Oskar Morgenstern, Game Theory: I. Theoretical Aspects (1968)

Oskar Morgenstern’s encyclopedia article presents game theory as a mathematics of strategic interdependence whose implications extend from recreational games to economic institutions and political organization. Moving from definitions and historical origins through two-person contests to coalitions, voting, composite games, and nonzero-sum interactions, it argues that social action requires concepts fundamentally different from individual maximization. The central difficulty is that each participant’s outcome depends on others who also pursue their own advantage. Game theory therefore both prescribes optimal conduct and supplies models for interpreting observed behavior.

In analyzing games the theory does not assume rational behavior; rather, it attempts to determine what "rational" can mean when an individual is confronted with the problem of optimal behavior in games and equivalent situations.

Rationality is thus a problem to be formulated, not an explanatory premise already secured. Morgenstern distinguishes strategic uncertainty from chance: nature is treated as indifferent, whereas opponents respond purposefully. Probability calculations alone cannot establish optimal strategic conduct. His historical account identifies von Neumann’s 1928 minimax theorem as the decisive foundation and their joint 1944 work as the extension toward multiplayer games and economics. The analogy with probability theory’s importance for natural science expresses a corresponding ambition for the social sciences.

The conceptual exposition distinguishes rules from particular plays and extensive-form game trees from their equivalent normalized representation. Rules delimit permissible action; in social applications, they include physical and legal constraints. A strategy specifies contingent conduct across an entire game, rather than merely a move. Payoffs are represented numerically, principally through money or transferable utility, although Morgenstern acknowledges modifications to these assumptions. The framework also explicitly assumes knowledge of alternative payoffs and the computational capacity needed to determine optimal behavior. These strong assumptions make the mathematical problem precise without eliminating practical difficulties of calculation.

A maximum (or minimum) of a function can be achieved only when all variables on which the maximum (minimum) depends are under the complete control of the would-be maximizer.

This claim supports Morgenstern’s challenge to conventional economic optimization: strategic interaction is a different conceptual structure, not simply maximization under additional constraints. Two-person zero-sum games provide the elementary case. A payoff matrix identifies each player’s strategies; where maximin and minimax coincide, a saddle point determines defensive optimal conduct. When no saddle point exists in pure strategies, deliberately randomized mixed strategies supply the solution guaranteed by von Neumann’s theorem. Randomization does not convert strategy into chance, because choosing the probability distribution remains a strategic act. Morgenstern stresses both the defensive character of this result and the gap between proving that optimal strategies exist and computing them, notably in chess.

With three or more players, cooperation changes the problem qualitatively. Essential games permit coalitions whose members can gain more together than separately; inessential games offer no such advantage. The characteristic function records coalition proceeds, while imputations specify distributions among players. Solutions become sets of imputations satisfying internal nondomination and domination of excluded alternatives. Stability belongs to this relational structure, not to a uniquely protected distribution.

Each solution may correspond to a specific mode of behavior of the players or a specific form of social organization.

Multiplicity is therefore sociologically meaningful. The same physical resources and technology may sustain different internally consistent institutions and income distributions. A socially accepted standard can select a relevant solution set, but the theory does not thereby determine which coalition or allocation within it will occur. Side payments, discriminatory exclusions, and unequal admission terms reveal how bargaining shapes distributions. Morgenstern’s preference for cooperative analysis also underlies his criticism of equilibrium accounts that suppress the gains from association. Yet he recognizes noncooperative equilibrium as important where communication cannot sustain cooperation, introducing Nash’s existence result and the possibility of multiple equilibria.

The subsequent sections extend this relational account. Simple voting games distinguish formal voting weights from effective power; a participant’s significance depends on coalition possibilities rather than an isolated contribution. Shapley’s approach evaluates average contributions across possible coalitions. Similarly, adding players does not necessarily produce statistical simplification, because coalition formation can turn many individuals into a few interacting groups. Composite games show how outside contributions or withdrawals can alter bargaining and destabilize existing social structures.

The fact that solutions of $n$-person games are not single numbers or single sets of numbers—but that the above-mentioned, more complicated structures emerge—is not an imperfection of the theory: it is a fundamental property of social organization that can be described only by game-theoretic methods.

Morgenstern finally presents nonzero-sum games through a fictitious balancing player, connecting productive cooperation with aggregate gains. His survey closes with applications across economics, politics, warfare, bargaining experiments, and statistical decision theory. Its significance lies in joining mathematical exposition to an argument about social organization: outcomes depend on coalition relations and institutional possibilities, not simply individual attributes. Its account remains historically situated, explicitly reporting unresolved existence and computational problems rather than presenting every branch as complete.