Scand. J.

of Economics 108(2), 185211, 2006

DOI: 10.1111/j.1467-9442.2006.00448.x
Robert Aumanns Game and Economic
Theory
Sergiu Hart

Hebrew University, Jerusalem 91904, Israel

hart@huji.ac.il
I. Introduction
Robert J. Aumann has been a central figure in developing game theory
and establishing its key role in modern economics. Aumann has shaped the
field through his fundamental and pioneering work. His contributions during
the past half century are profoundconceptually, and often also mathe-
maticallyand at the same time crystal clear, elegant and illuminating.
Among Aumanns main scientific contributions, three areas stand out:
(i) Repeated Games: the study of long-term interactions and the account-
ing for cooperative and other patterns of behavior in terms of the
classical selfish utility-maximizing paradigm.
(ii) Knowledge, Rationality and Equilibrium: the analysis of knowledge
in multi-agent environments and the laying of the foundations of
rationality and equilibrium.
(iii) Perfect Competition: the modeling of perfect competition by a con-
tinuum of agents and the relations between price equilibria and co-
operative outcomes.
The next three sections discuss each of these topics in turn (with special
emphasis on the area of repeated games, the main topic for which Aumann
was awarded the 2005 Nobel Memorial Prize in Economic Sciences). We
conclude with a short overview of some of his other contributions.
II. Repeated Games
Most relationships between decision-makers last a long time. Competi-
tion between firms in markets, insurance contracts, credit relationships and

Partially supported by a grant of the Israel Science Foundation. The author thanks Elchanan
Ben-Porath, Avinash Dixit, Andreu Mas-Colell, Eyal Winter and Shmuel Zamir for their
comments. A presentation is available at http://www.ma.huji.ac.il/hart/abs/aumann-p.html. The
author is affiliated to the Center for the Study of Rationality, the Department of Economics,
and the Einstein Institute of Mathematics, The Hebrew University of Jerusalem. Web page:
http://www.ma.huji.ac.il/hart.
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Oxford, OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
186 S. Hart
negotiations are often long-term affairs. The same is true of employer
employee, lawyerclient and firmsubcontractor relationships, of conflicts
and agreements between political parties and nations, of evolutionary pro-
cesses in biology.
In such long-term interactions, the different stages are naturally inter-
dependent. Decision-makers react to past experience, and take into ac-
count the future impact of their choices. Many interesting and important
patterns of behaviorlike rewarding and punishing, cooperation and threats,
transmitting information and concealing itcan only be seen in multi-stage
situations.
The general framework for studying strategic interaction is game theory.
The players are decision-makersbe they individuals, collectives, com-
puter programs or geneswhose actions mutually affect one another. Game
theory studies interactions (games) from a rational decision-theoretic
point of view, and develops models and methodologies that are universal
and widely applicableparticularly in economics.
Foremost among the multi-stage models are repeated games, where the
same game is played at each stage. Such models allow us to untangle the
complexities of behavior from the complexities of the interaction itself,
which here is simply stationary over time.
The Classical Folk Theorem
The simplest setup is as follows. Given an n-person game G, let G

be the
supergame of G: the same n players are repeatedly playing the game G,
at periods t = 1, 2, . . . . (It is customary to call G the one-shot game,
and G

the repeated game; to avoid confusion, the choices of the players

in G are referred to as actions, and the choices in G

as strategies.)
At the end of each period, every player is informed of the actions chosen
by all players in that period; thus, before choosing his action at time t,
each player knows what everyone did in all the previous periods 1, 2, . . . ,
t 1.
The payoffs in G

are defined as an appropriate average of the payoffs

received in each period. (There are technical difficulties here; however, any
reasonable definition that is consistent with the long-term idea turns out
to lead to essentially the same result; see Aumann, 1959 and 1989, Ch. 8,
and the survey of Sorin, 1992.)
The question is, What are the Nash
1
(1951) equilibria of G

, i.e., those
strategy combinations for the n players such that no player can increase
his payoff by unilaterally changing his strategy? What are the resulting
1
John F. Nash, Jr., was awarded the 1994 Nobel Memorial Prize in Economics for this work.
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Robert Aumanns game and economic theory 187
outcomes? The answer is given in the following result, which emerged in
the late fifties. Its precise authorship is unknown, and it is part of the
folklore of game theory; it is thus known as the Folk Theorem.
2
The Folk Theorem. The set of Nash equilibrium outcomes of the repeated
game G

is precisely the set of feasible and individually rational outcomes

of the one-shot game G.
An outcome
3
is represented as an n-dimensional vector a = (a
1
,
a
2
, . . . , a
n
), whose ith coordinate a
i
is the payoff (or utility) of player
i. Such an outcome is feasible in G if it can be obtained from some com-
bination of actions in the game G, or more generally from a probability
distribution over such action combinations.
4
For example, if G is the two-
person Battle of the Sexes game
5
L R
T 3, 1 0, 0
B 0, 0 1, 3
then the set of feasible outcomes is the triangle with vertices (3, 1), (1, 3)
and (0, 0) (see Figure 1). Note that some feasible outcomes require a joint
(or correlated) randomization: for instance, (2, 2) corresponds to the two
action combinations TL and BR being played with equal probabilities of
1
2
each (thus (2, 2) is not a Nash equilibrium outcome in G; in fact, it cannot
be obtained from any pair of independently randomized actions of the two
players).
6
A payoff of a player i is individually rational in G if player i can-
not be prevented by the other players from getting it. Formally, let r
i
be
the minimax value of the zero-sum game where i wants to maximize, and
the other players want to minimize, the payoff of player
7
i. An outcome
a is individually rational in G if a
i
r
i
for all players i = 1, 2, . . . , n.
2
See the sections The Import of the Folk Theorem and A Historical Note below.
3
Or payoff vector.
4
Formally, an outcome is feasible if it lies in the convex hull of the vectors (u
1
(s), u
2
(s), . . . ,
u
n
(s)), where s ranges over all pure action combinations (s
1
, s
2
, . . . , s
n
), and u
i
denotes the
payoff function of player i.
5
Here and in the other examples, the actions are called T(op), B(ottom), L(eft) and
R(ight).
6
It would thus be more appropriate (but unwieldy) to call this coordinated feasibility or
cooperative feasibility or joint feasibility.
7
Take player 1; then r
1
= min
x
2
,...,x
n
max
x
1
u
1
(x
1
, x
2
, . . . , x
n
), where each x
j
ranges over the
randomized actions of player j, and u
1
is the payoff function of player 1.
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188 S. Hart
Payo to player 1
Payo to player 2
(0, 0)
(3, 1)
(1, 3)
(2, 2)
r
2
=
r
1
=
3

