A popular way to relate probabilistic information to binary rational beliefs is the Lockean Thesis, which is usually formalized in terms of thresholds. This approach seems far from satisfactory: the value of the thresholds is not well-specified and the Lottery Paradox shows that the model violates the Conjunction Principle. We argue that the Lottery Paradox is a symptom of a more fundamental and general problem, shared by all threshold-models that attempt to put an exact border on something that is intrinsically (...) vague. We propose application of the language of relative analysis—a type of non-standard analysis—to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’ and satisfies a moderately weakened form of the Conjunction Principle. We also propose an adaptation of the model that is able to deal with beliefs that are less firm than ‘almost certainty’. The adapted version is also of interest for the epistemicist account of vagueness. (shrink)

The literature of economic theory, like that of philosophy, abounds in prefaces and prolegomena. Methodology and analysis of concepts take an important place in a science which has not found the sure path of development. But there is no sure path for methodology either. The selfconscious methodology of social science has been largely a borrowing from that of physical science, where procedures have developed to a stage of considerable maturity. But the analogy falls down where guidance is most needed, at (...) the points where social science is most likely to develop new concepts and new types of structure. Philosophers have not been lacking, indeed, to belittle the entire enterprise, and to deny the possibility of anything that could strictly be called social science. (shrink)

The aim of the paper is to draw a connection between a semantical theory of conditional statements and the theory of conditional probability. First, the probability calculus is interpreted as a semantics for truth functional logic. Absolute probabilities are treated as degrees of rational belief. Conditional probabilities are explicitly defined in terms of absolute probabilities in the familiar way. Second, the probability calculus is extended in order to provide an interpretation for counterfactual probabilities--conditional probabilities where the condition has zero probability. (...) Third, conditional propositions are introduced as propositions whose absolute probability is equal to the conditional probability of the consequent on the antecedent. An axiom system for this conditional connective is recovered from the probabilistic definition. Finally, the primary semantics for this axiom system, presented elsewhere, is related to the probabilistic interpretation. (shrink)

The rational price of the Pasadena and Altadena games, introduced by Nover and Hájek (2004 ), has been the subject of considerable discussion. Easwaran (2008 ) has suggested that weak expectations — the value to which the average payoffs converge in probability — can give the rational price of such games. We argue against the normative force of weak expectations in the standard framework. Furthermore, we propose to replace this framework by a bounded utility perspective: this shift renders the problem (...) more realistic and accounts for the role of weak expectations. In particular, we demonstrate that in a bounded utility framework, all agents, even if they have different value functions and disagree on the price of an individual game, will finally agree on the rational price of a repeated, averaged game. Thus, we explain the intuitive appeal of weak expectations, while avoiding both trivialization of the original paradox and the drawbacks of previous approaches. (shrink)

In this paper I add a new twist to Colyvan's version of the Petrograd paradox.

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)

In this paper we explore the connections between ethics and decision theory. In particular, we consider the question of whether decision theory carries with it a bias towards consequentialist ethical theories. We argue that there are plausible versions of the other ethical theories that can be accommodated by “standard” decision theory, but there are also variations of these ethical theories that are less easily accommodated. So while “standard” decision theory is not exclusively consequentialist, it is not necessarily ethically neutral. Moreover, (...) even if our decision-theoretic models get the right answers vis-`a-vis morally correct action, the question remains as to whether the motivation for the non-consequentialist theories and the psychological processes of the agents who subscribe to those ethical theories are lost or poorly represented in the resulting models. (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.

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)

The 'best-system' analysis of lawhood [Lewis 1994] faces the 'zero-fit problem': that many systems of laws say that the chance of history going actually as it goes--the degree to which the theory 'fits' the actual course of history--is zero. Neither an appeal to infinitesimal probabilities nor a patch using standard measure theory avoids the difficulty. But there is a way to avoid it: replace the notion of 'fit' with the notion of a world being typical with respect to a theory.

This paper revisits the Pasadena game (Nover and Háyek 2004), a St Petersburg-like game whose expectation is undefined. We discuss serveral respects in which the Pasadena game is even more troublesome for decision theory than the St Petersburg game. Colyvan (2006) argues that the decision problem of whether or not to play the Pasadena game is ‘ill-posed’. He goes on to advocate a ‘pluralism’ regarding decision rules, which embraces dominance reasoning as well as maximizing expected utility. We rebut Colyvan’s argument, (...) offering several considerations in favour of the Pasadena decision problem being well posed. To be sure, current decision theory, which is underpinned by various preference axioms, leaves indeterminate how one should value the Pasadena game. But we suggest that determinacy might be achieved by adding further preference axioms. We conclude by opening the door to a far greater plurality of decision rules. We suggest how the goal of unifying these rules might guide future research. (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)

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)

Kolmogorov''s axiomatization of probability includes the familiarratio formula for conditional probability: 0).$$ " align="middle" border="0">.

