- Person
- Robert Aumann
- Lived
- b. 1930
- Field
- Game theory
- From
- Israeli-American
The story
What does it mean for something to be common knowledge? Not just that everyone in a group knows it, but that everyone knows that everyone knows it, and that everyone knows that, and so on without end. The philosopher David Lewis introduced the idea in his 1969 study Convention, and the sociologist Morris Friedell defined it the same year. But no one had yet turned it into precise mathematics.
Robert Aumann, a mathematician at the Hebrew University of Jerusalem, did so in 1976. In a short paper titled Agreeing to Disagree, he gave the first formal, set-theoretic definition of common knowledge. In his model, each person's information is represented by how finely they can tell apart the possible states of the world, and an event is common knowledge when it can be recognised from the information that all of them share at every level of the chain of knowing.
He used this definition to prove a striking result, now called Aumann's agreement theorem. Suppose two rational people start from the same prior beliefs and update them correctly in the light of what they learn. If their resulting opinions about some event are common knowledge between them, then those opinions must be the same. In other words, such people cannot agree to disagree.
The theorem surprised many readers, because in real life people disagree all the time. It forced economists, philosophers and computer scientists to think carefully about where disagreement comes from: different starting assumptions, private information that has not been fully shared, or departures from strict rationality. Later researchers showed, for example, that two people who keep telling each other their latest estimates will eventually come to the same view, and others studied how quickly such dialogues converge.
Aumann's formal account of common knowledge proved even more influential than the theorem itself. Computer scientists took up the subject from the 1980s, as part of a wider interest in the logic of knowledge, and it became a standard tool for reasoning about coordination. Puzzles built on it, such as the famous problem of the islanders with blue eyes, show how a public announcement of something everyone already knows can still change what people do. His work on repeated games and conflict and cooperation earned him the 2005 Nobel Memorial Prize in Economic Sciences.
Why it matters
Aumann's formal idea of common knowledge became a basic tool in economics, philosophy and computer science, used in the study of markets, coordination and distributed computer systems.
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 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 and Lloyd Shapley analysed games of the manyWith Lloyd Shapley, Robert Aumann wrote Values of Non-Atomic Games in 1974, extending game theory to markets and elections with huge numbers of players and giving rise to the Aumann–Shapley value.
- 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
- Aumann's agreement theorem Wikipedia
- Common knowledge (logic) Wikipedia
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.