Another note from info economics class. Recall Aumann’s common knowledge: fix an event $E$ and consider the two agents’ posteriors of event $E$ after agents each conduct their own deterministic partition experiment. (see this post for a recap)

Theorem (Aumann 1976) If A’s posterior is $q_A$ and B’s posterior is $q_B$ is common knowledge for A and B, then $q_A = q_B$.

But how come $q_A$ and $q_B$ become common knowledge in the first place? Geanakoplos and Polemarchakis’s paper We Can’t Disagree Forever (1982) gives a way that the two people can use (non-strategic) communication to arrive at a common understanding of each other’s posterior:

  • $t = -1$ agent A and B has partition experiment $P_A, P_B$.

  • $t = 0$: $w\in \Omega$ realizes. A learns from $P_A$, B learns from $P_B$ and forms their posterior about event $E$: $q_A, q_b$.

  • $t = 1$: A reports truthfully “my posterior is $q_A$” B hears and update his posterior

  • $t = 2$: B reports truthfully “my posterior is $q_B$” A hears and update his posterior

  • …

The process terminates, and A, B would have the same posterior.

Reference

John D Geanakoplos, Heraklis M Polemarchakis. We can’t disagree forever, Journal of Economic Theory, Volume 28, Issue 1, 1982.