John von Neumann was one of the greatest mathematicians of the twentieth century. For mathematical psychology, his most consequential achievement was the 1944 book Theory of Games and Economic Behavior, co-authored with Oskar Morgenstern, which established the axiomatic foundations of rational choice under risk and invented game theory. Every subsequent model of decision making -- from Savage's subjective expected utility to Kahneman and Tversky's prospect theory -- is defined in relation to the von Neumann-Morgenstern framework.
Expected Utility Theory
If preferences satisfy these axioms, there exists u such that:
L1 preferred to L2 iff Sum p_i * u(x_i) >= Sum q_i * u(x_i)
u is unique up to positive affine transformation
The representation theorem proved that if a decision maker's preferences over lotteries satisfy four axioms, then there exists a utility function such that the decision maker always prefers the lottery with the higher expected utility. This was a mathematical proof that rational preferences must be representable in this way, providing the normative benchmark against which all subsequent descriptive theories of choice have been evaluated.
Von Neumann's 1928 minimax theorem proved that every finite two-person zero-sum game has an optimal mixed strategy for each player such that neither can improve by unilateral deviation. This established game theory as a rigorous mathematical discipline and provided the foundation for strategic decision making, bargaining theory, and mechanism design.
Game Theory and Strategic Interaction
The theory of games extended decision theory from individual choice to strategic interaction -- situations where each actor's optimal decision depends on what others do. Von Neumann and Morgenstern classified games by number of players, information structure, and possibility of cooperation, establishing a taxonomy that organizes strategic analysis to this day.
Legacy and Impact
Von Neumann's axiomatization of rational choice provided mathematical psychology with its most fundamental theoretical framework. Expected utility theory is the starting point for understanding risk aversion, the Allais paradox, prospect theory, and all modern behavioral decision theory. The mathematical standards he set -- precise axioms, rigorous proofs, clear assumptions -- established the template for formal theorizing in the behavioral sciences.