An interview with

H: Perhaps it is time now to ask, what is game theory?

A: Game theory is the study of interactions from a rational viewpoint. Even though the rationality does not have to be conscious, it is still there in the background. So we are interpreting what we see in the world from a rational viewpoint.

In other words, we ask, what is best for people to do when there are other people, other decision-makers, other entities who also optimize their decisions? Game theory is optimal decision-making in the presence of others with different objectives.

H: And where everyone’s decision influences everyone’s outcomes. One takes into account that everyone is doing his own optimization and everyone is trying to advance his own objectives.

Game theory started formally with the von Neumann and Morgenstern book in the 1940s. Probably the war had a lot to do with the fact that many people got interested. Just to see how it developed, in the first international game theory workshop in 1965 in Jerusalem there were seventeen people.

A: There were three conferences on game theory in Princeton in the fifties: ’53, ’55, and ’57. Those were attended by more than seventeen people. The seventeen people in 1965 were seventeen selected people.

H: The discipline has really grown—from a few dozen people in the fifties and sixties, to more than six hundred at the last game theory congress in Marseille.

This is a good point to discuss the universality of game theory. In the Preface to the first volume of the Handbook of Game Theory [iv] we wrote that game theory may be viewed as a sort of umbrella or unified field theory.

A: It’s a way of talking about many sciences, many disparate disciplines. Unlike other approaches to disciplines like economics or political science, game theory does not use different, ad-hoc constructs to deal with various specific issues, such as perfect competition, monopoly, oligopoly, international trade, taxation, voting, deterrence, animal behavior, and so on. Rather, it develops methodologies that apply in principle to all interactive situations, then sees where these methodologies lead in each specific application.

But rather than being an umbrella for all those disciplines, it’s perhaps better to think of it as a way of thinking about a certain aspect of each—the interactively rational aspect. There are many things in these disciplines that have nothing to do with this aspect. In law, in computer science, in mathematics, in economics, in politics, there are many things that have nothing to do with game theory. It is not like a unified field theory, which would cover all of gravitation, magnetism, and electricity.

H: Perhaps it is like mathematics applied to other sciences, which is a tool, a language for formalizing and analyzing.

A: That’s an interesting analogy. Mathematics helps in certain aspects of many sciences—those given to formalization. Game theory is similar in that respect: it helps in many disciplines, specifically in their interactively rational parts. Figure 1 is a stylized representation.

H: The Game Theory Society was established in ’99. You were the first, founding president, up to 2003. By now you should have a good overview of what game theory is, and of what the Game Theory Society is.

A: Game theory has become a big discipline, or rather a big interdiscipline. It is time to have a tool for gathering game theorists in all kinds of senses. Conferences, journals, the Web. When discussing my education, I mentioned that at City College there were a couple of tables reserved for the more dedicated math students. People would come between classes, sit down, have an ice cream soda, and talk about math. The Game Theory Society is the game theory table in the cafeteria that’s called the world. It is a place where people can discuss game theory and exchange ideas, in various senses and various ways.

Figure 1. The blooming of game theory

H: Do you have any thoughts on where game theory is going?

A: It is difficult to tell. It is very hard to know where things are going. In the presidential address at the Game Theory Society Congress in Bilbao in 2000 [81], I discussed some directions for research in the future.

Let me say something of a more general nature. People are pushing in different directions; we are going to find a spreading of the discipline among different people. Some people go in a very strongly mathematical direction, very deep mathematics. We will see a separation of the more mathematical branches from the more applied branches like economic applications. We’ll see a lot of experimental and engineering application of game theory. People in game theory will understand each other less in the future.

H: Do you expect a Tower of Babel syndrome to develop?

A: It is not something that I would like, but it’s a sign of maturity. Tower of Babel syndrome is a very good way of putting it.

H: What is definitely true is that from a small community where essentially everybody could understand everybody else, game theory has grown to a big “city,” where people are much more specialized. As in any developing discipline, it’s natural that everybody goes deeper into one of the aspects and understands less and less of the others. Nevertheless, at this point there is still interplay between the various aspects and approaches, so everybody benefits from everybody else. Take physics or mathematics. I wouldn’t understand what algebraic topology does nowadays. Somebody in combinatorics may understand something about probability, but wouldn’t understand some of the things we do in game theory. Nevertheless, mathematics is a single discipline.

