For some reason I’ve always had the damnedest time reading payoff matrices when I try to learn game theory; this has made learning the subject rather hard. My studies get to Nash equilibria (named after famous schizophrenic John Nash, of A Beautiful Mind fame), and something just locks up. I’m not sure what it is. The last time I tried to study game theory rather abstractly — namely Gintis’s Game Theory Evolving — this is exactly what happened. The study ended after a few pages.
(A brief explanation of my vague understanding of the terms follows.)
Samuel Bowles is doing something quite different here, and it works. The concepts are fairly straightforward: a Nash equilibrium in a game between two people is a set of choices of strategies such that neither person has any incentive to switch strategies unilaterally. (Maybe they’re allowed to talk about it — e.g., we agree that you’ll put your gun down and I’ll do the same with mine — but many of the elementary problems don’t allow coordination, and that example probably suggests why it’s valuable not to model the parties as though they trusted one another.) A party will switch if he can make his position better by doing so. If I know every strategy available to you, and a particular strategy available to me is a loser no matter which strategy you pick, then it’s reasonable to suppose that I won’t pick the loser; the loser here is called a “dominated” strategy. So one approach to figuring out what strategy to play is to work out the full tree of possible games that we could play, eliminate dominated strategies from the final move, and keep pruning the tree of possible games until we get back to decide which move we should play at the start of the game. I believe this is called “backward induction.”
That much is fairly interesting, but it’s always struck me as abstract and a bit of a yawn — particularly when I’ve read cases that seem far more interesting, like the iterated prisoner’s dilemma, and when much of my behavioral-economics reading suggests that fully rational game theory isn’t a very good model of how actual actors behave.
But of course I should just keep reading. They start with the less interesting stuff to build to the more interesting stuff, of course. I guess my attention just doesn’t stick around as long as it should.
Bowles has kept it, though, mostly because he’s trying as hard as he can to get to the interesting cases right away. First of all, he wants to return economics to a broader base of interest. Rather than defining economics rather narrowly as “the study of allocating scarce means amongst unlimited wants,” Bowles wants to understand how individual behaviors can produce an outcome that’s good for everyone; he calls this the “coordination problem.” The Prisoner’s Dilemma is the most famous small-scale coordination problem, where individually optimal behaviors lead to an outcome that everyone would have chosen to avoid had he been able to dictate the group’s outcome rather than just his own. The point is to build institutions in such a way that people are motivated to “do the right thing.” Taxation and the price system are probably the most obvious institutions that help in this direction: charge people enough for gas, and they will drive their cars less. The failure of the price system to charge people for the damage they cause is known as a “negative externality,” which Bowles is building towards. Rather than start with orthodox economics and eventually show where it breaks, Bowles realizes that the breakages are what we’re actually interested in.
The trick that Bowles uses to keep my interest here is to very quickly study the evolution of toy strategies in a given population, when there are two strategies available. He starts by modeling conformity: suppose some fraction of the population (call it p) believes a certain proposition (call it x), whereas I believe some other competing proposition y. Suppose in a given period of time (a day, say), that I am randomly paired with someone else from the population — someone who may believe x or y. Add some other assumptions about x and y, specifically about the benefits accruing to those who believe x and y, and the benefits that accrue to someone who “plays” the x “strategy” against the y strategy.
Without much complexity in the model, you can start looking at the time evolution of conformity in the population. You can ask about equilibria: assuming the probability that people believe x starts at p, where will p end up given sufficient time? If p = 0, in particular, then everyone believes y at the start of the game and no one will ever switch unless the system gets “pushed” somehow.
So then let’s talk about pushing the system. Suppose the system is suddenly “invaded” by someone playing a different strategy. In the case of modeling conformity, this means that a new idea comes to town. Intuitively, the new strategy will disappear if it doesn’t confer any benefits on people who adopt it — specifically, if I play the new strategy against you in this round of the game, and the payoff to my doing so is lower than the payoff to my switching (the normal verb here is “defecting”), then I will defect by the next round.
A strategy (again, think “idea” here if it helps) that is resistant to invaders is an Evolutionarily Stable Strategy. Bowles reaches these very quickly as well, and starts modeling with them right away.
The pedagogy is as smooth as I could ask for, with mathematical derivations that fall right out of his setup and are surrounded by enough explanatory text that you’ll never feel lost in a thicket of equations. He’s always grounded in the concrete. He never wants his fellow-economists to lose sight of what their discipline has to offer, and he’ll never let the reader lose sight either.
More reports as I dive in further. I was just so pleased with the coherent picture I’ve finally built of evolutionary game theory (vague and inaccurate as it probably is), that I felt it necessary to gush.