The No-Trade Problem and Common Knowledge (revised Feb. 18)
Larry Samuelson’s “Modelling Knowledge…” in JEL 2004 is a very good article about common knowledge, no trade theorems, and such.
The article doesn’t discuss forward induction. Is forward induction equilibrium in a complete information game demanded by rationality. That may be so. But if trembles are possible, it is not. Then, there could be an equilibrium in which the first player must make a costly communication to show he will play his preferred action in the succeeding battle of the sexes. That is because if he does not make the communication, the second player can think that it was because of a tremble, and that they are then in the subgame in which the second player’s preferred action is played in the battle of the sexes. Without trembles, the second player’s rational response may well be to think that the first player is simply dispensing with an unnecessary communication.
That is a case of trembles introducing a new equilibrium, rather than eliminating an old one as in the Selten 1975 perfectness article.
What about trembles in the Bertrand game? Trembles eliminate weakly dominated strategies such as P=MC. I’m not sure what happens, but conceivably not much. There would be a mixed strategy equilibrium with prices from MC on up to the reservation price. But if most of the probability is down near MC, prices are mostly near MC, just as in the original game. If the mixing range really does go all the way down to MC, then the expected payoff, equalling any pure-strategy payoff in the mixing range, must be arbitrarily close to zero.
Samuelson uses as a running example Alice and Bob, who each interview a different employee about the company’s chances of survival. If Alice offers to sell the stock to Bob after the interviews, what will Bob do?
In the Law Lunch today we discussed an example similar to Samuelson’s first one. Jeff owns land worth V= J+E, the sum of its oil, J, and its gas, E. J is either 1 or 2, and E is either 0 or 4, with equal independent probabilities. Jeff knows J and Eric knows E. If Eric makes an offer to buy the land from Jeff, making the offer costs Eric a penny, .01.
No trade will occur. The land is worth at least 1.00. If Eric offers a price of 1.00 or more, Jeff could conclude that E=4 and he would refuse to sell, because if E=0 and Jeff sold at that price, Eric’s payoff would be 1+0-.01 = .99. If Eric offers P=.99 or lower, Jeff does better by refusing regardless of the value of E.
This is double adverse selection. Standing alone, it is not so unreasonable, but there are some subtleties. What should Jeff really think if Eric, out of equilibrium, offers 1.01? If that shows Eric is trembling, then maybe Jeff’s expected value is 3 (1 + .5(0)+ .5(4)) and he should accept. But then Eric would pretend to tremble. The equilibrium would be in mixed strategies, with Eric sometimes trembling, sometimes having value E=4,and Jeff sometimes accepting, sometimes not, but in that equilibrium Eric would offer P=1.01 only with tiny, tremble-level probability.
The real problem is that we often see speculators trade in situations like Jeff and Eric’s. How can both sides think they will win? It seems incompatible with the Harsanyi Doctrine.