Two essential papers by APS

Original article (link) posted: 20/08/2005

What's the difference between APS (1986) and APS (1990)? The distinction is similar to that of Abreu (1986) and Abreu (1988). The earlier papers (both published in JET), APS (1986) and Abreu (1988), analyze the optimal strategies in actual oligopoly models by using powerful properties in repeated games established by them. The later papers (both in Econometrica) extend the results in generalized situations. Although these two Econometrica papers are more general and sophisticated than those of JET, they may have some drawbacks such as, lack of motivation, too concise explanation (to understand), and more seriously, showing no actual optimal strategies. Two papers in JET are strongly motivated by the actual collusion problems in IO, and hence you can see how useful the properties they found are.

Two APS papers heavily rely on the technique used in dynamic programming. Using those technique, they reduce the repeated game to a static structure from which can be extracted the optimal equilibria in question. That is,they construct a new game by truncating the discounted supergame as follows:
after each first-period history, replace the sequential equilibrium successor by the payoffs associated with that successor.

APS (1990) summarize the distinction of the two papers.

First, it relaxes the restriction of symmetry, showing the theory capable of embracing both asymmetric equilibria of symmetric games and arbitrary asymmetric games. Secondly, the sufficiency of using bang-bang reward functions in efficiently collusive equilibria is strengthened to a necessity theorem. Finally, we provide an algorithm useful in computing the sequential equilibrium value set.
(APS (1990), p.1044)

Finally, I would like to remind you of the difference between Abreu and APS. These papers are essentially different in the sense that the former analyze perfect monitoring cases whereas the latter consider imperfect monitoring cases. Moreover, even though all these papers are inspired by dynamic programming technique, their focus is different more or less. The argument in Abreu is about equilibrium paths, but that in APS is about equilibrium payoff sets.


Abreu (1986) "Extremal Equilibria of Oligopolistic Supergames" JET, 39
Abreu (1988) "On the Theory of Infinitely Repeated Games with Discounting" Econometrica, 56
Abreu, Pearce and Stacchetti (1986) "Optimal Cartel Equilibria with Imperfect Monitoring" JET, 39
Abreu, Pearce and Stacchetti (1990) "Toward a Theory of Discounted Repeated Games with Imperfect Monitoring" Econometrica, 58

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