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:
“Those graceful acts,
Those thousand decencies, that daily flow
From all her words and actions, mixed with love
And sweet compliance, which declare unfeigned
Union of mind, or in us both one soul.”
—John Milton (16081674)
“It is the sin of omission, the second kind of sin,
That lays eggs under your skin.”
—Ogden Nash (19021971)
“There is a relation between the hours of our life and the centuries of time. As the air I breathe is drawn from the great repositories of nature, as the light on my book is yielded by a star a hundred millions of miles distant, as the poise of my body depends on the equilibrium of centrifugal and centripetal forces, so the hours should be instructed by the ages and the ages explained by the hours.”
—Ralph Waldo Emerson (18031882)