Subjective Expected Utility with Nonadditive Probabilities

In this chapter we shall adopt the terminology of decision making under uncertainty. We shall characterize, in Theorem VI.5.1, subjective expected utility maximization with continuous utility for the case where the probability measure no longer has to be additive. The main characterizing condition will be the ‘nonrevelation of comonotoniccontradictory tradeoffs’. The ‘nonadditive probability measures’ will be called capacities. Choquet(1953–54, 48.1) has indicated, for a special class of capacities, a way to extend the Lebesgue integral to this class of capacities. We shall adopt Choquet’s way of integrationrf2. Capacities play a role in cooperative game theory with side payments, where I is a set of ‘players’, subsets S of I are ‘coalitions’, and the capacity is a ‘characteristic function’, or ‘game’, indicating productivity, power etc.rf3 Capacities also play a role in the study of robustness in statisticsrf4.