Aumann: It is always misleading to sum it up in a few words, but here

goes: in the long run, you cannot use information without revealing it;

you can use information only to the extent that you are willing to reveal

it. A player with private information must choose between not making

use of that information”and then he doesn™t have to reveal it”or mak-

ing use of it, and then taking the consequences of the other side ¬nding

it out. That™s the big picture.

Hart: In addition, in a non-zero-sum situation, you may want to pass

information to the other side; it may be mutually advantageous to reveal

your information. The question is how to do it so that you can be

trusted, or in technical terms, in a way that is incentive-compatible.

340 Sergiu Hart

Aumann: The bottom line remains similar. In that case you can use

the information, not only if you are willing to reveal it, but also if you

actually want to reveal it. It may actually have positive value to reveal the

information. Then you use it and reveal it.

Hart: You mentioned something else and I want to pick up on that: the

Milnor“Shapley paper on oceanic games. That led you to another major

work, “Markets with a Continuum of Traders” [Aumann (1964)]:

modeling perfect competition by a continuum.

Aumann: As I already told you, in ™60“™61, the Milnor“Shapley paper

“Oceanic Games” caught my fancy. It treats games with an ocean”

nowadays we call it a continuum”of small players, and a small number

of large players, whom they called atoms. Then in the fall of ™61, at the

conference at which Kissinger and Lloyd Shapley were present, Herb

Scarf gave a talk about large markets. He had a countable in¬nity of

players. Before that, in ™59, Martin Shubik had published a paper called

“Edgeworth Market Games,” in which he made a connection between

the core of a large market game and the competitive equilibrium. Scarf ™s

model somehow wasn™t very satisfactory, and Herb realized that himself;

afterwards, he and Debreu proved a much more satisfactory version, in

their International Economic Review 1963 paper. The bottom line was

that, under certain assumptions, the core of a large economy is close to

the competitive solution, the solution to which one is led from the law

of supply and demand. I heard Scarf ™s talk, and, as I said, the formula-

tion was not very satisfactory. I put it together with the result of Milnor

and Shapley about oceanic games, and realized that that has to be the

right way of treating this situation: a continuum, not the countable

in¬nity that Scarf was using. It took a while longer to put all this to-

gether, but eventually I did get a very general theorem with a continuum

of traders. It has very few assumptions, and it is not a limit result. It

simply says that the core of a large market is the same as the set of

competitive outcomes. This was published in Econometrica in 1964

[Aumann (1964)].

Hart: Indeed, the introduction of the continuum idea to economic

theory has proved indispensable to the advancement of the discipline. In

the same way as in most of the natural sciences, it enables a precise and

rigorous analysis, which otherwise would have been very hard or even

impossible.

Aumann: The continuum is an approximation to the “true” situation,

in which the number of traders is large but ¬nite. The purpose of the

continuous approximation is to make available the powerful and elegant

methods of the branch of mathematics called “analysis,” in a situation

An Interview with Robert Aumann 341

where treatment by ¬nite methods would be much more dif¬cult or even

hopeless”think of trying to do ¬‚uid mechanics by solving n-body prob-

lems for large n.

Hart: The continuum is the best way to start understanding what™s

going on. Once you have that, you can do approximations and get limit

results.

Aumann: Yes, these approximations by ¬nite markets became a hot

topic in the late sixties and early seventies. The ™64 paper was followed by

the Econometrica ™66 paper [Aumann (1966)] on existence of competitive

equilibria in continuum markets; in ™75 came the paper on values of such

markets, also in Econometrica [Aumann (1975)]. Then there came later

papers using a continuum, by me with or without coauthors [Aumann

(1973, 1980), Aumann and Kurz (1977a,b), Aumann, Gardner, and

Rosenthal (1977), Aumann, Kurz, and Neyman (1983, 1987)], by Werner

Hildenbrand and his school, and by many, many others.

Hart: Before the ™75 paper, you developed, together with Shapley,

the theory of values of nonatomic games [Aumann and Shapley (1974)];

this generated a huge literature. Many of your students worked on

that. What™s a nonatomic game,

by the way? There is a story about

a talk on “Values of nonatomic

games,” where a secretary thought

a word was missing in the title, so

it became “Values of nonatomic

war games.” So, what are non-

atomic games?

Aumann: It has nothing to do

with war and disarmament. On

the contrary, in war you usually

have two sides. Nonatomic means

the exact opposite, where you

have a continuum of sides, a very

large number of players.

Hart: None of which are

atoms.

Aumann: Exactly, in the sense

that I was explaining before. It is

like Milnor and Shapley™s oceanic

games, except that in the oceanic

games there were atoms”“large”

players”and in nonatomic games

Figure 15.3 Werner Hildenbrand

with Bob Aumann, Oberwolfach, 1982. there are no large players at all.

342 Sergiu Hart

There are only small players. But unlike in Milnor“Shapley, the small

players may be of different kinds; the ocean is not homogeneous. The

basic property is that no player by himself makes any signi¬cant contribu-

tion. An example of a nonatomic game is a large economy, consisting of

small consumers and small businesses only, without large corporations or

government interference. Another example is an election, modeled as

a situation where no individual can affect the outcome. Even the 2000

U.S. presidential election is a nonatomic game”no single voter, even in

Florida, could have affected the outcome. (The people who did affect the

outcome were the Supreme Court judges.) In a nonatomic game, large

coalitions can affect the outcome, but individual players cannot.

Hart: And values?

Aumann: The game theory concept of value is an a priori evaluation

of what a player, or group of players, can expect to get out of the game.

Lloyd Shapley™s 1953 formalization is the most prominent. Sometimes,

as in voting situations, value is presented as an index of power (Shapley

and Shubik 1954). I have already mentioned the 1975 result about

values of large economies being the same as the competitive outcomes

of a market [Aumann (1975)]. This result had several precursors, the ¬rst

of which was a ™64 RAND Memorandum of Shapley.

Hart: Values of nonatomic games and their application in economic

models led to a huge literature.

Another one of your well-known contributions is the concept of cor-

related equilibrium ( Journal of Mathematical Economics, ™74 [Aumann,

1974] ). How did it come about?

Aumann: Correlated equilibria are like mixed Nash equilibria, except

that the players™ randomizations need not be independent. Frankly, I™m

not really sure how this business began. It™s probably related to repeated

games, and, indirectly, to Harsanyi and Selten™s equilibrium selection.

These ideas were ¬‚oating around in the late sixties, especially at the very

intense meetings of the Mathematica ACDA team. In the Battle of the

Sexes, for example, if you™re going to select one equilibrium, it has to be

the mixed one, which is worse for both players than either of the two pure

ones. So you say, “Hey, let™s toss a coin to decide on one of the two pure

equilibria.” Once the coin is tossed, it™s to the advantage of both players

to adhere to the chosen equilibrium; the whole process, including the coin

toss, is in equilibrium. This equilibrium is a lot better than the unique

mixed strategy equilibrium, because it guarantees that the boy and the

girl will de¬nitely meet”either at the boxing match or at the ballet

”whereas with the mixed strategy equilibrium, they may well go to

different places.

An Interview with Robert Aumann 343