Construction by

Jean-François Mertens Jean-François Mertens (11 March 1946 – 17 July 2012) was a Belgian game theorist and mathematical economist. Mertens contributed to economic theory in regards to order-book of market games, cooperative games, noncooperative games, repeated ga ...

and Zamir implementing with

John Harsanyi John Charles Harsanyi ( hu, Harsányi János Károly; May 29, 1920 – August 9, 2000) was a Hungarian-American economist and the recipient of the Nobel Memorial Prize in Economic Sciences in 1994. He is best known for his contributions to the ...

'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. 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 examples of the usage of the construction are the

induction puzzles Induction puzzles are logic puzzles, which are examples of multi-agent reasoning, where the solution evolves along with the principle of induction. A puzzle's scenario always involves multiple players with the same reasoning capability, who go ...

. And so is

Robert Aumann Robert John Aumann (Hebrew name: , Yisrael Aumann; born June 8, 1930) 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 ...

's construction of

common knowledge Common knowledge is knowledge that is publicly known by everyone or nearly everyone, usually with reference to the community in which the knowledge is referenced. Common knowledge can be about a broad range of subjects, such as science, literat ...

References

{{game theory Game theory