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, |
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)))