[Ward Farnsworth (guest-blogging), August 1, 2007 at 9:06am] Trackbacks
The Stag Hunt.

The Legal Analyst has a section on cooperation and other problems for groups. Some of the chapters in that section involve game theory, and one of them covers the game known as the stag hunt. Let me explain it here. It's fun to think about, and I've found that it's less well-known than the prisoner's dilemma (which is also discussed in the book, of course).

The two of us are hunters. If we cooperate we can take down a stag and have it for dinner; if we work alone, we'll only be eating hare. My hope, of course, is that we'll cooperate and eat stag. But I wouldn't want to hunt stag if you don't; that's my last choice. It will be a waste of time and leave me hungry. Of course you don't want to go after stag, either, unless you're sure I will, too.

So maybe we can agree to work jointly and have a stag hunt, but there is risk in it. Unless we have good reason to trust each other, the temptation will be great for either of us to give up on the stag and give chase to any hare that goes by. At least if I do get a hare I no longer have to worry that I'll end up with nothing if you decide to go after a hare because you think it's good insurance against the danger that I'll go after a hare because I'm worried that—and so forth.

We owe this parable to Rousseau's Discourse on the Origin and Basis of Inequality among Men. There are many situations like this where players in a drama do best by all cooperating, but otherwise had better all defect; if they don't all cooperate (or if some critical mass won't cooperate), in other words, then nobody should — so perhaps nobody does. A mutiny can be another simple example, or (more vividly in our times) efforts to overpower hijackers on an airplane.

The stag hunt may remind some readers of the prisoner's dilemma, but it works differently. In a prisoner's dilemma it's best for you to defect no matter what I do. In a stag hunt that's not true; if I know you will cooperate, I'll want to cooperate, too. Sometimes a stag hunt is said to be an assurance game because that is the true issue: if everyone has good assurances that everyone else will cooperate, they will be glad to do the same.

For understanding these games, the idea of a Nash equilibrium helps. The players of any game have arrived at a Nash equilibrium if, given what everyone else is doing, none can do better by changing their strategies. Sometimes there is more than one equilibrium the parties can reach, and a stag hunt is an example of this. Do you see why?

Eating hare is an equilibrium. If we take as given that you're eating hare, there's no sense in my getting up to chase stag; if we take as given that I'm eating hare, there is no point in your chasing stag, either. The situation is unappetizing but stable. We all could do better by changing, but it wouldn't be rational for either one of us to be the first. So we just keep eating hare. Yet hunting stag also is an equilibrium if we somehow can get there. Once we're doing it — once we're cooperating confidently — there's no reason for anyone to go back to having hare.

Efforts to solve stag hunts, like the problem of rent seeking discussed yesterday, help to explain some laws and legal doctrines; examples are discussed in the book, and include deposit insurance for banks, "circuit breakers" in stock markets, some cases of neighborhood segregation, and maybe even some of the rules about when judges are supposed to interview and hire law clerks. The value of understanding the stag hunt (or any game) is that it allows you to perceive common patterns in situations like these. That's interesting in itself, and it also can then help you make sense out of new problems that aren't solved yet and just look like an uncooperative mess.

Perhaps you can suggest some other "stag hunt" situations that law is used to address (sometimes the tricky part is separating these from the prisoner's dilemmas, where your first ("rational") choice is for everyone to cooperate except you; those tend to be more common than stag hunts and to call for different sorts of solutions). Or maybe you would enjoy speculating about some situations that might be stag hunts might not be. Part of what makes a stag hunt interesting is figuring out why the game works that way — in other words, what makes people reluctant to do X unless others do it, too. Why are some people so much more willing to support redistributive tax policies that bind everyone than they are to just voluntarily write checks that redistribute their own income to others? Why did so many hockey players resist wearing helmets until it was made a rule? Etc.

martinned (mail) (www):

Stag hunt games are great for researching trust, as well as folk theorems. Since the one shot game does not have a dominant strategy, it is much easier to observe a wide variety of outcomes for multi-period games...

