Coherent Choice Functions under Uncertainty

We discuss several features of coherent choice functions – where the admissible options in a decision problem are exactly those which maximize expected utility for some probability/utility pair in fixed set S of probability/utility pairs. In this paper we consider, primarily, normal form decision problems under uncertainty – where only the probability component of S is indeterminate. Coherent choice distinguishes between each pair of sets of probabilities. We axiomatize the theory of choice functions and show these axioms are necessary for coherence. The axioms are sufficient for coherence using a set of probability/almost-state-independent utility pairs. We give sufficient conditions when a choice function satisfying our axioms is represented by a set of probability/state-independent utility pairs with a common utility. Keywords. Choice functions, coherence, Γ-Maximin, Maximality, uncertainty, state-independent utility.

Shared Preferences and State-Dependent Utilities

This investigation combines two questions for expected utility theory: 1. When do the shared preferences among expected utility maximizers conform to the dictates of expected utility? 2. What is the impact on expected utility theory of allowing preferences for prizes to be state-dependent? Our principal conclusion (Theorem 4) establishes very restrictive necessary and sufficient conditions for the existence of a Pareto, Bayesian compromise of preferences between two Bayesian agents, even when utilities are permitted to be state-dependent and identifiable. This finding extends our earlier result (Theorem 2, 1989a) which applies provided that all utilities are state-independent. A subsidiary theme is a decision theoretic analysis of common rules for "pooling" expert probabilities. Against the backdrop of "horse lottery" theory (Anscombe and Aumann 1963) and subject to a weak Pareto rule, we show, generally, that there is no Bayesian compromise between two Bayesian agents even when state-dependent utilities are entertained in an identifiable way. The word "identifiable" is important because, if state-dependence is permitted merely by dropping the Anscombe-Aumann axiom (Axiom 4 here) for "state-independence," though a continuum of possible Bayesian compromises emerges, also it leads to an extreme underdetermination of an agent's personal probability and utility given the agent's preferences. Instead, when state-dependence is monitored through (our version of) the approach of Karni, Schmeidler, and Vind (1983), the general impossibility of a Bayesian, Pareto compromise in preferences reappears.consensus, horse lotteries, prize-state lotteries, subjective expected utility

The Fundamental Theorems of Prevision and Asset Pricing

We explore the connections between the concepts of coherence, as defined by deFinetti, and arbitrage in financial markets. 1. Introduction. Let Ω be a set of states with a σ-field of subsets A. Let X stand for a set of measurable real-valued functions defined on Ω. Whether X contains unbounded functions will be made clear in each context. The elements of X will be called gambles, risky assets, or random variables. Functions of elements of X will also be called by those same names