Mixed Strategies | Lecture 13

1. What are Mixed Strategies?

Mixed Strategy

A probability distribution over a player's pure strategies.

σᵢ : Sᵢ → [0, 1] with Σσᵢ(s) = 1

Pure Strategy

Choose one specific action with probability 1.
Example: "Always play Rock"

Mixed Strategy

Randomize over actions with specific probabilities.
Example: "Rock 1/3, Paper 1/3, Scissors 1/3"

Why Mix?

No pure NE exists: Some games have no stable pure strategies
Unpredictability: In competitive games, being predictable is exploitable
Nash's Theorem: Every finite game has at least one (possibly mixed) NE

2. Interactive: Rock-Paper-Scissors

Play Against Different AI Strategies

Your Choice

Computer's Choice

❓

AI Strategy: Optimal mixed strategy - cannot be exploited!

0

Wins

0

Draws

0

Losses

Your Move Distribution:

Payoff Analysis:

Your Win Rate: 0%

Expected Payoff vs Mixed AI: 0

Against Nash strategy (⅓,⅓,⅓), no strategy beats 50% win rate long-term.

3. Computing Mixed Nash Equilibrium

The Indifference Principle

At a mixed NE, each player must be indifferent between all strategies they play with positive probability. Otherwise, they would prefer one over the other!

Example: Matching Pennies

	H	T
H	1, -1	-1, 1
T	-1, 1	1, -1

P1 wins if same, P2 wins if different

Finding Mixed NE:

Let P2 play H with prob q, T with prob (1-q).

P1's expected payoff for H: 1·q + (-1)·(1-q) = 2q - 1

P1's expected payoff for T: (-1)·q + 1·(1-q) = 1 - 2q

Indifference: 2q - 1 = 1 - 2q → q = ½

Mixed NE: Both play (½ H, ½ T)

General Method for 2×2 Games

Let P1 play strategy 1 with prob p, P2 with prob q
Write expected payoffs for each player's strategies
Set expected payoffs equal (indifference)
Solve for p and q

4. Properties of Mixed Strategies

Existence Guaranteed

Every finite game has at least one mixed strategy Nash Equilibrium (Nash 1950).

Interpretation Debate

Do players actually randomize? Or does mixing represent beliefs about opponent behavior?

Unexploitable

Playing the Nash mixed strategy guarantees at least the value of the game - opponent cannot exploit you.

Sports Applications

Penalty kicks, tennis serves, play calling - mixed strategies observed in competitive sports!