Two stable outcomes, not one
Call a pair of choices an equilibrium if neither hunter could do better by changing only their own choice. (This is a Nash equilibrium, after John Nash.) Equilibria are where situations settle, because no one has a reason to move.
Whenever S > H > L, the stag hunt has exactly two equilibria in pure choices: both hunt the stag, and both hunt a hare.
- (Stag, Stag). If A switches to hare, A gets H instead of S. Since S > H, A won't. The same goes for B. Equilibrium.
- (Hare, Hare). If A switches to stag, A hunts alone and gets L instead of H. Since H > L, A won't. The same goes for B. Equilibrium.
- (Stag, Hare). A gets L, but could get H > L by switching to hare. A wants to move. Not an equilibrium.
- (Hare, Stag). The mirror image: B wants to move. Not an equilibrium.
Those are all four possibilities, so there are exactly two. QED
Look at what this means. Both outcomes are self-sustaining, yet one is clearly better for both hunters: S > H. Nobody has to be greedy or dishonest for the group to end up eating rabbit. The rules alone can't pick the outcome. Expectations pick it.
Why this isn't the Prisoner's Dilemma
| Prisoner's Dilemma | Stag Hunt | |
|---|---|---|
| Is the best joint outcome stable? | No: each side gains by betraying | Yes: no one gains by leaving |
| Is one move always best? | Yes: defect | No: it depends on your partner |
| What gets in the way | Temptation | Doubt |
| What fixes it | Enforcement and punishment | Assurance and trust |
How sure do you need to be?
In real life you never know for certain what your partner will do. So let p be how likely you think it is that they hunt the stag. Now compare your two options on average:
Hunting the stag is the better bet exactly when
Stag is better when pS + (1 − p)L > H. Subtract L from both sides to get p(S − L) > H − L. Since S > L, divide by S − L to get p > (H − L)/(S − L). QED
With our numbers, p* = 0.75. You should only go for the stag if you're more than 75% sure your partner will too, even though the stag is the better meal. Drag p below and watch which line is on top.
Here's the intuition in one line. Rewrite the threshold as
Hunting the stag risks 3 (falling from the hare's 3 to 0) to gain 1 (rising from 3 to 4). When the downside is big and the upside is modest, you need near-certainty. That's why the good outcome is so hard to reach.
The knife's edge
If each hunter goes for the stag with probability exactly p*, neither can do better by changing. This is an equilibrium too, but it pays only H = 3, no better than playing safe, and the slightest push knocks it over.
If your partner hunts the stag with probability p*, Theorem 2's calculation gives E[stag] = p*S + (1 − p*)L = H = E[hare]. You're exactly indifferent, so every mix, including p*, is a best response. By symmetry the same holds for your partner. Both payoffs equal H. QED
Now zoom out from two hunters to a whole community. People pair up at random, and whatever earns more tends to spread as people copy what works. If a share x of people hunt stag, stag hunters average xS + (1 − x)L, while hare hunters get H. So stag hunting spreads exactly when x > p*, and dies out when x < p*.
This is why cultures of trust and cultures of cynicism both persist. A group that starts out trusting stays trusting; a group that starts out wary stays wary. Where you start decides where you end up.