Mixed Nash Equilibrium
Coordination games also have mixed strategy Nash equilibria. In the generic coordination game above, a mixed Nash equilibrium is given by probabilities p = (d-b)/(a+d-b-c) to play Up and 1-p to play Down for player 1, and q = (D-C)/(A+D-B-C) to play Left and 1-q to play Right for player 2. Since d > b and d-b < a+d-b-c, p is always between zero and one, so existence is assured (similarly for q).
The reaction correspondences for 2×2 coordination games are shown in Fig. 6.
The pure Nash equilibria are the points in the bottom left and top right corners of the strategy space, while the mixed Nash equilibrium lies in the middle, at the intersection of the dashed lines.
Unlike the pure Nash equilibria, the mixed equilibrium is not an evolutionarily stable strategy (ESS). The mixed Nash equilibrium is also Pareto dominated by the two pure Nash equilibria (since the players will fail to coordinate with non-zero probability), a quandary that led Robert Aumann to propose the refinement of a correlated equilibrium.
Read more about this topic: Coordination Game
Famous quotes containing the words mixed, nash and/or equilibrium:
“Let us not deny it up and down. Providence has a wild, rough, incalculable road to its end, and it is of no use to try to whitewash its huge, mixed instrumentalities, or to dress up that terrific benefactor in a clean shirt and white neckcloth of a student of divinity.”
—Ralph Waldo Emerson (1803–1882)
“The witch’s face was cross and wrinkled,
The witch’s gums with teeth were sprinkled.”
—Ogden Nash (1902–1971)
“They who feel cannot keep their minds in the equilibrium of a pair of scales: fear and hope have no equiponderant weights.”
—Horace Walpole (1717–1797)