McGee 1999), and 'The two envelopes puzzle' (Broome 1995). Others are new. A unified resolution of the puzzles shows that Dutch book arguments have no force in infinite cases. It thereby provides evidence that reasonable utility functions may be unbounded and that reasonable credence functions need not be countably (...) additive. The resolution also shows that when infinitely many decisions are involved, the difference between making the decisions simultaneously and making them sequentially can be the difference between riches and ruin. Finally, the resolution reveals a new way in which the ability to make binding commitments can save perfectly rational agents from sure losses. (shrink)

Standard decision theory has trouble handling cases involving acts without finite expected values. This paper has two aims. First, building on earlier work by Colyvan (2008), Easwaran (2014), and Lauwers and Vallentyne (2016), it develops a proposal for dealing with such cases, Difference Minimizing Theory. Difference Minimizing Theory provides satisfactory verdicts in a broader range of cases than its predecessors. And it vindicates two highly plausible principles of standard decision theory, Stochastic Equivalence and Stochastic Dominance. The second aim is to (...) assess some recent arguments against Stochastic Equivalence and Stochastic Dominance. If successful, these arguments refute Difference Minimizing Theory. This paper contends that these arguments are not successful. (shrink)

Challenging, with several powerful arguments, some of our deepest beliefs about rationality, morality, and personal identity, Parfit claims that we have a false view about our own nature. It is often rational to act against our own best interersts, he argues, and most of us have moral views that are self-defeating. We often act wrongly, although we know there will be no one with serious grounds for complaint, and when we consider future generations it is very hard to avoid conclusions (...) that most of us will find very disturbing. (shrink)

Set Theory has experienced a rapid development in recent years, with major advances in forcing, inner models, large cardinals and descriptive set theory. The present book covers each of these areas, giving the reader an understanding of the ideas involved. It can be used for introductory students and is broad and deep enough to bring the reader near the boundaries of current research. Students and researchers in the field will find the book invaluable both as a study material and as (...) a desktop reference. (shrink)

Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters. The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It (...) discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes. (shrink)

This is the classic work upon which modern-day game theory is based. What began as a modest proposal that a mathematician and an economist write a short paper together blossomed, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry (...) it yielded--game theory--has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences. (shrink)

Which is better: a guarantee of a modest amount of moral value, or a tiny probability of arbitrarily large value? To prefer the latter seems fanatical. But, as I argue, avoiding such fanaticism brings severe problems. To do so, we must decline intuitively attractive trade-offs; rank structurally identical pairs of lotteries inconsistently, or else admit absurd sensitivity to tiny probability differences; have rankings depend on remote, unaffected events ; and often neglect to rank lotteries as we already know we would (...) if we learned more. Compared to these implications, fanaticism is highly plausible. (shrink)

This paper presents a new solution to the problems for orthodox decision theory posed by the Pasadena game and its relatives. I argue that a key question raised by consideration of these gambles is whether evaluative compositionality (as I term it) is a requirement of rationality: is the value that an ideally rational agent places on a gamble determined by the values that she places on its possible outcomes, together with their mode of composition into the gamble (i.e. the probabilities (...) assigned to them)? The paper first outlines a certain simple response to the Pasadena game and identifies two problems with this response, the second of which is that it leads to a wholesale violation of evaluative compositionality. I then argue that rationality does not require decision makers to factor in outcomes of arbitrarily low probability. A method for making decisions which flows from this basic idea is then developed, and it is shown that this decision method (Truncation) leads to a limited — as opposed to wholesale — violation of evaluative compositionality. The paper then argues that the truncation method yields solutions to the problems posed by the Pasadena game and its relatives that are both attractive in themselves and superior to those yielded by alternative proposals in the literature. (shrink)

A de minimis risk is defined as a risk that is so small that it may be legitimately ignored when making a decision. While ignoring small risks is common in our day-to-day decision making, attempts to introduce the notion of a de minimis risk into the framework of decision theory have run up against a series of well-known difficulties. In this paper, I will develop an enriched decision theoretic framework that is capable of overcoming two major obstacles to the modelling (...) of de minimis risk. The key move is to introduce, into decision theory, a non-probabilistic conception of risk known as normic risk. (shrink)

