Professors for a Strong Israel

Photo: Professor Robert J. Aumann.

Professors for a Strong Israel congratulates Prof. Robert Aumann, long-time member of PSI, on being awarded the Nobel Memorial Prize in Economics for 2005.  Prof. Aumann's research and the awards it has earned bring honor and prestige to the state of Israel and its academic community. Written by Dr. Ron Breiman, Chairman of PSI, www.professors.org.il

Robert J. Aumann's has been one of the leading figures in the mathematical surge that has characterized Neo-Walrasian economics and game theory in the past forty years. Aumann entered into economics via cooperative game theory -

In Neo-Walrasian theory, Robert Aumann is perhaps best known for his theory of core equivalence in a "continuum" economy. Aumann introduced measure theory into the analysis of economies with an infinite number of agents - formalizing the "perfectly competitive" scenario. In his classical 1964 paper, Aumann proved the equivalence of the Edgeworthian core and Walrasian equilibrium allocations when there are an uncountable infinite number of agents - thereby providing the limit case for future work on core convergence. In order to prove this result was not vacuous, Aumann went on to prove the existence of equilibrium (1966) in this "perfectly competitive" scenario. On his way, he contributed to mathematics itself by providing a definition of the "integral" of a correspondence (1965), which was previously absent. 

Photo: Game Theory Summer Festival 2004 / Robert Aumann and Yair Tauman , July, 2004.

Previously, Aumann (1962) had swung Ockham's razoe and helped remove the axiom of completeness of preferences from the Walrasian theory of choice. In another classical paper with F.J. Anscombe in 1964, Aumann formalized the notion of "subjective probability", a concept that had been earlier forwarded by Leonard Savage, that profoundly changed the theory of choice under uncertainty.  His contributions to game theory have perhaps been no less path-breaking. Aumann entered game theory in 1959 to carefully distinguish between infinitely and finitely repeated games. With Bezalel Peleg in 1960, Aumann formalized the notion of a coalitional game without transferable utility (NTU) - one of the organizing beacons of his later research. With Michael Maschler (1963), he introduced the concept of a "bargaining set". In 1974, Aumann went on to identify "correlated equilibrium" in Bayesian games. In 1975, Aumann went on to prove a convergence theorem for the Shapley value. In 1976, he formally defined the concept of "Common Knowledge". Also in 1976, in an unpublished paper with Lloyd Shapley, Aumann provided the perfect folk theorem using the limit of means criterion.  For Aumann, game theory is clearly the more "general theory". His ruminations on the role of game theory in economic analysis are wonderfully laid out in Aumann (1985).

Major Works of Robert J. Aumann

• "Acceptable Points in General Cooperative n-Person Games", 1959, in Contributions to the Theory of Games IV, Annals of Math. Study

• "Von Neumann-Morgenstern Solutions to Cooperative Games Without Side Payments" with B. Peleg, 1960, Bulletin of the American Mathematical Society.

• "Acceptable Points in Games of Perfect Information", 1960, Pacific Journal of Mathematics

• "Linearity of Unrestrictedly Transferable Utilities", 1960, Naval Research Logistics Quarterly

• "Spaces of Measurable Transformations", 1960, Bulletin of AMS

• "The Core of a Cooperative Game Without Side Payments", 1961, Transactions of the American Mathematical Society

• "Almost Strictly Competitive Games", 1961, Journal of the Society for Industrial and Applied Mathematics.

• "Utility Theory Without the Completeness Axiom", 1962, Econometrica.

• "A Definition of Subjective Probability" with F.J. Anscombe, 1963, Annals of Mathematical Statistics

• "The Bargaining Set for Cooperative Games", with M. Maschler, 1964, in Advances in Game Theory,

• "Mixed and Behavior Strategies in Infinite Extensive Games", 1964, in Advances in Game Theory.

• Values of Non-Atomic Games, with L.S. Shapley, 1964.

• "Markets with a Continuum of Traders", 1964, Econometrica.

• "A Variational Problem Arising in Economics", with M. Perles, 1965, Journal of Mathematical Analysis and Applications.

• "Integrals of Set-Valued Functions", 1965, Journal of Mathematical Analysis and Applications.

