SMU H3 Notes Game TheoryGamesSMU H3

Stag Hunt Game

Game theory analysis: Stag Hunt Game.

1 May 2026

Content map: SMU H3 Game Theory Map

Setup

Definition:

Stag Hunt Game

Players: Hunter 1 and Hunter 2.

Strategies: Each hunter chooses Stag or Hare.

Rules: Players move simultaneously; coordinating on Stag reaches the payoff-dominant outcome.

Payoff Matrix

	Stag	Hare
Stag	5, 5	0, 3
Hare	3, 0	4, 4

Derivation (Best Response Analysis)

If Hunter 2 chooses Stag, Hunter 1 compares $5$ with $3$ , so $BR_1(\text{Stag})=\text{Stag}$ .
If Hunter 2 chooses Hare, Hunter 1 compares $0$ with $4$ , so $BR_1(\text{Hare})=\text{Hare}$ .
By symmetry, $BR_2(\text{Stag})=\text{Stag}$ and $BR_2(\text{Hare})=\text{Hare}$ .
Mutual best responses occur at (Stag,Stag) and (Hare,Hare).

Nash Equilibrium

Result:

The pure-strategy Nash equilibria are:

\{\text{H1},\text{H2}\}\mapsto\{\text{Stag},\text{Stag}\}, \qquad \{\text{H1},\text{H2}\}\mapsto\{\text{Hare},\text{Hare}\}

Total payoff at (Stag,Stag) is $10$ .
Total payoff at (Hare,Hare) is $8$ .
(Stag,Stag) is socially optimal and Pareto-dominates (Hare,Hare).

Insights

Insight:

Stag is payoff-dominant because it gives both players $5$ .

With a $1/2$ belief on each action by the other hunter:

E(\text{Hare})=\frac{3+4}{2}=3.5, \qquad E(\text{Stag})=\frac{5+0}{2}=2.5

Hare is risk-dominant because it gives the higher expected payoff under equal uncertainty.

The game separates efficiency from safety.

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