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Nash equilibrium

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Nash Equilibrium
A solution concept in game theory
Subset of: Rationalizability, Correlated equilibrium
Superset of: Evolutionary stable strategy, Subgame perfect equilibrium, Perfect Bayesian equilibrium, Trembling hand perfect equilibrium
Proposed by: John Forbes Nash
Used for: All non-cooperative games
Example: Prisoner's dilemma

In game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it) is a kind of solution concept of a game involving two or more players, where no player has anything to gain by changing only his or her own strategy unilaterally. If each player has chosen a strategy and no player can benefit by changing his or her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

The concept of the Nash equilibrium (NE) is not exactly original to Nash (e.g., Antoine Augustin Cournot showed how to find what we now call the Nash equilibrium of the Cournot duopoly game). Consequently, some authors refer to it as a “Cournot-Nash equilibrium” (or as a “Nash-Cournot equilibrium”). However, Nash showed for the first time in his dissertation, Non-cooperative games (1950), that Nash equilibria must exist for all finite games with any number of players. Until Nash, this had only been proven for 2-player zero-sum games by John von Neumann and Oskar Morgenstern (1947).

Formal definition Edit

Let (S, f) be a game, where S is the set of strategy profiles and f is the set of payoff profiles. Let \sigma_{-i} be a strategy profile of all players except for player i. When each player i \in \{1, ..., n\} chooses strategy x_i resulting in strategy profile x = (x_1, ..., x_n) then player i obtains payoff f_i(x). Note that the payoff depends on the strategy profile chosen, i.e. on the strategy chosen by player i as well as the strategies chosen by all the other players. A strategy profile x^* \in S is a Nash equilibrium (NE) if no deviation in strategy by any single player is profitable, that is, if for all i



Proof sketchEdit

As above, let \sigma_{-i} be a mixed strategy profile of all players except for player i. We can define a best response correspondence for player i, b_i. b_i is a relation from the set of all probability distributions over opponent player profiles to a set of player i's strategies, such that each element of


is a best response to \sigma_{-i}. Define


One can use the Kakutani fixed point theorem to prove that b has a fixed point. That is, there is a \sigma^* such that \sigma^* \in b(\sigma^*). Since b(\sigma^*) represents the best response for all players to \sigma^*, the existence of the fixed point proves that there is some strategy set which is a best response to itself. No player could do any better by deviating, and it is therefore a Nash equilibrium.

Examples Edit

Competition game Edit

A competition game

Consider the following two-player game: both players simultaneously choose a whole number from 0 to 3. Both players then win the smaller of the two numbers in points. In addition, if one player chooses a larger number than the other, then s/he has to give up two points to the other. This game has a unique Nash equilibrium: both players choosing 0. Any other choice of strategies can be improved if one of the players lowers his number to one less than the other player's number. In the table to the left, for example, when starting at the green square it is in player 1's interest to move to the purple square by choosing a smaller number, and it is in player 2's interest to move to the blue square by choosing a smaller number. If the game is modified so that the two players win the named amount if they both choose the same number, and otherwise win nothing, then there are 3 Nash equilibria.

Coordination game Edit

Main article: Coordination game
A coordination game
Player 2 adopts strategy 1 Player 2 adopts strategy 2
Player 1 adopts strategy 1 A, A B, C
Player 1 adopts strategy 2 C, B D, D



The driving game
Drive on the Left Drive on the Right
Drive on the Left 100, 100 0, 0
Drive on the Right 0, 0 100, 100


Prisoner's dilemma Edit

Main article: Prisoner's dilemma (but watch out for differences in the orientation of the payoff matrix)



Nash equilibria in a payoff matrix Edit


We can apply this rule to a 3x3 matrix:

A Payoff Matrix
Option A Option B Option C
Option A 0, 0 25, 40 5, 10
Option B 40, 25 0, 0 5, 15
Option C 10, 5 15, 5 10, 10


This said, the actual mechanics of finding equilibrium cells is obvious: find the maximum of a column and check if the second member of the tuple has maximum of the row. If yes - you've got a Nash Equilibrium. Check all columns this way to find all NE cells. An NxN matrix may have between 0 and N pure strategy Nash equilibria.

Stability Edit







Note that stability of the equilibrium is related to, but distinct from, stability of a strategy.

Occurrence Edit

If a game has a unique Nash equilibrium and is played among players with certain characteristics, then it is true (by definition of these characteristics) that the NE strategy set will be adopted. Sufficient conditions to be met by the players are:

  2. The players are flawless in execution.
  3. The players have sufficient intelligence to deduce the solution.

Where the conditions are not met Edit


  2. Pong has an equilibrium which can be played perfectly by a computer, but to make human vs. computer games interesting the programmers add small errors in execution, violating the second condition.
  3. In many cases, the third condition is not met because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in Chinese chess[1]. Or, if known, it may not be known to all players, as when playing tic-tac-toe with a small child who desperately wants to win (meeting the other criteria).

Where the conditions are met Edit

Due to the limited conditions in which NE can actually be observed, they are rarely treated as a guide to day-to-day behaviour, or observed in practice in human negotiations. However, as a theoretical concept in economics, and evolutionary biology the NE has explanatory power. The payoff in economics is money, and in evolutionary biology gene transmission, both are the fundamental bottom line of survival. Agents failing to maximize these for whatever reason will be competed out of the market or environment, which are ascribed the ability to test all strategies. This conclusion is drawn from the "stability" theory above. In these situations the assumption that the strategy observed is actually a NE has often been borne out by research.[verification needed]

See also Edit

References Edit

  • Nash, John (1951) "Non-Cooperative Games" The Annals of Mathematics 54(2):286-295.

Notes Edit


cs:Nashova rovnováha

da:Nash-ligevægt de:Nash-Gleichgewicht es:Equilibrio de Nash fa:تعادل نش fr:Équilibre de Nash gl:Equilibrio de Nashhe:שיווי משקל נאש nl:Nash-evenwichtno:Nash-likevektro:Echilibru Nash ru:Равновесие Нэшаzh:納什均衡點


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