Part IV - Uncertainty and mixed strategies

Summary

INTRODUCTION

In Chapter 10 we start addressing the issue of uncertainty. When a player is unsure what strategies his rivals will choose, we will assume that the player attaches a probability to each of the possible choice combinations. Each choice of a strategy of his own then defines the probability with which each profile of all the players’ choices would be realized. Overall, each of the player’s strategies defines a lottery over the strategy profiles in the game.

In order to decide which strategy to choose, the player then has to figure which of the lotteries induced by his strategies he prefers. We will assume that the player’s preference over these lotteries is expressed by the expected utility accrued by this lottery – the weighted average of his utilities from his choice and the others’ choices, weighted by the probabilities that he ascribes to the other players’ choices. This assumption means that the utility levels now have a cardinal (rather than just ordinal) interpretation. Preferences over lotteries which can be represented by an expected utility over outcomes are named after von Neumann and Morgenstern, who isolated four axioms on the preference relation which obtain if and only if an expected-utility representation of the preferences is feasible.

These axioms do not always obtain. We bring in the example of the Allais Paradox for preferences that seem “reasonable” but which nevertheless cannot be represented by an expected utility.