Appendix: Mathematical Details of Implementation
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Derivation of an Agent's a Posteriori Knowledge
From the definitions of S and p for a given agent, we can infer that
Thus, we have derived the a posteriori probability of H from the point of view of a given agent, based on his a priori probability qn-1 and the probability rn based on the private signal. The agent will make his decision based on this probability.
Note the following: when |qn-1 - 0.5| > |p - 0.5|, P(H | AS) "points" in the same direction as qn-1, in other words, it's on the same side of 0.5. In this case, the agent makes a decision that's consistent with the available public information, regardless of its private signal. On the other hand, when |qn-1 - 0.5| < |p - 0.5|, the agent makes a decision consistent with its private signal, thus adding to the public pool of knowledge.
Finally, when |qn-1 - 0.5| = |p - 0.5|, there are two possibilities: when qn-1 and S (and rn) "point" in the same direction, the agent decides in that direction. Otherwise, if they conflict, the agent flips a coin to decide.(10)
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