Thursday, October 2, 2014

The Folk Theorem

We have talked a fair bit about coordination games and the forces shaping just what happens to be coordinated upon. In that light, it's important to realize how the presence of dynamic considerations, i.e. repeated game settings, fundamentally transforms essentially all games into coordination games. In our usual example to illustrate how to take advantage of dynamics to build relational contracts (self-enforcing repeated games equilibrium), we studied a simplified version of Bertrand competition. Firms could price high or low, low was a dominant strategy in the one-shot game, yet, with the right implicit contract, we can maintain high prices. The key, if you'll recall, was to balance a sufficient future punishment against the temptation to defect.

While coordinating on full cooperation seems the obvious course of action, if available, it is far from the only equilibrium to this game. For instance, it is fairly obvious that coordinating on the low price in every period is also an equilibrium--a really simple one, it turns out. Unlike the high price equilibrium with its nice and fight feedback loops, the no cooperation equilibrium requires no such machinery. The equilibrium consists of simply choosing a low price in every period regardless of past actions, and that's it.

There are many more equilibria in this game. For instance, if maintaining high prices yields each player $3 per period whereas fighting in every period yields $2 per period, then payoffs of any amount in between can also be sustained merely by interweaving the two. Suppose we wished to support payoffs that are high 5/6 of the time and low 1/6. In that case, we need only follow our high priced strategy whenever the period is not a multiple of 6 or if we're in a punishment phase and follow the low price strategy for periods that are a multiple of 6. Any fraction of high and low payoffs may be similarly supported.

This observation that just about any set of per period payoffs can be supported as an equilibrium is known as the Folk Theorem. It is so-named since, like many folk tales, no one was quite sure who first had the idea, but a Nobel winning game theorist, Robert Aumann, was the first to write the argument down (in a much more general and abstract way than my simple sketch above. Notice what the folk theorem implies in terms of our list of archetypal games:

All repeated games are coordination games
The theorem tells us that, regardless of the original form of the game, be it prisoner's dilemma, hawk-dove, matching pennies and so on, its repeated version amounts to a pure coordination game.

No comments: