Battle of the Sexes
A coordination game with asymmetric preferences over the two coordinated outcomes.
Definition
Payoff matrix representing two activities each player prefers differently but both prefer coordinating.
Two pure Nash equilibria and one mixed Nash equilibrium.
Intuition
Both want to be together; they disagree on where.
Highlights equilibrium selection challenges in coordination games.
Worked example
Pure equilibria: and where one player prefers one and the other prefers the other.
Mixed equilibrium has player 1 choose with probability and player 2 choose with probability .
The math
Payoff-dominance selects for player 1 and for player 2 — they disagree. Risk-dominance and communication can break the tie.
Correlated equilibrium expands the set of welfare-improving outcomes.
Where it is used
The VHS versus Betamax format war of the 1980s was a Battle of the Sexes: both Sony (Betamax) and JVC (VHS) preferred different standards, but consumers desperately needed a single coordinated outcome. VHS won despite Betamax's technical superiority.
The choice between iOS and Android for app developers is a Battle of the Sexes: developers prefer the platform with more users, and users prefer the platform with more apps. Network effects lock in coordination on whichever platform gains initial momentum.
More in Game theory
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