The Shapley value for cooperative games meets a set of useful axioms.

Axiomatic Basis for Shapley Value

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

University of Michigan - Ann Arbor

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

Mathematical Models of Collective Intelligence

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

The Model Thinker

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 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

The Diversity Prediction Theorem

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.

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