TUGLab (version 0.0.1)

weightedmajoritygame: Weighted majority game

Description

This function returns the characteristic function of the described weighted majority game.

Usage

weightedmajoritygame(q, w, binary = FALSE)

Value

The characteristic function of the weighted majority game associated with the described situation, as a vector in binary order if binary=TRUE and in lexicographic order otherwise.

Arguments

q

A quota, as a number between 0 and the sum of player weights.

w

The player weights, as a vector of non-negative numbers.

binary

A logical value. By default, binary=FALSE.

Details

Given a situation in which a number of agents have to vote for or against a certain measure, let \(N =\{1,\ldots,n\}\) be the set of voters, \(w\) be a non-negative vector of voter weights (the weight of each voter is the number of votes or the proportion of total votes they hold), and \(q \in [0,\sum_{i \in N}w_{i}]\) be the quota (the minimum number of votes or the minimum proportion of total votes needed to pass the measure). The corresponding weighted majority game, \(v\), is defined by $$v(S)=1 \text{ if } \sum_{i \in S}w_{i} \geqslant q \text{ and } v(S)=0 \text{ otherwise, for each }S\in 2^N.$$

Examples

q <- 39
w <- c(rep(7,5),rep(1,10))
weightedmajoritygame(q,w)