Strategy for Information Markets/Information Cascades

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Conditional Probability[edit]

If  Pr(A|B) = "Probability of A given B" or "Probability of A conditioned on B"


 Pr(A|B) = \frac {Pr(A\ AND\ B)}{Pr(B)}

Bayes' Rule[edit]

 Pr(B|A) = \frac { Pr(A|B)Pr(B) } { Pr(A|B)Pr(B)+Pr(A|NotB)Pr(NotB) }

Condorcet Jury Theorem[edit]

Binomial Distribution[edit]

If the probability of one success is \text{ }p, then

  Pr(k\text{ successes in}\ n\ \text{trials}) = 	\binom{n}{k} p^k (1-p)^{(n-k)}


  •  p^k\ stands for the probability of a particular \text{  }k trial being a success
  •   (1-p)^{(n-k)} stands for the probability of a particular \text{  }(n-k) trial being a failure

and in math,

  •  \binom{n}{k} = \frac {n!}{k!(n-k)!}

Group Decision/Voting[edit]

In order to determine if a group decision/voting is correct, the number of successes \text{  }C needs to be more than half of \text{  }n. The following formula derived from the Binomial Distribution Function tells the chance of the right group decision.

In the case here, by eliminating the situation that the vote is a tie, let's assume that the number of votes \text{  }n is odd so that \text{  }C could be more than half of \text{  }n.


 Pr(C) = \sum_{k=\frac{n+1}{2}}^{n} \binom{n}{k} p^k (1-p)^{(n-k)}

Influence-Dependent Model of Group Decision/Voting[edit]

In daily lives, people usually make votes with other influences, instead of absolutely independent decision making. Let's derive another model to determine the probability of correct group decision on other influences.


  •  p = the probability of being correct
  •   C = the group makes the correct decision (more than half of the votes are correct)
  •   P_I = the probability of the influence being correct
  •  \alpha =  the probability of the voter following the influence to make decision
  •  Pr(\text{Voter votes correctly}) = (1- \alpha)p + \alpha P_I
  •   P_T = Pr(\text{Voter Correct}|\text{Influence Correct})= (1- \alpha)p + \alpha = the probability of the voter being correct if the influence is correct
  •  P_F = Pr(\text{Voter Correct}|\text{Influence Wrong})= (1- \alpha)p = the probability of the voter being correct if the influence is wrong


 Pr(\text{Group Correct}) = Pr(\text{Influence Correct})Pr(\text{Group Correct}|\text{Influence Correct})

+ Pr(\text{Influence Wrong})Pr(\text{Group Correct}|\text{Influence Wrong})

 Pr(C) = P_I\sum_{k=\frac{n+1}{2}}^{n} \binom{n}{k} P_T^k (1-P_T)^{(n-k)}+(1-P_I)\sum_{k=\frac{n+1}{2}}^{n} \binom{n}{k} P_F^k (1-P_F)^{(n-k)}

Central Limit Theorem[edit]

Let x_1,x_2,...,x_n be a series of independently and identically distributed random variables. The mean of these variables is  \mu_x and the variance is \sigma_x^2.

Let y = \frac {1}{n} (x_1+x_2+...+x_n) .

When n gets larger, y gets closer to be a random variable that is normally distributed and has mean  \mu_y = \mu_x and variance \sigma_y^2 = \frac{\sigma_x^2}{n}