4
3

4
Fig. 1. The Folk Theorem for the Battle of the Sexes game
Returning to the Battle of the Sexes example, we have
8
r
1
= r
2
=
3
4
,
and the set of feasible and individually rational outcomes in G is the
darkly shaded area in Figure 1. The Folk Theorem says that this is pre-
cisely the set of outcomes of all Nash equilibria of the infinitely repeated
game G

.
An informal proof of the Folk Theorem is as follows (complete formal
proofs require many technical details; see for instance Aumann, 1959, and
1989, Ch. 8).
One direction consists of showing that the payoff vectors of the Nash
equilibria of G

are feasible and individually rational in G. First, any payoff

vector in G

is an average of payoff vectors of G, and thus feasible in G.

And second, any Nash equilibrium of G

(and of G) must yield individually

rational payoffs; otherwise, if some player, say player i, were to get less
than r
i
, then, given the other players strategies, i could find a reply that
gives him a payoff of at least r
i
in each period and thus also in the long
runwhich means that player i has a profitable deviation, in contradiction
to the Nash equilibrium requirement.
8
The row player can guarantee himself a payoff of
3
4
by playing T with probability
1
4
and
B with probability
3
4
: his payoff is then
1
4
0 +
3
4
1 =
3
4
when the column player plays
L, and
1
4
3 +
3
4
0 =
3
4
when the column player plays R (and so it is also
3
4
when the
column player plays a randomized action).
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Robert Aumanns game and economic theory 189
The other direction introduces two basic ingredients of repeated games:
a plan, and punishments (or threats). Let a be a feasible and individ-
ually rational payoff vector of G; from the definitions of these two notions
we immediately obtain:
(i) There exists a sequence of pure action combinations in G whose payoff
in G

is precisely a (in the above example, to get (2, 2), alternate

between TL in odd periods, t = 1, 3, 5, . . . , and BR in even periods,
t = 2, 4, 6, . . . )this is the plan.
(ii) For each player i there exist (mixed) actions of the other players in
G such that no matter what i plays his payoff in G cannot exceed
r
i
these constitute the punishment of i.
Now define for each player j a strategy in G

as follows: play according to

the plan (i.e., play the sequence of actions in (i) above), so long as everyone
did so in the past; if someone deviated from the plan, say player i was the
first to do so, at some period
9
d, then at every period thereafter, t = d +
1, d + 2, . . . , punish player i (i.e., play the mixed action as defined in (ii)
above which guarantees that, whatever player i does, his payoff will be at
most r
i
).
The combination of these strategies results in all players following the
plan, and so the payoff is indeed a by (i). Moreover, it constitutes a Nash
equilibrium of G

since any deviation by a player i will make the punish-

ments against him go into effect,
10
which will reduce his long-run payoff
in G

to at most r
i
(which is less than or equal to the payoff a
i
that he is
getting under the plan)so no increase in payoff is possible. Thus the threat
of punishment ensures that each player fulfills his part of the joint plan.
We will refer to the special Nash equilibria constructed above as
canonical equilibria.
The Import of the Folk Theorem
By their very nature, repeated games are complex objectsfor players to
play, and for theorists to analyze. There are a huge number of possible
strategies, even when the game is repeated just a few times;
11
in addition,
many of these strategies are extremely complex. This makes the equilibrium
analysis appear, at first, unmanageable.
The Folk Theorem shows that in fact this is not so. The resulting
geometric characterization of the equilibrium outcomes is extremely
9
If more than one player deviated at the same time, choose one of them.
10
Since the plan consists of a sequence of pure actions, any deviation is immediately detected.
11
The number of strategies is doubly exponential in the number of repetitions.
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simple (again, see Figure 1). What is more important however is the re-
sulting behavioral characterization: every equilibrium of G

is outcome-
equivalent to a canonical equilibrium, which consists of a coordinated plan,
supported by the threat of appropriate punishments.
12
The most valuable insight from the analysis is the connection that is
established between the so-called non-cooperative and cooperative the-
ories. The Folk Theorem relates non-cooperative behavior in the repeated
game (i.e., equilibrium in G

) to cooperative behavior in the one-stage game

(i.e., joint feasibility in G). This is best described in a survey of Aumann
(1981, Sec. 1):
The theory of repeated games of complete information is concerned with
the evolution of fundamental patterns of interaction between people (or for
that matter, animals; the problems it attacks are similar to those of social
biology). Its aim is to account for phenomena such as cooperation, altruism,
revenge, threats (self-destructive or otherwise), etc.phenomena which may
at first seem irrationalin terms of the usual selfish utility-maximizing
paradigm of game theory and neoclassical economics. [. . .]
The significance of the Folk Theorem is that it relates cooperative
behavior in the game G to non-cooperative behavior in its supergame G

