- Person
- Robert Aumann
- Lived
- b. 1930
- Field
- Game theory
- From
- Israeli-American
The story
Classic game theory deals with a handful of players. Many important situations, however, involve thousands or millions of participants, each with almost no influence alone: buyers and sellers in a market, or voters in a national election. In such games individuals matter only when they combine into large groups. Robert Aumann, a mathematician at the Hebrew University of Jerusalem, set out to build a mathematics for these games of the many.
His partner in the work was Lloyd Shapley of the RAND Corporation in Santa Monica, California. In 1951 Shapley had introduced what became known as the Shapley value, a method for dividing the gains or costs of a joint project fairly among the people who took part. It works by averaging how much each participant adds when they join every possible group of the others, and it is the only method that meets a short list of basic fairness conditions.
Together, Aumann and Shapley wrote Values of Non-Atomic Games, published by Princeton University Press in 1974. The book extended the Shapley value to games with a continuum of players, in which no single participant is large enough to matter on their own. Its central result, now known as the Aumann–Shapley value, includes a formula, often called the diagonal formula, that makes the calculation workable for such games. It was later extended further by Jean-François Mertens and Abraham Neyman, who wrote his own doctoral thesis on values of games with a continuum of players under Aumann's supervision in 1977.
The work helped connect cooperative game theory with economic theory about large competitive markets. It showed that ideas about fairness and bargaining developed for small groups could be carried over, with care, to whole economies. Aumann and Shapley also collaborated on perfect equilibria in repeated games while working together at RAND in 1976, research that the Nobel committee cited when Aumann shared the 2005 Nobel Memorial Prize in Economic Sciences.
Shapley went on to win a Nobel Prize of his own in 2012, for the theory of stable allocations and the practice of market design. The ideas behind the Shapley value, and its extension to very large games, are now used well beyond economics, from dividing profits in business partnerships to measuring how much each input contributes to the predictions of machine-learning models.
Why it matters
The Aumann–Shapley approach is used to allocate shared costs and gains fairly among many contributors, for example among the users of a common service or the parts of a business.
About Robert Aumann
German-born Orthodox Jewish mathematician in Jerusalem whose theory of repeated games showed how cooperation can last between rivals; Nobel Prize in Economics 2005.
More about Robert Aumann
- Robert Aumann co-founded Jerusalem's Center for RationalityAround 1990–91 Robert Aumann helped found the Center for Rationality at the Hebrew University of Jerusalem, a game theory centre drawing members from more than a dozen departments.
- Robert Aumann proved rational people can't agree to disagreeIn his 1976 paper Agreeing to Disagree, Robert Aumann gave the first formal definition of common knowledge and proved that rational people sharing the same starting beliefs cannot agree to disagree.
- Robert Aumann invented correlated equilibriumRobert Aumann was the first to define correlated equilibrium, a broader idea than the Nash equilibrium that shows how a shared signal or mediator can help rivals coordinate.
- A nuclear defence puzzle drew Robert Aumann to game theoryRobert Aumann, trained in pure mathematics, turned to game theory in the 1950s to solve a practical problem: how to defend a city against bombers when most of them are decoys.
- Robert Aumann showed mathematically why cooperation lastsRobert Aumann proved that repeated encounters can make cooperation between rivals rational, work on conflict and cooperation that earned him the 2005 Nobel Prize in Economic Sciences.
- Robert Aumann's game theory served nuclear arms controlIn the mid-1960s Robert Aumann and Michael Maschler developed the theory of repeated games with incomplete information for a US agency preparing arms control talks with the Soviet Union.
- Robert Aumann trained a generation of game theoristsRobert Aumann's doctoral students at the Hebrew University include Sergiu Hart, Abraham Neyman, David Schmeidler and Yair Tauman, each a leading figure in game theory.
Sources
- Robert Aumann Wikipedia
- Shapley value Wikipedia
- Robert J. Aumann, Biographical NobelPrize.org
Another act of tikkun olam. This is part of tikkun olam, the Jewish idea of repairing the world, and every one of us can add to it.