Nash Equilibrium

Nash equilibrium is a central solution concept in game theory describing a collection of strategies, one for each player, such that no player can obtain a higher payoff by changing their own strategy while the others’ strategies remain fixed. It expresses mutual consistency of individual choices rather than collective optimality. Introduced by John Forbes Nash Jr. in 1950, the concept applies to games involving both conflicting and shared interests and is widely used in economics. (doi.org)

Mathematical definition

A game in normal form specifies a set of players $N$, a strategy set $S_i$ for each player $i$, and a payoff function $u_i$. Payoffs represent the player’s preferences, often expressed as utility. A strategy profile $s=(s_1,\ldots,s_n)$ belongs to the Cartesian product $S_1\times\cdots\times S_n$. Write $s_{-i}$ for the strategies of every player except $i$. (live.ocw.mit.edu)

A profile $s^*$ is a Nash equilibrium if

Thus every player’s equilibrium strategy is a best response to the others. Equality is permitted: a player may have alternative strategies yielding the same payoff. The condition excludes profitable unilateral deviations, not changes by several players together. It also does not require a strategy to be optimal against every possible opponent strategy, as a dominant strategy would be. (live.ocw.mit.edu)

Pure and mixed strategies

A pure strategy selects one available strategy with certainty. A mixed strategy assigns a probability distribution over pure strategies. In the standard mixed-strategy formulation, players randomize independently and evaluate outcomes through expected payoffs. A pure strategy is the special case assigning probability one to a single choice. (doi.org)

For finite games, expected payoff under a mixed profile $\sigma$ is

The equilibrium inequality then applies to $U_i$ and all alternative mixed strategies. Every pure strategy receiving positive probability in an equilibrium mixture must give the same maximal expected payoff against the opponents’ mixtures. Otherwise, shifting probability toward a better strategy would be profitable. Strategies outside that mixture must not offer a higher payoff. (doi.org)

Illustrative games

The prisoner’s dilemma demonstrates why equilibrium need not be socially desirable. Consider the following payoff matrix, with the row player’s payoff listed first:

Defection gives each player a higher payoff regardless of the other’s choice. Consequently, mutual defection is the unique Nash equilibrium. Yet mutual cooperation gives both players higher payoffs. The equilibrium therefore fails Pareto efficiency: another outcome makes someone better off without making anyone worse off. (live.ocw.mit.edu)

Randomization matters in matching pennies, a zero-sum game where one player wins when the coins match and the other wins when they differ. Every pure profile gives one player an incentive to switch. An equilibrium instead has each player independently choose heads and tails with probability one-half, leaving neither with a profitable deviation. (live.ocw.mit.edu)

Equilibria can also be multiple. If two players receive one unit each for choosing the same action and zero for different actions, both coordinated pure profiles are equilibria. Independent equal-probability mixing is another equilibrium. The equilibrium condition alone does not select among them. (live.ocw.mit.edu)

Existence and historical development

Nash proved that every game with finitely many players and finitely many pure strategies has at least one equilibrium when mixed strategies are allowed. The theorem does not guarantee a pure-strategy equilibrium, uniqueness, or convergence of actual play. His 1950 proof used Kakutani’s fixed-point theorem: the mathematical structure of the mixed-strategy spaces and best-response correspondence ensures an equilibrium point exists. (doi.org)

For two-player zero-sum games, this result connects with the minimax theorem developed by John von Neumann. Nash’s formulation extends equilibrium analysis beyond strictly opposed interests. In 1994, Nash, John Harsanyi, and Reinhard Selten jointly received the Nobel Memorial Prize in Economic Sciences for their analysis of equilibria in non-cooperative games. (arxiv.org)

Computation and interpretation

In small games, equilibria can be found by identifying mutual best responses or solving indifference conditions for mixed strategies. Existence, however, does not imply easy computation. In computational complexity, finding an equilibrium of a general two-player finite game with rational payoffs is PPAD-complete. By contrast, two-player zero-sum equilibria can be computed through linear programming. An approximate equilibrium allows each player a bounded potential gain from deviation. (ocw.mit.edu)

Nash equilibrium characterizes incentives within a specified model; it is not a guarantee of observed behavior or a description of how players learn. Experimental choices can differ from equilibrium predictions, and the concept does not itself establish fairness, efficiency, or robustness to coordinated deviations. (doi.org)

For sequential games, subgame-perfect equilibrium requires Nash equilibrium within every subgame, excluding certain noncredible threats. For games involving private information, Bayesian Nash equilibrium applies best-response reasoning to strategies contingent on players’ information, evaluated using their beliefs about others. These concepts address features that a basic strategic-form equilibrium does not distinguish. (ocw.mit.edu)

References

1. The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1994nobelprize.org
2. 15 / 6.207 Networks, Lecture 13: Game Theory 1: Static Games with Complete Informationlive.ocw.mit.edu
3. Recitation 8: Introduction to Game Theoryocw.mit.edu
4. Settling the Complexity of Computing Two-Player Nash Equilibriaarxiv.org
5. 810S21 Game Theory, Lecture Slides 2: Games in Strategic Form and Nash Equilibriumlive.ocw.mit.edu
6. Recitation 11 Notesocw.mit.edu
7. Game Theory, Lecture Notesocw.mit.edu