Nash equilibrium

In game theory, the Nash equilibrium (named after John Nash) is a kind of optimal strategy for games involving two or more players, where no player has anything to gain by changing only one's own strategy. If there is a set of strategies for a game with the property that no player can benefit by changing his strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute a Nash equilibrium.

The concept of the Nash equilibrium (NE) was originated by Nash in his dissertation, Non-cooperative games (1950). Nash showed that the various solutions for games that had been given earlier all yield Nash equilibria.

Contents

1 Existence of Nash equilibria

2 Examples

2.1 Competition game
2.2 Coordination game
2.3 Prisoner's dilemma

3 Stability

4 Occurrence

4.1 Where the conditions are not met
4.2 Where the conditions are met

5 See also

6 Notes

Existence of Nash equilibria

A game may have many Nash equilibria, or none. The Brouwer fixed point theorem provides the sufficient, though not necessary, conditions for existence of a Nash equilibrium. Brouwer proved that for a continuous function f: mapping S→S, where S is a non-empty and convex compact set onto itself, there exists x* such that x*=f(x*)(x* is a fixed point). In a game context, if the set of strategies by player i, is a compact and continuous set, the payoff functions for all players are quasi-concave and continuous, then the game has a Nash Equilibrium (NE).

A game can have a pure strategy NE or a NE in its mixed extension (of choosing a pure strategy stochastically with a fixed frequency). Nash was able to prove that, if we allow mixed strategies (players choose strategies randomly according to pre-assigned probabilities), then every n-player game in which every player can choose from finitely many strategies admits at least one Nash equilibrium.

Examples

Competition game

Consider the following two-player game: both players simultaneously choose a whole number from 0 to 10. Both players then win the minimum of the two numbers in dollars. In addition, if one player chooses a larger number than the other, then he has to pay $2 to the other. This game has a unique Nash equilibrium: both players have to choose 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. 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 11 Nash equilibria.

Coordination game

Main article: Coordination game

The coordination game is a classic (symmetric) two player, two strategy game, with payoff matrix

	Player 2 adopts strategy 1	Player 2 adopts strategy 2
Player 1 adopts strategy 1	A	B
Player 1 adopts strategy 2	C	D

where the payoffs are according to A>C and D>B. The players should thus cooperate on either of the two strategies to receive a high payoff. Players in the game have to agree on one of the two strategies in order to receive a high payoff. If the players do not agree, a lower payoff is rewarded. An example of a coordination game is the setting where two technologies are available to two firms with compatible products, and they have to elect a strategy to become the market standard. If both firms agree on the chosen technology, high sales are expected for both firms. If the firms do not agree on the standard technology, few sales result. Both strategies are Nash equilibria of the game.

Driving on a road, and having to choose either to drive on the left or to drive on the right of the road, is also a coordination game. For example, with payoffs 100 meaning no crash and 0 meaning a crash, the coordination game can be defined with the following payoff matrix:

	Drive on the Left:	Drive on the Right:
Drive on the Left:	100	0
Drive on the Right:	0	100

In this case there are two pure strategy Nash equilibria, when both choose to either drive on the left or on the right. If we admit mixed-strategies (where a pure strategy is chosen at random, subject to some fixed probability), then there are three Nash equilibria for the same case: two we have seen from the pure-strategy form, where the probabilities are (0%,100%) for player one, (0%, 100%) for player two; and (100%, 0%) for player one, (100%, 0%) for player two respectively. We add another where the probabilities for each player is (50%, 50%).

Prisoner's dilemma

Main article: Prisoner's dilemma

The Prisoner's dilemma has one Nash equilibrium: when both players defect. However, "both defect" is inferior to "both cooperate", in the sense that the total jail time served by the two prisoners is greater if both defect. The strategy "both cooperate" is unstable, as a player could do better by defecting while their opponent still cooperates. Thus, "both cooperate" is not an equilibrium. As Ian Stewart put it, "sometimes rational decisions aren't sensible!"

Stability

The concept of stability, useful in the analysis of many kinds of equilibrium can also be applied to Nash equilibria.