Would you like to say anything about the different approaches in game theory? For example, mathematical vs. conceptual; axiomatic and cooperative vs. strategic and non-cooperative. Why is it that there are so many approaches? Are they contradictory or are they just different? And how about people who think that some approaches in game theory are valid, and other approaches are not?

A: You are quite right that there is a group of people, working in non-cooperative, strategic games, who think that cooperative (coalitional) game theory is less important, not relevant, not applicable.

Let me backtrack and describe what we mean by non-cooperative or strategic game theory, vis-à-vis cooperative or coalitional game theory. Strategic game theory is concerned with strategic equilibrium—individual utility maximization given the actions of other people, Nash equilibrium and its variants, correlated equilibrium, that kind of thing. It asks how people should act, or do act. Coalitional game theory, on the other hand, concentrates on division of the payoff, and not so much on what people do in order to achieve those payoffs.

Practically speaking, strategic game theory deals with various equilibrium concepts and is based on a precise description of the game in question. Coalitional game theory deals with concepts like the core, Shapley value, von Neumann–Morgenstern solution, bargaining set, nucleolus. Strategic game theory is best suited to contexts and applications where the rules of the game are precisely described, like elections, auctions, internet transactions. Coalitional game theory is better suited to situations like coalition formation or the formation of a government in a parliamentary democracy or even the formation of coalitions in international relations; or, what happens in a market, where it is not clear who makes offers to whom and how transactions are consummated. Negotiations in general, bargaining, these are more suited for the coalitional, cooperative theory.

H: On the one hand, negotiations can be analyzed from a strategic viewpoint, if one knows exactly how they are conducted. On the other hand, they can be analyzed from a viewpoint of where they lead, which will be a cooperative solution. There is the “Nash program”—basing cooperative solutions on non-cooperative implementations. For example, the alternating offers bargaining, which is a very natural strategic setup, and leads very neatly to the axiomatic solution of Nash—as shown by Rubinstein and Binmore.

A: These “bridges” between the strategic and the coalitional theory show that these approaches are not disparate. In order to make a bridge like that you have to define precisely the non-cooperative situation with which you are dealing. One of the bridges that we discussed earlier in this interview is the Folk Theorem for repeated games. There, the non-cooperative setup is the repeated game. When you have a bridge like that to the non-cooperative theory, the strategic side must be precisely defined. The big advantage of the cooperative theory is that it does not need a precisely defined structure for the actual game. It is enough to say what each coalition can achieve; you need not say how. For example, in a market context you say that each coalition can exchange among its own members whatever it wants. You don’t have to say how they make their offers or counteroffers. In a political context, it is enough to say that any majority of parliament can form a government. You don’t have to say how they negotiate in order to form a government. That already defines the game, and then one can apply the ideas of the coalitional theory to make some kind of analysis, some kind of prediction.

You asked about the sociology of game theorists, rather than game theory. There is a significant group of people in strategic game theory who have an attitude towards coalitional game theory similar to that of pure mathematicians towards applied mathematics fifty years ago. They looked down their noses and said, this is not really very interesting; we’re not going to sully our hands with this stuff.

There is no justification for this in the game-theoretic sociology, just as there was no justification for it in the mathematics sociology. Each one of these branches of the discipline makes its contribution. In many ways, the coalitional theory has done better than the strategic theory in giving insight into economic and other environments. A prime example of this is the equivalence theorem, which gave a game-theoretic foundation for the law of supply and demand. There has been nothing of that generality or power in strategic game theory. Strategic game theory has made important contributions to the analysis of auctions, but it has not given that kind of insight into economics, or into any other discipline.