Abstract: This paper defends a constraint that any satisfactory decision theory must satisfy. I show how this constraint is violated by all of the decision theories that have been endorsed in the literature that are designed to deal with cases in which opinions or values are represented by a set of functions rather than a single one. Such a decision theory is necessary to account for the existence of what Ruth Chang has called “parity” (as well as for cases in (...) which agents have incomplete preferences or imprecise credences). The problem with the all of the decision theories that have been defended to account for parity is that they are committed to a claim I call unanimity: when all of the functions in the set agree that an agent ought to do A, then an agent ought to do A. A decision theory committed to unanimity violates the constraint I defend in this paper. Thus, if parity exists, a new approach to decision theory is necessary. (shrink)

We introduce a St. Petersburg-like game, which we call the ‘Pasadena game’, in which we toss a coin until it lands heads for the first time. Your pay-offs grow without bound, and alternate in sign (rewards alternate with penalties). The expectation of the game is a conditionally convergent series. As such, its terms can be rearranged to yield any sum whatsoever, including positive infinity and negative infinity. Thus, we can apparently make the game seem as desirable or undesirable as we (...) want, simply by reordering the pay-off table, yet the game remains unchanged throughout. Formally speaking, the expectation does not exist; but we contend that this presents a serious problem for decision theory, since it goes silent when we want it to speak. We argue that the Pasadena game is more paradoxical than the St. Petersburg game in several respects. We give a brief review of the relevant mathematics of infinite series. We then consider and rebut a number of replies to our paradox: that there is a privileged ordering to the expectation series; that decision theory should be restricted to finite state spaces; and that it should be restricted to bounded utility functions. We conclude that the paradox remains live. (shrink)

Is Bayesian decision theory a panacea for many of the problems in epistemology and the philosophy of science, or is it philosophical snake-oil? For years a debate had been waged amongst specialists regarding the import and legitimacy of this body of theory. Mark Kaplan had written the first accessible and non-technical book to address this controversy. Introducing a new variant on Bayesian decision theory the author offers a compelling case that, while no panacea, decision theory does in fact have the (...) most profound consequences for the way in which philosophers think about inquiry, criticism and rational belief. The new variant on Bayesian theory is presented in such a way that a non-specialist will be able to understand it. The book also offers new solutions to some classic paradoxes. It focuses on the intuitive motivations of the Bayesian approach to epistemology and addresses the philosophical worries to which it has given rise. (shrink)

Newcomb's problem and similar cases show the need to incorporate causal distinctions into the theory of rational decision; the usual noncausal decision theory, though simpler, does not always give the right answers. I give my own version of causal decision theory, compare it with versions offered by several other authors, and suggest that the versions have more in common than meets the eye.

The existence of a transitive, complete, and weakly independent relation on the full set of gambles implies the existence of a non-Ramsey set. Therefore, each transitive and weakly independent relation on the set of gambles either is incomplete or does not have an explicit description. Whatever tools decision theory makes available, there will always be decision problems where these tools fail us. In this sense, decision theory remains incomplete.

:We address the question, in decision theory, of how the value of risky options should be assessed when they have no finite standard expected value, that is, where the sum of the probability-weighted payoffs is infinite or not well defined. We endorse, combine and extend the proposal of Easwaran to evaluate options on the basis of their weak expected value, and the proposal of Colyvan to rank options on the basis of their relative expected value.

We show that it is not possible to extend the ranking of one-stage lotteries based on their weak-expectation to a reflexive and transitive relation on the collection of one- and two-stage lotteries that satisfies two basic axioms, the minimal value axiom and the reduction axiom. We propose an extension that satisfies only the first axiom. This ranking takes payoffs, their probabilities, and the tree structure into account.

Is Bayesian decision theory a panacea for many of the problems in epistemology and the philosophy of science, or is it philosophical snake-oil? For years a debate had been waged amongst specialists regarding the import and legitimacy of this body of theory. Mark Kaplan had written the first accessible and non-technical book to address this controversy. Introducing a new variant on Bayesian decision theory the author offers a compelling case that, while no panacea, decision theory does in fact have the (...) most profound consequences for the way in which philosophers think about inquiry, criticism and rational belief. The new variant on Bayesian theory is presented in such a way that a non-specialist will be able to understand it. The book also offers new solutions to some classic paradoxes. It focuses on the intuitive motivations of the Bayesian approach to epistemology and addresses the philosophical worries to which it has given rise. (shrink)

This book defends the view that any adequate account of rational decision making must take a decision maker's beliefs about causal relations into account. The early chapters of the book introduce the non-specialist to the rudiments of expected utility theory. The major technical advance offered by the book is a 'representation theorem' that shows that both causal decision theory and its main rival, Richard Jeffrey's logic of decision, are both instances of a more general conditional decision theory. The book solves (...) a long-standing problem for Jeffrey's theory by showing for the first time how to obtain a unique utility and probability representation for preferences and judgements of comparative likelihood. The book also contains a major new discussion of what it means to suppose that some event occurs or that some proposition is true. The most complete and robust defence of causal decision theory available. (shrink)

"[This book] proposes new foundations for the Bayesian principle of rational action, and goes on to develop a new logic of desirability and probabtility."—Frederic Schick, _Journal of Philosophy_.

