[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.