The expected utility of playing a game
Alvin E. Roth · Cambridge University Press eBooks · 1988
Introduction This chapter is concerned with how the Shapley value can be interpreted as an expected utility function, the consequences of interpreting it in this way, and with what other value functions arise as utility functions representing different preferences. These questions brought themselves rather forcefully to my attention when I first taught a graduate course in game theory. After introducing utility theory as a way of numerically representing sufficiently regular individual preferences, and explaining which comparisons involving utility functions are meaningful and which are not, I found myself at a loss to explain precisely what comparisons could meaningfully be made using the Shapley value, if it was to be interpreted as a utility as suggested in the first paragraph of Shapley's 1953 paper. In order to state the problem clearly, it will be useful to remark briefly on some of the familiar properties of utility functions. First, utility functions represent preferences, so individuals with different preferences will have different utility functions. When preferences are measured over risky as well as riskless prospects, individuals who have the same preferences over riskless prospects may nevertheless have different preferences over lotteries, and so may have different expected utility functions. Second, there are some arbitrary choices involved in specifying a utility function, so the information contained in an individual's utility function is really represented by an equivalence class of functions.