Consider a game where player 1 has three strategies (T, M, B) and player 2 has two strategies (L, R). Player 2 plays L with probability q, and consequently R with probability (1 − q). Player 1 is trying to compare the payoff from playing the pure strategy B with the payoff from playing T with probability p and M with probability (1 − p). The payoffs to player 1 are given below:
L R
T 3, _ 0, _
M 0, _ 3, _
B 2, _ 2, _
With player 2 playing a (q, 1 − q) mix of strategies L and R, when will
player 1 receive a higher payoff from a (p, 1 − p) mix of strategies T and M than from strategy B? To answer this question in the most general way possible, you should
Calculate an expression for player 1’s payoff as a function of p and q. Remember that player 1 is using a (p, 1 − p) mix of strategies T and M and player 2 is using a (q, 1 − q) mix of strategies L and R.
Use this expression to obtain conditions under which the (T, M) mix will provide player 1 with a higher payoff than B will as a pure strategy. These conditions involve comparing p to a certain function f(q).
Graph f(q) vs. q with relative accuracy. Use your graph to show what combinations of p and q will result in a higher payoff to player 1 from the (T, M) mix than from the pure strategy B.
Provide a specific combination of p and q for which player 1 receives a lower payoff from the (T, M) mix than from the strategy B.