Assurance Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Assurance Game

• Players: Player 1 and Player 2.
• Strategies: Each player chooses Red or Black.
• Rules: Players move simultaneously and coordinate successfully when their colours match.

Payoff Matrix

Derivation (Best Response Analysis)

• If Player 2 chooses Red, Player 1 compares $2$ with $0$, so $BR_1(\text{Red})=\text{Red}$.
• If Player 2 chooses Black, Player 1 compares $0$ with $5$, so $BR_1(\text{Black})=\text{Black}$.
• By symmetry, $BR_2(\text{Red})=\text{Red}$ and $BR_2(\text{Black})=\text{Black}$.
• Mutual best responses occur at (Red,Red) and (Black,Black).
• The off-diagonal outcomes are not Nash equilibria because each player wants to match the other player’s colour.

Nash Equilibrium

Result:

The pure-strategy Nash equilibria are:

Social Optimum

• Total payoff at (Red,Red) is $4$.
• Total payoff at (Black,Black) is $10$.
• (Black,Black) is socially optimal and Pareto-dominates (Red,Red).

Insights

Insight:

• Multiple equilibria create a coordination-selection problem.
• The payoff-dominant equilibrium may fail if players expect cautious coordination.
• Bank runs and panic buying use the same self-fulfilling logic.