Another example of an important insight yielded by coalitional game theory is the theory of matching markets. This whole branch of game theory—and it is highly applied—grew out of the ’62 paper of Gale and Shapley “College Admissions and the Stability of Marriage.” It is not quite as fundamental as the equivalence theorem, but it is a very important application, certainly of comparable importance to the work on auctions in strategic game theory, which is very important. There is no reason to denigrate the contributions of coalitional game theory, either on the applied or the theoretical level.

H: Indeed, Adam Brandenburger said that his students at Harvard Business School found cooperative game theory much more relevant to them than the non-cooperative theory.

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Let’s switch to another topic. You have had an enormous impact on the profession by influencing many people. I am talking first of all about your students. By now you have had thirteen doctoral students. I think twelve of them are by now professors, in Israel and abroad, who are well recognized in the field and also in related fields.

A: Almost all the students eventually ended up in Israel, after a short break for a post-doc or something similar abroad.

H: That’s not surprising since most of them—all except Wesley—started in Israel and are Israelis.

A: There is quite a brain drain from Israel. A large proportion of prominent Israeli scientists who are educated in Israel end up abroad—a much larger proportion than among my students.

These are my doctoral students up until now: Bezalel Peleg, David Schmeidler, Shmuel Zamir, Binyamin Shitovitz, Zvi Artstein, Elon Kohlberg, Sergiu Hart, Eugene Wesley, Abraham Neyman, Yair Tauman, Dov Samet, Ehud Lehrer, and Yossi Feinberg. Of these, three are currently abroad—Kohlberg, Wesley, and Feinberg. Also, there are about thirty or forty masters students.

Each student is different. They are all great. In all cases I refused to do what some people do, and that is to write a doctoral thesis for the student. The student had to go and work it out by himself. In some cases I gave very difficult problems. Sometimes I had to backtrack and suggest different problems, because the student wasn’t making progress. There were one or two cases where a student didn’t make it—started working and didn’t make progress for a year or two and I saw that he wasn’t going to be able to make it with me. I informed him and he left. I always had a policy of taking only those students who seemed very, very good. I don’t mean good morally, but capable as scientists and specifically as mathematicians. All of my students came from mathematics. In most cases I knew them from my classes. In some cases not, and then I looked carefully at their grades and accepted only the very best. I usually worked quite closely with them, meeting once a week or so at least, hearing about progress, making suggestions, asking questions. When the final thesis was written I very often didn’t read it carefully. Maybe this is news to Professor Hart, maybe it isn’t. But by that time I knew the contents of the work because of the periodic meetings that we would have.

H: Besides, you don’t believe anything unless you can prove it to yourself.

A: I read very little mathematics—only when I need to know. Then, when reading an article I say, well, how does one prove this? Usually I don’t succeed, and then I look at the proof.

But it is really more interesting to hear from the students, so, Professor Hart, what do you think?

H: Most doctoral students want to finish their thesis and get out as soon as possible. Aumann’s students usually want to continue—up to a point, of course. This was one of the best periods in my life—being immersed in research and bouncing ideas back and forth with Professor Aumann; it was a very exciting period. It was very educating for my whole life. Having a good doctoral advisor is a great investment for life. There is a lot to say here, but it’s your interview, so I am making it very short. There are many stories among your students, who are still very close to one another.




Picture 6. At the GAMES 1995 Conference in Honor of Aumann’s 65^th birthday, Jerusalem, June 1995: Abraham Neyman, Bob Aumann, John Nash, Reinhard Selten, Ken Arrow, Sergiu Hart (from left to right)

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Next, how about your collaborators? Shapley, Maschler, Kurz, and Drèze are probably your major collaborators. Looking at your publications I see many other coauthors—a total of twenty—but usually they are more focused on one specific topic.

A: I certainly owe a lot to all those people. Collaborating with other people is a lot of work. It makes things a lot more difficult, because each person has his own angle on things and there are often disagreements on conceptual aspects. It’s not like pure mathematics, where there is a theorem and a proof. There may be disagreements about which theorem to include and which theorem not to include, but there is no room for substantive disagreement in a pure mathematics paper. Papers in game theory or in mathematical economics have large conceptual components, on which there often is quite substantial disagreement between the coauthors, which must be hammered out. I experienced this with all my coauthors.

