Games and Economic Behavior 66 (2009) 594–597

Contents lists available at ScienceDirect

Games and Economic Behavior www.elsevier.com/locate/geb

In Memoriam

David Gale in Paris For the three of us – of different mathematical generations and backgrounds – David Gale evokes different memories and different impacts, though joint admiration and strong agreement on his principal characteristic: he had the ability to look at each new problem as a babe – and he posed himself a large variety of questions in a large variety of domains – shedding all of his impressive knowledge and technical abilities in favor of the problem itself. Michel Balinski For me, Michel Balinski, David Gale was one of life’s constants – like π or e – or so it seems. Very soon after I moved permanently to Paris, David began his affair with Paris: first, of a year’s duration; then, short term visits; finally, the purchase of an apartment and extended, repeated living. David has, in one way or another, always been present. I repeatedly heard about and so learned about David from my first encounters with Albert W. Tucker at Princeton in 1956. However, when I actually first met him in person is lost in the mists of my memory. David’s thesis advisor at Princeton was Al Tucker: he was also my advisor of record. I say “of record” because, while he agreed on the problem I was to work on, he was away on sabbatical leave while I wrote my thesis, so the person to whom I really turned to for advice and counsel was Ralph Gomory (by this reckoning, Ralph tells me that I can claim to be his one student, a distinction of which I am proud). I moved to Paris in 1980. Within a year or two David got in touch with me because he had been awarded a Guggenheim fellowship and he wished to spend the year in Paris. I suggested and arranged that he be welcomed at the Laboratoire d’Économétrie of the École Polytechnique. This began an association that continued forever after. Whenever we could give him an office, he had one; otherwise, he shared; sometimes, he had to hop around. But he always had eating privileges at the canteen, and he loved to come and eat with the various members of the Laboratoire. The very first year, he lived in a small studio down the hallway from me and my family, on the grounds of the old École Polytechnique in the heart of Paris. He was always puzzling over new questions. I remember his coming to borrow a broom: he needed straws for the experiments he was conducting about how the eyes perceive parallel lines. Several years later, he began looking for an apartment to buy, and he would often ask me to accompany him to look at one or another, and when he finally made his choice, I went with him – to serve as an interpreter – for the signing of the elaborate French documents in the office of a Notaire. That apartment is the scene for an event that I cherish. David, to show his thanks and appreciation to our group, gave a wine-tasting party to which he invited every member of the Laboratoire. Some ten red wines from vastly different regions in France were wrapped in paper and carried numbers. We were all given a list of the names, years, and origins of the wines. The game was (and David always wanted to play games!): identify them. Some liked the game, others were a bit nervous. But David offered me a day of everlasting glory: it turned out that only one person identified them all: me. I only collaborated on one paper with David (Balinski and Gale, 1990): I learned how punctilious he could be on clarity and precision of exposition. This is apparent in his book on linear programming (Gale, 1960) reissued in 1989. It remains one of the best books in the area. David was also a strong defender of his ideas and work. The “stable marriage (and university admissions) problems” (Gale and Shapley, 1962) was also one of his ideas: each woman of a finite number of women has a list of strict preferences among the available men she is willing to marry and, symmetrically, each man of a finite number of men has a list of strict preferences among the available women he is willing to marry. Is there a matching of men to women that is stable, i.e., where there is no man and woman not matched who would prefer to be matched than to be in their current situations (matched to others or celibates)? David formulated the model and question, but could not find the answer. As was his want, he sent the question to the various friends and colleagues that might be interested: Lloyd Shapley furnished the answer, and the rest is history. I also came to work on this problem, mostly with my friend and colleague Mourad Baïou. Among other things, Baïou and Balinski (2002a), (2002b) generalized the stable marriage problem to real numbers: each man and woman has preferences as before, but now each of them has a certain number of hours to allocate in total and upper bounds on the number of 0899-8256/2009 Published by Elsevier Inc. doi:10.1016/j.geb.2009.04.006