In many other circumstances, though, it is easier to simplify matters by modeling the situation as a prisoners' dilemma instead of a stag hunt, so as to reduce the number of equilibria.
8.1.2007 10:15am
Jimmy Blue:
Respectfully, I have a difficult time seeing how any of this has any practical relevance. I realize it may be useful in law review articles, but I renew the objection. ;-)
8.1.2007 10:31am
Duffy Pratt (mail):
I may be misunderstanding the situation, but it seems to me that a whole bunch of laws are in place to create a situation of trust that allows a "stag hunt" to take place. Three obvious examples: Enforceability of contract, negotiable instruments, and corporate law allowing the raising of capital.
8.1.2007 10:43am
Sean O'Hara (mail) (www):
Perhaps you can suggest some other "stag hunt" situations that law is used to address

I'd say "marriage" but there's always the possibility that a woman I'm dating will one day stumble across this.
8.1.2007 11:32am
Political efforts to address climate change are a classic example of a stag hunt. If every country controlled carbon emissions, we'd all be better off, but no one wants to be the only one to take corrective measures for fear that they'll end up at a competitive disadvantage.
8.1.2007 11:44am
Aw, come on. I've asked politely three times and nobody seems to want to tell me whether Ward is related to E. Allan.
8.1.2007 12:13pm
A Guest:

I don't know that this is a classic example of a Stag hunt. It is a classic example of a public goods problem, a type of prisoners' dilemma. In order for this to be a Stag Hunt, it would have to be the case that (1) given that all other countries don't control emissions, a country is best off not controlling its own emissions, and (2) given that all other countries control emissions, a country is best off controlling emissions. (1) seems convincing, but this is an element of the PD too. (2) might be true, but maybe not in an obvious enough way to make this a classic example.
8.1.2007 12:17pm
Houston Lawyer:
I see a little of this in law firm behavior. I stated my position the other day to a partner in my section that quickly firing underperforming associates improves the morale of the good associates. It reinforces the idea that good behavior is rewarded and bad behavior is punished. It's a way to ensure that we are all hunting stag together and that those who are not interested or capable of hunting stag are culled from the hunt.
8.1.2007 12:26pm
A Guest:
I should have said the same as in my previous comment about Farnsworth's tax example:

Why are some people so much more willing to support redistributive tax policies that bind everyone than they are to just voluntarily write checks that redistribute their own income to others?

If everyone else voluntarily wrote a check, would you voluntarily write a check? Maybe some people would, but I feel certain that many people wouldn't and that there wouldn't be an equilibrium in which most people voluntarily wrote checks. If this is so, this would be an example of a prisoners' dilemma, not a stag hunt.
8.1.2007 12:29pm
Steve P. (mail):
I have always thought that non-smoking bars were a stag hunt problem. Bar owners often don't want to make their bars non-smoking, even if a majority of their clientele are non-smokers, simply because they can lose a few people. But once all bars are forced into non-smoking compliance, the equilibrium is stable again.

Of course, smokers like me are a little perturbed, but it's arguably a better social situation. And there are plenty of exceptions to this, like niche martini bars that don't allow smoking, but in general this is the behavior I've experienced.
8.1.2007 12:34pm
Steve P. (mail):
My previous example can be thought of as a prisoner's dilemma, however, as bars may have an incentive to defect (allow smoking) if they could. Hmm, disappointing.
8.1.2007 12:36pm
A Guest:
How about following traffic laws. (I'm thinking of rules at intersections and direction of travel, not speed limits.) Irrespective of enforcement, if everybody else follows traffic laws you have a strong incentive to follow them. If no one else does, you don't. Here, enforcement can be seen as maintaining a relatively stable cooperative equilibrium rather than imposing order in the face of individual incentives.
8.1.2007 12:40pm
A Guest:
Steve P.,

I dunno. I thought your example was pretty good. There might be a few bars that would have the incentive to defect, but mainly defection would cost bars their non-smoking clients. In the all-smoking equilibrium, you could allow smoking w/o losing non-smokers. In the potential no-smoking equilibrium this wouldn't be the case.
8.1.2007 12:45pm
Arms races are another. Looking at the Germany/Britain dreadnought race leading up to WWII, or the American/Soviet atomic bomb buildup, it would have been better for both sides to agree to do neither (hunt stag), but once one side was arms racing the other had to keep pace (rabbit eating).

For that matter, the entire Cold War was one big instance of rabbit-eating. Both sides would have been better off without investing in the large armies, covert ops, and foreign aid to third parties that they had to do to match and exceed the other.
8.1.2007 1:04pm
Daniel San:
With income taxation, the law itself changes the equilibrium. If no one cooperates, I know that enforcement is not likely to reach me, so incentive to cooperate is less than the incentive to keep my money. If cooperation is high, enforcement creates and incentive to cooperate.

