Matthew Rabin was here last week presenting his work with Erik Eyster about social learning. The most memorable theme of their their papers is what they call “anti-imitation.” It’s the subtle incentive to do the opposite of someone in your social network even if you have the same preferences and there are no direct strategic effects.
You are probably familiar with the usual herding logic. People in your social network have private information about the relative payoff of various actions. You see their actions but not their information. If their action reveals they have strong information in favor of it you should copy them even if you have private information that suggests doing the opposite.
Most people who know this logic probably equate social learning with imitation and eventual herding. But Eyster and Rabin show that the same social learning logic very often prescribes doing the opposite of people in your social network. Here is a simple intuition. Start with a different, but simpler problem. Suppose that your friend makes an investment and his level of investment reveals how optimistic he is. His level of optimism is determined by two things, his prior belief and any private information he received.
You don’t care about his prior, it doesn’t convey any information that’s useful to you but you do want to know what information he got. The problem is the prior and the information are entangled together and just by observing his investment you can’t tease out whether he is optimistic because he was optimistic a priori or because he got some bullish information.
Notice that if somebody comes and tells you that his prior was very bullish this will lead you to downgrade your own level of optimism. Because holding his final beliefs fixed, the more optimistic was his prior the less optimistic must have been his new information and its that new information that matters for your beliefs. You want to do the opposite of his prior.
This is the basic force behind anti-imitation. (By the way I found it interesting that the English language doesn’t seem to have a handy non-prefixed word that means “doing the opposite of.”) Suppose now your friend got his prior beliefs from observing his friend. And now you see not only your friend’s investment level but his friend’s too. You have an incentive to do the opposite of his friend for exactly the same reason as above.
This assumes his friend’s action conveys no information of direct relevance for your own decision. And that leads to the prelim question. Consider a standard herding model where agents move in sequence first observing a private signal and then acting. But add the following twist. Each agent’s signal is relevant only for his action and the action of the very next agent in line. Agent 3 is like you in the example above. He wants to anti-imitate agent 1. But what about agents 4,5,6, etc?
Cheap Talk 
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October 15, 2013 at 12:07 am
Alex F
You should have a category for prelim question ideas.
October 15, 2013 at 2:53 am
Lones Smith
In what might be my longest R&R (first submitted to REStud in 1997!!), “Informational Herding, Optimal Experimentation, and Contrarianism” (with Peter Sorensen and now also my brilliant UW PhD student Jianrong Tian), we show that it is forwardly rational to be “contrarian” — yes, I think that *that* is the word you want, rather than anti-imitation. The result is technically tricky — needing the latest and greatest in monotone comparative statics. Specifically, if a social planner were dictating behavior to social learners, he would ask that the more the public beliefs myopically favor taking an action, the more individuals should lean against taking that action. You do not need divergent prior beliefs, and in fact, we assume a common prior. Contrarianism also manifests itself in shrinking cascade sets as the social planner grows more patient.
We have now almost finished the third revision after over a year of proving and polishing, but a rough earlier draft is at:
http://tinyurl.com/contrarianism
October 15, 2013 at 8:45 am
jeff
We need the verb to be transitive so contrarian doesn’t do the job. I can’t contraryizate my friends.
October 15, 2013 at 11:43 am
Alex F
Antonyms for imitate: differ, oppose, clash, reverse. You could also maybe use avoid or contrast. If you’re looking for a counterpoint to herding, I’d use scattering. “Herding and Scattering” or maybe just “Scattering” would be a good Jeff-Ely-paper-title by the way.
October 16, 2013 at 8:08 am
Erik Eyster
Matthew and I thank Jeff for his post, and Lones for pointing us to it as well as his comment. We welcome discussion of our work on the limits to imitation in social learning, which is best presented in our more recent paper at http://elsa.berkeley.edu/~rabin/rewind.pdf
Although Jeff’s example uses non-common priors, our underlying logic does not. We assume common priors in our work, and indeed assume the same preferences and use the same solution concepts as the literature. One key difference between our results and what Lones describes above is that we analyze the traditional question of equilibrium behavior, instead of the planner’s problem. As Jeff put it well, we emphasize that very often the familiar logic of imitation sits side by side with anti-imitation.