Aumann Awarded Nobel Prize Abraham Neyman Robert J. Aumann, professor of math- application in the analysis of various specific issues. ematics at the Hebrew University and Some of his contributions require very deep and member of the Interdisciplinary Cen- complex technical analysis; others are (as he says ter for Rationality there, shares (with at times) “embarrassingly trivial” mathematically, Thomas C. Schelling) the 2005 Nobel but very profound conceptually [12, p. 1]. He has Prize in Economics [13]. influenced and shaped the field through his pio- Aumann was born in Frankfurt, Ger- neering work. There is hardly an area of game the- many, in 1930, and moved to New ory today where his footsteps are not readily ap- York with his family in 1938. In 1955 parent. Most of Aumann’s research is intimately he completed his Ph.D. in mathemat- connected to central issues in economic theory: on ics at MIT under the supervision of the one hand, these issues provided the motivation Robert J. Aumann George Whitehead. His thesis, in knot and impetus for his work; on the other, his results theory, was published in the Annals of produced novel insights and understandings in Mathematics [1]. economics. In addition to his own pioneering work, In 1955 Aumann joined the Princeton University Aumann’s indirect impact is no less important— group that worked on industrial and military ap- through his many students, collaborators, and col- plications, where he realized the importance and leagues. He inspired them, excited them with his relevance of game theory, then in its infancy. In vision, and led them to further important results 1956 Aumann joined the Institute of Mathematics [12, p. 2]. at the Hebrew University. The limited space allotted to this article does not Over the past half-century, Aumann has played begin to allow a comprehensive account of Au- an essential and indispensable role in shaping mann’s extensive contributions. Thus, the article game theory, and much of economic theory, to be- must confine itself to brief commentary touching come the great success it is today. He promotes a on only a small part of his work. It is important to unified view of the very wide domain of rational note that the scope of each description is not in- behavior, a domain that encompasses areas of dicative of the importance of the contribution. Fur- many apparently disparate disciplines, like eco- ther and more detailed accounts of Aumann’s con- nomics, political science, biology, psychology, math- tributions may be found in [12]. ematics, philosophy, computer science, law, and sta- We start with Aumann’s study of long-term in- tistics. Aumann’s research is characterized by an teractions, which had a most profound impact on unusual combination of breadth and depth. His sci- the social sciences. The mathematical model en- entific contributions are path-breaking, innovative, abling a formal analysis is a supergame G∗, con- comprehensive, and rigorous—from the discovery sisting of an infinite repetition of a given one-stage and formalization of the basic concepts and prin- game1 G. A pure strategy in G∗ assigns a pure ciples, through the development of the appropri- strategy in G to each period/stage, as a function ate tools and methods for their study, to their 1A game G in strategic form consists of a set of players Abraham Neyman is professor of mathematics at the He- N, pure strategy sets Ai for each player i, and payoff brew University of Jerusalem. His email address is functions gi, which describe the payoff to player i as a func- [email protected]. tion of the strategy profiles a ∈ A := ×i∈N Ai . 44 NOTICES OF THE AMS VOLUME 53, NUMBER 1 of the history of play up to that stage. A profile of information to your disadvantage; re- supergame strategies, one for each player, defines vealing, to use the information your- the play, or sequence of stage actions. The payoff self, and to permit the other players to associated with a play of the supergame is essen- use it to your advantage [8, pp. 46–47]. tially an average of the stage payoffs. In 1959 Aumann [2] defined the notion of a The stress here is on the strategic use strong equilibrium—a strategy profile where no of information—when and how to reveal group of players can gain by unilaterally changing and when and how to conceal, when to their strategies—and characterized the strong equi- believe revealed information and when librium outcomes of the supergame by showing that not, etc. [7, p. 23]. it coincides with the so-called β-core of G. When Aumann’s 1959 methodology is applied to Nash This problem of the optimal use of information is equilibrium—a strategy profile where no single solved in an explicit and elegant way in [9]. player can gain by unilaterally changing his strat- Another substantial line of contributions of Au- egy—the result is essentially the so-called Folk