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## Wednesday, January 21, 2015

### Experimental Game Theory Series #1 of ???

I'd like to try an experiment: methodological propaganda/skillsharing in a series of blog posts. I had originally planned a fairly large number of these and essentially an internet course in basic game theory, but then the 20th of the month snuck up on me, and there's very little chance the whole thing gets out before my blogging residency (such as it is) runs out. So let's get as far as we can, and see how people like these posts; if they prove popular, perhaps they can continue somewhere else. (I'm also totally hijacking the "games" category" on the blog for this. Because, obvs.)

With no further ado: an introduction to game theory for lawyers/law professors, post 1 of N: why?

I'm a huge fan of game theory as an intellectual tool. It provides a surprising amount of analytical punch at reasonable, lawyer level, amounts of math---you can make useful models that shed important light on the social world with nothing beyond basic algebra. (It also scales up to math way above my pay grade, including alarming things like linear programming.) And while many involved in the legal enterprise make use of the tool---mostly L&E folks---there are many who could benefit from it but find the jargon and the formalization intimidating, or are put off by the policy agendas associated with many who use the tool ("efficiency!").

So I will spend part of my blawggey bandwith this month setting out some basics of game theory that, I hope, will be suitable for academic readers with no training in the subject to put to immediate use in their own scholarship (with suitable feedback from specialists, of course), and potentially also introduce to students in courses that can benefit from it (e.g., anything with negotiation or regulation on the agenda). The intended audience is academics with no formal training in formal modeling beyond the sorts of references that may appear in undergraduate economics (at the introductory level) and political science courses. Readers with training in the subject will find this series of posts terribly uninteresting. Also, the posts will begin quite elementary and become more fancy over time; today, I will begin with the absolute basics: what game theory is, and why you might want to use it.

**1. What is Game Theory?**

Game theory is, in essence, the study of rational strategic preference-optimizing behavior. Its space is best carved out in contrast to its counterpart, decision theory, which has the same subject, minus the "strategic."

Decision theory is just math about how to get what you want. More wonkily, it imagines an actor who views a set of probabilistically arranged states of the world, and can rank-order those states in terms of his or her preferences (i.e., I prefer a .5 chance at state A to a certain chance of state B, and so forth). Thanks to the math underlying Von Neumann-Morgenstern utility, we know that a minimally rational ordering of such states allows us to get numbers on an interval scale, which we can call "utility." (Philosophers, economists, and the like have started a bunch of fights about what the notion of utility actually means; we can mostly leave them aside until such time as we want to use it to talk about things like "efficiency.") Then, essentially, decision theory is a fancy set of tools to flesh out the prediction that people will take the actions that lead them to their highest expected utility---the actions that bring it about that the sum of the utility numbers, weighted by the probability of the states that yield them, is largest.

Game theory takes the same actor, with the same properties, and introduces another player---a second agent, also with those properties. It then makes the paths from those agents' actions to the states of the world over which they have preference orderings dependent on one another's behavior. It is this interdependency to which we refer when we say that the subject of game theory is "strategic" action.

To see strategic action, consider the classic tale, "The Gift of the Magi." We all know it, it's the one about the poor husband and wife who give self-sacrifical gifts to one another: the husband sells his watch to buy the wife fancy combs for her hair, and the wife sells her hair to a wigmaker to buy the husband a fancy chain for his watch. In addition to being a heartwarming Christmas love story, yadda yadda, it's also a story of strategic action gone wrong: the outcomes of their gifts depended on what one another did, but neither took that into account in making his/her own decision: some game theory could have helped this couple.

**Why?**

The question "why game theory" is really the question "why (formal, sorta-mathematical-but-not-as-mathey-as-those-weirdos-in-econ) modeling?" The short version is that it generates predictions about how people will behave, and those predictions (flawed though they may be; abstractions from reality always are) as generated by game theory as opposed to by something else can be useful to your scholarly enterprise in several different ways:

*1. Sometimes you can just take a game off the shelf.*

Surprisingly often, having a basic familiarity with the classic games---the prisoners' dilemma, the battle of the sexes (being invented by a bunch of cold warriors and mathematicians in the 20th century, the legacy of game theory includes some unfortunate names with sexist connotations), the stag hunt, chicken---can spark a flash of insight. You might be considering a policy problem, and notice that the people involved have payoffs that resemble those assumed in one of the classic games. Great: you have a pretty good first-pass prediction about what's going to happen, and a pretty good first-pass idea about what might need to be changed to change the outcome. You might also be able to generate more insight by stating the conditions under which the payoffs actually resemble the game in question. [SHAMELESS SELF-PROMOTION ALERT:] My wonderful colleague Maya Steinitz and I have a paper that does just that in the context of corrupt transnational litigation.

