## Wednesday, April 3, 2013

### Cho and Kreps: "Signaling games and stable equilibria" (1987)

Because games often have several equilibria and no obvious way of choosing between them, it is always good sport to try to come up with some new, stronger refinement of the concept of the Nash equilibrium. This paper investigates a number of ways of doing so, mainly motivated by a single type of equilibrium that the authors find unintuitive.

The game that Cho and Kreps investigates is given by the following tree:

They motivate the game by the following annoyingly ridiculous story:
• A can have two types, "wimp" (with 10% probability) and "surly" (with 90%).
• A prefers quiche for breakfast when he's "wimp" and beer when he's "surly." He gets 1 point for having the breakfast he prefers, and 0 otherwise.
• In addition, A prefers not to duel B, and he gets 2 points for avoiding a duel.
• B prefers to duel A iff A is a "wimp" and not to duel him iff he is "surly." B gets 1 point for making the right decision and 0 otherwise.
Under these assumptions, the game has the following two equilibria:
1. A has beer for breakfast regardless of his type; B duels A iff he has quiche for breakfast.
2. A has quiche for breakfast regardless of his type; B duels A iff he has beer for breakfast.
So in the  first equilibrium, the game ends in one of the two lower branches of the subtrees on the left, with payoffs (2, 0) or (3, 1). In the other equilibrium, the game ends in one of the two lower branches of the subtrees on the right, with payoffs (3, 0) or (2, 1).

It is equilibrium no. 2 that Cho and Kreps find unintuitive and spend the most of the paper combatting. Their main focus in this effort is the notion of "stable equilibria" as defined by Elon Kohlberg and Jean-Francois Mertens in a 1986 paper—a concept that Cho and Kreps state that they have "mixed feelings" about (p. 181).