Nash Equilibrium

Two hunters head into the forest. Each must choose, alone and without consulting the other, what to hunt. Stag is the big prize — feeds them for a week — but it takes both of them to bring one down. Hare is the small prize — feeds one person for a day — but you can catch one alone. If one hunter goes for stag while the other goes for hare, the stag-hunter comes home empty.

This is the Stag Hunt, the canonical coordination game. Cooperation pays best, but only if both players commit. The hunters share a payoff matrix and need to predict each other’s move.

How do we predict what happens? The standard answer is the Nash equilibrium, named after John Nash. A Nash equilibrium is a set of strategies — one per player — where nobody can improve their payoff by changing their own strategy alone. It is a stability check, not an optimality claim. At a Nash equilibrium, everyone is reacting correctly to everyone else. Nobody regrets their own move given what the others did.

To find Nash equilibria, check each cell. Given everyone else’s choice, is the player’s current move their best response? If yes for every player at once, the cell is a Nash equilibrium. A game can have zero, one, or several. The Stag Hunt has two, and they are not equally good.