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 - q

According to equation (3),

P(S | HA) = r

And, using reasoning similar to that of eqn. (3)

P(S | LA) = P(S | L) = P(L | S) = 1 - r

Now, substituting (5), (6), and (7) into (4), we get

P(S | A) = q

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 q_{n-1} and the probability r_{n} based on the private signal. The agent will make his decision based on this probability.

Note the following: when |q_{n-1} - 0.5| > |p - 0.5|, P(H | AS) "points" in the same direction as q_{n-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 |q_{n-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 |q_{n-1} - 0.5| = |p - 0.5|, there are two possibilities: when q_{n-1} and S (and r_{n}) "point" in the same direction, the agent decides in that direction. Otherwise, if they conflict, the agent flips a coin to decide.(10)