Expected Utility in Two-Person Games
Published Online:1 May 1979https://doi.org/10.1287/moor.4.2.186
Abstract
Let P and Q be nonempty families of probability measures on the sets X and Y respectively of pure strategies which may be defined on different σ-algebras of subsets of X and Y respectively. For each q ∈ Q let (⊲, q) be a preference relation on P and for each p ∈ P let (p, ⊲) be a preference relation on Q. Necessary and sufficient conditions are given for the existence of a two-person zero-sum game (X, Y; a) such that (P, Q; A). A(p, q) = ∬a(x, y)dp(x)dq(y), is a randomization of (X, Y; a) which represents the preferences in the following sense:
$$p_{1}(\vartriangleleft, q)p_{2}\quad \hbox{iff}\ A(p_{1}, q) \lt A(p_{2}, q), \quad \hbox{for all}\ p_{1}, p_{2}\in P,\enspace q \in Q$$
$$q_{1}(p, \vartriangleleft)q_{2}\quad \hbox{iff}\ A(p, q_{2}) \lt A(p, q_{1}),\quad \hbox{for all}\ p \in P, q_{1},\enspace q_{2} \in Q.$$
This result is applied to obtain an axiomatization of expected utility for general two-person games in normal form.
