“On the Evolution of Attitudes Toward Risk in Winner-Take-All Games,” E. Dekel & S. Scotchmer (1999)

How about a couple of posts about evolution of preferences? Informal evolutionary arguments are everywhere in economics. People will do X in a market because if they didn’t, they would lose money and be forced out. Firms will profit maximize because if they don’t, they will be selected away. Many of these informal arguments are probably wrong: rare is the replicator dynamic with random matching that gives a trivial outcome! But they are important. If I have one heterodox crusade, it’s to get profit maximization by firms replaced by selection arguments: if you think firms in some sense luck into optimal pricing, or quantity setting, or marketing, rather than always minimizing costs, then you will be much more hesitant to support policies like patents that lead to monopoly power. I heard second-hand that a famed micro professor used to teach that he was more worried about the “big rectangles” of efficiency loss when monopolies don’t cost minimize than the “small triangles” of deadweight loss; the irony is that when I heard the story, the worried professor was Harberger of the Harberger Triangle himself!

But back to the Dekel and Scotchmer paper. The question here is whether, in a winner take all world, preference for risk will come to dominate. This is an informal argument both for what people will do in general situations (men, in particular, take a lot of risks and there are casual evobiology arguments that this is a result of winner-take-all mating in our distant past) and for what firms will survive situations like a patent race. This makes intuitive sense: if only the best of group survive to the next generation, and we can choose the random variable that represents our skill, we should choose one with high variance. What could be wrong with that argument?

Quite a bit, it turns out. I use “men” from now on to mean whatever agent is being selected in winner take all contests each generation. Each man is genetically programmed to choose some lottery from a finite set. In each period, groups of size m meet. Each man realizes an outcome from his lottery, and the highest outcome “wins” and reproduces in the next period. Here’s the trick. If a distribution (call it F) FOSD another distribution, then it is “favored,” meaning that measure of distribution F players will be higher next period. But risk loving behavior has to do with second order stochastic dominance; distributions that are second order stochastically dominated are more risky. And here the ranking is much less straightforward. Consider groups of size 2. Let F give 1 with probability 1. Let G give 1/4 with probability 2/3 and give 2.5 with probability 1/3. F SOSD G – F and G have the same mean, while G in a specific sense has more “spread” – but F is also favored in evolution over G.

The intuition of that example is that increasing risk in a way that just expands the tails is not very useful: in a contest, winning by epsilon is just as good as winning by a million. So you might imagine that some condition on the possible tail distributions is necessary to make risk loving evolutionarily dominant. And indeed there is. This condition requires the group size to be sufficiently large, though, so if the contests are played in small groups, even restricting the possible lotteries may not be enough to make risk loving dominate over time.

What if everybody plays in a contest against everybody else? Without mutations, this game will end in one period (whichever type draws the highest number in the one period the game is played with make it to the next generation). Adding a small number of mutations in the normal way allows us to examine this scenario, though. And surprisingly, it’s even harder to get risk loving behavior to dominate than in the cases where contests were in small groups. The authors give an example where a distribution first order stochastically dominates and yet is still not successful. The exact condition needed for SOSD to be linked to evolutionary success when contests are played among the whole population turns out to be a strengthening of the tail condition described above.

I don’t know that there’s a moral about evolution here, but there certainly is a good warning against believe informal evolutionary arguments! More on this point in tomorrow’s post, on a new and related working paper.

http://socrates.berkeley.edu/~scotch/wta.pdf (Final JET version; big thumbs up to Suzanne Scotchmer for putting final, published versions of her papers online.)

Advertisement