Games - Nash equilibrium

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	Games - Nash equilibrium

In game theory, the Nash equilibrium (named after John Nash, who proposed it) is a kind of optimal collective strategy in a game involving two or more players, where no player has anything to gain by changing only his or her own strategy. 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). 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 proved for 2-player zero-sum games by John von Neumann and Oskar Morgenstern (1947).

Formal definition and existence of Nash equilibria

Let (S, f) be a game, where S is the set of strategy profiles and f is the set of payoff profiles. When each player

i \in

chooses strategy

x_i \in S_i

resulting in strategy profile

x

= (x_1, ..., x_n)

then player

i

obtains payoff

f_i(x)

. 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

f_i(x*) \geq f_i(x_i, x*_{-i}).

A game can have a pure strategy NE or a NE in its mixed extension (that of choosing a pure strategy stochastically with a fixed frequency). Nash proved 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.

Proof sketch

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 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

:

b_i(\sigma_{-i})

is a best response to

\sigma_{-i}

. Define

:

b(\sigma) = b_1(\sigma_{-1}) \times b_2(\sigma_{-2}) \times \cdots \times b_n(\sigma_{-n})

.

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

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 s/he has to pay $2 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. 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

The coordination game is a classic (symmetric) two player, two strategy game, with the payoff matrix shown to the right, 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:

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