Shmuel Zamir is 70, and the international conference at Stony Brook dedicated today to this event. Shmuel’s past and present co-authors presented papers that are related to his work. Bob Aumann described some of Shmuel’s contribution to game theory, starting from his Ph.D. thesis on the rate of convergence of the value of n-stage repeated games with incomplete information on one side to the value of the corresponding infinitely repeated game. That is, we know from the work of Aumann and Maschler (1967, 1995) that the value v_n of the n-stage repeated game with incomplete information on one side converges to a limit v_*. We also know that v_* is the value of the same game that is infinitely repeated. The question that was open back then, when Shmuel was young, is the rate of convergence: what is the order of magnitude of the difference v_* – v_n. It was known that this difference is bounded by O(1/sqrt(n)), but it was not known whether this is the best bound. The first part of Shmuel’s thesis solves this problem, and proves that indeed O(1/sqrt(n)) is the best bound, and that for some classes of games better bounds are possible. Shmuel told me that back then he solves the value of 8-stage games, just to get the feeling of the problem. As someone who studied 3-stage games, I am truly impressed by this confession.
We also heard about the work that Shmuel did with Jean-Francois Mertens: the existence of the limit of the value of n-stage repeated games with incomplete information on both sides, and the existence of the universal belief space. I will dedicate another post to the universal belief space and to recent results relating to it.
Shmuel’s contributions to auction theory and to experimental game theory were not left behind either.
Undoubtedly, Shmuel Zamir’s contributions are fundamental, and the research of many among us, me included, was affected by his work. Mazal Tov, Shmuel. May the next 70 years be as fruitful as the last ones!