Robert Aumann, one of two new Nobel Laureates

by on October 10, 2005 at 8:43 am in Economics | Permalink

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 Also in 1976, in an unpublished paper with Lloyd Shapley, Aumann provided the perfect folk theorem using the limit of means criterion.

Here is a strange but fascinating interview with Aumann.  It covers John Nash, religion, and the Cold War, among other matters.  Here is his home page, with links to articles.

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…

1 Ethan October 10, 2005 at 11:02 am

I have the luck of being a visiting fellow at the Center for the Study of Rationality at Hebrew U this year. Aumann’s office is a few doors down. So, I was able to be at Aumann’s press conference, which just ended. I thought you might be interested to know that — in addition to some nice remarks about game theory, his students, colleagues, and teachers, and the Israeli school of economics — the point that Aumann made most strongly was that he deeply regreted that Lloyd Shapley didn’t share the prize. Aumann is a deep thinker and a class act. Great choice by the committee.

2 dsquared October 10, 2005 at 1:17 pm

Randall is exactly right and I am afraid that you and Robin are wrong on this one. What you are picking up is simply that people work on more complicated models of the world than the one that gets selected as “the” model for these comparisons, and the set of priors that they share is a small subset of their total prior information. (Quite apart from anything else, on almost all important questions, my prior information will include the fact that you and I can game the system by biasing what we tell each other, and we both know this, a fact which Aumann of all people would be aware of).

3 Robin October 10, 2005 at 7:56 pm

Randall Parker and dsquared, I’m glad to see your interest in the topic, but you seem to be making assumptions about our assumptions. We need not assume that actual thinking is mathematical, that people explicitly construct models they agree on, that models have any particular level of complexity, or that people do not “game the system.” And I don’t understand how to find a subset of prior (priors aren’t sets). Happy to discuss this further in a forum of your choice.

4 Robin October 11, 2005 at 9:17 am

Joshq, yes, the theory only applies to disagreements about which of many possible worlds is our world. So it need not apply to preferences, thought it should apply to questions of what acts are moral, as usually conceived.

5 Tom October 13, 2005 at 11:12 am

“[I]f you disagree with your ‘epistemic peers,’ you are probably not a truth-seeker.”

Or you are, you just don’t have confidence that THEY are. 😉

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