Solution
Making the Other Side Indifferent
Show the problem again
Row chooses Top or Bottom and Column chooses Left or Right, simultaneously and once. Each cell is (Row's payoff, Column's payoff). Top-Left: (2, 5). Top-Right: (6, 0). Bottom-Left: (6, 2). Bottom-Right: (0, 6). The game has no pure-strategy Nash equilibrium and exactly one Nash equilibrium overall. What is Row's expected payoff in it? Give a reduced fraction.
Worked solution
The answer is 18/5. With no pure equilibrium, each player must randomize so that the other is indifferent. If Column plays Left with probability q, Row's Top earns 2q + 6(1 − q) and Bottom earns 6q, so 6 − 4q = 6q and q = 3/5. Row's payoff is then 6(3/5) = 18/5, whichever row she plays. For completeness, Row plays Top with probability p such that Column is indifferent: 5p + 2(1 − p) = 6(1 − p), giving p = 4/9. Notice that Row's equilibrium payoff is fixed by Column's mixing probability, which in turn is chosen using Row's own payoffs.
Source: Standard 2×2 mixed-equilibrium exercise; payoffs original to AxiomIQ.