21.20 Mixed Strategies

Games with finite number, and finite strategy for each player, there will always be a Nash equilibrium (might not be pure Nash, but always mixed)

Extended game

• Suppose we allow each player to choose randomizing strategies

• For example, the server might serve left half of the time, and right half of the time

• In general, suppose the server serves left a fraction $p$ of the time

• What is the receiver’s best response?

Best Responses

If $p = 1$, the receiver should defend to the left

$p = 0$, the receiver should defend to the right

The expected payoff to the receiver is

$p \times 3/4 + (1-p) \times 1/4$ if defending left

$p \times 1/4 + (1-p) \times 3/4$ if defending right

Hence, she should defend left if

$p \times 3/4 - (1-p)\times 1/4 > p \times 1/4 + (1-p) \times 3/4$

We said to defend left whenever

$$p \times 3/4 - (1-p)\times 1/4 > p \times 1/4 + (1-p) \times 3/4$$

Server’s Best response

• Suppose that the receiver goes left with probability $q$

• if $q = 1$, the server should serve right

• If $q = 0$, the server should server left

• Hence, serve left if $1/4 \times q + 3/4 \times (1-q) > 3/4\times q + 1/4 \times (1-q)$

• Simplifying, he should serve left if $q < 1/2$

Mixed strategy equilibrium:

• A mixed strategy equilibrium is a pair of mixed strategies that are mutual best responses

• In the tennis example, this occurred when each play chose a 50-50 mixture of left and right

• Your best strategy is when you make the option given to your opponent is obsolete.

• A player chooses his strategy to make his rival indifferent

• A player earns the same expected payoff for each pure strategy chosen with positive probability

• Important property: When player’s own payoff form a pure strategy goes up (or down), his mixture does not change.