Appendix: Mathematical Details of Implementation

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**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 q_{n}, 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 r_{n}, because |q_{n-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:

Where
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 q_{n-1}, because |q_{n-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:
q_{n} = q_{n-1}
Case 3: the other agents see that for agent n, |q_{n-1} - 0.5| = |p - 0.5|, but that the agent's action "points" in the direction opposite from q_{n-1}. Then they can infer this agent's signal (as opposite to q_{n-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, |q_{n-1} - 0.5| = |p - 0.5|, and that the agent's action "points" in the same direction as q_{n-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:
Where
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.
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