Decision Theory

Every day, a multitude of decisions are made that affect not only a small number of individuals, but also potentially millions of people. These decisions, which take place in hospitals, pharmaceutical companies, government offices, investing companies, and so on, [Page 237]are made based on incomplete information and under various conditions of uncertainty as to the ability of the decision maker(s) to follow through with the commitments made. Thus, nations declare wars with incomplete information as to the capabilities of the others involved in the war, and with uncertainty as to what the impact of these actions might be on their own economies and citizens. A pharmaceutical company must decide whether to market a new drug based on limited information resulting from the clinical trials and economic uncertainty as to whether the drug can compete in the market with other drugs.

Decision theory consists of techniques, ideas, and methodologies that are appropriate for helping the decision maker to reach a decision in an optimal fashion in the face of uncertainty. Given the universality of decision theory in corporate life, government action, and everyday life, it is not surprising to find that decision theory has been embraced by almost every scientific discipline. Thus, game theory permeates the theory and applications in economics. Psychologists know game theory as the theory of social interactions, and political scientists study rational choice theory.

All of these approaches to decision making have several essential elements in common that will be discussed below in the context of statistical decision theory. Game theory served as the precursor to most of the ideas in decision theory, and its place in modern decision making was cemented by John von Neumann and Oskar Morgenstern's fundamental work on the Theory of Games and Economic Behavior (1944). Although decision theory developed from game theory, there is a fundamental difference between the two. Informally, whereas in game theory, players make decisions based on their beliefs of what other players—whose interests may be diametrically opposed to theirs—will do, decision theory concerns itself with the study of decisions of individuals unconcerned with the plans of others—their “opponent” being nature.

Wald unified at once ideas from game theory and Neyman's and Pearson's mathematical developments in the theory of statistics in his elegant work Statistical Decision Functions. It is this approach on which the rest of the discussion focuses.