With repeated games, one gets a similar result by alternating: one

evening boxing, the next ballet. Of course, that way one only gets to the

convex hull of the Nash equilibria.

This is pretty straightforward. The next step is less so. It is to go to

three-person games, where two of the three players gang up on the third

”correlate “against” him, so to speak [Aumann (1974), Examples 2.5

and 2.6]. This leads outside the convex hull of Nash equilibria. In writing

this formally, I realized that the same de¬nitions apply also to two-

person games; also there, they may lead outside the convex hull of the

Nash equilibria.

Hart: So, correlated equilibria arise when the players get signals

that need not be independent. Talking about signals and information”

how about common knowledge and the “Agreeing to Disagree” paper?

Aumann: The original paper on correlated equilibrium also discussed

“subjective equilibrium,” where different players have different probabil-

ities for the same event. Differences in probabilities can arise from differ-

ences in information; but then, if a player knows that another player™s

probability is different from his, he might wish to revise his own prob-

ability. It™s not clear whether this process of revision necessarily leads to

the same probabilities. This question was raised”and left open”in

Aumann (1974) [Section 9j]. Indeed, even the formulation of the ques-

tion was murky.

I discussed this with Arrow and Frank Hahn during an IMSSS summer

in the early seventies. I remember the moment vividly. We were sitting in

Frank Hahn™s small of¬ce on the fourth ¬‚oor of Stanford™s Encina Hall,

where the economics department was located. I was trying to get my

head around the problem”not its solution, but simply its formula-

tion. Discussing it with them”describing the issue to them”somehow

sharpened and clari¬ed it. I went back to my of¬ce, sat down, and

continued thinking. Suddenly the whole thing came to me in a ¬‚ash”

the de¬nition of common knowledge, the characterization in terms of

information partitions, and the agreement theorem: roughly, that if the

probabilities of two people for an event are commonly known by both,

then they must be equal. It took a couple of days more to get a coherent

proof and to write it down. The proof seemed quite straightforward. The

whole thing”de¬nition, formulation, proof”came to less than a page.

Indeed, it looked so straightforward that it seemed hardly worth pub-

lishing. I went back and told Arrow and Hahn about it. At ¬rst Arrow

wouldn™t believe it, but became convinced when he saw the proof. I

expressed to him my doubts about publication. He strongly urged me to

publish it”so I did [Aumann (1976)]. It became one of my two most

widely cited papers.

344 Sergiu Hart

Six or seven years later I learned that the philosopher David Lewis

had de¬ned the concept of common knowledge already in 1969, and,

surprisingly, had used the same name for it. Of course, there is no

question that Lewis has priority. He did not, however, have the agree-

ment theorem.

Hart: The agreement theorem is surprising”and important. But your

simple and elegant formalization of common knowledge is even more

important. It pioneered the area known as “interactive epistemology”:

knowledge about others™ knowledge. It generated a huge literature”in

game theory, economics, and beyond: computer science, philosophy,

logic. It enabled the rigorous analysis of very deep and complex issues,

such as what is rationality, and what is needed for equilibrium. Interest-

ingly, it led you in particular back to correlated equilibrium.

Aumann: Yes. That™s Aumann (1987). The idea of common know-

ledge really enables the “right” formulation of correlated equilibrium. It™s

not some kind of esoteric extension of Nash equilibrium. Rather, it says

that if people simply respond optimally to their information”and this is

commonly known”then you get correlated equilibrium. The “equilib-

rium” part of this is not the point. Correlated equilibrium is nothing

more than just common knowledge of rationality, together with com-

mon priors.

Hart: Let™s talk now about the Hebrew University. You came to the

Hebrew University in ™56 and have been there ever since.

Aumann: I™ll tell you something. Mathematical game theory is a branch

of applied mathematics. When I was a student, applied mathematics was

looked down upon by many pure mathematicians. They stuck up their

noses and looked down upon it.

Hart: At that time most applications were to physics.

Aumann: Even that”hydrodynamics and that kind of thing”was

looked down upon. That is not the case anymore, and hasn™t been for

quite a while; but in the late ¬fties when I came to the Hebrew Univer-

sity that was still the vogue in the world of mathematics. At the Hebrew

University I did not experience any kind of inferiority in that respect, nor

in other respects either. Game theory was accepted as something worth-

while and important. In fact, Aryeh Dvoretzky, who was instrumental in

bringing me here, and Abraham Fr¤nkel (of Zermelo“Fr¤nkel set theory),

who was chair of the mathematics department, certainly appreciated this

subject. It was one of the reasons I was brought here. Dvoretzky himself

had done some work in game theory.

Hart: Let™s make a big jump. In 1991, the Center for Rationality was

established at the Hebrew University.

An Interview with Robert Aumann 345

Aumann: I don™t know whether it was the brainchild of Yoram Ben-

Porath or Menahem Yaari or both together. Anyway, Ben-Porath, who

was the rector of the university, asked Yaari, Itamar Pitowsky, Motty

Perry, and me to make a proposal for establishing a center for rationality.

It wasn™t even clear what the center was to be called. Something having

to do with game theory, with economics, with philosophy. We met many

times. Eventually what came out was the Center for Rationality, which

you, Sergiu, directed for its ¬rst eight critical years; it was you who really

got it going and gave it its oomph. The Center is really unique in the

whole world in that it brings together very many disciplines. Throughout

the world there are several research centers in areas connected with game

theory. Usually they are associated with departments of economics: the

Cowles Foundation at Yale, the Center for Operations Research and

Econometrics in Louvain, Belgium, the late Institute for Mathematical

Studies in the Social Sciences at Stanford. The Center for Rationality at

the Hebrew University is quite different, in that it is much broader. The

basic idea is “rationality”: behavior that advances one™s own interests.

This appears in many different contexts, represented by many academic

disciplines. The Center has members from mathematics, economics, com-

puter science, evolutionary biology, general philosophy, philosophy of

science, psychology, law, statistics, the business school, and education.

We should have a member from political science, but we don™t; that™s a

hole in the program. We should have one from medicine too, because

medicine is a ¬eld in which rational utility-maximizing behavior is very

important, and not at all easy. But at this time we don™t have one. There

is nothing in the world even approaching the breadth of coverage of the

Center for Rationality.

It is broad but nevertheless focused. There would seem to be a contra-

diction between breadth and focus, but our Center has both”breadth

and focus. The breadth is in the number and range of different disci-

plines that are represented at the Center. The focus is, in all these disci-

plines, on rational, self-interested behavior”or the lack of it. We take all

these different disciplines, and we look at a certain segment of each one,

and at how these various segments from this great number of disciplines

¬t together.

Hart: Can you give a few examples for the readers of this journal?

They may be surprised to hear about some of these connections.

Aumann: I™ll try; let™s go through some applications. In computer

science we have distributed computing, in which there are many different

processors. The problem is to coordinate the work of these processors,

which may number in the hundreds of thousands, each doing its own

work.

346 Sergiu Hart

Hart: That is, how processors that work in a decentralized way reach

a coordinated goal.

Aumann: Exactly. Another application is protecting computers against

hackers who are trying to break down the computer. This is a very grim

game, just like war is a grim game, and the stakes are high; but it is a

game. That™s another kind of interaction between computers and game

theory.

Still another kind comes from computers that solve games, play games,

and design games”like auctions”particularly on the Web. These are

applications of computers to games, whereas before, we were discussing

applications of games to computers.

Biology is another example where one might think that games don™t

seem particularly relevant. But they are! There is a book by Richard

Dawkins called The Sel¬sh Gene. This book discusses how evolution makes