The stag hunt also illustrates the strength of voluntary associations (service clubs, churches, etc.). Members with mutual goals and loyalties are more likely to believe their individual efforts will not be wasted. The same applies to families and tribes.

In the legal profession, attorneys who are likely to see each other again frequently have an incentive to cooperate (or at least be civil). The advantage I gain by incivility is usually small, but may be significant if I know that payback is unlikely.
8.1.2007 1:13pm
A Guest:

Maybe I'm being picky here, but arms races, like public goods problems, are generally thought of as types of prisoners' dilemmas. The idea is that if the other player doesn't stock arms, you have the unilateral incentive to stock arms yourself. The cooperative outcome (no arms) is preferred by all (over the arms race), but as in the prisoners' dilemma, individual incentives don't make the cooperative outcome an equilibrium.

I actually don't think I'm being picky in trying to separate the PDs from the Stag Hunts onaccounta the cooperative outcome in the PD needing serious enforcement and the cooperative outcome in the Stag Hunt being (largely) self-reinforcing. This distinction seems critical when you're thinking about the law.
8.1.2007 1:17pm
I.I (mail) (www):, the Free State Project, and other "I commit to do X if Y other people also commit" deals are all based on this exact principle.
8.1.2007 1:21pm
David M. Nieporent (www):
A Guest: if one looks at the arms race primarily through the prism of a national budget, as JB was seemingly doing, then it's a stag hunt scenario. But if one looks at an arms buildup (more correctly, IMO) as actually getting a country something useful -- namely, the ability to impose its will on others -- then it's a prisoner's dilemma situation.
8.1.2007 1:40pm
ras (mail):


Hockey players know that if everyone wears low-protection gear (e.g. no helmets or shoulder pads) the hitting will be cleaner because the hitter is almost as likely to be hurt as the hittee. But once you allow the extra equipment, one or two guys with it will run wild and force everyone else to wear it.

This was a serious discussion in my college beer-league games, btw, not a hypothetical. We opted for less equipment, not more, and there were never any serious injuries.


You asked about other examples of the stag hunt and one comes to mind that repeats itself over and over: the election of porkers.

In an election, pork-loving reps (e.g. in the US it'd be in the House of Reps or the Senate) often win despite the public's distaste for pork.

Why? Well, if we in our district think that most other districts are likely to vote for a pork cleanup then we too can vote that way. We can hunt stag together.

But if a critical mass of anti-pork candidates looks unlikely to win (i.e. not enough other districts will hunt stag with us) then our most rational strategy becomes the opposite: to vote for the biggest porker we can find. That's our hare.

If we had instead stuck to the stag hunt despite the lack of critical mass, meaning that our district elected an anti-pork candidate to sit with a majority of porkers, our own district would "go hungry" as our principled electee eschewed pork but our taxes still subsidized it in other districts.
8.1.2007 2:44pm
Vivictius (mail):
Other one is government "pork". We would all be better off if we got rid of it but right now if you vote in politicians that wont the money will just go to some other state. So you elect people that will send back as much cash as possible.
8.1.2007 3:01pm
Doug Sundseth (mail):
If I understand the terminology correctly, desertion on the field of battle has stag hunt properties.

If everybody on your side runs away from battle, your army will lose, and casualties will be high. In addition, your ancillary losses (from pillage, etc.) are likely to be devastating. But, the first to run are the least likely to be killed at the battle and the most likely to be able to rescue chattels from the victorious enemy. The result is a stable equilibrium on running away immediately and an enemy army pillaging your homeland.

If nobody on your side runs away from the battle, your military strength stays high and your chances for winning are greatly increased. And winners' casualties are very low. The result is a moderately stable equilibrium on not running away and pillaging the enemy's homeland. To achieve this latter equilibrium, each participant needs to have trust in his comrades' fighting ability and morale.

(This scenario is particularly the case in pre-modern warfare, where most casualties are the result of close combat, and the casualty rates of winner and loser are usually wildly disproportionate -- not uncommonly multiple orders of magnitude different.)

