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:
“To brew up an adult, it seems that some leftover childhood must be mixed in; a little unfinished business from the past periodically intrudes on our adult life, confusing our relationships and disturbing our sense of self.”
—Roger Gould (20th century)
“I don’t mind their having a lot of money, and I don’t care how they employ it,
But I do think that they damn well ought to admit they enjoy it.”
—Ogden Nash (1902–1971)
“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 (1803–1882)