.
This is the fundamental message of the theory of repeated games of com-
plete information; that cooperation may be explained by the fact that the
games people playi.e., the multiperson decision situations in which they
are involvedare not one-time affairs, but are repeated over and over. In
game-theoretic terms, an outcome is cooperative if it requires an outside
enforcement mechanism to make it stick. Equilibrium points are self-
enforcing; once an equilibrium point is agreed upon, it is not worthwhile
for any player to deviate from it. Thus it does not require any outside
enforcement mechanism, and so represents non-cooperative behavior. On
the other hand, the general feasible outcome does require an enforcement
mechanism, and so represents the cooperative approach. In a sense, the
repetition itself, with its possibilities for retaliation, becomes the enforce-
ment mechanism.
Thus, the Folk Theorem shows, first, that one can succinctly analyze
complex repeated interactions; second, that simple, natural and familiar
behaviors emerge; and third, how non-cooperative strategic behavior brings
about cooperation. This emergence of cooperation from a non-cooperative
setup makes repeated games a fascinating and important topic.
The result of the Folk Theorem has turned out to be extremely
robust. The extensions and generalizations are concerned with varying the
12
The Revelation Principle in mechanism design is of a similar type: everything that can
be implemented can also be implemented by a simple direct mechanism.
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Robert Aumanns game and economic theory 191
equilibrium concept, the long-run evaluation of the payoffs, imposing
restrictions on strategies, bounded rationality, modifying the structure of
the game, introducing asymmetric information, imperfect monitoring, and
so on. It should be emphasized that many of the results in this vast literature
are not just simple extensions; they almost always embody new ideas and
important insights, while overcoming the many conceptual, and at times
also technical, complexities of the models.
A Historical Note
The Folk Theorem was essentially known to most people working in the
area in the late fifties. However, it had not been published. Perhaps it
was considered too simple a result; perhaps, too complicated to write
rigorously.
13
The discounted repeated Prisoners Dilemma with discount factor close
enough to 1 is mentioned in Luce and Raiffa (1957, p. 102) as having
equilibria yielding efficient outcomes; Shubik (1959b, Sec. 10.4) presents
a more detailed analysis.
Aumann (1959, 1960, 1961) was the first to provide an extensive analysis
of infinitely repeated games. After setting up the model in an explicit
and rigorous manner, he showed that the two approachesnon-cooperative
for the repeated game G

, and cooperative for the one-shot game G

lead to the same solution.
14
The 1959 paper of Aumann is the funda-
mental paper on repeated games; while it goes beyond the classical Folk
Theorem, it addresses and resolves the basic issues that were necessary for
the development of this area.
Credibility and Perfect Equilibria
The canonical equilibria of G

(recall the end of the section The Classi-

cal Folk Theorem above) entail punishments that are constructed so as to
decrease the payoff of a deviating player. However, these punishment actions
may hurt some of the punishers as well. In such a case, after a deviation
of a player i, a punishing player j may find it to his own advantage not to
punish i, since js own payoff would decrease as a consequence. The threat
of punishment appears not to be rational, and thus not credible.
15
13
Luce and Raiffa (1957, top of p. 102) point out that there are difficulties here in translating
intuition into precise arguments.
14
See the section Coalitions and Strong Equilibria below.
15
Aumann (1959, beginning of Sec. 10) points out that there are difficulties with equilibria
that entail unrelenting punishment, which moreover may also hurt the punishers, and that
there are other, more reasonable equilibrium points, which give deviating players a chance
to return to the fold.
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192 S. Hart
But wouldnt the equilibrium requirements take care of this problem?
After all, in equilibrium any player, in particular player j, cannot increase
his payoff by modifying his strategy. But the payoff is determined by the
plan alone; deviations do not occur along the equilibrium play. Therefore
changes in the strategy of player j following a deviation by player i do not
affect player js payoff. This (perhaps slightly technical) argument shows that
the equilibrium requirements have no implication for the behavior of the
players in the punishment phase. In equilibrium, the threat of punishment
deters player i from deviating; but once i has in fact deviated, a punishing
player j may benefit by not carrying out the punishment.
What this suggests is a need to supplement the notion of equilibrium
with additional requirements, namely, that after every history of play
whether or not it occurs in equilibriumno player should be able to increase
his payoff in the continuation of the game by unilaterally changing his
strategy there. These conditions are embodied in the concept of (subgame-)
perfect equilibrium (Selten,
16
1965, 1975), which requires the combination
of strategies to constitute an equilibrium not only in the whole game but
also in every continuation game (subgame).
17
A strengthening of the Folk Theorem, due to Aumann and Shapley (1976)
and to Rubinstein (1976, 1979), in independent work, shows that in the
repeated game G

, the equilibrium outcomes and the perfect equilibrium

outcomes in fact coincide.
The Perfect Folk Theorem, Aumann and Shapley (1976) and Rubinstein
(1976). The set of perfect Nash equilibrium outcomes of the repeated game
G

is precisely the set of feasible and individually rational outcomes of the

one-shot game G.
The idea is to refine the construction of equilibrium, so that the result-
ing canonical perfect equilibria entail punishments that last only finitely
many periods (during which the one-period gain made by a deviation from
the cooperative plan is essentially wiped out), after which everyone returns
to the plan. The return to the plan makes it rational for each player to
punish all deviations by other playersand the equilibrium is indeed per-
fect (there are some subtleties here). In short: a plan, punishmentsand
also forgiveness; punishing is not forever, but just long enough to make
deviations unprofitable.
16
Reinhard Selten was awarded the 1994 Nobel Memorial Prize in Economics for this work.
17
In the special case where the punishments form a Nash equilibrium of the one-shot game,
like the Bertrand equilibrium in a duopoly model, or the Battle of the Sexes game, the
canonical equilibria of the classical Folk Theorem turn out to be in fact perfect (since
punishing is now an equilibrium in the continuation game); see Friedman (1971).
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Robert Aumanns game and economic theory 193
The importance of the Perfect Folk Theorem is that the threats now
become crediblethey will indeed be carried out if neededwhich makes
the cooperative plan much more reasonable and robust.
Coalitions and Strong Equilibria
Another strengthening of the equilibrium requirements considers deviations
by groups of players (rather than just single players). A strong equilibrium,
a concept introduced by Aumann (1959), is a combination of strategies
from which no group of players can profitably deviate. Aumann (1959)
shows that the outcomes of the strong equilibria of the repeated game
G

constitute a cooperative solution of the one-shot game G, namely, the

appropriate core
18
of G.
The Strong Folk Theorem, Aumann (1959). The set of strong Nash equi-
librium outcomes of the repeated game G

is precisely the core of the

one-shot game G.
Like the classical Folk Theorem, this result demonstrates the connection
between the non-cooperative approach (the strong equilibria of G

