An Asymmetric Shapley–Shubik Power Index

Abstract

Shapley–Shubik power index, Banzhaf index, Simple game, Voting, C710, D710, D720, 91A12, 91A40, 91B12,

Generalized Roll-Call Model for the Shapley-Shubik Index

“…The proof in [3] uses the axiomatization of the Shapley-Shubik index as a indirect approach and reports of combinatorial difficulties for the direct approach. Here we give an easy combinatorial proof and weaken the assumptions, i.e., we assume that the probability for s "yes"-and n − s "no"-votes only depends on the number s. Note that the same result was previously obtained in [10,Proposition 4], as we found out recently.…”

Axiomatizations for the Shapley–Shubik power index for games with several levels of approval in the input and output

“…In [16] it is mentioned that the model also yields the same result if we assume that all players independently vote yes with a xed probability p ∈ [0, 1]. This was further generalized to probability measures p on {0, 1} n where vote vectors with the same number of yes votes have the same probability, see [11].…”

Importance in Systems With Interval Decisions

“…possible orders π : N → N in which the agents are called are assumed to be equiprobable and the votes of each agent are independent with expectation 0 ≤ p ≤ 1 for voting 1, i.e., the probability for voting 1 is exactly p. For a given simple game v the pivotal agent i is determined by the unique index i such that {j ∈ N : π(j) < π(i)} is losing and {j ∈ N : π(j) ≤ π(i)} is winning in v. Interestingly enough, the Shapley-Shubik index of agent i in v equals the probability that agent i is pivotal in the above roll-call model. Note that this statement is independent of p. The assumptions on the model can be even further weakened to correlated agents still maintaining the coincidence between the Shapley-Shubik index and pivot probabilities, see [23].…”