If we consider the class of congestion game more broadly, again considering the case of aid agencies rushing to get assistance to a remote region, there are incentives for the aid agencies to coordinate in advance: for one agency to send in aid via the road, for the other agency to send in aid via the airport.
But a distributive tension exists here as well, despite the altruistic aims of the agencies.
A Nash equilibrium exists when none of the players can do better (i.e., receive a higher payoff) by changing to another strategy given the strategy the other player is playing.
I make the players’ payoffs cardinal rather than ordinal simply for concreteness; we could use any numbers consistent with the players’ ordinal ranking of the outcomes.
So, in the scene, a group of women come into the bar in which Nash sits with a group of his friends who are all men.
The group of women include one blonde woman, and the rest are brunettes.If both men approach the blonde, both men will remain alone.If both men approach a (different) brunette, then each will enjoy the company of that person.To wit, there are two men, who are the strategic actors in Nash’s story, and a group of women, one of whom is blonde, the others are brunettes.Both men prefer the blonde to the brunettes, but prefer a brunette to being alone.Both of these strategies are (Pareto) efficient, that is, there is no outcome in the game in which one player can do better without the other player doing worse.The mixed-strategy equilibrium exists with both players “mixing” between the strategies with probability one half (0.5).(Strategies in pure strategy equilibria are played with probability 1 or zero; strategies in mixed-strategy equilibria are played with probabilities less than one but greater than zero.) The two pure strategy equilibria are mirrors of each other.One of the boys pairs with the blonde, the other boy pairs with a brunette.(This is the “p” and the “q” in the figures.) Expected payoffs for each player in the mixed-strategy equilibrium is “1.” So this is an inefficient equilibrium.It is the nature of the equilibria that presents the problem that the film’s narrative attempts to solve. The boys might agree that one of them should end up with the blonde, and so agree they should coordinate their actions.