This is the most comprehensive book ever published on philosophical methodology. A team of thirty-eight of the world's leading philosophers present original essays on various aspects of how philosophy should be and is done. The first part is devoted to broad traditions and approaches to philosophical methodology. The entries in the second part address topics in philosophical methodology, such as intuitions, conceptual analysis, and transcendental arguments. The third part of the book is devoted to essays about the interconnections between philosophy (...) and neighbouring fields, including those of mathematics, psychology, literature and film, and neuroscience. (shrink)

How all philosophical explanations of human consciousness and the fundamental structure of the cosmos are bizarre—and why that’s a good thing Do we live inside a simulated reality or a pocket universe embedded in a larger structure about which we know virtually nothing? Is consciousness a purely physical matter, or might it require something extra, something nonphysical? According to the philosopher Eric Schwitzgebel, it’s hard to say. In The Weirdness of the World, Schwitzgebel argues that the answers to these fundamental (...) questions lie beyond our powers of comprehension. We can be certain only that the truth—whatever it is—is weird. Philosophy, he proposes, can aim to open—to reveal possibilities we had not previously appreciated—or to close, to narrow down to the one correct theory of the phenomenon in question. Schwitzgebel argues for a philosophy that opens. According to Schwitzgebel’s “Universal Bizarreness” thesis, every possible theory of the relation of mind and cosmos defies common sense. According to his complementary “Universal Dubiety” thesis, no general theory of the relationship between mind and cosmos compels rational belief. Might the United States be a conscious organism—a conscious group mind with approximately the intelligence of a rabbit? Might virtually every action we perform cause virtually every possible type of future event, echoing down through the infinite future of an infinite universe? What, if anything, is it like to be a garden snail? Schwitzgebel makes a persuasive case for the thrill of considering the most bizarre philosophical possibilities. (shrink)

In response to cases that involve tiny probabilities of huge payoffs, some argue that we ought to discount small probabilities down to zero. However, this paper shows that doing so violates Independence and Continuity, and as a result of these violations, those who discount small probabilities can be exploited by money pumps. Various possible ways of avoiding exploitation will be discussed. This paper concludes that the money pump for Independence undermines the plausibility of discounting small probabilities. Much of the discussion (...) on Independence generalizes to other views that also violate Independence. (shrink)

Countenancing unbounded utility in ethics gives rise to deep puzzles in formal decision theory. Here, these puzzles are taken as an invitation to assess the most fundamental principles relating probability and value, with the aim of demonstrating that unbounded utility in ethics is compatible with a desirable decision theory. The resulting theory frames further discussion of Expected Utility Theory and of principles concerning symmetries of utility.

In slogan form, the thesis of this dissertation is that shaping the far future is overwhelmingly important. More precisely, I argue that: Main Thesis: From a global perspective, what matters most is that we do what is best for the general trajectory along which our descendants develop over the coming millions, billions, and trillions of years. The first chapter introduces some key concepts, clarifies the main thesis, and outlines what follows in later chapters. Some of the key concepts include: existential (...) risk, the world's development trajectory, proximate benefits and ripple effects, speeding up development, trajectory changes, and the distinction between broad and targeted attempts to shape the far future. The second chapter is a defense of some methodological assumptions for developing normative theories which makes my thesis more plausible. In the third chapter, I introduce and begin to defend some key empirical and normative assumptions which, if true, strongly support my main thesis. In the fourth and fifth chapters, I argue against two of the strongest objections to my arguments. These objections come from population ethics, and are based on Person-Affecting Views and views according to which additional lives have diminishing marginal value. I argue that these views face extreme difficulties and cannot plausibly be used to rebut my arguments. In the sixth and seventh chapters, I discuss a decision-theoretic paradox which is relevant to my arguments. The simplest plausible theoretical assumptions which support my main thesis imply a view I call fanaticism, according to which any non-zero probability of an infinitely good outcome, no matter how small, is better than any probability of a finitely good outcome. I argue that denying fanaticism is inconsistent with other normative principles that seem very obvious, so that we are faced with a paradox. I have no solution to the paradox; I instead argue that we should continue to use our inconsistent principles, but we should use them tastefully. We should do this because, currently, we know of no consistent set of principles which does better. (shrink)

