Aumann model of incomplete information with beliefs

Definition (Aumann model of incomplete information with beliefs)

An Aumann model of incomplete information with beliefs (over a set of states of nature $S$) consists of five elements $(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}, \mathbf{P})$

• the four elements from the Aumann model of incomplete information
  • $N$ is a finite set of players
  • $Y$ is a finite set of elements called states of the world
  • $\mathcal{F}_i$ is a partition of $Y$, for each $i \in N$
  • $\mathfrak{s}: Y \to S$ is a function associating each state of the world with a state of nature
• additionally, $\mathbf{P}$ is a probability distribution over $Y$ such that $\mathbf{P}(\omega) > 0$ for each $\omega \in Y$

Theorem

Let $(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s}, \mathbf{P})$ be an Aumann model of incomplete information with beliefs. Then $\forall \omega \in Y$, $\forall i \in N$ (players), $\forall A \subseteq Y$ (events), player $i$ knows event $A$ in state of the world $\omega$ if and only if he attributes probability $1$ to that event:

See also

• Aumann model of incomplete information
• Aumann's agreement theorem

References

1. M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, pp. 323-324.
2. https://cet.econ.northwestern.edu/dekel/pdf/hierarchies-beliefs-common-knowledge.pdf