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But there is a distributive tension between the players.
The point being that even among altruistically motivated agents, distributive tensions can threaten coordination.
Given this tension, perhaps the mixed-strategy equilibrium is the most plausible of the three equilibria for the game, despite being the only inefficient equilibrium.
If one man approaches the blonde while the other approaches a brunette then, again, each will enjoy the company of that individual.
The Figure below represents this as a simple 2×2 game.
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.
Or they might just flip a coin and easily coordinate on a single equilibrium.
They need a coordination mechanism of some sort, however.
Nash and all his friends prefer to pair with the blonde relative to the brunettes, but prefer to pair with one of the brunettes relative to not pairing up at all.
As Nash explains it, if all the men approach the blonde first, none of the men will pair off: The blonde will reject them all as a crowd, and her brunette friends will reject them all individually because none of the women will accept being second choice to her friend.