Alan Hajek

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

Arguably, Hume's greatest single contribution to contemporary philosophy of science has been the problem of induction (1739). Before attempting its statement, we need to spend a few words identifying the subject matter of this corner of epistemology. At a first pass, induction concerns ampliative inferences drawn on the basis of evidence (presumably, evidence acquired more or less directly from experience)—that is, inferences whose conclusions are not (validly) entailed by the premises. Philosophers have historically drawn further distinctions, often appropriating the term “induction” to mark them; since we will not be concerned with the philosophical issues for which these distinctions are relevant, we will use the word “inductive” in a catch-all sense synonymous with “ampliative”. But we will follow the usual practice of choosing, as our paradigm example of inductive inferences, inferences about the future based on evidence drawn from the past and present. A further refinement is more important. Opinion typically comes in degrees, and this fact makes a great deal of difference to how we understand inductive inferences. For while it is often harmless to talk about the conclusions that can be rationally believed on the basis of some..

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.

David Lewis claims that a simple sort of anti-Humeanism-that the rational agent desires something to the extent he believes it to be good-can be given a decision-theoretic formulation, which Lewis calls 'Desire as Belief' (DAB). Given the (widely held) assumption that Jeffrey conditionalising is a rationally permissible way to change one's mind in the face of new evidence, Lewis proves that DAB leads to absurdity. Thus, according to Lewis, the simple form of anti-Humeanism stands refuted. In this paper we investigate whether Lewis's case against DAB can be strengthened by examining how it fares under rival versions of decision theory, including other conceptions of rational ways to change one's mind. We argue that the anti-Humean may escape Lewis's argument either by adopting a version of causal decision theory, or by claiming that the refutation only applies to hyper-idealised rational agents, or by denying that the decision-theoretic framework has the expressive capacity to formulate anti-Humeanism.

This is the sequel to my “Fifteen Arguments Against Finite Frequentism” ( Erkenntnis 1997), the second half of a long paper that attacks the two main forms of frequentism about probability. Hypothetical frequentism asserts: The probability of an attribute A in a reference class B is p iff the limit of the relative frequency of A ’s among the B ’s would be p if there were an infinite sequence of B ’s. I offer fifteen arguments against this analysis. I consider various frequentist responses, which I argue ultimately fail. I end with a positive proposal of my own, ‘hyper-hypothetical frequentism’, which I argue avoids several of the problems with hypothetical frequentism. It identifies probability with relative frequency in a hyperfinite sequence of trials. However, I argue that this account also fails, and that the prospects for frequentism are dim.

Bayesianism is our leading theory of uncertainty. Epistemology is defined as the theory of knowledge. So “Bayesian Epistemology” may sound like an oxymoron. Bayesianism, after all, studies the properties and dynamics of degrees of belief, understood to be probabilities. Traditional epistemology, on the other hand, places the singularly non-probabilistic notion of knowledge at centre stage, and to the extent that it traffics in belief, that notion does not come in degrees. So how can there be a Bayesian epistemology?