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A: 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 infinity 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 formulation 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 infinity that Scarf
was using. It took a while longer to put all this together, 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 [16].
H: 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.
A: The continuum is an approximation to the “true” situation, in which the
number of traders is large but finite. 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 where treatment by finite methods would be much
more difficult or even hopeless—think of trying to do fluid mechanics by solving
n
-body problems for large n.
H: 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.
A: Yes, these approximations by finite markets became a hot topic in the late
sixties and early seventies. The ’64 paper was followed by the Econometrica ’66
paper [23] on existence of competitive equilibria in continuum markets; in ’75
came the paper on values of such markets, also in Econometrica [32]. Then there
came later papers using a continuum, by me with or without coauthors [28, 37,
38, 39, 41, 44, 52], by Werner Hildenbrand and his school, and by many, many
others.
H: Before the ’75 paper, you developed, together with Shapley, the theory of
values of nonatomic games [i]; 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
696
SERGIU HART
F
IGURE
3. Werner Hildenbrand with Bob Aumann, Oberwolfach, 1982.
was missing in the title, so it became “Values of nonatomic war games.” So, what
are nonatomic games?
A: 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.
H: None of which are atoms.
A: 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 there are no large players at all. 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 significant contribution. 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.
H: And values?
INTERVIEW WITH ROBERT AUMANN
697
A: 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 [32]. This result had several precursors, the first
of which was a ’64 RAND Memorandum of Shapley.
H: 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 correlated
equilibrium (Journal of Mathematical Economics, ’74 [29]). How did it come
about?
A: 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 floating 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 definitely meet—either at the boxing
match or at the ballet—whereas with the mixed strategy equilibrium, they may
well go to different places.
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 [29, Examples 2.5 and 2.6]. This leads outside the convex hull of
Nash equilibria. In writing this formally, I realized that the same definitions apply
also to two-person games; also there, they may lead outside the convex hull of the
Nash equilibria.
H: So, correlated equilibria arise when the players get signals that need not be
independent. Talking about signals and information—how about common knowl-
edge and the “Agreeing to Disagree” paper?
A: The original paper on correlated equilibrium also discussed “subjective
equilibrium,” where different players have different probabilities for the same
event. Differences in probabilities can arise from differences in information; but
then, if a player knows that another player’s probability is different from his, he
might wish to revise his own probability. It’s not clear whether this process of