Battle of the sexes
Also known as: Bach or stravinsky
Both want to be together, but each prefers a different place to meet.
What it means
A coordination game with conflicting preferences: two players most want to coordinate on the same activity, yet each favors a different option, so there are two pure equilibria that reward them unequally. Coordination is jointly better than going separate ways, but the partners disagree on which equilibrium to pick, blending cooperation with distributional conflict. The game highlights the role of focal points, communication, commitment, and turn-taking in resolving who gets their preferred outcome. It matters as the standard model of situations where parties need each other but have divergent interests — partnerships, alliances, and standard-setting among rivals.
The three equilibria
Formally the game has three Nash equilibria. Two are pure: both pick the first activity, or both pick the second, each leaving one partner happier than the other. The third is mixed, with each player randomizing to make the other indifferent. This mixed equilibrium is the trap. Because the two choose independently, they miscoordinate more than half the time and end up apart, so both earn less in expectation than at either pure outcome. Rationality alone does not tell either player which equilibrium to expect, and that indeterminacy is the problem the game poses: nothing internal to the game breaks the symmetry between two unequal but equally valid solutions.
What the evidence shows
Laboratory play confirms that coordination is genuinely hard. One-shot games without communication produce frequent miscoordination and lost payoffs. Non-binding messages help: Cooper and colleagues found even a single round of cheap talk raised coordination sharply, and one-way messages worked better than two-way, because when both speak, each claims the equilibrium that favours them and the messages cancel. Salient labels help too, but only so far. Crawford, Gneezy and Rottenstreich showed that a focal label yields near-perfect matching under symmetric payoffs, yet introducing even a minute asymmetry, nudging the game toward battle of the sexes, sharply degrades coordination, because each player now hopes the other will be the one to yield.
Correlation and taking turns
Two mechanisms reliably restore efficiency. The first is a correlated equilibrium, due to Aumann: a shared random signal, a coin or a scheduler, tells each player what to do, and because the recommendation is itself an equilibrium, neither wants to deviate once they trust the other to obey. This gives full coordination and, across draws, equal expected payoffs. The second appears when the game repeats: players alternate, one taking the preferred outcome now and conceding next time. Lau and Mui show such turn-taking can be sustained as an equilibrium and splits the surplus evenly. Both work by adding structure, a signal or a shared history, that a single anonymous play lacks.
Where it shows up
The pattern recurs wherever parties need each other but rank the joint outcomes differently. Technology standards are the classic case: rival firms both prefer one format to a fragmented market, yet each pushes the version it owns, and history is full of costly format wars. Treaties and military alliances share the shape, as do household divisions of labour and the choice of a common platform between merging teams. The loss from failing to agree is large and mutual, which keeps everyone at the table; the gap between the two agreements is what makes the bargaining bitter. The wider that gap, the harder coordination becomes.
Related but distinct
Battle of the sexes sits among three neighbours. In a pure coordination game players agree on which equilibrium is best, so any focal hint solves it; battle of the sexes adds distributional conflict, which is why communication and labels lose force. In chicken the danger is the opposite corner, where both playing aggressively is the disaster, so the pressure is to yield rather than to match. In the stag hunt the tension is risk against reward: the good equilibrium is efficient but unsafe if you doubt your partner. Battle of the sexes isolates what the others blur, players who fully agree they must coordinate and fully disagree on the terms.
Examples
A couple wants to spend the evening together but one prefers the opera and the other the game; either shared choice beats going alone.
Two firms both want a single industry charging standard, since either beats a split market, but each pushes the version built around its own patents.
Two departments must pick one shared meeting slot, and no meeting helps nobody, but sales wants Monday morning and engineering wants Thursday afternoon, each insisting theirs is the obvious choice.
Two researchers agree to co-author one paper rather than publish separately, but the economist wants to submit it to an economics journal and the psychologist to a psychology one.
Two neighbouring towns both want a single shared hospital rather than none, yet each insists it be built within its own limits to capture the jobs and the tax base.
First described in Luce & Raiffa (1957).
Key references
- Crawford, V. P., Gneezy, U., & Rottenstreich, Y. (2008). The Power of Focal Points Is Limited: Even Minute Payoff Asymmetry May Yield Large Coordination Failures. American Economic Review, 98(4), 1443-1458. doi.org/10.1257/aer.98.4.1443
- Lau, S.-H. P., & Mui, V.-L. (2008). Using Turn Taking to Mitigate Coordination and Conflict Problems in the Repeated Battle of the Sexes Game. Theory and Decision, 65(2), 153-183. doi.org/10.1007/s11238-008-9100-8
- Cooper, R., DeJong, D. V., Forsythe, R., & Ross, T. W. (1989). Communication in the Battle of the Sexes Game: Some Experimental Results. RAND Journal of Economics, 20(4), 568-587. doi.org/10.2307/2555734
- Aumann, R. J. (1974). Subjectivity and Correlation in Randomized Strategies. Journal of Mathematical Economics, 1(1), 67-96. www.dklevine.com/archive/refs4389.pdf