How ought you to evaluate your options if you're uncertain about what's fundamentally valuable? A prominent response is Expected Value Maximisation (EVM)—the view that under axiological uncertainty, an option is better than another if and only if it has the greater expected value across axiologies. But the expected value of an option depends on quantitative probability and value facts, and in particular on value comparisons across axiologies. We need to explain what it is for such facts to hold. Also, EVM (...) is by no means self-evident. We need an argument to defend that it’s true. This book introduces an axiomatic approach to answer these worries. It provides an explication of what EVM means by use of representation theorems: intertheoretic comparisons can be understood in terms of facts about which options are better than which, and mutatis mutandis for intratheoretic comparisons and axiological probabilities. And it provides a systematic argument to the effect that EVM is true: the theory can be vindicated through simple axioms. The result is a formally cogent and philosophically compelling extension of standard decision theory, and original take on the problem of axiological or normative uncertainty. (shrink)

This paper presents a new solution to the well-known exchange paradox, or what is sometimes referred to as the two-envelope paradox. Many recent commentators have analyzed the paradox in terms of the agent’s biased concern for the contents of his own arbitrarily chosen envelope, claiming that such bias violates the manifest symmetry of the situation. Such analyses, however, fail to make clear exactly how the symmetry of the situation is violated by the agent’s hypothetical conclusion that he ought to switch (...) envelopes on the assumption that his own envelope contains some specific amount of money. In this paper, I offer an explanation of this fact based on the idea that the agent’s deliberations are not only constrained by the epistemic symmetry reflected in the agent’s uniform ignorance as to the contents of the two envelopes, but also by a deeper methodological symmetry that manifests itself in the intuition that the two envelopes constitute equally legitimate sources of potential information. I interpret this intuition as implying that the agent’s final decision should reflect a point of rational equilibrium arrived at through an iterated process of counterfactual self-reflection, whereby the agent takes account of what he would have thought had he instead chosen the other envelope. I provide a formal model of this method of counterfactual self-reflection and show, in particular, that it cannot issue in the paradoxical conclusion that the agent ought to switch regardless of how much money is in his envelope. In this way, by correcting for the bias in the agent’s reasoning, the paradox is resolved. (shrink)

In the growing literature on decision-making under moral uncertainty, a number of skeptics have argued that there is an insuperable barrier to rational "hedging" for the risk of moral error, namely the apparent incomparability of moral reasons given by rival theories like Kantianism and utilitarianism. Various general theories of intertheoretic value comparison have been proposed to meet this objection, but each suffers from apparently fatal flaws. In this paper, I propose a more modest approach that aims to identify classes of (...) moral theories that share common principles strong enough to establish bases for intertheoretic comparison. I show that, contra the claims of skeptics, there are often rationally perspicuous grounds for precise, quantitative value comparisons within such classes. In light of this fact, I argue, the existence of some apparent incomparabilities between widely divergent moral theories cannot serve as a general argument against hedging for one's moral uncertainties. (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)

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)

I have argued for a picture of decision theory centred on the principle of Rationally Negligible Probabilities. Isaacs argues against this picture on the grounds that it has an untenable implication. I first examine whether my view really has this implication; this involves a discussion of the legitimacy or otherwise of infinite decisions. I then examine whether the implication is really undesirable and conclude that it is not.

People with the kind of preferences that give rise to the St. Petersburg paradox are problematic---but not because there is anything wrong with infinite utilities. Rather, such people cannot assign the St. Petersburg gamble any value that any kind of outcome could possibly have. Their preferences also violate an infinitary generalization of Savage's Sure Thing Principle, which we call the *Countable Sure Thing Principle*, as well as an infinitary generalization of von Neumann and Morgenstern's Independence axiom, which we call *Countable (...) Independence*. In violating these principles, they display foibles like those of people who deviate from standard expected utility theory in more mundane cases: they choose dominated strategies, pay to avoid information, and reject expert advice. We precisely characterize the preference relations that satisfy Countable Independence in several equivalent ways: a structural constraint on preferences, a representation theorem, and the principle we began with, that every prospect has a value that some outcome could have. (shrink)