This paper explores the consequences of applying two natural ideas from epistemology to decision theory: (1) that knowledge should guide our actions, and (2) that we know a lot of non-trivial things. In particular, we explore the consequences of these ideas as they are applied to standard decision theoretic puzzles such as the St. Petersburg Paradox. In doing so, we develop a “knowledge-first” decision theory and we will see how it can help us avoid fanaticism with regard to the St. (...) Petersburg puzzle and related puzzles. The result will be a decision theory that gives a novel, but well-motivated, reason for discounting small probabilities when making decisions. We examine the merits and demerits of such a decision theory. (shrink)

A decade ago, Harris Nover and I introduced the Pasadena game, which we argued gives rise to a new paradox in decision theory even more troubling than the St Petersburg paradox. Gwiazda's and Smith's articles in this volume both offer revisionist solutions. I critically engage with both articles. They invite reflections on a number of deep issues in the foundations of decision theory, which I hope to bring out. These issues include: some ways in which orthodox decision theory might be (...) supplemented; the role of simulations of such infinite games; the role of small probabilities, and of idealization, in decision theory; tolerance about practical norms; and alternative ways of understanding decision theory's job description. (shrink)

We argue that indeterminate probabilities are not only rationally permissible for a Bayesian agent, but they may even be rationally required . Our first argument begins by assuming a version of interpretivism: your mental state is the set of probability and utility functions that rationalize your behavioral dispositions as well as possible. This set may consist of multiple probability functions. Then according to interpretivism, this makes it the case that your credal state is indeterminate. Our second argument begins with our (...) describing a world that plausibly has indeterminate chances. Rationality requires a certain alignment of your credences with corresponding hypotheses about the chances. Thus, if you hypothesize the chances to be indeterminate, your will inherit their indeterminacy in your corresponding credences. Our third argument is motivated by a dilemma. Epistemic rationality requires you to stay open-minded about contingent matters about which your evidence has not definitively legislated. Practical rationality requires you to be able to act decisively at least sometimes. These requirements can conflict with each other-for thanks to your open-mindedness, some of your options may have undefined expected utility, and if you are choosing among them, decision theory has no advice to give you. Such an option is playing Nover and Hájek’s Pasadena Game , and indeed any option for which there is a positive probability of playing the Pasadena Game. You can serve both masters, epistemic rationality and practical rationality, with an indeterminate credence to the prospect of playing the Pasadena game. You serve epistemic rationality by making your upper probability positive-it ensures that you are open-minded. You serve practical rationality by making your lower probability 0-it provides guidance to your decision-making. No sharp credence could do both. (shrink)

(No abstract is available for this citation).

In some games, the products of the probabilities times the payouts result in a series that is conditionally convergent, which means that the sum can vary based on the order in which the products are summed. The purpose of this paper is to address the question: How should such games be valued? We first show that, contrary to widespread belief, summing in the order determined by the mechanism of the game does not lead to the correct value. We then consider (...) other attempts to value such games. We suggest that games with conditionally convergent series are properly valued by summing the products in increasing order of the absolute values of the game’s payouts. (shrink)

Morgenbesser's Coin is a thought experiment that exemplifies a widespread disposition to infer counterfactual independence from causal independence. I argue that this disposition is mistaken by analysing a closely related thought experiment.

By recourse to the fundamentals of preference orderings and their numerical representations through linear utility, we address certain questions raised in Nover and Hájek 2004, Hájek and Nover 2006, and Colyvan 2006. In brief, the Pasadena and Altadena games are well-defined and can be assigned any finite utility values while remaining consistent with preferences between those games having well-defined finite expected value. This is also true for the St Petersburg game. Furthermore, the dominance claimed for the Altadena game over the (...) Pasadena game, and that would have been claimed for the St Petersburg game over the Altadena, can be contradicted without fear of inconsistency with the axioms of utility theory. However, insistence upon dominance can be made to yield a contradiction of the Archimedean axiom of utility theory. CiteULike Connotea Del.icio.us What's this? (shrink)

