Mixed-Strategy Chicken Game / Mixed-Strategy Anti-Coordination Game

Content map: SMU H3 Game Theory Map

Setup

Definition:

Mixed-Strategy Chicken Game / Mixed-Strategy Anti-Coordination Game

Players: Player 1 and Player 2.

Strategies: Each player mixes over Straight (ST) and Swerve (SW).

Rules:

Players move simultaneously.

Player 1 uses $(p,1-p)$ and Player 2 uses $(c,1-c)$ over (ST,SW).

Payoff Details

ST wins against SW.
In mixed equilibrium, each player is indifferent between ST and SW.

Payoff Matrix

	ST $(c)$	SW $(1-c)$
ST $(p)$	-1, -1	2, 1
SW $(1-p)$	1, 2	0, 0

Derivation (Mixed-Strategy Indifference)

Let $c$ be the probability that Player 2 chooses ST.
Player 1’s expected payoff from ST:

E(\pi_1 \mid \text{ST})=-c+2(1-c)=2-3c

Player 1’s expected payoff from SW:

E(\pi_1 \mid \text{SW})=c

Player 1 is willing to mix only if:

2-3c=c \quad \Rightarrow \quad c=\frac{1}{2}

Let $p$ be the probability that Player 1 chooses ST.
Player 2’s expected payoff from ST:

E(\pi_2 \mid \text{ST})=-p+2(1-p)=2-3p

Player 2’s expected payoff from SW:

E(\pi_2 \mid \text{SW})=p

Player 2 is willing to mix only if:

2-3p=p \quad \Rightarrow \quad p=\frac{1}{2}

Nash Equilibrium

Result:

The mixed-strategy Nash equilibrium is:

\{\text{P1},\text{P2}\}\mapsto \left\{ \left(\frac{1}{2},\frac{1}{2}\right), \left(\frac{1}{2},\frac{1}{2}\right) \right\}

where the strategy order is (ST,SW) for both players.

Total payoff at (ST,SW) is $3$ .
Total payoff at (SW,ST) is $3$ .
Total payoff at (SW,SW) is $0$ .
Total payoff at (ST,ST) is $-2$ .
The socially optimal outcomes are the asymmetric anti-coordination outcomes.

Insights

Insight:

Mixed play keeps each player indifferent between standing firm and yielding.

Each player chooses ST with probability $\frac{1}{2}$ .

Randomisation reflects strategic unpredictability, not confusion.

Mixed-Strategy Chicken Game / Mixed-Strategy Anti-Coordination Game

Setup

Payoff Details

Payoff Matrix

Derivation (Mixed-Strategy Indifference)

Nash Equilibrium

Social Optimum

Insights