Battle of the Sexes
The Battle of the Sexes is a two-player coordination game in which both players want to choose the same option but each prefers a different one. It has two pure Nash equilibria, each favoring one player, and a mixed equilibrium in which miscoordination can occur.
Some textbooks call it Bach or Stravinsky.
Payoff matrix
| Row player (rows) / Column player (columns) | Row player's preferred option | Column player's preferred option |
|---|---|---|
| Row player's preferred option | (5, 3) Nash equilibrium | (0, 0) |
| Column player's preferred option | (0, 0) | (3, 5) Nash equilibrium |
Payoffs are ordinal scores from the DecisionPlay engine's template for this game, 0 to 5, higher is better. They rank outcomes and are not dollar values. [MODELED]
Nash equilibria
- Pure: Row player plays Row player's preferred option, Column player plays Row player's preferred option. Payoffs (5, 3).
- Pure: Row player plays Column player's preferred option, Column player plays Column player's preferred option. Payoffs (3, 5).
- Mixed: Row player plays Row player's preferred option 62.5%, Column player's preferred option 37.5%; Column player plays Row player's preferred option 37.5%, Column player's preferred option 62.5%. Expected payoffs (1.88, 1.88).
Computed by the DecisionPlay solver from the matrix above. [MODELED]
Where it shows up
Illustrative
- A couple choosing an evening out. The textbook origin: two people each prefer a different event but would rather attend the same one than go separately.
- Agreeing on a standard that favors each side differently. Two firms need to adopt one common specification, and each would rather it be the one it has already built.
Common misreadings
- There is one predicted outcome. There are two pure equilibria, each favoring a different player, and the model alone does not say which one will occur.
- The mixed equilibrium is a good compromise. In the mixed equilibrium the players sometimes fail to coordinate, which leaves both with lower expected payoffs than either pure equilibrium gives them.
Try it
Map your own scenario or edit a payoff matrix in the solver.