The Origin Story: When Did Game Theory First Formalize One-Shot Games?

Game theory has become an indispensable tool for understanding strategic decision-making across countless fields. At its heart lies a deceptively simple concept: the one-shot game, where players make decisions simultaneously without the prospect of future interaction. But when and how did mathematicians first formalize this fundamental building block of strategic thinking? The answer takes us back to the mid-20th century and the revolutionary work that would transform how we understand conflict, cooperation, and rational choice.

The Pre-History: Strategic Thinking Before Formalization

While game theory as a mathematical discipline emerged in the 20th century, humans have engaged in strategic reasoning for millennia. Military strategists, merchants, and diplomats have long understood that their optimal choices depend on anticipating others’ actions. Ancient texts like Sun Tzu’s “The Art of War” reveal sophisticated strategic thinking, even if they lacked mathematical rigor.

In the 18th and 19th centuries, mathematicians began applying formal analysis to games of chance. Pioneers like Pierre de Fermat and Blaise Pascal developed probability theory partly through analyzing gambling problems. However, these early efforts focused on games against nature or chance, not strategic interactions between rational players with conflicting interests. The critical leap—recognizing that rational players must account for each other’s reasoning—would wait until the 20th century.

The 1928 Watershed: Von Neumann’s Minimax Theorem

The true birth of game theory, and with it the implicit formalization of one-shot games, occurred in 1928 when John von Neumann published his landmark paper “Zur Theorie der Gesellschaftsspiele” in the journal Mathematische Annalen. This work represented a quantum leap in mathematical sophistication applied to strategic interactions.

Von Neumann’s central achievement was proving the minimax theorem for zero-sum games. In essence, he demonstrated that in any two-player zero-sum game, there exists an equilibrium where each player adopts a mixed strategy that minimizes their maximum possible loss. This was revolutionary because it showed that even in purely adversarial situations, rational players could identify optimal strategies through mathematical analysis.

Though von Neumann’s 1928 paper didn’t explicitly categorize games as “one-shot” versus repeated, his framework inherently dealt with single-play scenarios. He analyzed the game in isolation, without considering how past play might influence current decisions or how current play might affect future interactions. This was the essence of one-shot analysis, even if the terminology came later.

The 1944 Breakthrough: Theory of Games and Economic Behavior

Game theory remained largely confined to mathematical circles until 1944, when John von Neumann collaborated with economist Oskar Morgenstern to publish “Theory of Games and Economic Behavior.” This monumental 600-page treatise represented the first comprehensive treatment of game theory and its potential applications to economics and social science.

Von Neumann and Morgenstern systematically developed the mathematical foundations for analyzing strategic interactions. They introduced crucial concepts including the extensive form (game trees) and normal form (payoff matrices) representations of games. The normal form, in particular, became the standard way to represent one-shot games: a simple matrix showing the payoffs for each combination of player strategies.

The book focused heavily on zero-sum games, where one player’s gain exactly equals another’s loss, and von Neumann’s minimax theorem remained central. However, von Neumann and Morgenstern also tackled the more complex problem of games with more than two players and the possibility of coalition formation. They introduced the concept of characteristic function form and developed solution concepts like the core and the Shapley value, though these would be refined by later researchers.

Crucially, “Theory of Games and Economic Behavior” established game theory as a field worthy of serious scholarly attention. It demonstrated that strategic interactions could be analyzed with mathematical rigor comparable to physics or engineering. The book’s influence extended far beyond economics, inspiring researchers in political science, biology, philosophy, or computer science to adopt game-theoretic reasoning.

Nash’s Revolution: The 1950-1951 Turning Point

While von Neumann and Morgenstern laid the groundwork, the formalization of one-shot games took its most decisive step forward between 1950 and 1951 with John Forbes Nash Jr.’s doctoral dissertation and subsequent publications. Nash, then a 21-year-old graduate student at Princeton, developed what would become the most influential solution concept in game theory: the Nash equilibrium.

Nash’s key insight was extending equilibrium analysis beyond zero-sum games to the general case of non-cooperative games. He proved that every finite game has at least one equilibrium point where no player can improve their payoff by unilaterally changing their strategy, assuming other players maintain their strategies. This equilibrium concept applied whether players used pure strategies (deterministic choices) or mixed strategies (probabilistic combinations).

The Nash equilibrium provided exactly what was needed to fully formalize one-shot games. Unlike repeated games, where players might condition their behavior on past play, or cooperative games, where players might form binding agreements, one-shot non-cooperative games required a solution concept based purely on simultaneous, independent decision-making. The Nash equilibrium captured this perfectly: each player chooses their strategy by correctly anticipating others’ choices, but without communication, coordination, or concern for future interaction.