In our 2004, we introduced two games in the spirit of the St Petersburg game, the Pasadena and Altadena games. As these latter games lack an expectation, we argued that they pose a paradox for decision theory. Terrence Fine has shown that any finite valuations for the Pasadena, Altadena, and St Petersburg games are consistent with the standard decision-theoretic axioms. In particular, one can value the Pasadena game above the other two, a result that conflicts with both our intuitions and (...) dominance reasoning. We argue that this result, far from resolving the Pasadena paradox, should serve as a reductio of the standard theory, and we consequently make a plea for new axioms for a revised theory. We also discuss a proposal by Kenny Easwaran that a gamble should be valued according to its 'weak expectation', a generalization of the usual notion of expectation. (shrink)

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)

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)

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)

The Pasadena paradox presents a serious challenge for decision theory. The paradox arises from a game that has well-defined probabilities and utilities for each outcome, yet, apparently, does not have a well-defined expectation. In this paper, I argue that this paradox highlights a limitation of standard decision theory. This limitation can be (largely) overcome by embracing dominance reasoning and, in particular, by recognising that dominance reasoning can deliver the correct results in situations where standard decision theory fails. This, in turn, (...) pushes us towards pluralism about decision rules. (shrink)

We provide a solution to the well-known “Shooting-Room” paradox, developed by John Leslie in connection with his Doomsday Argument. In the “Shooting-Room” paradox, the death of an individual is contingent upon an event that has a 1/36 chance of occurring, yet the relative frequency of death in the relevant population is 0.9. There are two intuitively plausible arguments, one concluding that the appropriate subjective probability of death is 1/36, the other that this probability is 0.9. How are these two values (...) to be reconciled? We show that only the first argument is valid for a standard, countably additive probability distribution. However, both lines of reasoning are legitimate if probabilities are non-standard. The subjective probability of death rises from 1/36 to 0.9 by conditionalizing on an event that is not measurable, or whose probability is zero. Thus we can sometimes meaningfully ascribe conditional probabilities even when the event conditionalized upon is not of positive finite (or even infinitesimal) measure. (shrink)

Among recent objections to Pascal's Wager, two are especially compelling. The first is that decision theory, and specifically the requirement of maximizing expected utility, is incompatible with infinite utility values. The second is that even if infinite utility values are admitted, the argument of the Wager is invalid provided that we allow mixed strategies. Furthermore, Hájek has shown that reformulations of Pascal's Wager that address these criticisms inevitably lead to arguments that are philosophically unsatisfying and historically unfaithful. Both the objections (...) and Hájek's philosophical worries disappear, however, if we represent our preferences using relative utilities rather than a one-place utility function. Relative utilities provide a conservative way to make sense of infinite value that preserves the familiar equation of rationality with the maximization of expected utility. They also provide a means of investigating a broader class of problems related to the Wager. (shrink)

We examine a distinctive kind of problem for decision theory, involving what we call discontinuity at infinity. Roughly, it arises when an infinite sequence of choices, each apparently sanctioned by plausible principles, converges to a ‘limit choice’ whose utility is much lower than the limit approached by the utilities of the choices in the sequence. We give examples of this phenomenon, focusing on Arntzenius et al.’s Satan’s apple, and give a general characterization of it. In these examples, repeated dominance reasoning (...) (a paradigm of rationality) apparently gives rise to a situation closely analogous to having intransitive preferences (a paradigm of irrationality). Indeed, the agents in these examples are vulnerable to a money pump set-up despite having preferences that exhibit no obvious defect of rationality. We explore several putative solutions to such problems, particularly those that appeal to binding and to deliberative dynamics. We consider the prospects for these solutions, concluding that if they fail, the examples show that money pump arguments are invalid. (shrink)

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)

This paper is partly a tribute to Richard Jeffrey, partly a reflection on some of his writings, The Logic of Decision in particular. I begin with a brief biography and some fond reminiscences of Dick. I turn to some of the key tenets of his version of Bayesianism. All of these tenets are deployed in my discussion of his response to the St. Petersburg paradox, a notorious problem for decision theory that involves a game of infinite expectation. Prompted by that (...) paradox, I conclude with some suggestions of avenues for future research. (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)

This book develops in detail the simple idea that assertion is the expression of belief. In it the author puts forward a version of 'probabilistic semantics' which acknowledges that we are not perfectly rational, and which offers a significant advance in generality on theories of meaning couched in terms of truth conditions. It promises to challenge a number of entrenched and widespread views about the relations of language and mind. Part I presents a functionalist account of belief, worked through a (...) modified form of decision theory. In Part II the author generates a theory of meaning in terms of 'assertibility conditions', whereby to know the meaning of an assertion is to know the belief it expresses. (shrink)

As a survey of many technical results in probability theory and probability logic, this monograph by two widely respected scholars offers a valuable compendium of the principal aspects of the formal study of probability. Hugues Leblanc and Peter Roeper explore probability functions appropriate for propositional, quantificational, intuitionistic, and infinitary logic and investigate the connections among probability functions, semantics, and logical consequence. They offer a systematic justification of constraints for various types of probability functions, in particular, an exhaustive account of probability (...) functions adequate for first-order quantificational logic. The relationship between absolute and relative probability functions is fully explored and the book offers a complete account of the representation of relative functions by absolute ones. The volume is designed to review familiar results, to place these results within a broad context, and to extend the discussions in new and interesting ways. Authoritative, articulate, and accessible, it will interest mathematicians and philosophers at both professional and post-graduate levels. (shrink)

When real-valued utilities for outcomes are bounded, or when all variables are simple, it is consistent with expected utility to have preferences defined over probability distributions or lotteries. That is, under such circumstances two variables with a common probability distribution over outcomes – equivalent variables – occupy the same place in a preference ordering. However, if strict preference respects uniform, strict dominance in outcomes between variables, and if indifference between two variables entails indifference between their difference and the status quo, (...) then preferences over rich sets of unbounded variables, such as variables used in the St. Petersburg paradox, cannot preserve indifference between all pairs of equivalent variables. In such circumstances, preference is not a function only of probability and utility for outcomes. Then the preference ordering is not defined in terms of lotteries. (shrink)