• "A Method of Computing the Kernel of n-Person Games", with B. Peleg and P. Rabinovitz, 1965, Mathematics of Computation.

• "Game-Theoretic Aspects of Gradual Disarmament", with M. Maschler, 1966, Mathematica

• "Existence of a Competitive Equilibrium in Markets with a Continuum of Traders", 1966, Econometrica.

• "A Survey of Cooperative Games Without Side Payments", 1967, in Essays in Mathematical Economics in Honor of Oskar Morgenstern.

• "Random Measure Preserving Transformations", 1967, in Proceedings of the Fifth Berkeley Symposium.

• "Measurable Utility and the Measurable Choice Theorem", 1969, in La D&eacut;cision.

• "Some Thoughts on the Minimax Principle" with M. Maschler, 1972, Management Science.

• "Disadvantegous Monopolies", 1973, JET.

• "Subjectivity and Correlation in Randomized Strategies", 1974, JMathE.

• Values of Non-Atomic Games, with L.S. Shapley, 1974.

• "A Note on Gale's Example", with B. Peleg, 1974, JMathE.

• "Cooperative Games with Coalition Structures", with J. Dréze, 1975, International Journal of Game Theory.

• "Values of Markets with a Continuum of Traders", 1975, Econometrica.

• "An Elementary Proof that Integration Preserves Uppersemicontinuity", 1976, JMathE.

• "Agreeing to Disagree", 1976, Annals of Statistics.

• "Solution Notions for Continuingly Competitive Situations", with L.S. Shapley, 1976.

• "Orderable Set Functions and Continuity III: Orderability and Absolute Continuity", with Uri Rothblum, 1977, SIAM Journal on Control and Optimization.

• "The St. Petersburg Paradox: A Discussion of some Recent Comments", 1977, JET.

• "Power and Taxes", with M. Kurz, 1977, Econometrica.

• "Core and Value for a Public Goods Economy: An example", with R.J. Gardner and R.W. Rosenthal, 1977, JET.

• "On the Rate of Convergence of the Core", 1979, IER.

• "Recent Developments in the Theory of the Shapley Value", 1978, in Proceedings of the International Congress of Mathematicians.

• "Survey of Repeated Games", 1981, in Essays in Game Theory and Mathematical Economics in Honor of Oskar Morgenstern

• "Approximate Purification of Mixed Strategies", with Katznelson, Radner, Rosenthal and Weiss, 1983, Mathematics of Operations Research

• "Voting for Public Goods", with M. Kurz and A. Neyman, 1983, RES.

• "An Axiomatization of the Non-Transferable Utility Value", 1985, Econometrica.

• "Game Theoretic Analysis of a Bankruptcy Problem from the Talmud", with M. Maschler, 1985, JET.

• "What is Game Theory Trying to Accomplish?", 1985, in Arrow and Honkaphola, editors, Frontiers in Economics.

• "On the Non-Transferable Utility Value: A Comment on the Roth-Shafer Examples", 1985, Econometrica.

• "Values of Markets with Satiation or Fixed Prices", with J. Dréze, 1986, Econometrica.

• "Game Theory", 1987, in New Palgrave.

• "Power and Public Goods", with M. Kurz and A. Neyman, 1987, JET.

• "Correlated Equilibrium as an Expression of Bayesian Rationality", 1987, Econometrica.

• "Endogenous Formation of Links between Players and of Coalitions: An Application of the Shapley value", with R.B. Myerson 1988, in Roth, editor, The Shapley Value.

• "Cooperation and Bounded Recall", with S. Sorin, 1989, GamesEB.

• Lectures on Game Theory, 1989.

• "Nash Equilibria are Not Self-Enforcing", 1990, in Gabszewicz et al, editors, Economic Decision-Making

• "Irrationality in Game Theory", 1992, in Dasgupta et al, editors, Economic Analysis of Markets and Games, Essays in Honor of Frank Hahn.

• "Long-Term Competition: A game-theoretic analysis", with L.S. Shapley, 1994, in Gale et al, editors, Essays in Game Theory

• "Backward Induction and Common Knowledge of Rationality", 1995, GamesEB.

• Repeated Games of Incomplete Information with M. Maschler, 1995.