You and I have written several joint papers, Sergiu. There wasn’t too much disagreement about conceptual aspects there.

H: The first of our joint papers [50] was mostly mathematical, but over the last one [82] there was some … perhaps not disagreement, but clarification of the concepts. The other two papers [69, 70], together with Motty Perry, involved a lot of discussion. I can also speak from experience, having collaborated with other people, including some longstanding collaborations. Beyond mathematics, the arguments are about identifying the right concept. This is a question of judgment; one cannot prove that this is a good concept and that is not. One can only have a feeling or an intuition that that may lead to something interesting, that studying this may be interesting. Everybody brings his own intuitions and ideas.

A: But there are also sometimes real substantive disagreements. There was a paper with Maschler—“Some Thoughts on the Minimax Principle” [27]—where we had diametrically opposed opinions on an important point that could not be glossed over. In the end we wrote, “Some experts think A, others think ‘Not A’.” That’s how we dealt with the disagreement. Often it doesn’t come to that extreme, but there are substantial substantive disagreements with coauthors. Of course these do not affect the major message of the paper. But in the discussion, in the conceptualization, there are nuances over which there are disagreements. All these discussions make writing a joint paper a much more onerous affair than writing a paper alone. It becomes much more time-consuming.

H: But it is time well consumed; having to battle for your opinion and having to find better and better arguments to convince your coauthor is also good for your reader and is also good for really understanding and getting much deeper into issues.

That is one reason why an interdisciplinary center is so good. When you must explain your work to people who are outside your discipline, you cannot take anything for granted. All the things that are somehow commonly known and commonly accepted in your discipline suddenly become questionable. Then you realize that in fact they shouldn’t be commonly accepted. That is a very good exercise: explain what you are doing to a smart person who has a general understanding of the subject, but who is not from your discipline. It is one of the great advantages of our Rationality Center. A lot of work here has been generated from such discussions. Suddenly you realize that some of the basic premises of your work may in fact be incorrect, or may need to be justified. The same goes for collaborators. When you think by yourself, you gloss over things very quickly. When you have to start explaining it to somebody, then you have to go very slowly, step by step, and you cannot err so easily.

A: That’s entirely correct, and I’d like to back it up with a story from the Talmud. A considerable part of the Talmud deals with pairs of sages, who consistently argued with each other; one took one side of a question and the other took the other side. One such pair was Rabbi Yochanan and Resh Lakish. They were good friends, but also constantly taking opposite sides of any given question. Then Resh Lakish died, and Rabbi Yochanan was inconsolable, grieved for many days. Finally he returned to the study hall and resumed his lectures. Then, for everything that Rabbi Yochanan said, one of the sages adduced thirty pieces of supporting evidence. Rabbi Yochanan broke down in tears and said, what good are you to me? You try to console me for the loss of Resh Lakish, but you do exactly the opposite. Resh Lakish would come up with thirty challenges to everything I said, thirty putative proofs that I am wrong. Then I would have to sharpen my wits and try to prove that he is wrong and thereby my position would be firmly established. Whereas you prove that I’m right. I know that I’m right; what good does it do that you prove that I am right. It doesn’t advance knowledge at all.

This is exactly your point. When you have different points of view and there is a need to sharpen and solidify one’s own view of things, then arguing with someone makes it much more acceptable, much better proved.

With many of my coauthors there were sharp disagreements and very close bargaining as to how to phrase this or that. I remember an argument with Lloyd Shapley at Stanford University one summer in the early seventies. I had broken my foot in a rock-climbing accident. Shapley came to visit me in my room at the Stanford Faculty Club, and I was hobbling around on crutches. This is unbelievable, but we argued for a full half hour about a comma. I don’t remember whether I wanted it in and Lloyd wanted it out, or the other way around. Neither do I remember how it was resolved. It would not have been feasible to say “some experts would put a comma here, others would not.” I always think that my coauthors are stubborn, but maybe I am the stubborn one.

