Here is one summary of Aumann’s work on game theory:
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.
Previously, Aumann (1962) had swung Ockham’s razor 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 Also in 1976, in an unpublished paper with Lloyd Shapley, Aumann provided the perfect folk theorem using the limit of means criterion.
My favorite Aumann paper is his 1976 piece on agreeing to disagree. He proved the startling result that if two rational, truth-seeking people have common "priors," in a Bayesian sense rigorously definable, then those two individuals should not disagree once they exchange opinions. Imagine I think there are 200 balls in the urn, but Robin Hanson thinks there are 300 balls in the urn. Once Robin tells me his estimate, and I tell him mine, we should converge upon a common opinion. In essence his opinion serves as a "sufficient statistic" for all of his evidence. (This analysis also led Aumann to clarify the important game-theoretic concept of "common knowledge.") Yet people disagree all the time. Does this mean that priors are rarely common? That we are rarely rational truth-seekers? A bit of both? Robin Hanson is doing much work on this topic. Here is my paper with Robin, we argue you that if you disagree with your "epistemic peers," you are probably not a truth-seeker.
Congratulations to both Aumann and Schelling, comments are open…