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?