Stag Hunt

Definition

Stag Hunt (SH)

A symmetric two-player game with actions C (hunt the stag) and D (hunt a hare), using the same payoff names as the Prisoner’s Dilemma ($R$, $S$, $T$, $P$), but ordered $R>T>P>S$

A stag is worth a lot but can only be caught if both hunters go for it. A hare can be caught alone, and is worth a little less if the other hunter also chases hares in the same field.

Intuition

Cooperation is safe only if you trust your partner

Mutual cooperation is the best outcome for both, and it is stable: if my partner hunts the stag, so should I ($R>T$). But hunting stag alone leaves me with nothing ($S$), while a hare is guaranteed. So whether I should cooperate comes down to whether I believe the other will. The Stag Hunt is a dilemma of coordination, or of trust.

Of the two motives for defecting, the Stag Hunt has fear ($P>S$) but no greed ($T<R$). That is the contrast with the Prisoner’s Dilemma, which has both: there the problem is incentives; here it is trust.

Two coding agents

	partner C	partner D
C	4, 4	0, 3
D	3, 0	2, 2

Two coding agents can coordinate changes to shared files to implement a big feature (C), or each work on an independent, low-impact task for a smaller but safe reward (D). The feature only lands if both coordinate.

Mathematical Formulation

An agent who believes its partner cooperates with probability $p$ compares

${\pi }_C(p)=pR+(1-p)S,{\pi }_D(p)=pT+(1-p)P$

Cooperating is the best response when ${\pi }_C(p)>{\pi }_D(p)$, that is, when the belief exceeds the threshold

$p^{\ast }=\frac{P-S}{(R-T)+(P-S)}$

where:

• $P-S$: what you lose by cooperating when your partner does not (the fear motive)
• $R-T$: what you gain by cooperating when your partner also does
• $p^{\ast }$: how much trust is needed before hunting the stag is worthwhile; larger $p^{\ast }$ means more trust required

For $R=4,S=0,T=3,P=2$: $p^{\ast }=2/(1+2)=2/3$.

Which equilibrium is played more often? With beliefs spread uniformly between 0 and 1, an agent cooperates with probability $1-p^{\ast }$. So (C,C) is the more frequent outcome when $p^{\ast }<1/2$, which is when $R-T>P-S$: the gain from successful cooperation outweighs the loss from being let down. With the numbers above, $R-T=1<P-S=2$, so (D,D) wins: (C,C) occurs with probability $(1/3{)}^2\approx 0.11$, (D,D) with $(2/3{)}^2\approx 0.44$, and the rest is miscoordination. Making the hare more attractive (raising $T$) shrinks $R-T$, raises $p^{\ast }$ and makes (C,C) rarer still. (Background beyond the course material: game theorists call the equilibrium favoured by this comparison the risk-dominant one.)

Key Properties / Variants

• Three Nash equilibria: (C,C) and (D,D) in pure strategies, plus a mixed one where each cooperates with probability $p^{\ast }$. See Nash Equilibrium.
• (C,C) is the only Pareto optimum and the social optimum; it Pareto-dominates (D,D). See Pareto Optimality.
• “Rational agents will coordinate on (C,C) because it is better for both” is false. Being better for both does not make an outcome the one reached; a rational agent cooperates only if its belief exceeds $p^{\ast }$.
• Population dynamics. In the Replicator Dynamics, $x=0$ and $x=1$ are stable and the interior rest point $x=p^{\ast }$ is unstable. A population with more than $p^{\ast }$ cooperators goes to all-cooperate; with fewer, to all-defect. With $p^{\ast }=2/3$, a population starting at 60% cooperators still collapses to all-defect, even though most agents cooperate at the start.
• Same threshold in both models. In a randomly mixed population the chance that your partner cooperates is the fraction of cooperators $x$, which plays the role of the belief $p$. So the rational threshold and the evolutionary tipping point coincide.
• What changes the outcome: communication, reputation or memory can raise beliefs above $p^{\ast }$ and let agents coordinate on the stag.

Connections

• Contrast: Prisoner’s Dilemma (greed and fear versus fear only)
• Solution concepts: Nash Equilibrium, Pareto Optimality
• Population view: Replicator Dynamics
• Setting: Multi-Agent System

Appears In

• AGAI-L01 - Agents, Multiagent Systems and Agentic AI
• AGAI-Lab01 - Simple Agents, Complex Dynamics
• AGAI-L02 - Game Theory and Population Dynamics
• AGAI-L03 - Intro to Multi-Agent Reinforcement Learning (gridworld stag hunt with independent Q-learners)