situation of incomplete information with beliefs

Definition (situation of incomplete information with beliefs)

An situation of incomplete information with beliefs is a vector $(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}, \mathbf{P}, \omega_*)$, where $(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}, \mathbf{P})$ is an Aumann model of incomplete information with beliefs and $\omega_* \in Y$ is a state of the world.

Theorem (knowledge hierarchy)

Every situation of incomplete information with beliefs $(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}, \mathbf{P}, \omega_*)$ uniquely determines a mutual belief hierarchy among the players over the set of states of the world $Y$, and therefore also a mutual belief hierarchy over the states of nature $S$.

See also

• Aumann situation of incomplete information
• Aumann model of incomplete information with beliefs

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, p. 337.