Aumann model of incomplete information

Definition (Aumann model of incomplete information)

Let $S$ be a finite set of states of nature. An Aumann model of incomplete information over set $S$ of states of nature consists of four components $(N, Y, (\mathcal{F}_i)_{i \in N}, \mathfrak{s})$ where

• $N$ 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 $i \in N$ (i.e. collection of disjoint nonempty subsets of $Y$ whose union is $Y$)
• $\mathfrak{s}: Y \to S$ is a function associating each state of the world with a state of nature

Other definitions

Definition (event)

An event is a subset of $Y$.

Event $A$ obtains in state of world $\omega$ if $\omega \in A$.
It follows that if event $A$ obtains in state of the world $\omega$ and if $A \subseteq B$, then event $B$ obtains in $\omega$.

Definition (knowledge)

Let $\omega \in Y$ be a state of the world, and let $A \subseteq Y$ be an event. Player $i$ knows $A$ in $\omega$ if

If $F_i(\omega) \subseteq A$, then in state of world $\omega$ player $i$ knows event $A$ obtains, even though he may not know the state of world is $\omega$, as according to his information, $F_i(\omega)$ are included in the event $A$.

Definition (operator)

Define operator $K_i : 2^Y \to 2^Y$ (where $2^Y$ is the collection of all subsets of $Y$) by

i.e. set of all states of world in which player $i$ knows event $A$; often denoted as $K_i A$.

Player $i$ knows event $A$ in state of the world $\omega_*$ iff $\omega_* \in K_i A$.

Theorem (states of world knowing an event are subsets of states of the world with the event)

$K_i A \subseteq A$ for every event $A \subseteq Y$ and every player $i \in N$.

Theorem (knowledge of an event which is contained in another event)

For every pair of events $A, B \subseteq Y$, and every player $i \in N$,

If event $A$ contained in event $B$, states of world in which player $i$ knows event $A$ form a subset of states of world in which player knows event $B$. In other worlds, in every state of the world in which a player knows event $A$, he also knows event $B$.

Theorem (knowledge of knowledge of event)

For every event $A \subseteq Y$ and every player $i \in N$, we have $K_i K_i A = K_i A$.

More generally, knowledge operator $K_i$ of player $i$ satisfies Kripke's S5 system.

Notes

Most widely accepted statistical approach to dealing with decision problems in situations of incomplete information is the Bayesian approach.

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
3. https://www.cambridge.org/core/journals/economics-and-philosophy/article/states-of-nature-and-the-nature-of-states/A527742E164FB1D451763C4BCEDA2DD4