) and the
cooperative approach (the core of G). In fact, this work led Aumann and
others to the development of the theory of general cooperative games
non-transferable utility games
19
which are most important in economic
theory; see Aumann (1967).
Asymmetric Information
A most important and fascinating extension of the Folk Theorem is to the
case of asymmetric information, where different players possess different
knowledge of the relevant parameters of the one-shot game that is repeatedly
played; for instance, a player may not know the utility functions of the other
players.
Now, information is valuable. But how should it be used advant-
ageouslyto gain a competitive edge, to attain mutual benefits through
cooperation, or both?
To illustrate the issues involved, assume for concreteness that one player,
call him the informed player, has private information which the other
players do not possess. In a one-shot interaction, the informed player will
clearly utilize his information so as to gain as much as possible; this
will often require him to play different actions depending on his private
18
Specifically, the so-called -core.
19
Prior to this, only transferable utility or side payment games were studied.
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194 S. Hart
information. However, if the situation is repeated, then the other players,
by observing the action taken by the informed player, may infer what his
information was. But then the informed player will no longer have an infor-
mational advantage in future periods. So, what good is private information
to the informed player if he cannot use it to his benefit? The problem here
is to find the right balance: to use the information as much as possible,
while revealing it as little as possible.
This is one side of the coin. The other is that there are situations where
the informed player would like to convey his information to the others, so
that their resulting actions will benefit him. But that is not always easy to
do; can the uninformed players trust him? (Should one trust a shop owner
who claims that buying from him is a good deal? Wouldnt he claim
the same even if it werent so?) The problem is how to make the revealed
information credible, so that everyone benefits. Of course, repetitioni.e.,
the long-term relationshipis essential here.
In the mid-sixties, following the Harsanyi
20
(19671968) model of
incomplete information games, Aumann and Michael Maschler founded
and developed the theory of repeated games with incomplete information.
In a series of path-breaking reports written in 1966, 1967 and 1968,
21
Aumann and Maschler set up the basic models, and showed how the
complex issues in the use of information alluded to above can actually
be resolved in an explicit and elegant way.
As suggested by the Folk Theorem, there are two issues that need to be
addressed: individual rationality and feasibility. We will deal with each in
turn.
Individual Rationality and the Optimal Use of Information
We start with individual rationality: determining how much a player can
guarantee (no matter what the other players do). We will illustrate the
issues involved with three simple examples. In each example there will
be two players who repeatedly play the same one-shot game, and we will
consider individual rationality from the point of view of the row player.
22
The row players payoffs are given by either the matrix M1 or the matrix
M2; the row player is informed of which matrix is the true matrix, but
the column player is nothe only knows that the two matrices are equally
20
John C. Harsanyi was awarded the 1994 Nobel Memorial Prize in Economics for this work.
21
These unpublished reports to the Mathematica Institute were widely circulated in the pro-
fession. They are now collected in one book, Aumann and Maschler (1995), together with
very extensive notes on the subsequent developments.
22
The game may thus be thought of as a zero-sum game, with the row player as the maximizer
of his payoffs, and the column player as the minimizer.
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Robert Aumanns game and economic theory 195
likely. After each stage both players observe the actions taken, but not the
payoffs.
23
The first example is
Game 1
M1

probability =
1
2

:
L R
T 4 0
B 0 0
M2

probability =
1
2

:
L R
T 0 0
B 0 4
Again, the question we ask is, How much can the row player guarantee?
If he were to play optimally in each matrix, i.e., T when the true matrix
is M1, and B when it is M2 (this is the only way he may get positive
payoffs), then his action would reveal which matrix is the true matrix.
Were the column player to play R after seeing T, and L after seeing B,
the row player would get a payoff of
24
0. If, instead, the row player were
to ignore his information, he would face the average non-revealing game
with payoff matrix
25
1
2
M1 +
1
2
M2:
L R
T 2 0
B 0 2
Here the best he can do is to randomize equally between T and B, which
would guarantee him a payoff of 1. This is better than 0, and it turns out
that 1 is in fact the most the row player can guarantee in the long run here
(the proof is by no means immediate). So, in Game 1, the row player can
23
Thus information is transmitted only through actions (if the column player were to observe
the payoffs, he could determine immediately which matrix is the true matrix). This is a
simplifying assumption that allows a clear analysis. Once these games are studied and well
understood, one goes on to the general model with so-called signaling matrices (where each
combination of actions generates a certain signal to each player; the signal could include the
payoff, or be more or less general; this is discussed already in Aumann (1959, Sec. 6, second
paragraph).
24
Except, perhaps, in the first period (which is negligible in the long run).
25
In each cell, the payoff is the average of the corresponding payoffs in the two matrices.
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196 S. Hart
guarantee the most by concealing his information and playing as if he did
not know which matrix is the true matrix.
We modify the payoffs and get the second example:
Game 2
M1

probability =
1
2

:
L R
T 4 4
B 4 0
M2

probability =
1
2

:
L R
T 0 4
B 4 4
In this game, if the row player ignores his information, then the average
game is
1
2
M1 +
1
2
M2:
L R
T 2 4
B 4 2
in which he can guarantee a payoff of 3 (by randomizing equally between
T and B). If, however, the row player were to play T when the true ma-
trix is M1 and B when it is M2, he would be guaranteed a payoff of 4,
which clearly is the most he can get in this game. So, in Game 2, the
row player can guarantee the most by usingand thus fully revealinghis
information.
The third example is
Game 3
M1

