Symmetric 3×3 simultaneous-move game with full-support and support-restricted mixing

Setup

Definition:

Symmetric $3\times 3$ simultaneous-move game with full-support and support-restricted mixing

• Players: Two players, Player 1 and Player 2.
• Strategies: Each player chooses $A$, $B$, or $C$.

Rules

• Start with a symmetric $3\times 3$ payoff matrix; Players simultaneously choose pure strategies or mix over supports.
• The player who makes the opponent indifferent over the relevant support reaches mixed equilibrium.
• Players move simultaneously.
• Diagonal outcomes give payoffs $1$, $2$, and $3$ at $(A,A)$, $(B,B)$, and $(C,C)$; The off-diagonal entries below are a compact formalisation consistent with the visible support-analysis results in the lecture notes.

Payoff Matrix

Derivation (Best Response Analysis)

• Against $A$, Player 1 compares $1,0,0$, so $BR_1(A)=A$.
• Against $B$, Player 1 compares $1,2,0$, so $BR_1(B)=B$.
• Against $C$, Player 1 compares $0,1,3$, so $BR_1(C)=C$.
• By symmetry, $BR_2(A)=A$, $BR_2(B)=B$, and $BR_2(C)=C$.
• Let Player 2 mix with probabilities $(p,q,1-p-q)$ on $(A,B,C)$. Then Player 1’s expected payoffs are:

• Pairwise indifference boundaries are:

• Hence, for $0\le p\le 1$ and $0\le q\le 1-p$,

• By symmetry, if Player 1 mixes with $(r,s,1-r-s)$, then Player 2’s expected payoffs are:

• Hence, for $0\le r\le 1$ and $0\le s\le 1-r$,

Diagram (Best Response Regions)

• Each graph uses the first two probabilities on the axes. The probability on $C$ is the residual term.

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Derivation (Nash Equilibrium)

• Pure mutual best responses give

• For support $\{A,B\}$, set the probability on $C$ equal to $0$ and impose

• This gives

• Since $q=1-p$ on this support,

• Check the omitted action:

• For support $\{A,C\}$, set the probability on $B$ equal to $0$ and impose

• Since $q=0$ on this support, this gives

• Check the omitted action:

• For support $\{B,C\}$, set the probability on $A$ equal to $0$ and impose

• Since $p=0$ on this support, this gives

• Check the omitted action:

• For a fully mixed equilibrium, impose

• Solving gives

• By symmetry, Player 1 uses the same probability vector in each mixed equilibrium.

Nash Equilibrium

Result:

The game has seven symmetric Nash equilibria:

• pure equilibria:

$$(A,A), \qquad (B,B), \qquad (C,C),$$

• two-action mixed equilibria:

$$\left\{\left(\tfrac{1}{2},\tfrac{1}{2},0\right),\left(\tfrac{1}{2},\tfrac{1}{2},0\right)\right\},$$ $$\left\{\left(\tfrac{3}{4},0,\tfrac{1}{4}\right),\left(\tfrac{3}{4},0,\tfrac{1}{4}\right)\right\},$$ $$\left\{\left(0,\tfrac{1}{2},\tfrac{1}{2}\right),\left(0,\tfrac{1}{2},\tfrac{1}{2}\right)\right\},$$

• a fully mixed equilibrium:

$$\left\{\left(\tfrac{1}{2},\tfrac{1}{4},\tfrac{1}{4}\right),\left(\tfrac{1}{2},\tfrac{1}{4},\tfrac{1}{4}\right)\right\},$$

with strategy order $(A,B,C)$ for both players.

Social Optimum

• Total payoff at $(A,A)$ is $2$.
• Total payoff at $(B,B)$ is $4$.
• Total payoff at $(C,C)$ is $6$.
• Every off-diagonal outcome gives total payoff $0$ or $1$.
• Therefore $(C,C)$ is the unique social optimum.

Insights

Insight:

• The new matrix supports one symmetric equilibrium for every nonempty support subset of $\{A,B,C\}$.
• Support restrictions change the indifference equations, so each two-action support generates a different mixed equilibrium.
• The Pareto-dominant equilibrium $(C,C)$ is also the unique social optimum.