The- mann is the introduction and study of the contin- orem for supergames: the set of Nash equilibria of uum idea in game theory and economic theory. the supergame G∗ coincides with the set of feasi- This includes modeling perfectly competitive ble and individual rational payoffs in the one-stage economies as economies with a continuum of game. In 1976 Aumann and Shapley [11] (and Ru- traders and proving the equivalence of the core and binstein2 in independent work) proved that the competitive equilibrium [3] as well as the equiva- equilibrium payoffs and the perfect equilibrium lence of the core (and competitive equilibrium) payoffs of the supergame G∗ coincide. and the value [5], proving the existence of the com- Supergames are repeated games of complete in- petitive equilibrium [4], and introducing and ex- formation; it is assumed that all players know pre- tensively developing the (Aumann-Shapley) value cisely the one-shot game that is being repeatedly of coalitional games with a continuum of players played. [10]. These models with a continuum of agents en- able precise analysis of economic and political sys- The theory of repeated games of com- tems where groups of participants have signifi- plete information is concerned with the cant influence over the outcome, but each evolution of fundamental patterns of in- individual’s influence is negligible. teraction between people (or for that Another fundamental contribution of Aumann matter, animals; the problems it attacks is “Agreeing to Disagree” [6]: it formalizes the no- are similar to those of social biology). tion of common knowledge and shows (the some- Its aim is to account for phenomena what unintuitive result) that if two agents start such as cooperation, altruism, revenge, with the same prior beliefs, and if their posterior threats (self-destructive or otherwise), beliefs (about a specific event), which are based on etc.—phenomena which may at first different private information, are common knowl- seem irrational—in terms of the usual edge, then these posterior beliefs coincide. This ‘selfish’ utility-maximizing paradigm of paper had a major impact; it led to the development game theory and neoclassical econom- of the area known as interactive epistemology and ics [7, p. 11]. has found many applications in different disci- The model of repeated games with incomplete plines like economics and computer science. information, introduced in 1966 by Aumann and Other fundamental contributions include the Maschler [9], analyzes long-term interactions in introduction and study of correlated equilibrium, which some or all of the players do not know which the study of bounded rationality, and many im- stage game G is being played. The game G = Gk de- portant contributions to cooperative game theory: pends on a parameter k; at the start of the game extending the theory of transferable utility (TU) a commonly known lottery q(k) with outcomes in games to general nontransferable utility (NTU) a product set S = × S is performed and player i is games, formulating a simple set of axioms that i i 3 informed of the i-th coordinate of the outcome. The characterize the NTU-value, and the “Game- repetition enables players to infer and learn in- theoretic analysis of a bankruptcy problem from 4 formation about the other players from their be- the Talmud”. havior, and therefore there is Robert J. Aumann has been a member of the U.S. National Academy of Sciences since 1985, a a subtle interplay of concealing and re- vealing information: concealing, to pre- 3Introduced in L. S. Shapley, Utility comparison and the vent the other players from using the theory of games, La Décision (Paris: Edition du C.N.R.S., 1969), pp. 251–263. 2A. Rubinstein, Equilibrium in Supergames, RM-26, Hebrew 4With M. Maschler, Journal of Economic Theory 36 (1985), University of Jerusalem, 1976. pp. 195–213. JANUARY 2006 NOTICES OF THE AMS 45 member of the Israel Academy of Sciences and Hu- [5] ——— , Values of markets with a continuum of traders, manities since 1989, a Foreign Honorary Econometrica 43 (1975), 611–646. Member of the American Academy of Arts and Sci- [6] ——— , Agreeing to disagree, Annals of Statistics 4 ences since 1974, and a corresponding fellow of the (1976), 1236–1239. British Academy since 1995. He received the Har- [7] ——— , Survey of repeated games, Essays in Game The- vey Prize in Science and Technology in 1983, the ory and Mathematical Economics in Honor of Oskar Morgenstern, Vol. 4 of Gesellschaft, Recht, Wirtschaft, Israel Prize in Economics in 1994, the Lanchester Wissenschaftsverlag, Bibliographisches Institut, Prize in Operations Research in 1995, the Nemmers Mannheim, 1981, pp. 11–42. Prize in Economics in 1998, the EMET prize in Eco- [8] ——— , What is game theory trying to accomplish?, nomics in 2002, the von Neumann prize in Oper- Frontiers of Economics (K.