probability =
1
2

:
L C R
T 4 2 0
B 4 2 0
M2

probability =
1
2

:
L C R
T 0 2 4
B 0 2 4
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Robert Aumanns game and economic theory 197
Using his information fullyplaying T when the true matrix is M1 and
B when it is M2guarantees only a payoff of 0 to the row player (the
column player plays R after seeing T, and L after seeing B). Ignoring the
information leads to the average game
1
2
M1 +
1
2
M2:
L C R
T 2 0 2
B 2 0 2
in which the row player can again guarantee only a payoff of 0 (the column
player plays C). At this point it may appear that the row player cannot
guarantee more than 0 in this game (since 0 is the most he is guaranteed,
whether concealing or revealing the information). But that turns out to be
false: by partially using his informationand thus partially revealing it
the row player can guarantee more. Indeed, consider the following strategy
for the row player:
Strategy of the row player
If M1:

play T forever, with probability

3
4
,
play B forever, with probability
1
4
.
If M2:

play T forever, with probability

1
4
,
play B forever, with probability
3
4
.
To see what happens, consider the case where the randomization yielded
play T forever. After the first period (in which T was played), the pos-
terior probability that the true matrix is M1 is now, by Bayes rule,
P(M1|T) =
P(T|M1) P(M1)
P(T|M1) P(M1) +P(T|M2) P(M2)
=
3
4

1
2
3
4

1
2
+
1
4

1
2
=
3
4
,
and, for matrix M2, it is P(M2|T) =
1
4
. The average game following T
in the first period is thus
3
4
M1 +
1
4
M2:
L C R
T 3 1 1
B 3 1 1
C
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198 S. Hart
in which playing T guarantees 1 to the row player. A similar computation
applies to the case of play B forever, where P(M1|T) =
1
4
and P(M2|T) =
3
4
, and playing B also guarantees 1. Therefore the strategy guarantees to
the row player a payoff of 1strictly more than either full revelation or
full concealmentwhich turns out to be the most he can guarantee in this
game. Note that the choice between the two options of play T forever
and play B forever is made by the row player only once, in the first
period, based on the information he has: the probabilities depend on which
matrix is the true matrix.
26
However, the action of the row player reveals his
information only partially: the prior probability of
1
2
on M1 now becomes
either the posterior probability
3
4
(after T) or
1
4
(after B).
27
So, in Game 3,
the partial use of informationand thus the ensuing partial revelationis
the only way that the row player guarantees the most.
To summarize the three examples: in some games the informed player
uses his information and reveals it fully; in others he does not use his
information at all and thus conceals it; and in still others he uses and
reveals it partially. When exactly does each case occur? The following
general result gives a precise answer. Some notations: let M1 and M2 be
two matrices of the same size, viewed as two-person zero-sum games;
28
one of them is chosen, with probabilities p and 1 p, respectively. Let
G( p) := p M1 +(1 p)M2 be the one-shot average game, and let G

( p)
be the supergame when the row player is informed of the chosen matrix
and the column player is not (this is called information on one side).
Theorem, Aumann and Maschler (1966). The minimax value function of
the repeated two-person zero-sum game with information on one side G

equals the concavification of the minimax value function of the one-shot

average game G:
val G

=

val G.
That is, for every prior probability p, let w( p) :=val G( p) be the mini-
max value of the one-shot average game G( p); let w denote the concavifi-
cation of w, i.e., the minimal concave function that is everywhere greater
than or equal to w (i.e., w( p) w( p) for all 0 p 1); the Aumann
Maschler Theorem says that val G

( p), the value of the repeated game

G

( p), is precisely w( p), the evaluation at p of the concavification w of w.

26
In all subsequent periods, he forgets his private information and just plays the same
action as in the first period.
27
The posteriors contain more information than the prior (they are closer to full information,
i.e., to 1 and 0, respectively).
28
The result applies to any number of matrices; we present it here for two matrices for
simplicity.
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Robert Aumanns game and economic theory 199
1 0 1
p
val G(p)
1
p
val G

(p)
1
prior
posteriors
1

2
1

4

3
4
1

2
Fig. 2. Value function of the one-shot average game (left), and of the repeated game (right,
bold line) in Game 3
Figure 2 illustrates this for our third example, Game 3 (note how the value
of G

( p) depends on the values of G( p

) for p

-s that are different from p).

This is a striking result. It tells the precise amount of information that
is optimal for the informed player to reveal. In some cases, he should use
the information fully, by playing differently according to his information: a
completely revealing or separating strategy. In other cases, he should
conceal the information and play the same actions no matter what his infor-
mation is: a non-revealing or pooling strategy. And there are still other
cases where it is strictly better for him to use the information partially, by
mixing it with an appropriate random noise; this yields posterior proba-
bilities that are more informative than the prior probabilities, but not fully
informative. The difficult problem turns out to have a precise, elegantand
at times surprisingsolution.
29
Feasibility and Strategic Communication
Unlike the complete information case, where the feasible outcomes were all
those obtained from an action combination or from a (weighted) average
of these, here one needs to take into account the informational issues. Not
every physically possible outcome can arise in equilibrium; for instance, if
an informed player is called upon to reveal his information, he needs to be
motivated to do so.
29
The AumannMaschler result provides optimal strategies also for the uninformed column
player; we do not discuss this here.
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200 S. Hart
Consider the following example, which has the same structure as the
three games above:
Game 4
M1