M. Balinski et al. / Games and Economic Behavior 66 (2009) 594–597

595

minutes he/she is willing to spend with each member of the opposite sex. Is there an allocation of minutes to women–men pairs that is stable? Generically there exists a unique stable matching. David seemed loathe to accept this. I told him about it the day I was writing up the result. He went off to his office and worked all afternoon, then notified me that the result was wrong: he had a counter-example that he handed to me on paper. His counter-example was written in the usual manner: lists of preferences of the two sexes. I immediately translated it into our graphical notation: the example failed. I phoned to give him the news and to repeat what I had told him before: use our graphical notation, everything becomes easy! Several days later, I received a second counter-example. This time I worried: was our proof wrong after all? I postponed looking at the example, worried, but facts cannot be denied. So I had to look. It was rendered, once again, in long lists. Thank God, his example failed once again. But this time David finally agreed that the graphical notation was the way to study the problem: the fight to persuade him had been long! The last five years Rida Laraki and I have been working on a new theory of social choice and voting. We kept it very much to ourselves before revealing our results. But then, we explained and discussed it with David several times, at the blackboard. I also well remember a long discussion about it one evening, with Sandra Gilbert, his companion, at Bofinger’s, a beautiful and very good Paris brasserie. These last years, David would only occasionally drop in at the Lab, usually at lunch time, after having called me to make sure that there I was free to join him. But he would always announce his arrival in Paris, and we would always dine together several times, usually at his place, together with Americans who lived in Paris or were visiting. Sometimes they were mathematicians, more often they were literary types, for Sandra Gilbert is a renowned poet and professor of literature (who cooks superb meals to boot). So David learned about our work over meals and with chalk. As usual, he questioned it, toyed with examples, admitted it was interesting, but cautiously withheld his judgment. Several months after having read our first publication, Balinski and Laraki (2007), he sent me the following email in his matchless style: “Hi Michel, I’ve now spent some more time looking at your grading scheme and until someone persuades me otherwise I’m convinced that majority ranking is the way to go, for all the reasons you give . . . ” But, as usual, David raised an important question, and proposed an answer: we had developed the theory and the method, but had not given a simple way of computing the results when there are many voters or judges who give their opinions, such as thousands or millions in elections. His question was very much to the point. His answer does not translate the basic rule correctly, but it is intuitive and interesting in its own right: we discuss this in our second publication on the majority judgment, Balinski and Laraki (2007), and call it Gale’s tie-breaking rule. There is a kind of emptiness now that I know I will no longer hear his friendly voice announcing his arrival in Paris. But then, his presence these many years leave memories, and his voice can always be heard on the wonderfully interesting website he created that explains What is Mathematics?1 and for which he won the Pirelli Award for “best multimedia science communication of mathematics” in 2006. How wonderful it is that on the last morning of his conscious life he was working on that web-site! David dedicated one of his books with a limpid “To Michel, with appreciation for his appreciation”: the phrase takes on all its importance when addressed to him. Rida Laraki It is an honor for me to be able to write about my encounters with so great a mathematician as David Gale. I first met him when I was a DEA or Masters student in the academic year 1996–1997. I had discovered his game of Chomp: two players take alternate turns on a checkerboard, (x, y ), where x = 1, . . . , m and y = 1, . . . , n. Each checks an uncovered square (x, y ), which implies covering all squares (x , y  ), for x  x and y   y. The player forced to cover the square (1, 1) loses. Gale had shown that the player who makes the first move has a winning strategy and gave the optimal rule when m = n. But when m = n, the problem is open, and Gale had offered a prize for its solution. So, student that I was, I devoted several weeks to try to solve it. To no avail. This reinforced my desire to write a thesis in the theory of games and to pursue work in the field. I was fortunate to be able to do so, writing my thesis with Sylvain Sorin as advisor. During the time that I was writing my Ph.D. thesis, David regularly visited the Laboratoire d’Économétrie. He was direct, open, accessible and very friendly. He would often dine and discuss with us, the thesis students. We were very impressed by him, and very respectful. After my appointment to the CNRS in 2001, I had the privilege to share my office with David intermittently over the span of several years. He had an unusual ability to listen to others and he often asked excellent, pertinent questions. But what most fascinated me was the diversity of the problems he chose to work on, very much the image of his entire career. Once he talked to me about his search for an optimal strategy for a naked woman in a swimming pool seeking to escape a man on the edge of the pool. The solution is known in certain cases – in particular, when the lady is at the center of a circular pool – but it is very difficult for other initial positions of the lady or pools of other shapes! The day that Michel forwarded to me David’s message that he was convinced by the majority judgement is unforgettable: I printed it and read and reread it!

1

http://mathsite.math.berkeley.edu/intro.html.

