Stag hunt
A game with two equilibria, all hunt the stag or all hunt hare, where the better one needs trust. Also called the assurance game.
Game with two equilibria: both hunt the stag (best outcome) and both hunt rabbits.
Stag Hare
Stag 10, 10 1, 8
Hare 8, 1 5, 5
Both similar to Prisoner's Dilemma and importantly different in that the cooperative equilibria is an equilibria (whereas in PD it is not: the only Nash equilibria is defect/defect).
Originally, mentioned by Rousseau in a Discourse on Inequality (and likely therefore older):
If it was a matter of hunting a deer, everyone well realized that he must remain faithful to his post; but if a hare happened to pass within reach of one of them, we cannot doubt that he would have gone off in pursuit of it without scruple…"
Interestingly Wikipedia https://en.wikipedia.org/wiki/Stag_hunt mentions that there are also examples in Hume:
In addition to the example suggested by Rousseau, David Hume provides a series of examples that are stag hunts. One example addresses two individuals who must row a boat. If both choose to row they can successfully move the boat. However, if one doesn't, the other wastes his effort. Hume's second example involves two neighbors wishing to drain a meadow. If they both work to drain it they will be successful, but if either fails to do his part the meadow will not be drained.
Hume's meadow is also read the other way: Laffont and Martimort (The Theory of Incentives, 2002, p. 23) cite it as an early statement of the free-rider problem. Which reading is right depends on whether one neighbor's work drains part of the meadow (closer to a Prisoner's Dilemma) or none of it.
Ordinal matrix
The same game with ordinal payoffs, 4 best to 1 worst, (row, column):
| Column stag | Column hare | |
|---|---|---|
| Row stag | 4, 4 | 1, 3 |
| Row hare | 3, 1 | 2, 2 |
Reward > temptation > punishment > sucker (R > T > P > S). In terms of Greed and fear: fear but no greed. If you hunt stag, I do best hunting stag too. If you hunt hare, I lose by hunting stag alone.
Equilibria
- Two pure equilibria: both stag, both hare. Both stag is better for both.
- A mixed equilibrium (with these numbers, each hunts stag with probability 1/2). It is unstable: any shift in beliefs tips play to one of the pure equilibria.
- Which pure equilibrium is the likelier depends on the cardinal payoffs. With a large loss from hunting stag alone, hare is "risk-dominant" (Harsanyi and Selten 1988 [?]).
Names
Assurance game. Heckathorn calls it the assurance game and gets it from the prisoner's dilemma by reversing the two top payoffs ("The Dynamics and Dilemmas of Collective Action", 1996, pp. 258–259). Ostrom (Governing the Commons, 1990, pp. 46–47) cites Runge and Taylor for CPR situations better modeled as assurance games than as prisoner's dilemmas. More aliases: Names of collective action problems.
Taylor counts assurance as not a collective action problem, on the assumption that rational players pick the better equilibrium (The Possibility of Cooperation, 1987, pp. 19, 39). Others treat it as a real but easier problem, which is how this wiki treats it.
Typical failure and fix
- Failure: everyone settles for hare because nobody is sure enough of the others. Bank runs are the clearest case: if others are withdrawing, I should too.
- Fix: assurance. Make others' commitment visible or guaranteed: deposit insurance, Assurance contracts that refund if a threshold isn't met, public pledges. Changing everyone's payoffs also works, but is more than needed.
Cases
- Bank runs (Northern Rock).
- Revolutions and protest turnout, which have a threshold.
- The hoplite battle line and arms races fit here and as a Prisoner's Dilemma.
Related
- Prisoner's dilemma vs stag hunt
- Stag hunt vs chicken
- Coordination game: the stag hunt with the risk removed.
- Aggregation technologies: weakest-link and threshold goods tend to be stag hunts.
- Patterns of collective action
Links
- https://sites.socsci.uci.edu/~bskyrms/bio/papers/StagHunt.pdf - excellent overview