Decision theorists widely accept a stochastic dominance principle: roughly, if a risky prospect A is at least as probable as another prospect B to result in something at least as good, then A is at least as good as B. Recently, philosophers have applied this principle even in contexts where the values of possible outcomes do not have the structure of the real numbers: this includes cases of incommensurable values and cases of infinite values. But in these contexts the usual (...) formulation of stochastic dominance is wrong. We show this with several counterexamples. Still, the motivating idea behind stochastic dominance is a good one: it is supposed to provide a way of applying dominance reasoning in the stochastic context of probability distributions. We give two new formulations of stochastic dominance that are more faithful to this guiding idea, and prove that they are equivalent. (shrink)

New arguments against Bayesian regularity and an otherwise plausible domination principle are offered on the basis of rotational symmetry. The arguments against Bayesian regularity work in very general settings.

By generalizing the Pasadena puzzle introduced by Nover and Hájek (2004) we show that the sum total of value produced by an act can be made to converge to any real number by applying the Riemann rearrangement theorem, even if the scenario faced by the decision maker is non-probabilistic and fully predictable. A wide range of solutions put forward in the literature for solving the original puzzle cannot solve this generalized version of the Pasadena puzzle.

Supererogatory acts are those that lie “beyond the call of duty.” There are two standard ways to define this idea more precisely. Although the definitions are often seen as equivalent, I argue that they can diverge when options are infinite, or when there are cycles of better options; moreover, each definition is acceptable in only one case. I consider two ways out of this dilemma.

Vann McGee has argued that, given certain background assumptions and an ought-implies-can thesis about norms of rationality, Bayesianism conflicts globally with computationalism due to the fact that Robinson arithmetic is essentially undecidable. I show how to sharpen McGee's result using an additional fact from recursion theory—the existence of a computable sequence of computable reals with an uncomputable limit. In conjunction with the countable additivity requirement on probabilities, such a sequence can be used to construct a specific proposition to which Bayesianism (...) requires an agent to assign uncomputable credence—yielding a local conflict with computationalism. (shrink)

Theories that use expected utility maximization to evaluate acts have difficulty handling cases with infinitely many utility contributions. In this paper I present and motivate a way of modifying such theories to deal with these cases, employing what I call “Direct Difference Taking”. This proposal has a number of desirable features: it’s natural and well-motivated, it satisfies natural dominance intuitions, and it yields plausible prescriptions in a wide range of cases. I then compare my account to the most plausible alternative, (...) a proposal offered by Arntzenius :31–58, 2014). I argue that while Arntzenius’s proposal has many attractive features, it runs into a number of problems which Direct Difference Taking avoids. (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)

Dozens of articles have addressed the challenge that gambles having undefined expectation pose for decision theory. This paper makes two contributions. The first is incremental: we evolve Colyvan's ``Relative Expected Utility Theory'' into a more viable ``conservative extension of expected utility theory" by formulating and defending emendations to a version of this theory proposed by Colyvan and H\'ajek. The second is comparatively more surprising. We show that, so long as one assigns positive probability to the theory that there is a (...) uniform bound on the utility of possible gambles (and assuming a uniform bound on the amount of utility that can accrue in a fixed amount of time), standard principles of anthropic reasoning (as formulated by Brandon Carter) place lower and upper bounds on the expected values of gambles advertised as having no expectation--even assuming that with positive probability, all gambles advertised as having infinite expected utility are administered faithfully. Should one accept the uniform bound premises, this reasoning thus dissolves (or nearly dissolves, in some cases) several puzzles in infinite decision theory. (shrink)

How should deontologists concerned with the ethics of killing apply their moral theory when we don’t know all the facts relevant to the permissibility of our action? Though the stakes couldn’t be higher, and uncertainty is endemic where killing is concerned, few deontologists have an answer to this question. In this paper I canvass two possibilities: that we should apply a threshold standard, equivalent to the ‘beyond a reasonable doubt’ standard applied for criminal punishment; and that we should fit our (...) deontological ethical theory into the apparatus of decision theory. I show that the first approach faces insurmountable obstacles, while the second holds much more promise for deontologists than they might first have assumed. (shrink)

