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


Now, from the point of view of a given agent, the a posteriori probability of H can be calculated using Bayes' Rule in the following way:

It is given that S is conditionally independent of A, given H. In other words,
P(S | HA ) = P(S | H) = r (3)
Now, let us work on the denominator of (2). Again, using Bayes' rule:
P(S | A) = P(SH | A) + P(SL | A) = P(S | HA)*P(H | A) + P(S | LA)*P(L | A) (4)
Now, according to the definitions,
P(L | A) = 1 - P(H | A) = 1 - qn-1 (5)
According to equation (3),
P(S | HA) = rn (6)
And, using reasoning similar to that of eqn. (3)
P(S | LA) = P(S | L) = P(L | S) = 1 - rn (7)
Now, substituting (5), (6), and (7) into (4), we get
P(S | A) = qn-1rn + (1 - qn-1)(1 - rn) (8)
Now, substituting (3) and (8) into (2), get:

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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