Editing Information cascade (section)

=== Quantitative description ===

A person's signal telling them to accept is denoted as {{mvar|H}} (a high signal, where high signifies he should accept), and a signal telling them not to accept is {{mvar|L}} (a low signal).  The model assumes that when the correct decision is to accept, individuals will be more likely to see an {{mvar|H}}, and conversely, when the correct decision is to reject, individuals are more likely to see an {{mvar|L}} signal.  This is essentially a [[conditional probability]] – the probability of {{mvar|H}}  when the correct action is to accept, or <math>P[H|A]</math>. Similarly <math>P[L|R]</math> is the probability that an agent gets an {{mvar|L}} signal when the correct action is reject. If these likelihoods are represented by ''q'', then ''q'' > 0.5.  This is summarized in the table below.<ref name="Easley">{{cite book|url=http://www.cs.cornell.edu/home/kleinber/networks-book/|title=Networks, Crowds and Markets: Reasoning about a Highly Connected World|last=Easley|first=David|publisher=Cambridge University Press|year=2010|pages=483–506}}</ref>

{| class="wikitable"
|-
! rowspan=2 | Agent signal
! colspan=2 | True probability state
|-
!                 Reject                 !! Accept
|-
|  ''L''       || ''q''                  || 1&minus;''q''
|-
|  ''H''       || 1&minus;''q''                || ''q''
|}
 
The first agent determines whether or not to accept solely based on his own signal.  As the model assumes that all agents act rationally, the action (accept or reject) the agent feels is more likely is the action he will choose to take.  This decision can be explained using [[Bayes' rule]]:

<math display="block">\begin{align}
  P\left(A|H\right) &= \frac{P\left(A\right) P\left(H|A\right)}{P\left(H\right)} \\
                    &= \frac{P\left(A\right) P\left(H|A\right)}{P\left(A\right) P\left(H|A\right) + P\left(R\right) P\left(H|R\right)} \\
                    &= \frac{pq}{pq + \left(1 - p\right)\left(1 - q\right)} \\
                    &> p
\end{align}</math>

If the agent receives an {{mvar|H}} signal, then the likelihood of accepting is obtained by calculating <math>P[A|H]</math>.  The equation says that, by virtue of the fact that ''q'' > 0.5, the first agent, acting only on his private signal, will always increase his estimate of ''p'' with an {{mvar|H}} signal.  Similarly, it can be shown that an agent will always decrease his expectation of ''p'' when he receives a low signal.  Recalling that, if the value, {{mvar|V}}, of accepting is equal to the value of rejecting, then an agent will accept if he believes ''p'' > 0.5, and reject otherwise.  Because this agent started out with the assumption that both accepting and rejecting are equally viable options (''p'' = 0.5), the observation of an {{mvar|H}} signal will allow him to conclude that accepting is the rational choice.

The second agent then considers both the first agent's decision and his own signal, again in a rational fashion.  In general, the ''n''th agent considers the decisions of the previous ''n''-1 agents, and his own signal.  He makes a decision based on Bayesian reasoning to determine the most rational choice.
 
<math display="block">P (A | \text{Previous}, \text{Personal signal}) = \frac{pq^a (1 - q)^b}{p q^a (1 - q)^b + (1 - p)(1 - q)^a q^b}</math>
 
Where {{mvar|a}} is the number of accepts in the previous set plus the agent's own signal, and {{mvar|b}} is the number of rejects.  Thus, {{tmath|1=a + b = n}}.  The decision is based on how the value on the right hand side of the equation compares with ''p''.<ref name="Easley"/>