Game Theory 1 – chap02. Interactive Knowledge

Interactive Knowledge Chapter 5

When a rational people makes a decision, he would reflect carefully on what he knows and what he does not, before choosing how to act.

What do I know, and what do he know

When two rational people interact, one would think about what the other knows and does not know, and what the other knows about what he knows. This interactive knowledge is a central element in economic theory.
An event is common knowledge (CK) among a group of people if each one knows it, each one knows that the others know it, each one knows that each one knows that the others know it, and so on.

e.g., Event E is CK
if P1 knows E, P2 knows E, P1 knows P2 knows E, P2 knows P1 knows E
P1 knows P2 knows P1 knows E, P2 knows P1 knows P2 knows E, …
→ event E is common knowledge

Hence CK is the limit of a potentially infinite chain of reasoning about knowledge.

Common Knowledge vs Public Knowledge

There is a subtle difference:
CK is a (much) stronger notion that public knowledge

Furthermore, this difference may lead to different outcomes in some cases.


Image 3 kids sitting in a circle, each wearing either a red or white hat. Each kid cannot observe her own hat but can observe the others.
Suppose all the hats are red.

When a teacher comes in and asks if any student can identify the color of her own hat, the answer would be “No”
But if the teacher happens to publicly announce
“there is at least one red hat in this room”,
then the student’s answer does change

\(\Omega\) = { RRR, RRW, RWR, WRR, RWW, WRW, WWR, WWW }

Information set

\(I_{A}\) = {{RRR, WRR}, {RRW, WRW}, {RWR, WWR}, {RWW, WWW}}
\(I_{B}\) = {{RRR, RWR}, {RRW, RWW}, {WWR, WRR}, {WWW, WRW}}
\(I_{C}\) = {{RRR, RRW}, {RWR, RWW}, {WRR, WRW}, {WWR, WWW}}

Lumping things together that students don’t recognize.
WWW is an information set and can be cleared (ignored)
Not answering sends a different message (students get a different set of information)

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