Solution
Battle of the Sexes
Show the problem again
A couple prefers being together but disagrees on the venue. Payoffs: both at Opera (2, 1), both at Football (1, 2), apart (0, 0). Besides the two pure equilibria, there is a mixed Nash equilibrium. With what probability does each player attend their own preferred venue in it? Express as a fraction.
Worked solution
The answer is 2/3. Let Player 2 attend Opera with probability q. Player 1 is indifferent when 2q = 1(1 − q), i.e., q = 1/3, so Player 2 attends their preferred Football with probability 2/3. By symmetry, Player 1 attends Opera with probability 2/3. In this equilibrium they miscoordinate with probability 5/9, which is why the mixed equilibrium is inefficient.
Source: Standard coordination game of the game-theory literature, named and analysed in Luce and Raiffa, 'Games and Decisions' (1957). Statement written for AxiomIQ.