The Nash equilibrium transformed game theory because it provided a unified solution concept applicable to virtually any strategic interaction. It made concrete the notion of rational play in a one-shot setting: players should choose strategies that are mutual best responses to each other. This circularity—my optimal choice depends on your choice, which depends on my choice—was resolved through the equilibrium concept.

The Prisoner’s Dilemma: Crystallizing the One-Shot Concept

The Prisoner’s Dilemma’s power lies in its simplicity and its paradox. Two prisoners, unable to communicate, must each decide whether to cooperate with their partner or defect and testify against them. The Nash equilibrium is both would be better off if they could somehow achieve mutual cooperation. This tension between individual rationality and collective benefit has made the Prisoner’s Dilemma endlessly fascinating to researchers.

The Prisoner’s Dilemma explicitly relies on the one-shot assumption. If the prisoners could communicate, make binding agreements, or expect future interactions, the strategic calculus changes dramatically. The game’s stark lesson—that rational individuals may produce collectively suboptimal outcomes—depends critically on the inability to establish trust through repeated interaction or enforceable promises.

Formalizing the Distinction: The 1950s and Beyond

Through the 1950s, game theorists increasingly recognized that the distinction between one-shot and repeated games was not merely technical but fundamental. Researchers began systematically exploring how repeated interaction changes strategic incentives.

The folk theorem, developed through contributions by multiple researchers in the late 1950s and 1960s, demonstrated that in infinitely repeated games, a vast range of outcomes (including cooperative ones in the Prisoner’s Dilemma) could be sustained as equilibria through the threat of future punishment. This stood in stark contrast to one-shot games, where only the immediate payoffs matter.

This formalization clarified that one-shot games represented a specific, well-defined category: strategic interactions where players move simultaneously or without knowledge of others’ choices, and where there is no prospect of future interaction that might enable reputation-building, punishment, or reward. The analytical tools for one-shot games—particularly the Nash equilibrium in normal form—were now fully established.

Institutional Context: RAND and the Cold War

The formalization of one-shot games didn’t occur in an intellectual vacuum. Much of the foundational work happened at or in connection with the RAND Corporation, established in 1948 to provide research and analysis for the United States armed forces. The Cold War created urgent demand for rigorous analysis of strategic interactions, particularly in nuclear deterrence scenarios.

Nuclear confrontation, paradoxically, often resembled a one-shot game. The decision to launch or not launch nuclear weapons would happen under time pressure, without negotiation, and with stakes so high that traditional deterrence mechanisms might fail. Game theory promised tools for analyzing such scenarios rationally. This practical motivation accelerated the theoretical development of one-shot game analysis.

Von Neumann himself consulted extensively on nuclear strategy, and RAND employed many leading game theorists. The Prisoner’s Dilemma emerged from Flood and Dresher’s RAND experiments. Thomas Schelling, later a Nobel laureate, developed his influential work on focal points and tacit coordination partly through RAND-sponsored research. The Cold War’s existential questions drove rapid progress in formalizing strategic interaction.

The Terminology Takes Hold

While the mathematical framework for one-shot games solidified in the 1950s, the specific terminology “one-shot game” gained currency gradually. Early papers typically referred to “the normal form” or “simultaneous move games” without explicitly contrasting them with repeated games. The term “one-shot” emphasized the crucial feature: players interact exactly once, without past history or future consequence.

By the 1960s and 1970s, textbooks and research papers routinely distinguished one-shot from repeated games as fundamental categories that continues to shape research across disciplines.

In economics, one-shot game models illuminate market competition, auction design, and bargaining.

In political science, they help analyze voting, international relations, and institutional design.

In biology, they’ve revolutionized our understanding of evolution and animal behavior through evolutionary game theory.

The one-shot assumption remains both powerful and controversial. It provides analytical clarity by stripping away the complexity of history and future interaction, allowing researchers to identify the pure logic of strategic interdependence. Critics argue this abstraction can mislead when real-world interactions involve repeated play, reputation, and evolving relationships. The debate continues about when one-shot models illuminate and when they oversimplify.

This formalization represented a genuine intellectual revolution. For the first time, scholars could rigorously analyze situations where intelligent agents with conflicting interests make simultaneous choices. The resulting insights provided crucial tools for navigating the Cold War’s strategic dilemmas.

Today, when researchers analyze an auction, a military confrontation, or an evolutionary contest as a one-shot game, they employ tools forged in those mid-century decades. The mathematics may have grown more sophisticated and the applications more diverse, but the core insight remains: sometimes the clearest path to understanding strategic interaction is to imagine a single play, with no yesterday to regret and no tomorrow to bargain for.