Appendix: Mathematical Details of Implementation

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Notation and Variables

The Informational Cascades model makes use of the formal concept of Perfect Bayesian Equilibrium. Agents update their belief of the state of reality according to Bayes' Rule, as actions of other agents are observed. The variables relevant to agent beliefs are stored in their knowledge vectors as described below.

This implementation is mathematically equivalent to the original model with one exception. The introduction of public beliefs about agent precision that can differ from the true values has the effect of making zero probability events possible. This requires a refinement of the original model, and is described below.

To begin describing the equilibrium, let us define appropriate variables with respect to a given agent n:
Event H - the true concept is V=1, i.e. adoption is desirable for each agent,
Event L - the true concept is V=-1, i.e. rejection is desirable for each agent
Event A - the event that all the actions of all the previous agents occur,
Event S - the private signal of this agent.

Some notational conventions:
P(event 1) means the probability of event 1;
P(event 1 | event 2) means the probability of event 1 given that event 2 is true.
Event1Event2 is the union of events 1 and 2.
q "points" in the same direction as r when
((q < 0.5) and (r < 0.5)) or ((q > 0.5) and (r > 0.5))
Note that, for instance, P(H) + P(L) = 1.

Furthermore, let us define that for the given agent n:
qn-1 = P(H | A) = probability of H given only the actions of the previous n-1 agents,
rn= P(H | S) = probability of H given only the private signal of the given agent n.
p = precision of the agent. For example, when p=0.6, the agent will get the correct signal 60% of the time.
Note that rn = P(H | S) = P(S | H), because P(S) = P(H).

Recall that for each agent the signal takes one of two values, out of the set {+1, -1}, and his action will be either adoption or rejection, which we will also designate with +1 and -1, respectively.

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