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Your best strategy is when you make the option given to your opponent is obsolete.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