Armies' response to the dilemma has historically been to raise the cost of running away by harshly punishing deserters, and to train in such a way as to increase interpersonal trust.
8.1.2007 3:22pm
A Guest:
David M. Nieporent,

I don't quite understand the distinction you make, but I'm not denying that a situation of arms build up could involve something like the SH. And of course J.B. could have been picturing such a situation. W/public goods, too, there could be a SH. (I.I. presents an interesting possible example.) BUT I do think it's useful to distinguish between SH &PD rather than just come up with examples in which cooperation can benefit all. Also, for what it's worth, if you mention arms races or public goods to someone who is used to thinking about games, they will picture a PD; If the specific situation is more like a SH, it might be worth explaining why. Maybe not for this thread. Dunno.
8.1.2007 3:30pm
A Guest:
Oh, and Daniel San's taxation example is a pretty cool public goods case in which SH is relevant (for complicated reasons that he explained).
8.1.2007 3:33pm
Pyrrhus (mail) (www):
pork -> PD. Its good for one to get pork even if others do not get pork.

Battle -> PD. Its good to run away even if others are not running away.
8.1.2007 4:43pm
Daniel San:
No necessarily. Pork - if I get pork, that is an immediate good, but I know that it leads to everyone getting pork, so in the long-term is bad for me. Battle - for similar reasons, (particularly in pre-modern warfare) if I run away, that greatly increases the odds that my army will collapse because a desertion erodes that trust that is essential to keeping everyone in place.
8.1.2007 5:03pm
A Guest:
Thanks Pyrrhus for joining me in an appointed task:

sometimes the tricky part is separating these [stag hunts] from the prisoner's dilemmas

But while both Battle and Pork could be PD, both are potentially SH. (And I think ras was explicit in arguing this.)

Battle: If everyone else runs away it's probably pointless not to. If no one else runs away, then it depends on all sorts of things. If it's not worth running away when no one else does (perhaps personal harm is unlikely or victory is a personal goal) then this would be a SH.

Pork: If the acquisition of pork depends on a critical mass of porky representatives (as ras suggested) then if no other constituency votes for pork, you don't have the incentive to either.
8.1.2007 5:05pm
Daniel San:
Sean: I'd say "marriage" but there's always the possibility that a woman I'm dating will one day stumble across this.

Cute, but I think, correct. A key reason for marriage is the creation of stable alliances. Your brother-in-law is more likely to hunt stag with you.
8.1.2007 5:11pm
John C:
Actually, hijackers on a plane doesn't really work as an example anymore; since everyone now assumes (at least, I now assume) that everyone will be dead if no one does anything (because one has to assume the terrorists are going to fly you into a building), everyone has incentive to resist, even if it is just you doing the resisting.
8.1.2007 5:59pm
ras (mail):
I presume that Pork is SH, not PD, because there is no concept of betrayal after the play; each voter/district has a pretty good idea what their neighbors are going to do.

In a prisoner's dilemma it's best for you to defect no matter what I do. In a stag hunt that's not true; if I know you will cooperate, I'll want to cooperate, too.

In the case of pork there is no benefit to you in "defecting" - .e. electing a porker when the rest of us don't. And, if the candidate's porkiness was his (or hers, but we'll presume either Byrd or Stevens fo sake of arg) key selling pt overiding some other important stance he had, such as on taking bribes, then you will be incurring the cost of trading down on that other selling pt with no corresponding benefit.

So it sounds a lot more like SH to me: if we know, or think we can predict which is much the same thing here, that our fellow voters are also going to elect anti-porkers, then we can stag hunt with them. If not, then the logical fallback is the hare.

Perhaps this is a contributor to why traditionally over the years presidents have averaged out as more conservative than the majority party in the House or the Senate: the voters do not face a SH dilemma when voting for a single person for pres, but they do for the others.
8.1.2007 7:03pm
Gregory Conen (mail):
In general, a great deal effort is spent turning Prisoner's Dilemmas into Stag Hunts. In the desertion example, the Stag Hunt aspect comes in because if you run when no one else does, you get punished by the military legal system.

Likewise, organized crime goes to great length to "break" the literal prisoner's dilemma by punishing rats. If a critical mass of people betray the mob, it falls apart, and those that stayed quiet end up with 100 consecutive 2 year terms, while those who turn get off light. But if just a few people turn State's Evidence, they get whacked.

It also seems as though an iterated series of Prisoner's Dilemmas ends up looking similar to a Stag Hunt, assuming that retaliation is assured.
8.1.2007 7:44pm