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)

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: abaker1@swarthmore.edu. (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.

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)

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)

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)

Common probabilistic fallacies and putative paradoxes are surveyed, including those arising from distribution repartitioning, from the reordering of expectation series, and from misconceptions regarding expected and almost certain gains in games of chance. Conditions are given for such games to be well-posed. By way of example, Bernoulli's "Petersburg Paradox" and Hacking's "Strange Expectations" are discussed and the latter are resolved. Feller's generalized "fair price, in the classical sense" is critically reviewed.

Cumulative Prospect Theory (CPT) does not explain the St. Petersburg Paradox. We show that the solutions related to probability weighting proposed to solve this paradox, (Blavatskyy, Management Science 51:677–678, 2005; Rieger and Wang, Economic Theory 28:665–679, 2006) have to cope with limitations. In that framework, CPT fails to accommodate both gambling and insurance behavior. We suggest replacing the weighting functions generally proposed in the literature by another specification which respects the following properties: (1) to solve the paradox, the slope at (...) zero has to be finite. (2) to account for the fourfold pattern of risk attitudes, the probability weighting has to be strong enough. (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)

In this paper I offer an account of the normative dimension implicit in D. Bernoulli’s expected utility functions by means of an analysis of the juridical metaphors upon which the concept of mathematical expectation was moulded. Following a suggestion by the late E. Coumet, I show how this concept incorporated a certain standard of justice which was put in question by the St. Petersburg paradox. I contend that Bernoulli would have solved it by introducing an alternative normative criterion rather than (...) a positive model of decision-making processes. (shrink)