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| Shapley–Shubik power index | |
|---|---|
| Name | Shapley–Shubik power index |
| Introduced | 1954 |
| Creators | Lloyd S. Shapley; Martin Shubik |
| Field | Game theory; Voting theory; Political science |
Shapley–Shubik power index is a measure in cooperative game theory that quantifies the distribution of power among players in a voting game. Developed by Lloyd S. Shapley and Martin Shubik in 1954, it assigns to each player the probability of being pivotal in a random ordering of voters, connecting concepts from John von Neumann's work, Nash equilibrium, and the Banzhaf power index. The index has influenced analyses by institutions such as the United Nations General Assembly, the European Union, and national United States Congress studies, and appears in literature by scholars affiliated with Harvard University, Princeton University, and Stanford University.
The index arose in a period marked by advances from figures like Kenneth Arrow and John Rawls in social choice and normative theory. Shapley and Shubik formalized power as the expected frequency a voter changes an outcome when votes are cast sequentially, relating to the Shapley value concept introduced by Lloyd S. Shapley and to cooperative solutions studied by John Harsanyi and Robert Aumann. Early applications examined voting in bodies such as the United Nations Security Council, the European Parliament, and the Council of the European Union, while later work linked the index to apportionment issues in the United States House of Representatives and judicial voting analyses in the Supreme Court of the United States.
Formally, consider a finite set of players identified in the literature with examples like delegations to the International Monetary Fund or parties in the Israeli Knesset. For any quota-qualified weighted voting game studied by analysts at London School of Economics or Yale University, the Shapley–Shubik power index assigns to player i the fraction of all permutations in which i is pivotal. This definition uses permutations central to work by Permutational group theory researchers and invokes combinatorial identities associated with scholars from University of Cambridge and Massachusetts Institute of Technology. Properties proven by Lloyd S. Shapley include efficiency (indices sum to one), symmetry (equal players receive equal power), null player (zero weight yields zero power), and additivity under union operations examined by Paul Samuelson and Kenneth Arrow-inspired frameworks.
The index satisfies axioms analogous to those used in characterizations by Ariel Rubinstein and John Harsanyi: it is uniquely determined by monotonicity and the pivot-probability interpretation, a result referenced in texts from Princeton University Press and journals such as those of American Economic Association contributors.
Exact computation enumerates all n! permutations, a method used in case studies of bodies like the African Union and Organization of American States by computational political scientists at University of Oxford. For moderate n, dynamic programming and generating-function techniques from researchers at Bell Labs and IBM Research reduce complexity. Monte Carlo sampling approaches, deployed in analyses of European Commission voting and by teams at Google and Microsoft Research, approximate indices for large electorates. Algorithmic optimizations leverage symmetry reductions inspired by group-theoretic work at Institut des Hautes Études Scientifiques and integer partition methods associated with École Normale Supérieure.
Advanced exact algorithms exploit integer linear programming and meet-in-the-middle strategies used in applied studies for the International Olympic Committee and the G20 modeling, while parallel implementations run on clusters at Lawrence Berkeley National Laboratory and Argonne National Laboratory to analyze weighted games with dozens of players.
Canonical examples include weighted voting in the United Nations Security Council and quota systems in the European Union Council of Ministers, where indices illuminate disparities between Germany, France, United Kingdom, and smaller member states. Political scientists have applied the index to party systems in the United Kingdom, coalition formation in Israel, and legislative bargaining in the United States Congress. Economists have used it to study shareholder voting at firms like Goldman Sachs and General Electric, and to assess corporate governance in contexts involving Berkshire Hathaway and Volkswagen AG.
Electoral reform debates in countries such as Australia and Canada have cited the index alongside alternative measures, while constitutional scholars comparing supreme courts like the Supreme Court of the United States and the Constitutional Court of South Africa use it to analyze swing justices. The index has also informed apportionment disputes in the United States and representation formulas in federations like India and Brazil.
Comparative studies contrast the index with the Banzhaf power index, the Johnston index, and the Deegan–Packel index, highlighting differences in pivot interpretations and sensitivity to coalitional structures analyzed by scholars at Columbia University and Brown University. Whereas the Banzhaf index counts swings across coalitions, the Shapley–Shubik index weights permutations, yielding different rankings in examples involving the European Parliament or corporate boards studied by researchers at Wharton School and INSEAD. Empirical papers in journals associated with MIT Press and Oxford University Press compare responsiveness under quota changes, coalition-size considerations by Joseph Stiglitz-influenced authors, and normative implications discussed in works connected to Harvard Kennedy School.
Critics associated with scholars from Princeton and London School of Economics note that the index presumes equally likely orderings, an assumption disputed in contexts like party discipline examined by Anthony Downs-influenced models and empirical voting studied at Stanford University. It abstracts from strategic voting dynamics central to work by Thomas Schelling and Robert Axelrod, and may misrepresent influence in repeated bargaining settings analyzed by Oliver Williamson and Douglass North. Computational tractability remains a concern for bodies with many players, prompting reliance on approximations criticized by statisticians affiliated with University of Chicago and Carnegie Mellon University. Finally, normative critiques from scholars influenced by Amartya Sen argue that pivot-based power does not capture notions of fairness advanced in welfare theory.
Category:Game theory Category:Voting theory