Consider the following claim: given the choice between saving a life and preventing any number of people from temporarily experiencing a mild headache, you should always save the life. Many moral theorists accept this claim. In doing so, they commit themselves to some form of ‘moral absolutism’: the view that there are some moral considerations that cannot be outweighed by any number of lesser moral considerations. In contexts of certainty, it is clear what moral absolutism requires of you. However, what (...) does it require of you when deciding under risk? What ought you to do when there is a chance that, say, you will not succeed in saving the life? In recent years, various critics have argued that moral absolutism cannot satisfactorily deal with risk and should, therefore, be abandoned. In this paper, we show that moral absolutism can answer its critics by drawing on—of all things—orthodox expected utility theory. (shrink)

How should deontologists approach decision-making under uncertainty, for an iterated decision problem? In this paper I explore the shortcomings of a simple expected value approach, using a novel example to raise questions about attitudes to risk, the moral significance of tiny probabilities, the independent moral reasons against imposing risks, the morality of sunk costs, and the role of agent-relativity in iterated decision problems.

: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.

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)

Bayesians have a seemingly attractive account of rational credal states in terms of coherence. An agent's set of credences are synchronically coherent just in case they conform to the probability calculus. Some Bayesians impose a further putative coherence constraint called regularity: roughly, if X is possible, then it is assigned positive probability. I look at two versions of regularity – logical and metaphysical – and I canvass various defences of it as a rationality norm. Combining regularity with synchronic coherence, we (...) have a set of constraints known as strict coherence.I argue that strict coherence is untenable. In particular, I attack regularity as a rationality norm. First, I rebut each of the various defences of regularity. Then I argue directly against regularity: it conflicts with the Bayesian decision‐theoretic treatment of rational action. Thus, seemingly plausible theoretical and pragmatic norms turn out to be inconsistent.[Barbossa is about to kill Will,but Jack Sparrow shows up:]Barbossa: It's not possible!Jack: Not probable.To be uncertain is to be uncomfortable,but to be certain is to be ridiculous. (shrink)

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)

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)

Many philosophers think that punishment is intentionally harmful and that this makes it especially hard to morally justify. Explanations for the latter intuition often say questionable things about the moral significance of the intent to harm. I argue that there’s a better way to explain this intuition.

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)

The principle of Anteriority says that prospects that are identical from the perspective of every possible person’s welfare are equally good overall. The principle enjoys prima facie plausibility, and has been employed for various theoretical purposes. Here it is shown using an analogue of the St Petersburg Paradox that Anteriority is inconsistent with central principles of axiology.

I trace the relationship between the view that knowledge is stakes sensitive and Laurie Paul’s account of the epistemology of transformative experience. The view that knowledge is stakes sensitive comes in different flavours: one can go for subjective or objective conceptions of stakes, where subjective views of stakes take stakes to be a function of an agent’s non-factive mental states, and objective views of stakes do not. I argue that there is a tension between subjective accounts of stakes sensitivity and (...) Paul’s epistemology of transformative experience. (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)

J.L. Schellenberg argues that divine hiddenness provides an argument for the conclusion that God does not exist, for if God existed he would not allow non-resistant non-belief to occur, but non-resistant non-belief does occur, so God does not exist. In this paper, I argue that the stakes involved in theistic considerations put pressure on Schellenberg’s premise that non-resistant non-belief occurs. First, I specify conditions for someone’s being a resistant non-believer. Then, I argue that many people fulfill these conditions because, given (...) some plausible assumptions, there is a very good pragmatic reason to be a theist rather than an atheist. I assume it is more likely that theists go to heaven than atheists, and I argue there is a non-zero probability that one can receive infinite utility and a method of comparing outcomes with infinite utilities in which the probability of each outcome affects the final expected values. Then, I show how this argument entails there is no good reason to think that there are very many non-resistant non-believers. (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)

