# shapleyValue: Shapley value of a cooperative game In cooptrees: Cooperative aspects of optimal trees in weighted graphs

## Description

Given a cooperative game, the `shapleyValue` function computes its Shapley value.

## Usage

 `1` ```shapleyValue(n, S = NULL, v) ```

## Arguments

 `n` number of players in the cooperative game. `S` vector with all the possible coalitions. If none has been specified the function generates it automatically. `v` vector with the characteristic function of the cooperative game.

## Details

The Shapley value is a solution concept in cooperative game theory proposed by Lloyd Shapley in 1953. It is obtained as the average of the marginal contributions of the players associated with all the posible orders of the players.

## Value

The `shapleyValue` functions returns a matrix with all the marginal contributions of the players (`contributions`) and a vector with the Shapley value (`value`).

## References

Lloyd S. Shapley. "A Value for n-person Games". In Contributions to the Theory of Games, volume II, by H.W. Kuhn and A.W. Tucker, editors. Annals of Mathematical Studies v. 28, pp. 307-317. Princeton University Press, 1953.

## Examples

 ```1 2 3 4 5``` ```# Cooperative game n <- 3 # players v <- c(4, 4, 4, 8, 8, 8, 14) # characteristic function # Shapley value shapleyValue(n, v = v) ```

### Example output

```Loading required package: igraph

Attaching package: 'igraph'

The following objects are masked from 'package:stats':

decompose, spectrum

The following object is masked from 'package:base':

union

\$contributions
[,1] [,2] [,3]
[1,]    4    4    6
[2,]    4    6    4
[3,]    4    4    6
[4,]    6    4    4
[5,]    4    6    4
[6,]    6    4    4

\$value
[1] 4.666667 4.666667 4.666667
```

cooptrees documentation built on May 30, 2017, 8:11 a.m.