Paradox-Proof Utility Functions for Heavy-Tailed Payoffs: Two Instructive Two-Envelope Problems (original) (raw)

Exceeding Expectations: Stochastic Dominance as a General Decision Theory

The principle that rational agents should maximize expected utility or choiceworthiness is intuitively plausible in many ordinary cases of decision-making under uncertainty. But it is less plausible in cases of extreme, low-probability risk (like Pascal's Mugging), and intolerably paradoxical in cases like the St. Petersburg and Pasadena games. In this paper I show that, under certain conditions, stochastic dominance reasoning can capture most of the plausible implications of expectational reasoning while avoiding most of its pitfalls. Specifically, given sufficient background uncertainty about the choiceworthiness of one's options, many expectation-maximizing gambles that do not stochastically dominate their alternatives "in a vacuum" become stochastically dominant in virtue of that background uncertainty. But, even under these conditions, stochastic dominance will generally not require agents to accept extreme gambles like Pascal's Mugging or the St. Petersburg game. The sort of background uncertainty on which these results depend looks unavoidable for any agent who measures the choiceworthiness of her options in part by the total amount of value in the resulting world. At least for such agents, then, stochastic dominance offers a plausible general principle of choice under uncertainty that can explain more of the apparent rational constraints on such choices than has previously been recognized.

An allais paradox for generalized expected utility theories

Economics Bulletin, 2008

This article reports the results of an experiment which aims at providing a test of ordinal independence, a necessary property of Generalized Expected Utility theories such as Rank-Dependent Expected Utility theory (RDEU). Our experiment is based on a modified version of the Allais paradox proposed by Machina, which allows testing ordinal independence restricted to simple lotteries, i.e. the tail-separability property. The results tend to support RDEU models since tail-separability is not violated by 71% of subjects while 73% violate the independence condition of classic Allais paradox. This confirms the relative theoritical soundness of RDEU models over Expected Utility model for the particular context of risk.

The Two-Envelope Paradox: An Axiomatic Approach

Mind, 2005

There has been much recent 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 expectationmaximizing 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.

Game Theory without Decision-Theoretic Paradoxes

Lecture Notes in Computer Science, 2009

Most work in game theory is conducted under the assumption that the players are expected utility maximizers. Expected utility is a very tractable decision model, but is prone to wellknown paradoxes and empirical violations , which may induce biases in game-theoretic predictions. La Mura (2009) introduced a projective generalization of expected utility (PEU) which avoids the dominant paradoxes, while remaining quite tractable. We show that every finite game with PEU players has an equilibrium, and discuss several examples of PEU games.

Unbounded Utility and Axiomatic Foundations

2021

In utility theory, certain infinite gambles pose problems, of which the St. Petersberg paradox is the most well-known. Various methods have been used to eliminate the paradoxes, including the supposition that utility is bounded, or that probabilities must decline faster than utility so as to always result in finite expectation values, or simply excluding the divergent gambles by definition. We present a unified resolution of the matter that allows for unbounded utility and accommodates convergent infinite gambles. We also argue that merely excluding the divergent gambles, by whatever method, does not entirely avoid the associated paradoxes, which in practical terms already show up for finite approximations to them.

The two envelope paradox and infinite expectations

Analysis 57(1) (1997): 42–50, 1997

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.