Strategy for Information Markets/Information Cascades

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Conditional ProbabilityEdit

If   "Probability of A given B" or "Probability of A conditioned on B"

then,

 


Bayes' RuleEdit

 

Condorcet Jury TheoremEdit

Binomial DistributionEdit

If the probability of one success is  , then

 

while

  •   stands for the probability of a particular   trial being a success
  •   stands for the probability of a particular   trial being a failure


and in math,

  •  


Group Decision/VotingEdit

In order to determine if a group decision/voting is correct, the number of successes   needs to be more than half of  . 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   is odd so that   could be more than half of  .

Therefore,

 

Influence-Dependent Model of Group Decision/VotingEdit

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.

Let

  •   the probability of being correct
  •   the group makes the correct decision (more than half of the votes are correct)
  •   the probability of the influence being correct
  •   the probability of the voter following the influence to make decision
  •  
  •   the probability of the voter being correct if the influence is correct
  •   the probability of the voter being correct if the influence is wrong


Therefore,


 

 


 

Central Limit TheoremEdit

Let   be a series of independently and identically distributed random variables. The mean of these variables is   and the variance is  .

Let  .

When   gets larger,   gets closer to be a random variable that is normally distributed and has mean   and variance