Chicken Game¶

Definition¶

The chicken game is a symmetric two-player normal-form game in which each player chooses $S$ (swerve, yield) or $T$ (go straight, stand firm), and each player's payoffs satisfy

Here $v_i(a_i,a_j)$ is player $i$'s payoff when she plays $a_i$ and the other player plays $a_j$. Standing firm is best if the other yields, yielding is best if the other stands firm, and mutual standing firm is the worst outcome. A standard numerical version:

Interpretation¶

Chicken is an anti-coordination game: each player wants to do the opposite of the other. It models standoffs such as two drivers heading for each other, price wars, or a confrontation in which backing down is costly but collision is disastrous. In biology the same structure is called the hawk–dove game.

In the figure $p$ and $q$ are the probabilities with which players 1 and 2 choose $T$ in the table above. Player 1's best response $BR_1$ is $T$ ($p=$$p=1$$1$) when $q<\tfrac{1}{3}$$\frac{1}{3}$, $S$ ($p=$$p=0$$0$) when $q>\tfrac{1}{3}$$\frac{1}{3}$, and any $p$ when $q=\tfrac{1}{3}$$\frac{1}{3}$; player 2's $BR_2$ is the mirror image. Because each player wants the opposite of the other, both curves step downwards and cross three times: at the two pure equilibria and at the mixed equilibrium $(\tfrac{1}{3},\tfrac{1}{3})$.

Compare it with the Prisoner's Dilemma. There, defecting is best whatever the other does, so there is a single equilibrium. In chicken, no action is dominant, $(T,T)$ is worse than being the one who yields, and the game has three equilibria. The theory alone does not say which one is played; the two pure equilibria require the players to know who yields.

Chicken is a standard example for purification: if each player's toughness, that is the payoff from standing firm, is slightly private, the mixed equilibrium can be reinterpreted as pure behaviour by different types, and the game becomes a Bayesian game.

Key Results¶

• The pure Nash equilibria are $(S,T)$ and $(T,S)$; $(S,S)$ and $(T,T)$ are not equilibria.
• There is exactly one further Nash equilibrium in mixed strategies, symmetric, in which each player chooses $T$ with the probability that makes the other indifferent.
• No action is strictly dominated, so iterated elimination of strictly dominated strategies removes nothing.
• The mixed equilibrium is Pareto-dominated by $(S,S)$, which is not an equilibrium: it is not Pareto efficient.

Examples¶

Mixed equilibrium in the table above. Let the opponent play $T$ with probability $q$. By the indifference principle, $S$ yields $-q$ and $T$ yields $(1-q)-3q=1-4q$$4q$; equating gives $q=\tfrac{1}{3}$$\frac{1}{3}$. By symmetry each player plays $T$ with probability $\tfrac{1}{3}$, and each earns $-\tfrac{1}{3}$.

• The crash $(T,T)$ occurs with probability $\tfrac{1}{9}$.
• Both players get $0>-\tfrac{1}{3}$ at $(S,S)$, which confirms the Pareto comparison.
• In the pure equilibria the payoffs are $(-1,1)$ and $(1,-1)$.

General parameters. With $v_i(S,S)=0$$0$, $v_i(T,S)=w>0$$0$, $v_i(S,T)=-\ell<0$ and $v_i(T,T)=-c$ with $c>\ell$$\ell$, indifference gives $-\ell q=w(1-q)-cq$$cq$, so $q=w/(w+c-\ell)$$\ell )$. A more costly crash (larger $c$) lowers the probability of standing firm. The table above has $w=1$$1$, $\ell=1$$1$, $c=3$$3$, so $q=1/3$$1/3$.

Prerequisites¶

• Normal-Form Game
• Nash Equilibrium
• Nash Equilibrium in Mixed Strategies
• Indifference Principle for Mixed Strategies

Related Concepts¶

• Harsanyi Purification Theorem
• Bayesian Game
• Pareto Efficiency
• Folk Theorem
• Prisoner's Dilemma

Sources¶

• Game Theory · Week 5

Aliases: game of chicken, chicken, hawk–dove game, hawk-dove game, snowdrift game Keywords: anti-coordination game, swerve or go straight, two pure equilibria and one mixed equilibrium

Last updated: 27 October 2026 11:52