probability =
1
3

:
L R
T 3, 3 0, 0
B 3, 3 0, 0
M2

probability =
2
3

:
L R
T 4, 0 3, 3
B 4, 0 3, 3
Two players repeatedly play a one-shot game with payoffs given by either
the bimatrix M1 or the bimatrix M2 (this is now a non-zero-sum game).
M2 is twice as likely as M1, and only the row player is informed of whether
M1 or M2 is the true bimatrix; finally, the actions, but not the payoffs, are
observed after each stage.
30
The outcome (3, 3) appears in both bimatrices; to obtain it, the informed
row player must communicate to the uninformed column player whether the
true bimatrix is M1 or M2, e.g. by playing T in the first period if it is M1,
and B if it is M2; after T the column player will play L, and after B he will
play R. However, this plan is not incentive-compatible: the row player will
cheat and play T in the first period also when the true bimatrix is M2,
since he prefers that the column player choose L rather than R also when it
is M2 (he gets a payoff of 4 rather than 3)! Thus the communication of the
row player cannot be trusted. It turns out that the row player has no way
to transmit his information credibly to the column player, and the outcome
(3, 3) is not reachable. (The only feasible outcome is for the column player
always to play R, which yields an expected payoff vector of only (2, 2).)
Thus the presence of asymmetric information hinders cooperationeven
when it is to the mutual advantage of both players. The informed player
wants to convey his information, but he cannot do so credibly.
The above example is due to Aumann, Maschler and Stearns (1968),
where this theory is first developed. That paper exhibits equilibria that
consist of a number of periods of information transmission (by the in-
formed player) interspersed with randomizations (by both players), which
30
The payoffs correspond to a standard signaling setup, as in principalagent interactions:
the row player possesses the information and the column player determines the outcome (the
two rows yield identical payoffs, so the informed players actions have no direct effect on
the outcome).
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Robert Aumanns game and economic theory 201
ultimately lead to certain combinations of actions (like (i) in the section The
Classical Folk Theorem, above); this whole plan (of communications and
actions) is sustained in equilibrium by appropriate punishments when a
deviation is observed (like (ii) in the same section, based on the
individual rationality results above). The complete characterization of all
equilibria, and their canonical representation in terms of communications,
joint plans and punishments, is provided by Hart (1985); see also Aumann
and Maschler (1995, Postscript to Ch. 5), and, for a related setup, Aumann
and Hart (2003).
The study of repeated games of incomplete informationwhich has
flourished since the pioneering work of Aumann and Maschlerclarifies
in a beautiful way the strategic use of information: how much to reveal,
how much to conceal, how much of the revealed information to believe.
SummaryRepeated Games
The model of repeated games is an extremely simple, fundamental and
universal model of multi-stage interactions. It allows participants truly to
interact and react to one another; their behavior may be simpleor highly
intricate and complex. The analysis is deep and challenging, both conceptu-
ally and technically. In the end, the results are elegant and most insightful:
simple and natural behavior emerges.
Aumann has been the leader in this area. The highlights of his contribu-
tion are (in chronological order):
(A) The initial study of repeated games: showing how to analyze
repeated games, and, most importantly, showing how repeated inter-
action yields cooperative outcomes (the classical Folk Theorem, and
Aumann, 1959).
(B) Asymmetric information: introducing the essential ingredient of infor-
mation, and showing how to use information optimally and rationally
in long-run strategic interactions (Aumann and Maschler, 1966, and
Aumann, Maschler and Stearns, 1968).
(C) Credible threats and perfectness: making the equilibria more natural
and robust and thus of much wider applicability, which paved the way
for their wide use, in particular in economics (Aumann and Shapley,
1976, and Rubinstein, 1976).
Each one of these three on its own is a landmark contribution;
taken together, they complement and strengthen one another, providing a
cohesive and significant big picture: the evolution of cooperative (and
other) patterns of behavior in repeated interactions between rational utility-
maximizing individuals.
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202 S. Hart
For further reading, see Aumanns highly influential surveys and lec-
ture notes throughout the years (1967, 1981, 1985, 1987b, 1992); and,
more recently, Mertens, Sorin and Zamir (1994), the Handbook of Game
Theory, with Economic Applications, in particular the chapters of Sorin
(1992), Zamir (1992) and Forges (1992), the extended postscripts of
Aumann and Maschler (1995), and the chapters in Part C of Hart and
Mas-Colell (1997).
III. Knowledge, Rationality and Equilibrium
In this section we discuss the two topics of correlated equilibrium and
interactive knowledge, together with a beautiful connection between them
all introduced by Aumann.
Correlated Equilibrium
Consider a given and known game, and assume that, before playing it, each
player receives a certain signal that does not affect the payoffs of the game.
Can such signals affect the outcome?
Indeed, they can: players may use these signals to correlate their
choices. Aumann (1974) defined the notion of correlated equilibrium: it
is a Nash equilibrium of the game with the signals. When the signals are
(stochastically) independent, this is just a Nash equilibrium of the given
game. At the other extreme, when the signals are perfectly correlated
for instance, when everyone receives the same public signal (such as
sunspots)it amounts to an average of Nash equilibria (this is called a
publicly correlated equilibrium). But in general, when the signals are private
and partially correlated, it leads to new equilibria, which may lie outside
the convex hull of the Nash equilibria of the game.
For example, take the two-person Chicken game:
LEAVE STAY
LEAVE 4, 4 2, 5
STAY 5, 2 0, 0
There are two pure Nash equilibria, (STAY, LEAVE) and (LEAVE, STAY), and
also a mixed Nash equilibrium where each player plays LEAVE with proba-
bility
2
3
and STAY with probability
1
3
; the payoffs are, respectively, NE1 =
(5, 2), NE2 = (2, 5) and NE3 = (3
1
3
, 3
1
3
); see Figure 3.
Consider now a public fair-coin toss (i.e., the common signal is either
H or T, with probabilities
1
2