*2. Intuitions are unreliable.*

We all have intuitions about the incentives that a given institutional structure or policy creates for those who interact with it. More than once, however, I've been pretty convinced that system X created outcome Y, tried to prove it formally, and found that I'm unable to formalize the intuition---sometimes because the result the analysis yields actually is the opposite of what I had predicted. This is obviously important, and shows how modeling can provide a rough and ready way of testing our beliefs about the world.

*3. Push the intuitions a step further.*

Even if your intuitions are reliable, how far out do they go? You may have an intuition about what happens when people interact once, but are the intuitions as strong or as reliable when they interact multiple times? Sometimes, the math can keep going when the intuitions run out.

*4. Generate and refine empirically testable hypotheses*

Another advantage of formal modeling is that it allows us to see a variety of candidate causal factors on a given behavioral outcome. By explicitly specifying the utility functions that generate agents' payoffs, we can identify what things to take to our data.

*5. Find policy levers.*

Back in #1 above, I said that stating the conditions under which the payoffs resemble a given game can add insight to the world. But "insight" is never enough for legal scholarship---we (well, someone---law review editors? tenure committees? John "I hate Kant and Bulgaria" Roberts?) always demand some kind of doctrinal or policy payoff at the end. Here's one way to get it: Now that you know the conditions under which the players have an incentive to do X, you have at least one if not several candidates for places that policymakers can intervene in order to bring about/abolish X-doing. But by having all the different things that feed into the incentives laid out before you in neat mathematical form (as per the previous item), it can allow you to see policy options that may previously have escaped notice. ("Congress will have an incentive to kick puppies so long as the price of tea in China is greater than the number of votes for Scottish independence, so down with the Union!")

So there's my brief for why game theory is worth caring about. The details are for subsequent posts.

Posted by Paul Gowder on January 21, 2015 at 09:58 AM in Games | Permalink

## Comments

I like the post and am looking forward to some more. I have read a bit about game theory but have never felt confident enough to use it in my work. Perhaps your posts will help.

Posted by: Stuart Ford | Jan 22, 2015 10:18:10 AM

Paul,

While my mathematical skills are insufficient for advanced game theory and I have little or no occasion to use it in my own work, Jon Elster has convinced me of its significance: in his words, "The invention of game theory may come to be seen as the most important single advance of the social sciences in the twentieth century." So, as is my wont, I've assembled a small list of titles by way of an introduction to the (I hope some of the best) literature. Perhaps your readers will find it helpful.

• Aumann, Robert, Sergiu Hart, H. Peyton Young, and Shmuel Zamir, eds. Handbook of Game Theory with Economic Applications. Amsterdam: North-Holland, 1992-2015.

• Baird, Douglas G., Robert H. Gertner, and Randal C. Picker. Game Theory and the Law. Cambridge, MA: Harvard University Press, 1994.

• Binmore, Ken. Game Theory and the Social Contract, Vol. 1: Playing Fair. Cambridge, MA: MIT Press, 1994.

• Binmore, Ken. Game Theory and the Social Contract, Vol. 2: Just Playing. Cambridge, MA: MIT Press, 1998.

• Binmore, Ken. Natural Justice. New York: Oxford University Press, 2005.

• Binmore, Ken (et al.) Does Game Theory Work? The Bargaining Challenge. Cambridge, MA: MIT Press, 2007.

• Camerer, Colin F. Behavioral Game Theory: Experiments in Strategic Interaction. Princeton, NJ: Princeton University Press/ New York: Russell Sage Foundation, 2003.

• Davis, Morton D. Game Theory: A Nontechnical Introduction. Mineola, NY: Dover, 1997 (1983)

• Dixit, Avinash K., Susan Skeath, and David H. Reiley Jr. Games of Strategy. New York: W.W. Norton & Co., 4th ed., 2014.

• Gintis, Herbert. The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences. Princeton, NJ: Princeton University Press, revised ed., 2014.

• Morrow, James D. Game Theory for Political Scientists. Princeton, NJ: Princeton University Press, 1994.

• Vega-Redondo, Fernando. Economics and the Theory of Games. Cambridge, UK: Cambridge University Press, 2003.

Posted by: Patrick S. O'Donnell | Jan 22, 2015 11:24:57 AM

I should have noted that the first entry above is 4 volumes.

Posted by: Patrick S. O'Donnell | Jan 22, 2015 11:27:48 AM