The Shapley–Shubik power index (SSPI) is a measure of a participant’s a priori voting power in a decision-making body. It was introduced by Lloyd S. Shapley and Martin Shubik in 1954 as an application of the Shapley value from cooperative game theory to simple yes–no voting games. The index assigns to each player the probability of being pivotal—that is, the first to turn a losing coalition into a winning one—over all possible orderings of voters.
Definition Let there be n voters with weights (possibly all 1) and a quota q for passage. For any permutation (ordering) of the voters, the pivotal voter is the first whose addition to the running total weakly meets q. The Shapley–Shubik index for voter i is the fraction of all n! permutations in which i is pivotal; indices sum to 1 across voters.
Mathematical formulation Let N {\displaystyle N} be the voter set and v ( S ) ∈ { 0 , 1 } {\displaystyle v(S)\in \{0,1\}} indicate whether coalition S ⊆ N {\displaystyle S\subseteq N} is winning. Under the assumption that v {\displaystyle v} is a simple, monotone voting function, the Shapley-Shubik value obeys the following equation. For player i {\displaystyle i} ,
φ i = 1 n ! ∑ π ∈ Π [ v ( P i π ∪ { i } ) − v ( P i π ) ] {\displaystyle \varphi _{i}={\frac {1}{n!}}\sum _{\pi \in \Pi }{\big [}\,v(P_{i}^{\pi }\cup \{i\})-v(P_{i}^{\pi })\,{\big ]}}
where Π {\displaystyle \Pi } is the set of all permutations of N {\displaystyle N} and P i π {\displaystyle P_{i}^{\pi }} is the set of players preceding i {\displaystyle i} in permutation π {\displaystyle \pi } . The bracketed term is 1 iff i {\displaystyle i} is pivotal in π {\displaystyle \pi } .
Historical background Shapley and Shubik’s original 1954 paper introduced the index as a way to quantify individual influence in committees, parliaments, and other decision-making bodies. Their work built on cooperative game theory and the Shapley value, adapting it for monotonic binary voting games. Subsequent research expanded the theory, connected it to other power indices, and applied it to legislative design, corporate governance, and international organizations.
Properties The SSPI satisfies standard axioms for power measures in simple, monotone games (also called simple voting games):
Efficiency: ∑ i ∈ N φ i = 1 {\displaystyle \sum _{i\in N}\varphi _{i}=1} . Symmetry: equally situated voters have equal indices. Dummy: a voter who never turns any losing coalition into a winner has index 0. Additivity/transfer: behavior is consistent under sums of games. It is sensitive to both the quota and the distribution of weights. Unlike the Banzhaf power index, which averages over coalitions, SSPI averages over orderings and therefore embeds an ordering-based notion of pivotality.
Worked examples
Four-member weighted body Let weights be A=3, B=2, C=1, D=1 with quota q=4. Enumerating the 24 permutations shows A pivotal in 12, and each of B, C, D pivotal in 4. Thus A has 0.50, while B=C=D=1/6.
Shareholder meeting With shares (40, 30, 30) and q=51, the pivotal-probability rule yields indices (0.50, 0.25, 0.25). The largest shareholder’s influence exceeds proportional ownership.
Supermajority example For five equal voters and q=4 (a 4/5 supermajority), each voter’s index equals 0.2: symmetry implies equal power even under a high quota, though actual passage rates drop.
Computational complexity and algorithms Direct enumeration uses n! permutations. For weighted majority games, dynamic programming over weight sums reduces computation to pseudo-polynomial time O(n·W), where W is the total weight, or O(n·2^n) over coalitions; sampling methods estimate indices with probabilistic error bounds. Computing Shapley-value-based indices in general cooperative games is #P-complete, implying no polynomial-time algorithm exists unless P=#P.
Applications Legislatures and councils. SSPI is used to quantify influence under different seat allocations and quotas, including supermajorities and veto rules. International organizations. Analyses of the United Nations Security Council and the Council of the European Union compare permanent-member veto power and weighted voting under treaty reforms. Corporate governance. Shareholder power under dual-class structures or staggered thresholds can diverge substantially from proportional ownership. Constitutional and institutional design. Designers assess how proposed rule changes (quotas, membership, weights) affect ex ante influence before adoption.
Variants and extensions Researchers have extended the index to settings with abstention, probabilistic approval, communication structures, and hierarchical or multiple-issue games; the SSPI appears as a special case of the Shapley value under simple games and uniform ordering probabilities. Methods based on the multilinear extension and marginal-contribution distributions provide efficient approximations in large electorates.
… excerpt ends here. Continue reading the full article.