1
2
); assume that after H the row player
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Robert Aumanns game and economic theory 203
Payo to player 1
Payo to player 2
NE3
NE1
NE2
CE2
CE1
Fig. 3. Nash equilibria and correlated equilibria for the game of Chicken (see text; figure
not drawn to scale)
plays STAY and the column player plays LEAVE, whereas after T they play
LEAVE and STAY, respectively. This is easily seen to constitute a Nash equi-
librium of the extended game (with the signals), and so it is a publicly
correlated equilibrium of the original Chicken game. The probability
distribution of outcomes is
LEAVE STAY
LEAVE 0
1
2
STAY
1
2
0
and the payoffs are CE1 = (3
1
2
, 3
1
2
) (see Figure 3; the set of payoffs of
all publicly correlated equilibria, which is the convex hull of the Nash
equilibrium payoffs, is the gray area there).
But there are other correlated equilibria, in particular, one that results in
each action combination except (STAY, STAY) being played with an equal
probability of
1
3
:
LEAVE STAY
LEAVE
1
3
1
3
STAY
1
3
0
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204 S. Hart
Indeed, let the signal to each player be L or S; think of this as a
recommendation to play LEAVE or STAY, respectively. When the row player
gets the signal L, he assigns a (conditional) probability of
1
2
to each one
of the two pairs of signals (L, L) and (L, S); so, if the column player
follows his recommendation, then the row player gets an expected payoff
of 3 =
1
2
4 +
1
2
2 from playing LEAVE, and only of 2
1
2
=
1
2
5 +
1
2
0 from
deviating to STAY. When the row player gets the signal S, he deduces that
the pair of signals is necessarily (S, L), so if the column player indeed
follows his recommendation and plays LEAVE then the row player is better
off choosing STAY. Similarly for the column player. So altogether both
players always follow their recommendations, and we have a correlated
equilibrium. The payoffs of this correlated equilibrium are CE2 = (3
2
3
, 3
2
3
),
which lies outside the convex hull of the Nash equilibrium payoffs (again,
see Figure 3).
The notion of correlated equilibrium is most natural and arises in many
setups. It is embodied in situations of communication and coordination,
when there are mediators or mechanisms, and so on. In fact, signals are all
around us, whether public or private; it is unavoidable that they will find
their way into the equilibrium notion.
Common Knowledge and Interactive Epistemology
In multi-agent interactive environments, the behavior of an agent depends
on what he knows. Since the behavior of the other agents depends on what
they know, it follows that what an agent knows about what the others know
is also relevant; hence the same is true of what one knows about what the
others know about what he knowsand so on. This leads to the notions
of interactive knowledge and interactive beliefs, which are fundamental to
the understanding of rationality and equilibrium.
An event E is common knowledge among a set of agents if every-
one knows E, and everyone knows that everyone knows E, and everyone
knows that everyone knows that everyone knows E, and so on. This concept
was introduced by the philosopher David Lewis (1969) and, independently,
by Aumann (1976). Aumanns work went far beyond this, formalizing the
concept and exploring its implications.
Formally, the information of an agent in a certain state of the world
is described by the set of all states that he considers possible when
is the true state, i.e., those states

