Hierarchy of beliefs

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Construction by Jean-François Mertens and Zamir implementing with John Harsanyi's proposal to model games with incomplete information by supposing that each player is characterized by a privately known type that describes his feasible strategies and payoffs as well as a probability distribution over other players' types. [1]

Jean-François Mertens Belgian game theorist

Jean-François Mertens was a Belgian game theorist and mathematical economist.

John Harsanyi hungarian economist

John Charles Harsanyi was a Hungarian-American economist.

Such probability distribution at the first level can be interpreted as a low level belief of a player. One level up the probability on the belief of other players is interpreted as beliefs on beliefs. A recursive universal construct is built—in which player have beliefs on their beliefs at different level—this construct is called the hierarchy of beliefs.

The result is a universal space of types in which, subject to specified consistency conditions, each type corresponds to the infinite hierarchy of his probabilistic beliefs about others' probabilistic beliefs. They also showed that any subspace can be approximated arbitrarily closely by a finite subspace.

Another popular example of the usage of the construction is the Prisoners and hats puzzle. And so is Robert Aumann's construction of common knowledge. [2]

Robert Aumann Israeli-American mathematician

Robert John Aumann is an Israeli-American mathematician, and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel. He also holds a visiting position at Stony Brook University, and is one of the founding members of the Stony Brook Center for Game Theory.

Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum.

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References

  1. Jean -François Mertens and Shmuel Zamir (1985-03-01). "Formulation of Bayesian analysis for games with incomplete information". INTERNATIONAL JOURNAL OF GAME THEORY. 14 (1): 1–29. doi:10.1007/BF01770224.
  2. Herbert Gintis (16 March 2009). The bounds of reason: game theory and the unification of the behavioral sciences. Princeton University Press. p. 158. ISBN   978-0-691-14052-0 . Retrieved 3 March 2012.