In a game without a pure strategy Nash equilibrium, a mixed strategy may result in a Nash equilibrium.
Consider a modified version of Benjamin Bunny and Al Paca in which there are just two fields, A and B. If Al sprays A and Ben chooses A for foraging then Al wins and Ben loses, and so on. Their wins and losses are shown in the payoff table.
|Ben||A||-1, 1||1, -1|
|B||1, -1||-1, 1|
The best responses in this game are
|bBen(AAl) = B||bAl(ABen) = A|
|bBen(BAl) = A||bAl(BBen) = B|
The game seems to have no Nash equilibrium. Suppose Al plays the mixed strategy (A,p). When Ben chooses to graze A then his expected payoff is p(-1)+(1-p)(1) = 1-2p. When Ben chooses B then his expected payoff is p(1) + (-1)(1-p) = 2p-1.
|Al plays mixed strategy (A, p)|
|Ben's payoff from playing A||Ben's payoff from playing B|
In response to Al's mixed strategy, Ben's expected payoff from grazing A is better than that from choosing B only when
When p = 1/2 Ben finds that the pure strategy of A and the pure strategy of B have the same expected payoff. By also playing a mixed strategy of (A, 1/2) Ben is playing a best response. In conclusion, while there was no Nash equilibrium when the two players were confined to pure strategies, there is a Nash equilibrium when they play mixed strategies.