weightedshapleyvalue {TUGLab}R Documentation

Positively weighted Shapley value

Description

Given a game, positive player weights and an ordered partition of the set of players, this function returns the corresponding weighted Shapley value.

Usage

weightedshapleyvalue(v, binary = FALSE, weights, partition = NULL)

Arguments

v

A characteristic function, as a vector.

binary

A logical value. By default, binary=FALSE. Should be set to TRUE if v is introduced in binary order instead of lexicographic order.

weights

The player weights, as a vector of positive numbers.

partition

An ordered partition of the set of players, as a list of vectors. When not specified, it is taken to be the partition whose only element is the set of all players.

Details

A weight system \omega is a pair \omega=(\lambda,\mathcal{S}) where \lambda=(\lambda_{i})_{i \in N} is a positive weight vector (\lambda_{i}>0 for each i \in N) and \mathcal{S}=(S_{1},\dots,S_{m}) is an ordered partition of N. The weighted Shapley value with weight system \omega=(\lambda,\mathcal{S}) is the linear map Sh^{\omega} that assigns to each unanimity game u_{T}, with T \in 2^{N} \setminus \emptyset, the allocations Sh^{\omega}_{i}(u_{T})=\frac{\lambda_{i}}{\lambda(T \cap S_{k})} if i \in T \cap S_{k} and Sh^{\omega}_{i}=0 if i \notin T \cap S_{k}, where k=\max\{i \in N : S_{i} \cap T \neq \emptyset\}. Then, for each v \in G^{N} and being c_{T} the Harsanyi dividend of coalition T \in 2^{N},

Sh^{\omega}(v)=\sum_{T \in 2^{N} \setminus \emptyset}c_{T}Sh^{\omega}(u_{T}).

Value

The positively weighted Shapley value of the game, as a vector.

References

Shapley, L. S. (1953). Additive and non-additive set functions. PhD thesis, Department of Mathematics, Princeton University.

See Also

coalitionweightedshapleyvalue, harsanyidividend, shapleyvalue

Examples

v <- c(0,0,0,0,0,0,1,0,0,1,3,4,6,8,10)
weightedshapleyvalue(v,binary=TRUE,weights=c(0.5,0.2,0.2,0.1))
w <- c(0,0,0,0,30,30,40,40,50,50,60,70,80,90,100)
weightedshapleyvalue(w,weights=c(1,2,3,4),partition=list(c(1,2),c(3,4)))

[Package TUGLab version 0.0.1 Index]