Nash Equilibrium

A Nash equilibrium is a set of strategies, one for each rational player, in which no player can improve their payoff by changing strategy alone. Each player's choice is a best response to the strategies chosen by all others, so unilateral deviation is unprofitable.

Core logic

Players choose strategies independently while taking the strategies of others as given. An equilibrium is reached when their choices are mutually consistent best responses, even if the resulting outcome is not the best collective outcome.

• The equilibrium concerns a complete strategy profile, not the isolated choice of one player.
• It is stable against unilateral deviation, but coordinated changes by several players may produce a better outcome.
• Payoffs represent whatever each player values, such as income, security, influence or political advantage.

Forms and limitations

A Nash equilibrium may involve pure strategies, where players choose definite actions, or mixed strategies, where they randomise among actions with specified probabilities.

• A game may have one equilibrium or multiple equilibria; every finite game has at least one Nash equilibrium when mixed strategies are allowed.
• A Nash equilibrium need not be Pareto efficient: all players may prefer another outcome, as illustrated by the Prisoner's Dilemma.
• Rationality alone does not guarantee that players will coordinate on a particular equilibrium when several equilibria exist.

Meaning in international relations

In non-cooperative international interaction, states can be treated as rational players whose strategies are shaped by the expected responses of other states. A strategic situation is in equilibrium when no state expects a unilateral policy change to improve its payoff.

• Non-cooperative does not necessarily mean hostile; it means that outcomes are not imposed through a binding joint decision mechanism.
• An equilibrium may sustain cooperation when compliance is individually rational, or sustain rivalry when unilateral restraint would leave a state worse off.
• Repeated interaction, credible commitments and changes in relative costs can alter incentives and produce a different equilibrium.