Zero-Sum Mixed-Strategy Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Zero-Sum Mixed-Strategy Game

• Players: Two players, Kicker and Goalie.
• Strategies: Kicker: shoot Left or Right; Goalie: dive Left or Right.

Rules

• Start with a kicker and goalkeeper choosing sides; Players simultaneously choose Left or Right.
• The player who correctly exploits the side mismatch or match wins the zero-sum contest.
• Players move simultaneously.
• The kicker’s payoff is scoring probability; The goalie’s payoff is save probability.

Payoff Matrix

Derivation (Best Response Analysis)

• Let $q$ be the probability that the goalie dives Left.
• Kicker’s expected payoff from shooting Left:

• Kicker’s expected payoff from shooting Right:

• Indifference for the kicker requires

• Let $p$ be the probability that the kicker shoots Left.
• Goalie’s expected payoff from diving Left:

• Goalie’s expected payoff from diving Right:

• Indifference for the goalie requires

Derivation (Nash Equilibrium)

• Each player chooses probabilities that make the opponent indifferent.
• Therefore, in the strategy orders (Left, Right) for both players, the mixed equilibrium profile is

• The value of the game to the kicker is

Nash Equilibrium

Result:

The unique mixed Nash equilibrium is

where the strategy order is (Left, Right) for both players. The kicker’s equilibrium scoring probability is $\frac{955}{12}\%\approx 79.58\%$.

Social Optimum

• Not applicable in the usual welfare sense because the game is zero-sum.
• The relevant aggregate concept is the equilibrium value of the game.

Insights

Insight:

• In zero-sum games, predictable pure strategies are exploitable.
• Observed frequencies in professional sport are often close to the minimax prediction.