596

M. Balinski et al. / Games and Economic Behavior 66 (2009) 594–597

David will always remain in my memory, and his ideas, his papers, his light humor, and the problems he posed and talked about will always remain in any course that I give in the theory of games: stable marriages, Chomp, and the algorithm for Hex, in particular, will always be present. Sylvain Sorin I first met David in Berkeley in 1986 during the special year at MSRI on Game Theory and Mathematical Economics. I was surprised to face a young person: I was expecting an old, famous mathematician (a winner of the John von Neumann Theory Prize . . . ). Was this the true David Gale? Was this the man whom I knew for infinite games with perfect information, Nim type games, Hex and fixed points, so many duality results, general equilibrium and growth models, global univalence, stable matchings, and so on and on . . . ? During that time, I was part of a lively French–Belgian group and David was one of the few Berkeley mathematicians (with Steve Smale) who enjoyed our company and was always suggesting new activities such as wine tasting in his garden, or jazz in Oakland . . . . He arrived at the Laboratoire d’Économétrie of the École Polytechnique when I had just left for the University Louis Pasteur in Strasbourg and we met again regularly when I came back to the Université Paris X around 1995. For me David did not change since our first meeting. He was permanently enthusiastic, full of new ideas, of recent results he had just obtained or heard about, and that he was excited to share. In addition, he showed the same curiosity, always listening, with amazing attention: he would always ask short and precise questions, quite innocent, always eager to hear about and understand new ideas. He could be aggressive too, but he was right to the point. There was no need to repeat the crux of an argument, only to make it more precise, exactly adequate to the problem; and whenever existence was the issue, he wanted to elaborate on an explicit algorithm. The discussions usually started in my office, continued at lunch and then started again over oysters parties that he liked so much. Obviously Paris was also a right place to go out for jazz. New Morning, Duc des Lombards, Sunset . . . , were among his favorite places were he used to take me. It is amusing to notice that I discovered one of his masterpieces, his book on linear economic models (Gale, 1960) when I started teaching a course on optimization and games at Université de Paris VI in 2000. This book is a wonderful model of depth, clarity and elegance that covers an enormous amount of material. David had the marvelous talent of starting by explaining simple phenomena, then going on to sophisticated elaborations, reducing the complexity, presenting new approaches – duality was one of his predilections – and ending with constructions and algorithms. A more recent event occurred when I had just taken the direction of the Equipe Combinatoire and Optimisation, founded by Claude Berge in 1975. I discovered in the proceedings of the 1976–1977 seminar a lecture dealing with the “Gale transform of polyhedra” (Gale, 1956) and noticed that Gale is quoted in all the works of Berge (topological games Gale and Stewart, 1953, graphs and networks Gale, 1957, and combinatorial games Gale, 1974). David was a kind of epicurian, a mathematical gourmet, always agreeable, smiling and enjoying similar people. He told me the pleasure he had in writing columns for the Mathematical Intelligencer and sent me the book he made of them with the dedication: “To Sylvain, for your amusement.” A wonderful game theorist.

Michel Balinski École Polytechnique, 1 rue Descartes, 75005 Paris, France E-mail address: [email protected] Rida Laraki C.N.R.S., 1 rue Descartes, 75005 Paris, France E-mail address: [email protected] Sylvan Sorin Université de Paris, VI, 175, rue du Chevaleret, 75013 Paris, France E-mail address: [email protected] 6 April 2009 Available online 15 April 2009

References Baïou, Mourad, Balinski, Michel, 2002a. The stable allocation (or ordinal transportation) problem. Math. Oper. Res. 27 (3), 485–503. Baïou, Mourad, Balinski, Michel, 2002b. Erratum: The stable allocation (or ordinal transportation) problem. Math. Oper. Res. 27 (4), 662–680. Balinski, Michel, Gale, David, 1990. On the core of the assignment game. In: Leifman, Lev J. (Ed.), Functional Analysis, Optimization, and Mathematical Economics: A Collection of Papers Dedicated to the Memory of Leonid Vital’evich Kantorovich. Oxford Univ. Press, pp. 274–289. Balinski, Michel, Laraki, Rida, 2007. Election by majority judgement: Experimental evidence. Technical report 2007–28, Laboratoire d’Économétrie, École Polytechnique, December 17. Balinski, Michel, Laraki, Rida, 2007. A theory of measuring, electing and ranking. Proc. Natl. Acad. Sci. USA 104, 8720–8725. Gale, David, 1956. Neighboring vertices on a convex polyhedron. In: Kuhn, Harold W., Tucker, Albert W. (Eds.), Linear Inequalities and Related Systems. In: Ann. of Math. Stud., vol. 38. Princeton Univ. Press, pp. 255–263. Chapter 15.

M. Balinski et al. / Games and Economic Behavior 66 (2009) 594–597

597

Gale, David, 1957. A theorem on flows in networks. Pacific J. Math. 7, 1073–1082. Gale, David, 1960. The Theory of Linear Economic Models. McGraw–Hill. Gale, David, 1974. A curious Nim-type game. Amer. Math. Monthly 81, 876–879. Gale, David, Shapley, Lloyd S., 1962. College admissions and the stability of marriage. Amer. Math. Monthly 69, 9–15. Gale, David, Stewart, Frank M., 1953. Infinite games with perfect information. In: Kuhn, Harold W., Tucker, Albert W. (Eds.), Contributions to the Theory of Games II. In: Ann. of Math. Stud., vol. 28. Princeton Univ. Press, pp. 245–266.