New Research In
Physical Sciences
Social Sciences
Featured Portals
Articles by Topic
Biological Sciences
Featured Portals
Articles by Topic
 Agricultural Sciences
 Anthropology
 Applied Biological Sciences
 Biochemistry
 Biophysics and Computational Biology
 Cell Biology
 Developmental Biology
 Ecology
 Environmental Sciences
 Evolution
 Genetics
 Immunology and Inflammation
 Medical Sciences
 Microbiology
 Neuroscience
 Pharmacology
 Physiology
 Plant Biology
 Population Biology
 Psychological and Cognitive Sciences
 Sustainability Science
 Systems Biology
Equilibrium points in nperson games

Communicated by S. Lefschetz, November 16, 1949
One may define a concept of an nperson game in which each player has a finite set of pure strategies and in which a definite set of payments to the n players corresponds to each ntuple of pure strategies, one strategy being taken for each player. For mixed strategies, which are probability distributions over the pure strategies, the payoff functions are the expectations of the players, thus becoming polylinear forms in the probabilities with which the various players play their various pure strategies.
Any ntuple of strategies, one for each player, may be regarded as a point in the product space obtained by multiplying the n strategy spaces of the players. One such ntuple counters another if the strategy of each player in the countering ntuple yields the highest obtainable expectation for its player against the n − 1 strategies of the other players in the countered ntuple. A selfcountering ntuple is called an equilibrium point.
The correspondence of each ntuple with its set of countering ntuples gives a onetomany mapping of the product space into itself. From the definition of countering we see that the set of countering points of a point is convex. By using the continuity of the payoff functions we see that the graph of the mapping is closed. The closedness is equivalent to saying: if P_{1}, P_{2}, … and Q_{1}, Q_{2}, …, Q_{n}, … are sequences of points in the product space where Q_{n} → Q, P_{n} → P and Q_{n} counters P_{n} then Q counters P.
Since the graph is closed and since the image of each point under the mapping is convex, we infer from Kakutani’s theorem^{1} that the mapping has a fixed point (i.e., point contained in its image). Hence there is an equilibrium point.
In the twoperson zerosum case the “main theorem”^{2} and the existence of an equilibrium point are equivalent. In this case any two equilibrium points lead to the same expectations for the players, but this need not occur in general.
Footnotes

↵*The author is indebted to Dr. David Gale for suggesting the use of Kakutani’s theorem to simplify the proof and to the A. E. C. for financial support.
Citation Manager Formats
More Articles of This Classification
Related Content
Cited by...
 Experimental evidence for tipping points in social convention
 John Nash and the Organization of Stroke Care
 Alternatives for domestic water tariff policy in the municipality of Chania, Greece, toward water saving using game theory
 Superhuman AI for headsup nolimit poker: Libratus beats top professionals
 Extinction rates in tumour public goods games
 DeepStack: Expertlevel artificial intelligence in headsup nolimit poker
 Adaptable history biases in human perceptual decisions
 Dynamics of prebiotic RNA reproduction illuminated by chemical game theory
 PNAS at 101: Heading into the next century
 Biological trade and markets
 Debreus social equilibrium existence theorem
 Stochastic games
 Escaping the tragedy of the commons through targeted punishment
 From Nash to CournotNash equilibria via the MongeKantorovich problem
 Virtual bargaining: a theory of social decisionmaking
 Neural correlates of strategic reasoning during competitive games
 Public goods in relation to competition, cooperation, and spite
 Aspiration dynamics of multiplayer games in finite populations
 The price of anarchy in mobilitydriven contagion dynamics
 A generalization of Nash's theorem with higherorder functionals
 Modigliani's and Simon's Early Contributions to Uncertainty (195261)
 Complex dynamics in learning complicated games
 Polyandry and alternative mating tactics
 The agencies method for coalition formation in experimental games
 The influence of altruism on influenza vaccination decisions
 The economic approach to 'theory of mind'
 Extortion and cooperation in the Prisoner's Dilemma
 The price of your soul: neural evidence for the nonutilitarian representation of sacred values
 Paths to climate cooperation
 Inequality, communication, and the avoidance of disastrous climate change in a public goods game
 On Hedge Fund Structures: Improving Allocation * Models for Illiquid Investments
 Economic aspects of global warming in a postCopenhagen environment
 Evolutionary dynamics in structured populations
 Qualitative criterion for interception in a pursuit/evasion game
 Trust and cooperation among economic agents
 PNAS will eliminate Communicated submissions in July 2010
 Saving for an uncertain future
 Behavioral and Neural Changes after Gains and Losses of Conditioned Reinforcers
 Role of the Superior Colliculus in Choosing MixedStrategy Saccades
 Cortical mechanisms for reinforcement learning in competitive games
 Neural correlates of economic game playing
 Integrating epidemiology, psychology, and economics to achieve HPV vaccination targets
 Gambling for global goods
 Social DecisionMaking: Insights from Game Theory and Neuroscience
 Understanding Neural Coding through the ModelBased Analysis of Decision Making
 Longstanding influenza vaccination policy is in accord with individual selfinterest but not with the utilitarian optimum
 Target Selection Signals for Arm Reaching in the Posterior Parietal Cortex
 An optimal brain can be composed of conflicting agents
 Essential equilibria
 From The Cover: Evolutionary cycles of cooperation and defection
 Nash Equilibrium
 Neuroeconomics: The Consilience of Brain and Decision
 The Nash equilibrium: A perspective
 NEUROSCIENCE: Predicting Future Rewards
 Stochastic game theory: For playing games, not just for doing theory
 The hard sciences
 Cooperation and selfinterest: Paretoinefficiency of Nash equilibria in finite random games
 Nonatomic games on Loeb spaces
 Creating a Context for Game Theory
 Game Theory at Princeton, 19491955: A Personal Reminiscence
 Speaking Axiomatically: Citation Patterns to Early Articles in General Equilibrium Theory