The theory of games against nature relies on complete preferences among all conceivable acts, i.e. among all potential assignments of consequeces to states of nature (case 1). Yet most decision problems call for choosing an element from a limited set of acts. And in games of strategy, the set of strategies available to a player is givent and not amenable to artificial extensions. In ÒAssessing Strategic RiskÓ,(ECON DP 2005-20) R.J. Aumann and J.H. Drze extend the basic result of decision theory (maximisation of subjectvely expected utility) to situations where preferences are defined only for a given set of acts, and for lotteries among these and sure consequences (case 2). In this paper, we provide a similar extension for two other situations : those where only the set of optimal elements from a given set of acts is known (case 3); and those where only a single optimal act is known (case 4). To these four cases correspond four nested sets of admissible subjective probabilities over the states or the opponentÕs strategies, namely a singleton in case 1 and increasing sets in cases 2-4. The results for case 3 and 4 also define the extent to which subjective probabilities must be specified in order to solve a given decision problem or play a given name.
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