Payoff-dominant vs risk-dominant
Stag Hunt
You and a partner can hunt a stag together, or each grab a hare alone. The stag is a feast — but only if both of you commit. A hare is smaller, and you can catch it by yourself. If you go for the stag and they go for a hare, you come home empty-handed.
Rousseau's stag hunt is the cleanest illustration of a coordination problem with a payoff-dominant and a risk-dominant equilibrium. Both hunting stag is better for everyone. Both hunting hare is safer if you doubt your partner. Unlike the Prisoner's Dilemma, cooperation is itself a Nash equilibrium — nobody wants to deviate if the other is already committed.
The remaining problem is assurance. If you are even slightly unsure they will show up, a hare starts to look rational. One defection can unravel a string of successful hunts. That is why communication, shared history, and repeated play matter so much here.
Play it against a Tit-for-Tat bot and you can feel the difference: start with stag and the hunt holds. Start with hare and you may never get back.
What theory predicts
There are two pure Nash equilibria: both hunt stag, and both hunt hare. Stag/stag pays more (payoff-dominant). Hare/hare is safer if you are not sure they will show up (risk-dominant). Unlike the Prisoner's Dilemma, cooperation here is also an equilibrium — the problem is trust, not dominance.
What people actually do
People often manage to hunt stag, especially after they have seen each other cooperate. A single hare can unravel the hunt. Communication, reputation, and repeated play all push toward the stag.