Comparing Banzhaf and Shapley-Shubik power indices

March 22, 2012
By

(This article was first published on R snippets, and kindly contributed to R-bloggers)

Last week I analyzed Shapley-Shubik power index in R. I got several requests to write a code calculating Banzhaf power index. Here is the proposed code.
Again I use data from Warsaw School of Economics rector elections (the details are in my last post). I give the code for calculation of Shapley-Shubik and Banzhaf power indices below.

# Constituency list
const <- c(30, 22, 27, 27, 41, 2 + 11, 38 + 5, 1 + 9)

# Shapley-Shubik power index
library(gtools)
perms <- permutations(8, 8)
outcome <- apply(perms, 1, function(x) {
    x[sum(cumsum(const[x]) < 107) + 1] })
sspi <- prop.table(table(outcome))

# Banzhaf power index
subsets <- length(const) - 1
subs <- matrix(FALSE, subsets, subsets)
for (i in 1:subsets) {
    subs[,i] <- rep(c(rep(FALSE, 2 ^ (i - 1)),
                      rep(TRUE, 2 ^ (i - 1))),
                    2 ^ (subsets - i))
}
banzhaf <- function(i) {
    other <- const[-i]
    part.sum <- apply(subs, 1, function(x) { sum(other[x]) } )
    sum((part.sum < 106.5) & ((part.sum + const[i]) > 106.5))
}
bpi <- prop.table(sapply(1:8, banzhaf))

# power index comparison
names(bpi) <- c("C_1","C_2", "C_3", "C_4", "C_5",
                "C_67", "C_89", "C_1011")
barplot(rbind(bpi, sspi, const / sum(const))[,order(const)],
        col=c(2,4,1), beside = TRUE,
        legend.text = c("Banzhaf", "Shapley-Shubik", "Votes"),
        args.legend list("top"))

Calculating Banzhaf power index is more complex to implement in R in comparison to Shapley-Shubik power index but the code is faster.

At the end of the code I plot comparison of both power indices. It is interesting to note that the results are very similar. Banzhaf power index slightly favors smaller constituencies but the difference is negligible. Again we can see that although constituency #2 has over 50% more votes than combined constituencies #6 and #7 (22 vs. 13) it has exactly the same power according to both indices.

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