Fine has shown that assigning any value to the Pasadena game is consistent with a certain standard set of axioms for decision theory. However, I suggest that it might be reasonable to believe that the value of an individual game is constrained by the long-run payout of repeated plays of the game. Although there is no value that repeated plays of the Pasadena game converges to in the standard strong sense, I show that there is a weaker sort of convergence (...) it exhibits, and use this to define a notion of ‘weak expectation’ that can give values to the Pasadena game and many others, though not to all games that fail to have a strong expectation in the standard sense. CiteULike Connotea Del.icio.us What's this? (shrink)

This paper evaluates a recent method proposed by Jeremy Gwiazda for calculating the value of gambles that fail to have expected values in the standard sense. I show that Gwiazda’s method fails to give answers for many gambles that do have standardly defined expected values. However, a slight modification of his method (based on the mathematical notion of the ‘Cauchy principal value’ of an integral), is in fact a proper extension of both his method and the method of ‘weak expectations’. (...) I show that this method gives an appropriate value when the ‘tails’ of the gambles that are eliminated in the truncation are ‘stable’, but that the value is not appropriate when the tails are not stable. I do not attempt to give an argument for the use of this method, but just note that it is more general than Gwiazda’s method, and is mathematically quite natural. (shrink)

There has been much discussion on the two-envelope paradox. Clark and Shackel (2000) have proposed a solution to the paradox, which has been refuted by Meacham and Weisberg (2003). Surprisingly, however, the literature still contains no axiomatic justification for the claim that one should be indifferent between the two envelopes before opening one of them. According to Meacham and Weisberg, "decision theory does not rank swapping against sticking [before opening any envelope]" (p. 686). To fill this gap in the literature, (...) we present a simple axiomatic justification for indifference, avoiding any expectation reasoning, which is often considered problematic in infinite cases. Although the two-envelope paradox assumes an expectation-maximizing agent, we show that analogous paradoxes arise for agents using different decision principles such as maximin and maximax, and that our justification for indifference before opening applies here too. (shrink)

Lara Buchak sets out a new account of rational decision-making in the face of risk. She argues that the orthodox view is too narrow, and suggests an alternative, more permissive theory: one that allows individuals to pay attention to the worst-case or best-case scenario, and vindicates the ordinary decision-maker.

Classic analysis of the subject and the development of personal probability; one of the greatest controversies in modern statistcal thought.

It is a commonplace that in making decisions agents often have to juggle competing values, and that no choice will maximise satisfaction of them all. However, the prevailing account of these cases assumes that there is always a single ranking of the agent's values, and therefore no unresolvable conflict between them. Isaac Levi denies this assumption, arguing that agents often must choose without having balanced their different values and that to be rational, an act does not have to be optimal, (...) only what Levi terms 'admissible'. This book explores the consequences of denying the assumption and develops a general approach to decision-making under unresolved conflict. Professor Levi discusses conflicts of value in several domains - those arising in moral dilemmas, the drawing of scientific inferences, decisions taken under uncertainty, and in social choice. In each of these he adapts his theoretical framework, showing how conflict may often be reduced though not always altogether eliminated. (shrink)

It is possible that the world contains infinitely many agents that have positive and negative levels of well-being. Theories have been developed to ethically rank such worlds based on the well-being levels of the agents in those worlds or other qualitative properties of the worlds in question, such as the distribution of agents across spacetime. In this thesis I argue that such ethical rankings ought to be consistent with the Pareto principle, which says that if two worlds contain the same (...) agents and some agents are better off in the first world than they are in the second and no agents are worse off than they are in the second, then the first world is better than the second. I show that if we accept four axioms – the Pareto principle, transitivity, an axiom stating that populations of worlds can be permuted, and the claim that if the ‘at least as good as’ relation holds between two worlds then it holds between qualitative duplicates of this world pair – then we must conclude that there is ubiquitous incomparability between infinite worlds. I show that this is true even if the populations of infinite worlds are disjoint or overlapping, and that we cannot use any qualitative properties of world pairs to rank these worlds. Finally, I argue that this incomparability result generates puzzles for both consequentialist and non-consequentialist theories of objective and subjective permissibility. (shrink)

Policy-makers must sometimes choose between an alternative which has somewhat lower expected value for each person, but which will substantially improve the outcomes of the worst off, or an alternative which has somewhat higher expected value for each person, but which will leave those who end up worst off substantially less well off. The popular ex ante Pareto principle requires the choice of the alternative with higher expected utility for each. We argue that ex ante Pareto ought to be rejected (...) because it conflicts with the requirement that, when possible, one ought to decide as one would with full information. We apply our argument in an analysis of US policy on screening for breast cancer. -/- . (shrink)