In 1944, as World War II reshaped borders and bureaucracies, it also accelerated a “mathematical turn” in the social sciences. Governments were mobilizing unprecedented resources, scientists were applying formal methods to logistics and weapons systems, and economists were grappling with strategic problems—rationing, procurement, bargaining, cartel behavior, and the design of institutions under uncertainty. It was in this milieu that John von Neumann (1903–1957), a mathematician deeply involved in wartime research, and Oskar Morgenstern (1902–1977), an economist trained in the Austrian tradition and skeptical of loose verbal theorizing, published Theory of Games and Economic Behavior (Princeton University Press, 1944). Their central claim was that much of economic life is not well described as impersonal markets moving toward equilibrium, but as strategic interaction among agents whose outcomes depend on one another’s choices.
The book addressed several interwar and wartime problems. Classical and early neoclassical economics had powerful tools for price-taking markets, but it struggled with oligopoly, bargaining, coalition formation, and conflict—settings in which the “invisible hand” story becomes indeterminate. The Great Depression had already undermined confidence in self-equilibrating markets, while wartime planning highlighted coordination and adversarial decision-making. Von Neumann and Morgenstern argued that the standard apparatus—marginal calculus and competitive equilibrium—was insufficient when a firm’s optimal price depends on rivals’ prices, or when a nation’s policy depends on an adversary’s expectations. Their answer was to import and generalize von Neumann’s earlier work on parlor games and minimax strategies into a unified framework for economics and the social sciences.
At the core of their approach is the representation of economic situations as games: formal models consisting of players, available actions (or strategies), information, and payoffs. The novelty was not merely the word “game,” but the insistence that economic agents are interdependent optimizers. A “solution” requires a concept of rational choice that is consistent when each agent anticipates others’ rationality.
One strand of their contribution was a rigorous foundation for rational choice under uncertainty, via expected utility. While probabilistic reasoning existed in economics, von Neumann and Morgenstern supplied axioms under which preferences over lotteries can be represented by the expected value of a utility function. In modern notation, if an agent faces outcomes (x_i) with probabilities (p_i), then the preference ranking can be represented as maximizing $$ \mathbb{E}[u(x)] = \sum_i p_i u(x_i). $$ This “von Neumann–Morgenstern utility” did not claim that utility is measurable in a psychological sense; rather, it is unique up to positive affine transformations (u' = a u + b) with (a>0). The key point is operational: when choices involve risk, rational behavior can be treated as maximizing an expectation, enabling both theory and empirical work (later, tests and refinements) to proceed with a clear target.
The other strand was game theory proper, beginning with zero-sum conflict. In a two-player zero-sum game, one player’s gain is the other’s loss; payoffs can be represented by a matrix (A), where (A_{ij}) is the payoff to Player 1 when Player 1 chooses row (i) and Player 2 chooses column (j). Von Neumann’s celebrated minimax theorem (proved in 1928 and made central in the 1944 book) states that under broad conditions, $$ \max_{p \in \Delta} \min_{q \in \Delta} \, p^\top A q \;=\; \min_{q \in \Delta} \max_{p \in \Delta} \, p^\top A q, $$ where (p) and (q) are mixed strategies (probability distributions over actions) and (\Delta) denotes the simplex of such distributions. This equality guarantees a game value (v) and optimal mixed strategies for both players: each can secure at least (v) regardless of the other’s actions. Technically, the result foreshadowed the deep connections between economics and convex analysis, fixed-point theorems, and linear programming. It also introduced economists to the disciplined use of randomization as a rational strategy, not merely as ignorance.
Moving beyond zero-sum games, von Neumann and Morgenstern developed cooperative game theory, in which binding agreements and coalitions are permitted. Here the primitive is a characteristic function (v(S)), assigning to each coalition (S) the total payoff it can guarantee itself. The question becomes how to distribute (v(N)) (the worth of the grand coalition) in a way that is stable against deviations by sub-coalitions. Their most influential cooperative solution concept was the core: the set of allocations such that no coalition (S) can improve upon it given its own worth (v(S)). In symbols, an allocation (x) is in the core if (\sum_{i \in N} x_i = v(N)) and for all coalitions (S), (\sum_{i \in S} x_i \ge v(S)). This provided economists with a language for bargaining, cartels, and coalition politics, and later became intertwined with general equilibrium theory via the idea that competitive equilibria can be seen as core allocations in large economies.
The immediate impact of Theory of Games and Economic Behavior was paradoxical: it was quickly recognized as a landmark, but slowly absorbed into mainstream economics. Many economists in the 1940s found it daunting in mathematical style and unsure how to apply cooperative concepts empirically. Critics also noted that the book’s emphasis on coalitions and binding agreements did not match many market settings where contracts are incomplete and enforcement limited. Moreover, the solution concepts for general non-zero-sum, non-cooperative interaction were not yet fully crystallized in 1944.
That missing piece arrived in 1950–51 with John Nash’s papers introducing what is now called Nash equilibrium, a non-cooperative solution concept that generalized strategic analysis well beyond the cooperative framework. In that sense, von Neumann and Morgenstern supplied the architecture—players, strategies, payoffs, rationality under risk—while Nash supplied a widely applicable equilibrium notion. Game theory then became central to industrial organization, bargaining theory, political economy, and international relations, particularly during the Cold War, when strategic deterrence and arms races made the language of conflict and credible threats newly salient.
The legacy today is both foundational and contested. Modern microeconomics routinely models firms and consumers as strategic agents; auction design, mechanism design, and matching theory all inherit the von Neumann–Morgenstern formalization of preferences and strategic reasoning. At the same time, empirical and behavioral research has pressed on their rationality assumptions. The Allais paradox (1953) and later work in behavioral economics documented systematic departures from expected utility, prompting refinements such as prospect theory. In applied work, the econometrics of games—estimating strategic models from data—has grown into a field of its own, grappling with identification, multiple equilibria, and counterfactual policy evaluation.
In the longer line of continuity, von Neumann and Morgenstern stand between the marginalist revolution’s optimization framework and the postwar formalization of equilibrium and information. They drew on earlier concerns about oligopoly and interdependence (from Cournot in 1838 through the imperfect competition debates of the 1930s), and they helped set the stage for Nash equilibrium, Harsanyi’s Bayesian games (1967–68), Selten’s refinements (1975), and the information economics of the 1970s onward. By translating strategic interaction into a precise mathematical language, Theory of Games and Economic Behavior did more than “add a tool”: it changed what counted as an economic problem and what counted as an answer—an enduring hallmark of the mathematical turn that wartime exigencies helped bring to maturity.