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The minmax in repeated games with imperfect monitoring can differ from the minmax of those games with perfect monitoring when two or more players are able to gain common information known only to themselves, and utilize this information at a later stage. Gossner and Tomala showed that in a class of such games, the minmax is given by a weighted average of the payoffs of two main strategies: one in which the information is gained, and the other in which the information is utilized. However, all examples analyzed to date require only one main strategy in which information is created and utilized simultaneously. We show that two strategies are indeed needed by providing and solving a concrete example of a three-player game.

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... For every correlation system c such that x is almost surely constant, H c ≥ 0 thus V intersects the half-plane x 1 ≥ 0. Since V is compact, so is its convex hull and the supremum is indeed a maximum. The set V need not be convex as shown in Goldberg [7]; the supremum in the definition of w above might not be achieved by a point in V , but might be achieved by a convex combination involving two points of V with nonzero weights. For computations, it is convenient to express the number w through the boundary of co V . ...

... The solution and its proof are rather involved and the reader is referred to Gossner et al. [10] for the statement of the solution. Building on this result, two examples of games and signalling structures have been completely resolved so far: one in Gossner et al. [10] and one in Goldberg [7]. Note that for each h ∈ , either cav uuhh = uuhh or cav u is linear on some interval containing h. ...

... The relative lengths of these phases are 1 − . Gossner et al. [10] showed that our main example is of the first kind and Goldberg [7] exhibited an example of the second. We consider once more the main example. ...

We characterize the maximum payoff that a team can guarantee against another in a class of repeated games with im- perfect monitoring. Our result relies on the optimal trade-off for the team between optimization of stage-payoffs and generation of signals for future correlation.

Glossary Definition of the Subject Introduction Games with Observable Actions Games with Non‐observable Actions Acknowledgments Bibliography

We characterize the maximum payoff that a team can guarantee against another in a class of repeated games with im- perfect monitoring. Our result relies on the optimal trade-off for the team between optimization of stage-payoffs and generation of signals for future correlation.

Let (xn)n be a process with values in a finite set X and law P, and let yn f(xn) be a function of the process. At stage n, the conditional distribution pn P(xnx1,,xn1), element of (X), is the belief that a perfect observer, who observes the process online, holds on its realization at stage n. A statistician observing the signals y1,,yn holds a belief enP(pnx1,,xn) () on the possible predictions of the perfect observer. Given X and f, we characterize the set of limits of expected empirical distributions of the process (en) when P ranges over all possible laws of (xn)n.