On Decision Theory Under Competition

Decision theory treats models in which a decision maker must choose an information function and a decision function, where the former transforms states of nature into messages, the latter transforms messages into actions, and the ultimate payoff depends upon the state of nature, the message, and the action. This paper treats similar models in which nature's role is played by a selfseeking rational opponent. These models fall into the purview of the theory of two-person games, and the main results concern the maximin and minimax theories of these games. The various games treated differ in the assumptions made about the decision maker's ability to probability mix his choices and about his opponent's information concerning these choices. Twenty different sets of assumptions are considered, but it is shown that the maximin and minimax theories of the resulting games are all reducible in substance to those of just three games: (i) in which the decision maker cannot probability mix his choices at all, and his opponent observes both choices; (ii) in which the decision maker can mix arbitrarily, and his opponent observes neither of his choices; or (iii) in which the decision maker can mix decision functions only, and his opponent observes only the information function. These games are represented in terms of matrix games, and the matrices are compressed by the deletion of dominated strategies for the decision maker. Further compressions are obtained for the case in which the cost of information is separable from the payoff resulting from the actions of the decision maker and his opponent. The game matrices are finally reduced to manageable proportions for the further specialized cases in which either the capacity of the decision maker's information channel is significantly limited and information cost is constant, or the capacity of the information channel is large and there is one low--cost "non-message" available. A numerical example is given, and the concluding comments point to directions for further research.