Why groups settle for rabbit
Game theorists John Harsanyi and Reinhard Selten gave the two equilibria names. (Stag, Stag) is payoff-dominant: better for everyone. (Hare, Hare) can be risk-dominant: safer when you're unsure. To compare them, multiply what each hunter would lose by leaving each equilibrium:
| Equilibrium | Each hunter's loss from leaving | Product |
|---|---|---|
| Both hunt hare | H − L = 3 | 9 |
| Both hunt stag | S − H = 1 | 1 |
The larger product is the risk-dominant equilibrium. With these numbers: . It's exactly the same as asking whether p* is above ½: whether you'd hunt hare if you thought your partner's choice was a coin flip.
This isn't just theory. In a well-known 1990 experiment by Van Huyck, Battalio and Beil, groups of 14 to 16 people played a many-player version of this game for real money, and they quickly slid to the safest, lowest-paying outcome.
More hunters, more trust needed
Now suppose the stag needs n hunters, and every one must hold their post. If you think each of the others shows up independently with probability q, the chance that all n − 1 of them do is qn−1.
With n hunters, the stag is the better bet only if qn−1 > p*, which means q > (p*)1/(n−1).
Apply Theorem 2 with "all the others show up" in place of "my partner shows up". That event has probability qn−1, so you need qn−1 > p*. Taking the (n−1)th root of both sides gives the threshold. QED
This is the hidden cost of size. Big groups only bring down stags when nearly everyone is extremely dependable. Small, reliable groups catch stags that big, loose ones can't.
The whole argument on one page
- No move is best in every case. Your best move depends on your partner's (Lemma 1).
- There are two stable outcomes: everyone hunts stag, or everyone hunts hare. The stag is better for all (Theorem 1).
- Hunt the stag only if you're more than p* = (H−L)/(S−L) sure your partner will (Theorem 2).
- There's a tipping point at p*. Groups slide toward trust or toward caution depending on where they start (Theorem 3).
- The trust required grows quickly with group size (Theorem 4).
So the obstacle to the best outcome is not greed. It is doubt. Remove the doubt and the stag is caught. QED
Four ways to catch the stag
The threshold p* = (H−L)/(S−L) tells you exactly which levers move the outcome:
A bigger stag
The more valuable the shared prize, the less certainty anyone needs to go for it.
A softer fall
If failing together costs less, trying together is safer. Limit what anyone can lose.
Fewer hunters
Every extra person who must show up compounds the risk. Keep the circle tight.
Visible trust
Repeated play, reputation, and members who give before they take make showing up the expected thing to do.