Working with Mike

Mike and I worked together for half a century. To date, I’ve had some twenty-five collaborators—co-authors of joint papers and books; there’s no question that Mike is the premier one. He and I wrote three joint papers—the original Bargaining Set paper (21), the one on the Minimax Principle (38), and the Talmud paper (50)—and one joint book (64), about repeated games of incomplete information. These works are among my own most important, and perhaps among Mike’s, too. To be sure, each of us also did other things, but these stand out. Our collaboration started in the late nineteen-fifties, two or three years after I’d come to the math department of the Hebrew University. Mike was working in complex function theory, in which he had done his thesis; I had already made the switch from my thesis topic—knots—to games. One autumn afternoon I spoke at the mathematics colloquium, a weekly gathering of the whole department where a faculty member or guest gives a talk that’s supposed to be of general interest to any mathematician, not only a specialist. I decided to speak about the von Neumann– Morgenstern (N-M) “solution,” a.k.a. stable set (von Neumann and Morgenstern, 1944). This is a very subtle and beautiful solution notion for coalitional games; at the time, I myself did not fathom its full beauty and subtlety, which became apparent only after subsequent work with Mike. During the question period after the lecture, Mike asked several questions that challenged its appropriateness. As the discussion lengthened, I suggested continuing in private, which we did. I did what I could to explain the N-M notion, but could not satisfy Mike. At last, a little exasperated, I said, well, let’s see if you can come up with something better. He said, OK, give me a couple of days. That started a lifetime of friendship and collaboration. Mike indeed came up with a proposal after a few days, which I promptly “shot down.” That is, I constructed a “counterintuitive example”: a game in which the proposed definition yields unacceptable results. This process continued for many months—Mike would propose a definition, and I would shoot it down. Finally, as the academic year was drawing to a close, Mike came up with a definition that I could not “shoot down.” I didn’t like his definition, and told him so; it seemed overly complex and arbitrary, lacking elegance and simplicity. There wasn’t even a general existence theorem; it was, indeed, sometimes empty. But, I could not shoot it down. Shortly thereafter, I left Israel for an extended trip to the United States, and occupied myself with other matters. To my surprise, some time later I received a manuscript from Mike entitled “The Bargaining Set for Cooperative Games”—containing Mike’s definition, some worked-out examples, and some additional analysis—by R.J. Aumann and M. Maschler! I wrote to him that