Definition (Aumann situation of incomplete information)

An Aumann situation of incomplete information over a set of states of nature SS is a quintuple (N,Y,(ℱi)i∈N,𝔰,ω∗)(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}, \omega_*), where (N,Y,(ℱi)i∈N,𝔰)(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}) is an Aumann model of incomplete information and ω∗∈Y\omega_* \in Y.

State ω∗\omega_* is the "true state of the world", each player knows the partition element Fi(ω∗)F_i (\omega_*) in his information partition that contains the true state.

Describes knowledge structure particular state of world, i.e. a particular reality. Compare models of incomplete information which allow us to analyze all possible situations.

Theorem (knowledge hierarchy)

Every situation of incomplete information (N,Y,(ℱi)i∈N,𝔰,ω∗)(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}, \omega_*) uniquely determines a knowledge hierarchy over the set of states of the world YY in state of the world ω∗\omega_*.

For every subset C⊆SC \subseteq S of the set of states of nature, can consider the event that contains all states of the world whose state of nature is an element of CC,

𝔰−1(C):={ω∈Y:𝔰(ω)∈C}\mathfrak{s}^{-1}(C) := \{ \omega \in Y : \mathfrak{s}(\omega) \in C \}

Corollary

Every situation of incomplete information (N,Y,(ℱi)i∈N,𝔰,ω∗)(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}, \omega_*) uniquely determines a knowledge hierarchy over the set of states of nature SS in state of world ω∗\omega_*.

References

  1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 324.