Many philosophers have become worried about the use of standard real numbers for the probability function that represents an agent's credences. They point out that real numbers can't capture the distinction between certain extremely unlikely events and genuinely impossible ones—they are both represented by credence 0, which violates a principle known as “regularity.” Following Skyrms 1980 and Lewis 1980, they recommend that we should instead use a much richer set of numbers, called the “hyperreals.” This essay argues that this popular (...) view is the result of two mistakes. The first mistake, which this essay calls the “numerical fallacy,” is to assume that a distinction that isn't represented by different numbers isn't represented at all in a mathematical representation. In this case, the essay claims that although the real numbers do not make all relevant distinctions, the full mathematical structure of a probability function does. The second mistake is that the hyperreals make too many distinctions. They have a much more complex structure than credences in ordinary propositions can have, so they make distinctions that don't exist among credences. While they might be useful for generating certain mathematical models, they will not appear in a faithful mathematical representation of credences of ordinary propositions. (shrink)

Bayesianism is a collection of positions in several related fields, centered on the interpretation of probability as something like degree of belief, as contrasted with relative frequency, or objective chance. However, Bayesianism is far from a unified movement. Bayesians are divided about the nature of the probability functions they discuss; about the normative force of this probability function for ordinary and scientific reasoning and decision making; and about what relation (if any) holds between Bayesian and non-Bayesian concepts.

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)

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)

In this paper I discuss the kinds of idealisations invoked in normative theories—logic, epistemology, and decision theory. I argue that very often the so-called norms of rationality are in fact mere idealisations invoked to make life easier. As such, these idealisations are not too different from various idealisations employed in scientific modelling. Examples of the latter include: fluids are incompressible (in fluid mechanics), growth rates are constant (in population ecology), and the gravitational influence of distant bodies can be ignored (in (...) celestial mechanics). Thinking of logic, epistemology, and decision theory as normative models employing various idealisations of these kinds, changes the way we approach the justification of the models in question. (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)

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)

ABSTRACT While pragmatic arguments are traditionally seen as supporting decision theory, recent discussions suggest the possibility of pragmatic arguments against this theory. I respond to two such arguments, and clarify what it would take for arguments of this sort to succeed.

In their paper, “Vexing Expectations,” Nover and Hájek (2004) present an allegedly paradoxical betting scenario which they call the Pasadena Game (PG). They argue that the silence of standard decision theory concerning the value of playing PG poses a serious problem. This paper provides a threefold response. First, I argue that the real problem is not that decision theory is “silent” concerning PG, but that it delivers multiple conflicting verdicts. Second, I offer a diagnosis of the problem based on the (...) insight that standard decision theory is, rightly, sensitive to order. Third, I describe a new betting scenario—the Alternating St. Petersburg Game—which is genuinely paradoxical. Standard decision theory is silent on the value of playing this game even if restrictions are placed on the order in which the various alternative payoffs are summed. †To contact the author, please write to: Department of Philosophy, Swarthmore College, Swarthmore, PA 19081; e-mail: [email protected]. (shrink)

Decision theory faces a number of problematic gambles which challenge it to say what value an ideal rational agent should assign to the gamble, and why. Yet little attention has been devoted to the question of what an ideal rational agent is, and in what sense decision theory may be said to apply to one. I show that, given one arguably natural set of constraints on the preferences of an idealized rational agent, such an agent is forced to be indifferent (...) among entire families of goods, and hence cannot choose among them. This result illustrates the dangers of speaking of the choices of an ?ideal rational agent? when one does not make precise the exact nature of the idealizing assumptions. The result may also be viewed as providing an upper bound on the kinds of idealizing assumptions which can be made for rational agents, beyond which the very concept of choice becomes attenuated. (shrink)

The Pasadena game is an example of a decision problem which lacks an expected value, as traditionally conceived. Easwaran (2008) has shown that, if we distinguish between two different kinds of expectations, which he calls ‘strong’ and ‘weak’, the Pasadena game lacks a strong expectation but has a weak expectation. Furthermore, he argues that we should use the weak expectation as providing a measure of the value of an individual play of the Pasadena game. By considering a modified version of (...) the Pasadena game, I argue that weak expectations may provide a very poor measure of the value of an individual play of the game, and hence should not be used to value individual plays unless further information is taken into consideration. (shrink)