An equation to calculate the collective accuracy of a crowd. It shows that diversity matters just as much as ability. (Acc = accuracy)
- Collective Acc Crowd = Avg Individual Acc - Collective Predictive Diversity

University of Michigan - Ann Arbor

The crowd's performance on some "intelligent" task is better than the best or average member of the crowd. 

Supperadditive Intelligence (Wisdom of Crowds)

- Information entropy equation
- Last-on-the-bus value
- Shapley value
- Diversity theorem

Mathematical Models of Collective Intelligence

Page, Scott E. The Difference: How the Power of Diversity Creates Better Groups, Firms, Schools, and Societies. Princeton University Press, 2008. 

The Difference by Scott Page

The Diversity Prediction Theorem

A crowd's prediction accuracy can't be worse than the average prediction accuracy of a crowd's members.
- Collective Prediction Error < Average Individual Error

The Crowd Beat Averages Law

Predictive ability can be understated when assuming independence in a small crowd. Contrarily, an assumption of independence may overstate the predictive ability of large crowds.

Assumptions of Independence

Crowd of models is used to refer to crowds of experts.

Crowd of Models

People often fixate on a variation of attributes even if they are viewing the same situation or tasked with solving the same problem. These fixations on attributes are each separate projection interpretations.

Projection interpretations

Provides platforms with the ability to compare the success of forecasting ability and training.

Forecasting Challenge Ensembles

A set of N individuals and a value function that assigns a value to every subset of individuals (subsets are referred to as coalitions). A coalition with no players has a value of 0, and a coalition with all players would have the total value.

Cooperative Games

Problems which add uncertainty to learning the best alternative. Rewards for potential alternatives are represented by distributions (not fixed outcomes). Bandit problems can model pharmaceutical drug trials, choice among technologies, where to place advertisements and anything else with uncertain payoffs. Can help figure out optimal explore-exploit tradeoff.

Bandit problems

A jury of voters classifies something correctly more than any member of the jury on average. As the number of jurors increases, the accuracy of the jury will approach (but never reach) 100% accuracy.

Condorcet Jury Theorem

The wisdom of crowd effect suggests that the median estimate of a group is more likely to have  closer accuracy to an answer than the estimate of an individual.

Wisdom of Crowd Effect

Proposed by Albert Bandura and other researchers, social learning theory is a perspective on learning that incorporates cognitive processes. It evolved from behaviorism but differed by asserting that internal mental states play a crucial role in learning. Unlike pure behaviorism, this theory can explain how learning occurs even without direct external reinforcement, emphasizing the importance of observation and mental processing.

Social Learning Theory

A measure of uncertainty based on the number of flips of a fair coin. Low entropy means little information is being revealed (i.e., sun rising in east) and high entropy means the outcomes are very uncertain but when actualized reveal a lot of information (i.e., a lottery ticket and reveal). For example, we would need to ask N questions to learn the sexes of N children. N questions would distinguish $2^N$ potential outcome sequences (N binary random events: N flips of a fair coin). Each outcome sequence has a probability of $$1\frac{1}{2^N}$$, which is converted to N = - $\log_2 (\frac{1}{2^N}) ^2$. If an outcome arises with probability p, it will have an uncertainty of $\log_2 (p) $ (an approximation of the # of yes/no questions needed to identify the outcome sequence).

Information Entropy

An Indvidual's marginal contribution in group that has been previously formed. If they are the last player to a join a group in a cooperative game, what value do they add.

Last-on-the-bus value

In a cooperative game, it is an individual's average marginal contribution across every possible coalition after being added to the group.

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