When studying dynamic interactions, economists like discounted games. Existence of equilibrium is assured because the payoff is a continuous function of the strategies of the players, and construction of equilibrium strategies often require ingenious tricks, and so are fun to think about and fun to read. Unfortunately in practice the discounted evaluation is often not relevant. In many cases players, like countries, firms, or even humans, do not know their discount factor. Since the discounted equilibrium strategies highly depend on the discount factor, this is a problem. In other cases, the discount factor changes over time in an unknown way. This happens, for example, when the discount factor is derived from the interest rate or the players monetary or familial situation. Are predictions and insights that we get from a model with a fixed and known discount still hold in models with changing and unknown discount factor?

To handle such situations, the concept of a **uniform equilibrium** was defined. A strategy vector is a **uniform ε-equilibrium **if it is ε-equilibrium in all discounted games, for all discount factors sufficiently close to 1 (that is, whenever the players are sufficiently patient). Thus, if a uniform ε-equilibrium exists, then the players can play an approximate equilibrium as soon as they are sufficiently patient. In our modern world, in which one can make zillions of actions in each second, the discount factor is sufficiently close to 1 for all practical purposes. A payoff vector x is a **uniform equilibrium payoff** if for every ε>0 there is a uniform ε-equilibrium that yields payoff that is ε-close to x.

In repeated games, the folk theorem holds for the concept of uniform equilibrium (or for the concept of uniform subgame perfect equilibrium). Indeed, given a feasible and strictly individually rational payoff x, take a sequence of actions such that the average payoff along the sequence is close to x. Let the players play repeatedly this sequence of actions while monitoring the others, and have each deviation punished by the minmax value. When the discount factor is close to 1, the discounted payoff of the sequence of actions is close to the average payoff, and therefore the discounted payoff that this strategy vector yields is close to x. If one insists on subgame perfection, then punishment is achieved by a short period of minmaxing followed by the implementation of an equilibrium payoff that yields the deviator a low payoff.

For two-player zero-sum stochastic games, Mertens and Neyman (1981) proved that the uniform value exists. Vieille (2000) showed that in two-player non-zero-sum stochastic games uniform ε-equilibria exist. Whether or not this result extends to any number of players is still an open problem.

Why do I tell you all that? This is a preparation for the next post, that will present a new and striking result by a young French Ph.D. student.

## 4 comments

May 31, 2014 at 3:33 am

Links for 05-31-14 | The Penn Ave Post[…] 4:33 on May 31, 2014 by Mark Thoma What Do We "Deserve" Anyway? – Brad DeLong Uniform Equilibrium – Theory of the Leisure Class […]

June 1, 2014 at 9:11 am

The Mertens’ Conjecture | The Leisure of the Theory Class[…] is a finite game, and therefore by backwards induction it has an equilibrium. As I mentioned in my previous post, the discounted game has an equilibrium (even a stationary equilibrium) because of […]

June 11, 2014 at 10:24 am

X. L.WHEN studying dynamic interactions, economists like discounted games … Are predictions and insights that we get from a model with a fixed and known discount still hold in models with changing and unknown discount factor? … To handle such situations, the concept of a uniform equilibrium was defined …

Dear Eilon, why do (most) economists not quite accept the notion of (\epsilon)-uniform equilibrium? Rather, most of recent literatures on supergames are about the limit of (exact)-discounted equilibrium payoff set?

Was there ever a debate in history and how was ended? It seems that the situation is rather embarrassing: economists and game theorists are separated from each other, and are doing research (on repeated games) with different notations: compare with the two books of Repeated games by MSZ and Samuelson & Mailath.

It’s abnormal in science.

June 13, 2014 at 7:11 am

EilonHello,

I have never heard of a debate on the subject. And as I am not an economist, I do not know why economists prefer exact equilibria in discounted games to uniform equilibria. Each model poses different theoretical difficulties, and so they are both challenging to work on.