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Nash proved that if mixed strategies (where a player chooses probabilities of using various pure strategies) are allowed, then every game with a finite number of players in which each player can choose from finitely many pure strategies has at least one Nash equilibrium, which might be a pure strategy for each player or might be a probability ...
However, many games do have pure strategy Nash equilibria (e.g. the Coordination game, the Prisoner's dilemma, the Stag hunt). Further, games can have both pure strategy and mixed strategy equilibria. An easy example is the pure coordination game, where in addition to the pure strategies (A,A) and (B,B) a mixed equilibrium exists in which both ...
This theorem is a combination of strategies in which no player can improve their payoff or outcome by changing their strategy, given the strategies of the other players. In other words, a Nash equilibrium is a set of strategies in which each player is doing the best possible, assuming what the others are doing to receive the most optimal ...
The two pure strategy Nash equilibria are unfair; one player consistently does better than the other. The mixed strategy Nash equilibrium is inefficient: the players will miscoordinate with probability 13/25, leaving each player with an expected return of 6/5 (less than the payoff of 2 from each's less favored pure strategy equilibrium).
There is a unique pure strategy Nash equilibrium. This equilibrium can be found by iterated elimination of weakly dominated strategies. [4] Intuitively, guessing any number higher than 2/3 of what you expect others to guess on average cannot be part of a Nash equilibrium. The highest possible average that would occur if everyone guessed 100 is ...
In game theory, the best response is the strategy (or strategies) which produces the most favorable outcome for a player, taking other players' strategies as given. [1] The concept of a best response is central to John Nash's best-known contribution, the Nash equilibrium, the point at which each player in a game has selected the best response (or one of the best responses) to the other players ...
In game theory, the purification theorem was contributed by Nobel laureate John Harsanyi in 1973. [1] The theorem justifies a puzzling aspect of mixed strategy Nash equilibria: each player is wholly indifferent between each of the actions he puts non-zero weight on, yet he mixes them so as to make every other player also indifferent.
It should be clear that mixed PoA ≥ pure PoA, because any pure Nash equilibrium is also a mixed Nash equilibrium (this inequality can be strict: e.g. when =, = =, =, and = =, the mixed strategies = = (/, /) achieve an average makespan of 1.5, while any pure-strategy PoA in this setting is /). First we need to argue that there exist pure Nash ...