I will say one thing about coauthorship. Mike Maschler is a wonderful person and a great scientist, but he is about the most stubborn person I know. One joint paper with Maschler is about the bargaining set for cooperative games [17]. The way this was born is that in my early days at the Hebrew University, in 1960, I gave a math colloquium at which I presented the von Neumann–Morgenstern stable set. In the question period, Mike said, I don’t understand this concept, it sounds wrongheaded. I said, okay, let’s discuss it after the lecture. And we did. I tried to explain and to justify the stable set idea, which is beautiful and deep. But Mike wouldn’t buy it. Exasperated, I finally said, well, can you do better? He said, give me a day or two. A day or two passes and he comes back with an idea. I shoot this idea down—show him why it’s no good. This continues for about a year. He comes up with ideas for alternatives to stable sets, and I shoot them down; we had well-defined roles in the process. Finally, he came up with something that I was not able to shoot down with ease. We parted for the summer. During that summer he wrote up his idea and sent it to me with a byline of Robert Aumann and Michael Maschler. I said, I will have no part of this. I can’t shoot it down immediately, but I don’t like the idea. Maschler wouldn’t take no for an answer. He kept at me stubbornly for weeks and months and finally I broke down and said, okay, I don’t like it, but go ahead and publish it. This is the original “Bargaining Set for Cooperative Games” [17]. I still don’t like that idea, but Maschler and Davis revised it and it eventually became, with their revision, a very important concept, out of which grew the Davis–Maschler kernel and Schmeidler’s nucleolus. Because of where it led more than because of what it is, this became one of my most cited papers. Maschler’s stubbornness proved justified. Maybe it should have waited for the Davis–Maschler revision in the first place, but anyway, in hindsight I’m not sorry that we published this. Michael has always been extremely stubborn. When he wants something, it gets done. As you say, Sergiu, coauthorship is much more exacting, much more painful than writing a paper alone, but it also leads to a better product.

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H: This very naturally leads us to what you view as your main contributions. And, what are your most cited papers, which may not be the same thing.

A: One’s papers are almost like one’s children and students—each one is different, one loves them all, and one does not compare them. Still, one does keep abreast of what they’re doing; so I also keep an eye on the citations, which give a sense of what the papers are “doing.”

One of the two most cited papers is the Equivalence Theorem—the “Markets with a Continuum of Traders” [16]—the principle that the core is the same as the competitive equilibrium in a market in which each individual player is negligible. The other one is “Agreeing to Disagree” [34], which initiated “interactive epistemology”—the formal theory of knowledge about others’ knowledge. After that come the book with Shapley, Values of Non-Atomic Games [i], the two papers on correlated equilibrium [29, 53], the bargaining set paper with Maschler [17], the subjective probability paper with Anscombe [14], and “Integrals of Set-Valued Functions” [21], a strictly mathematical paper that impacted control theory and related areas as well as mathematical economics. The next batch includes the repeated games work—the ’59 paper [4], the book with Maschler [v], the survey [42], and the paper with Sorin on “Cooperation and Bounded Recall” [57]; also, the Talmud paper with Maschler [46], the paper with Drèze on coalition structures [31], the work with Brandenburger on “Epistemic Conditions for Nash Equilibrium” [65], the “Power and Taxes” paper with Kurz [37], some of the papers on NTU-games [10, 24], and others.

That sort of sums it up. Correlated equilibrium had a big impact. The work on repeated games, the equivalence principle, the continuum of players, interactive epistemology—all had a big impact.

Citations do give a good general idea of impact. But one should also look at the larger picture. Sometimes there is a body of work that all in all has a big impact, more than the individual citations show. In addition to the above-mentioned topics, there is incomplete information, NTU-values and NTU-games in general—with their many applications—perfect and imperfect competition, utilities and subjective probabilities, the mathematics of set-valued functions and measurability, extensive games, and others. Of course, these are not disjoint; there are many interconnections and areas of overlap.

There is a joint paper with Jacques Drèze [51] on which we worked very, very hard, for very, very long. For seven years we worked on it. It contains some of the deepest work I have ever done. It is hardly cited. This is a paper I love. It is nice work, but it hasn’t had much of an impact.