Coordination Game with Mixed Equilibrium

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Coordination Game with Mixed Equilibrium

• Players: Two players, Player 1 and Player 2.
• Strategies: Player 1: $U$ or $D$; Player 2: $L$ or $R$.

Rules

• Start with a coordination payoff matrix.
• Players simultaneously choose one of two actions or mix between them.
• The player who coordinates or mixes at the indifference probability is optimal.
• Players move simultaneously.
• Each player prefers to match on a different diagonal outcome, but both prefer coordination to mismatch.

Payoff Matrix

Derivation (Best Response Analysis)

• Let $q$ be the probability that Player 2 chooses $L$.
• Player 1’s expected payoff from $U$ is $3q$.
• Player 1’s expected payoff from $D$ is $4(1-q)$.
• Therefore:
  • if $3q > 4(1-q)$, Player 1 prefers $U$,
  • if $3q < 4(1-q)$, Player 1 prefers $D$,
  • if $3q = 4(1-q)$, Player 1 is indifferent.
• Solving $3q = 4(1-q)$ gives $q = \frac{4}{7}.$

• Let $p$ be the probability that Player 1 chooses $U$.
• Player 2’s expected payoff from $L$ is $4p$.
• Player 2’s expected payoff from $R$ is $3(1-p)$.
• Therefore:
  • if $4p > 3(1-p)$, Player 2 prefers $L$,
  • if $4p < 3(1-p)$, Player 2 prefers $R$,
  • if $4p = 3(1-p)$, Player 2 is indifferent.
• Solving $4p = 3(1-p)$ gives

Derivation (Nash Equilibrium)

• Pure-strategy mutual best responses occur at:
  • $(U,L)$,
  • $(D,R)$.
• A mixed equilibrium requires both players to be indifferent:

• The best-response correspondences intersect at $(0,0)$, $(1,1)$, and $\left(\frac{4}{7},\frac{3}{7}\right)$ in $(q,p)$-space.
• Therefore, in the strategy orders $(U,D)$ for Player 1 and $(L,R)$ for Player 2, the mixed equilibrium profile is

Diagram (Best Response Functions)

Nash Equilibrium

Result:

The game has three Nash equilibria:

• two pure equilibria: $(U,L)$ and $(D,R)$,
• one mixed equilibrium:

where the strategy orders are $(U,D)$ for Player 1 and $(L,R)$ for Player 2.

Social Optimum

• $(U,L)$ yields total payoff $7$.
• $(D,R)$ yields total payoff $7$.
• Off-diagonal outcomes yield total payoff $0$.
• Both pure equilibria are socially optimal relative to mismatch.

Insights

Insight:

• Mixed equilibrium arises because each player uses probabilities to keep the other indifferent.
• The mixed equilibrium is not a compromise outcome. It is a strategic randomisation over competing coordination points.