A Nash equilibrium for a mixed strategy game is stable if a small change (specifically a infinitesimal change) in probabilities for one player leads to a situation where two conditions hold:

the player who did not change has no better strategy in the new circumstance
the player who did change is now playing with a strictly worse strategy

If these cases are both met, then a player with the small change in their mixed-strategy will return immediately to the Nash equilibrium. The equilibrium is said to be stable. If condition one does not hold then the equilibrium is unstable. If only condition one holds then there are likely to be an infinite number of optimal strategies for the player who changed. John Nash showed that the latter situation could not arise in a range of well-defined games.

We have both stable and unstable equilibria in the Coordination game example above.

The equilibria involving mixed-strategies with 100% probabilities are stable. If either player changes their probabilities slightly, they will be both at a disadvantage, and their opponent will have no reason to change their strategy in turn.

In the case of the (50%,50%) equilibrium, there is instability. If either player changes their probabilities, then the other player immediately has a better strategy at either (0%, 100%) or (100%, 0%).

Stability is crucial in practical applications of Nash equilibria, since the mixed-strategy of each player is not perfectly known, but has to be inferred from statistical distribution of their actions in the game. In this case unstable equilibria are very unlikely to arise in practice, since any minute change in the proportions of each strategy seen will lead to a change in strategy and the breakdown of the equilibrium.

Note that stability of the equilibrium is connected to, but not the same thing as the stability of a strategy.

Occurrence

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. The necessary and sufficient conditions to be met by the players are:

Each player believes all other participants are rational.
The game correctly describes the utility payoff of all players.
The players are flawless in execution.
The players have sufficient intelligence to deduce the solution.
Each player is rational.

The reasoning behind this realization of the NE is that the first four conditions make playing the NE strategy optimal for each player, and that since the fifth condition identifies each player as an optimizing agent each will take the personally optimizing strategy.

Where the conditions are not met

Examples of game theory problems in which these conditions are not met:

In “Chicken” or an arms race a major consideration is the possibility that the opponent is irrational. This criterion may not be met even where the fifth criteria actually is true (so that players wrongly distrusting each others rationality adopt counter-strategies to expected irrational play on their opponents behalf).
The prisoner’s dilemma is not a dilemma if either player is happy to be jailed indefinitely.
Pong has a NE which can be played perfectly by a computer, but to make human vs. computer games interesting the programmers add small errors in execution.
If playing tic-tac-toe with a small child who desperately wants to win (meeting the other criteria) the NE strategy is often not optimal because your young opponent will not themselves adopt an optimal strategy. When playing Chinese chess most people are uncertain of the NE strategy since they haven’t the deductive ability to produce it.Template:Ref
Even if every player believes that all the others are rational one of them may subvert this assumption and opt for a (possibly self-destructive) strategy. This occasionally happens in top-level poker, when an expert player surprisingly goes "on tilt".

Where the conditions are met

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 practise in human negotiations. However, as a theoretical concept in economics, and evolutionary biology the NE has great explanatory power: In these cases the conditions are generally met, for the following reasons:

In these long-run cases the ‘average’ agent can be assumed to act ‘as if’ they were rational, because agents who don’t are competed out of the market or environment (in standard theory). This conclusion is drawn from the “stability” theory above.
The payoff in economics is money, and in evolutionary biology gene transmission, both are the fundamental bottom line of survival (agents ignoring these will not appear in the long run).
The assumption of rationality among all participants is based on the long-run time scale arguments.
‘The market’ or ‘evolution’ are ascribed the ability to test all strategies.
As point one, an irrational agent is presumed to disappear.

In these situations the assumption that the strategy observed is actually a NE has often been born out by research.

Notes

Template:Note Nash has proven that a perfect NE exists for this type of finite extensive form game – it can be represented as a strategy complying with his original conditions for a game with a NE. Such games may not have unique NE, but at least one of the many equilibrium strategies would be played by players having perfect knowledge of all 10¹⁵⁰ game trees.

Topics in game theory
Evolutionarily stable strategy - Mechanism design - No-win - Winner's curse - Zero-sum
Games: Prisoner's dilemma - Chicken - Stag hunt - Ultimatum game - Matching pennies ...
Related topics: Mathematics - Economics - Behavioral economics - Evolutionary biology - Evolutionary game theory - Population genetics - Behavioral ecology
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