Question

My question is about game theory. Say we have a game with mixed equilibria, but no pure Nash equilibria. How does the strategy of one player affect the strategy of the other player in a mixed equilibrium?

Answer 1

Mixed strategy Nash equilibrium is the probability distribution that is given to each of the actions with the likelihood of being selected.

This type of strategy is randomized ie atleast one player randomly chooses the strategy and no player can increase his expected payoff by playing an alternate strategy.

In simple words, if one player would randomly choose our randomize between strategies only when he has the same expected payoff from both. If one gives more expected payoff than the other then he won't be indifferent between them.

We consider the example of matching pennies game

Player 2

Player 1

	Head	tail
Head	(1,-1)	(-1,1)
Tail	(-1,1)	(1,-1)

Here suppose that player one knows that player 2 plays head with probability p which implies that player one when chooses head would get payoff 1 with probability p and-1 with probability(1-p).

So the expected payoff is 1(P)+(-1)(1-p) = 2p-1

Similarly player 1' s expected utility when he chooses tail given player 2 is choosing head with probability p

Expected payoff is (-1)P+(1)(1-p) = 1-2p

Now we know that player 1 would be indifferent only when the payoffs give same value.

So this means that the payoffs are equal ie

1-2p= 2p-1

4p= 2

P= 1/2

Thus Row randomizes to make Column indifferent and vice versa. That would lead to each player playing the best. One player needs to randomize in order to let other be indifferent would be the strategy in mixed equilibrium.

(You can comment for doubts)