POLI 488: Nash Equilibrium
Nash Equilibrium
- A Nash equilibrium corresponds to a steady-state: if, whenever the game is played, the action profile is the same Nash equilibrium a*, then no player has a reason to choose any action different from her component of a* - Players' beliefs about each other's actions are (assumed to be) correct. - ui (a*) ≥ ui (ai , a*-i) for every action ai of player i
Best Response Function
- Function giving the payoff-maximizing choice for one player for each of a continuum of strategies of the other player. - Any action in Bi (a-i) is at least as good for player i as every other action of player i when the other players' actions are given by a-i.
Beliefs about Players in Nash Equilibrium
When solving the game through iterated elimination of dominated (strictly or weakly) strategies, we assume that rationality is common knowledge. The definition of Nash Equilibrium is made up of two components; 1. players are rational : each player's strategy maximizes his payoff, given his belief about the strategies used by other players 2. beliefs are accurate : each player's belief about the strategies used by the other players are true (correct)
Symmetric Games
A game is symmetric when: (1)All players have the same strategy sets, (2)Players receive the same payoff when they choose the same strategy, (3)If two players' strategies are switched, then their payoffs switch as well. • In a symmetric game, if a strategy profile is a Nash equilibrium, then so is a strategy profile that has players swap strategies.
More on Nash Equilibrium
A strategy profile is a Nash Equilibrium if each player's strategy maximizes his or her payoff, given the strategies used by the other players
Nash Equilibrium and Dominant Strategies
If a player has a dominant strategy, a Nash Equilibrium requires that the player use it. If all players have a dominant strategy, then there is a unique Nash equilibrium in which each player uses his or her dominant strategy.
Nash Equilibrium Example
Detailed explanation (Fink, Fink) is a Nash equilibrium because: given that player 2 chooses Fink, player 1 is better off choosing Fink than Quiet given that player 1 chooses Fink, player 2 is better off choosing Fink than Quiet No other action profile is a Nash equilibrium. Eg, (Quiet, Quiet) is not a Nash equilibrium because: - if player 2 chooses Quiet, player 1 is better off choosing Fink; - (moreover), if player 1 chooses Quiet, player 2 is also better off choosing Fink
Why Is Nash Equilibrium Relevant to the above example?
The incentive to free ride eliminates the possibility that the mutually desirable outcome (Quiet, Quiet) occurs.
Nash Equilibrium for Sneetches Game
The sneetches is a symmetric game. • There are two Nash equilibria. • Nash equilibrium 1: (n - 1)/2 sneetches have stars and (n + 1)/2 sneetches do not. Sneetches with stars have larger payoffs than those without stars. • Nash equilibrium 2: (n + 1)/2 sneetches have stars and (n - 1)/2 sneetches do not have stars. Sneetches without stars have larger payoffs than those with stars.
Game of Sneetches
There are n sneetches and n is odd. • A sneetch can get a star painted on her belly. • If a fewer number of sneetches have stars painted on their belly than those who do not, then sneetches with stars are esteemed. • If more sneetches have stars painted on their belly than those who do not, then sneetches without stars are esteemed. • How many sneetches will have stars painted on their belly?