that his information does not allow

him to distinguish from . An event E is identified with the set of states
where it occurs. The agent knows an event E at a state if all states that
he considers possible at lie in E (i.e., E occurs at all these states). The
event E is common knowledge at a state if, at , all agents know E,
and all agents know (the event) that all know E, and so on (as Aumann
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Robert Aumanns game and economic theory 205
showed, all this can be succinctly expressed using the algebra of
partitions).
The fascinating result in Aumanns paper is
The Agreement Theorem, Aumann (1976). Consider two people who start
with the same prior beliefs and may receive different information. If their
posterior probabilities for some event A are common knowledge, then these
posteriors must be identical.
That is, in a state where each one knows the other ones posterior, and
knows that the other one knows his own posterior, and knows that the other
one knows that he knows the other ones posterior, and so on (and all this
has been taken into account when computing the posteriors), the agents
must have identical posteriors. Formally, let and be two numbers, and
consider all states where the posterior probability for A of the first agent is
, and that of the second agent is ; this set of states is an eventcall it
E. The Agreement Theorem says that if E is common knowledge at some
state, then necessarily = .
This result was the starting point of a whole new area, called interactive
epistemology; it generated a huge literature, in economics (for instance,
the no trade results), computer science, logic, philosophy, and beyond.
Aumann went on to provide solid foundations (the so-called syntactic ap-
proach) of this whole area; see Aumann (1999a,b).
31
It is deep work that
deals with basic questions, such as, why can one assume without loss of
generality that the information structures are commonly known?
Rationality and Equilibrium
In 1987 Aumann established an intriguing connection between two notions:
correlated equilibrium and common knowledge of rationality.
Theorem, Aumann (1987a). Consider a game whose players start with the
same prior beliefs and may receive different information. If it is common
knowledge that all players are Bayesian rational, then they are playing a
correlated equilibrium of the game.
A player is Bayesian rational if his action is optimal given his infor-
mation. Let R denote the set of all states where all players are Bayesian
rational (formally, R is an event), and assume that R is common knowledge
at some state. The Aumann (1987a) Theorem says that the prior distribution
yields a correlated equilibrium of the given game.
31
These papers were circulated as lecture notes in the eighties.
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206 S. Hart
It is remarkable that correlated equilibria are obtained from common
knowledge of rationality and a common prior only; no assumptions of equi-
librium behavior are needed.
32
IV. Perfect Competition
Perfect competition has always been a central theme in economics. It refers
to a situation where there are many participants and the influence of each
individual on the economy as a whole is negligible. But how should perfect
competition be modeled?
Following various attempts,
33
Aumann (1964) introduced the general
model of a continuum of agents:
The influence of an individual participant on the economy cannot be math-
ematically negligible, as long as there are only finitely many participants.
Thus a mathematical model appropriate to the intuitive notion of perfect
competition must contain infinitely many participants. We submit that the
most natural model for this purpose contains a continuum of participants,
similar to the continuum of points on a line or the continuum of particles in
a fluid. [. . .]
The continuum can be considered an approximation to the true situa-
tion in which there is a large but finite number of particles (or traders, or
strategies, or possible prices). The purpose of adopting the continuous ap-
proximation is to make available the powerful and elegant methods of the
branch of mathematics called analysis, in a situation where treatment by fi-
nite methods would be much more difficult or even hopeless (think of trying
to do fluid mechanics by solving n-body problems for large n). [Aumann,
1964, Sec. 1]
The continuum of agents idea has turned out to be indispensable to the
advancement of economic theory. As in the natural sciences, it has enabled
precise and rigorous modeling and analysis, which otherwise would have
been very hard or even impossible. Once the continuum model was studied
and understood, one could go back and examine the true situations with
finitely many agents, using appropriate approximations and limits.
The foremost result is the equivalence between competitive (or Walrasian)
equilibria and core allocations in perfectly competitive markets: individual
price-taking behavior and social stability yield the same outcomes (Aumann,
1964, and also the survey of Anderson, 1994). This may be viewed as a
modern and rigorous formalization of the invisible hand of Adam Smith
and the limit contract curve of Edgeworth.
32
For Nash equilibrium, the precise epistemic requirements are studied in Aumann and
Brandenburger (1995) and Aumann (1995).
33
For instance, Shubik (1959a) and Debreu and Scarf (1963).
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Robert Aumanns game and economic theory 207
The core is a classical concept of cooperative game theory (recall also
the Strong Folk Theorem above). Another central cooperative concept is
that of value, originally introduced by Shapley (1953). The notion of value
embodies standard economic ideas such as expected outcome and
marginal contribution. Applying it to markets with a continuum of traders
yielded a new equivalence result, this time between the price equilibria and
the value allocations (Aumann and Shapley, 1974, Aumann, 1975; see the
survey of Hart, 2002).
All this led to an extensive study of the relationships between competitive
equilibria and various game-theoretic concepts, which led to a very general
equivalence principle. To quote Aumann (1987b, Sec. 19601970, v):
Perhaps the most remarkable single phenomenon in game and economic the-
ory is the relationship between the price equilibria of a competitive market
economy, and all but one
34
of the major solution concepts for the corre-
sponding game. [. . .]
Intuitively, the equivalence principle says that the institution of market
prices arises naturally from the basic forces at work in a market, (almost) no
matter what we assume about the way in which these forces work.
The end result of the introduction of the continuum model in economics
is that it has led to a much better understanding of perfect competition,
its implications and limitations, from many disparate but converging view-
points.
V. Other Contributions
In the previous three sections we presented three areas of Aumanns
landmark contributions. But Aumanns pioneering work goes far beyond
these. We can only list here some of the topics: cooperative games (non-
transferable utility games, value, core, bargaining sets, nucleolus, consis-
tency, bankruptcy and the Talmud, coalition formation, axiomatization),
subjective probability and utility, power and taxes, games in extensive form
and mathematics (for example, set-valued functions).
But that is not all. Not only did Aumann lay down and deepen the
micro foundations, he always espoused and emphasized the macro out-
look. Aumann has been highly influential in providing the philosophical
and conceptual groundwork for game theory; see, e.g., Aumann (1985 and
1987b). Of particular significance is his consistent promotion of the view
of a unified game theory:
34
The one exception is the stable set; see Hart (1974) (more recently, additional exceptions
are certain NTU values; see Hart, 2002, Sec. 5.4).
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208 S. Hart
Game Theory may be viewed as a sort of umbrella or unified field theory
for the rational side of social science, where social is interpreted broadly, to
include human individuals as well as other kinds of players (collectives such
as corporations and nations, animals and plants, computers, etc.). 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. [Aumann and
Hart, 1992, Preface, pp. xixii]
VI. Aumann Inspires and Teaches
An overview of Aumanns contributions would not be complete were it
to cover just his own papers. Indeed, Aumann has greatly influenced and
inspired generations of researchers, directly and indirectly: his students,
collaborators, colleagues, and the many people who read his work, listened
to his lectures, or conducted discussions with him. His papers are masterly
written, combining deep analysis with informal explanations and intuitions;
his surveys on game theory in general, and on various specific topics, have
largely shaped the field. Aumann has been the driving force behind the
development of many areas (such as those we have presented above). He
has always applied the most rigorous standards, to his as well as others
work; at the same time, he never lost sight of the emerging big picture:
he aimed for it, and directed everyone toward it. (For a fascinating glimpse
of Aumanns scientific as well as personal world, see his interview in Hart,
2005.)
Aumann has had, to date, thirteen doctoral students: Bezalel Peleg (Ph.D.,
1964), David Schmeidler (1969), Shmuel Zamir (1971), Elon Kohlberg
(1973), Benyamin Shitovitz (1974), Zvi Artstein (1974), Eugene Wesley
(1975), Sergiu Hart (1976), Abraham Neyman (1977), Yair Tauman (1979),
Dov Samet (1981), Ehud Lehrer (1987) and Yossi Feinberg (1997). All but
one (who pursued a non-academic career) are now leading researchers in
their own right. Aumann is very proud of his scientific familyas he
calls itand we are all proud to be part of it.
VII. A Short Biography of Aumann
Robert John Yisrael Aumann
35
was born in 1930 in Frankfurt, Germany.
In 1938 the Aumann family left Germany and came to America. Robert
Aumann studied at City College in New York, and then at the Massachusetts
35
Aumanns curriculum vitae and publications are available at http://www.ma.huji.ac.il/
raumann.
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Robert Aumanns game and economic theory 209
Institute of Technology (MIT), where he got his doctoral degree in pure
mathematics in 1955. After two years as a post-doc at Princeton University,
he immigrated to Israel in 1956, and has been at the Hebrew University of
Jerusalem since then. Though now officially retired, Aumann continues his
scientific work, teaches and supervises students.
In 1991 the multi-disciplinary Center for the Study of Rationality was
established at the Hebrew University of Jerusalem; the two founding
fathers were Aumann and Menahem Yaari. Today Aumann chairs its Aca-
demic Committee. Aumann was also instrumental in the foundation of the
international Game Theory Society, and served as its first president from
1999 to 2003.
Aumann has received many awards and prizes, including the Harvey
Prize, the Israel Prize, the Lanchester Prize, the Nemmers Prize, the EMET
Prize, the von Neumann Prize, and the Nobel Memorial Prize.
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