The beer-quiche game is a common way to illustrate how pooling equilibria work. We therefore use it to practice what we have learned in the last couple of lectures.
In the game, Nature draws Player 1 as a “real man” type or a “wimp” type. Player 1 then chooses whether to drink beer or eat quiche. Player 2 observes Player 1’s food choice and chooses whether to fight him or not.
Player 2’s goal is to fight a wimp and avoid a fight with a real man. Player 1’s main goal is to avoid a fight. But the types differ in their food preferences: real men prefer beer, whereas wimps prefer quiche.
Aside: If you think this is a weird setup, you are not alone. Its origins trace back to an important early paper on signaling models and equilibrium selection. Around that time, there was a popular press book called Real Men Don’t Eat Quiche, which satirizes masculinity. The game has survived much longer than the book.
- Because off-the-path beliefs can take any value, when pooling equilibria exist, there is usually a set of infinitely many perfect Bayesian equilibria. The strategies are identical, but the beliefs are different because they fall within a range of values.
- When multiple equilibria exist, a single belief usually makes the receiver indifferent between some strategies. The equilibria outside that belief tend to be much more straightforward because the receiver will have a single optimal best response.
- For the indifferent belief, the equilibria are more complicated. The indifferent receiver can mix freely among the strategies. The mixtures are limited to strategies that still induce the signaling types to follow the pooling strategies.
- Multiple pooling strategies can produce equilibria. And more generally, the perfect Bayesian equilibrium solution concept can produce multiple equilibria with different signaling strategies. Sometimes the off-the-path beliefs necessary to sustain such equilibria are strange. In later lectures, we will formally define what “strange” means and how we can filter those out.