Appendix: Mathematical Details of Implementation
Updating Rules for Public Knowledge
Now, the question stands, how does the public information get updated? In other words, from the point of view of the next agent, what is qn, the a priori probability of H based on the actions of agents 1...n-1 as well as that of agent n? The possibilities correspond to the three possibilities described above.
Case 1: agent n bases his decision purely on rn, because |qn-1 - 0.5| < |p - 0.5|. In this case, the purely rational agent number n+1 can infer S because he knows that in this case the nth agent's action pointed in the same direction as his signal. So, in this case, the update rule is:
That is, the information that's passed on is the same as the information agent n himself possesses, because everyone can infer his private signal (which is in the same direction as his action). The only modification is that we use p' - the public estimate of the agent's precision, to calculate rn, whereas the agent himself used p - the value he knows to be his own true precision.
Case 2: agent n bases his decision purely on qn-1, because |qn-1 - 0.5| > |p - 0.5|. In this case, no one can infer anything about agent n's signal. As a result, q remains unchanged:
qn = qn-1
Case 3: the other agents see that for agent n, |qn-1 - 0.5| = |p - 0.5|, but that the agent's action "points" in the direction opposite from qn-1. Then they can infer this agent's signal (as opposite to qn-1 - see (10)). In this case, the update rule is the same as in Case 1.
Case 4: the other agents see that for agent n, |qn-1 - 0.5| = |p - 0.5|, and that the agent's action "points" in the same direction as qn-1. In this case, we have to perform algebra similar to that which led us to eqn. 9. The end result is the following update rule:
In other words, the update rule is the same as the case in which the action implies the signal, only a signal of 1/3 the usual strength is used. We can confirm this by looking at the expectation values of S.