Mathematical Properties of the Banzhaf Power Index

Dubey, Pradeep; Shapley, Lloyd S.

doi:10.1287/moor.4.2.99

Cited by 597 publications

(433 citation statements)

References 55 publications

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“…Instead of looking at random orders of players joining already existing coalitions, as Shapley did, the Banzhaf index takes into account only such coalitions S that S -{i} is losing, while S is winning; the full mathematical treatment of the Banzhaf index can be found in (Dubey and Shapley, 1979). Despite the fact that the Shapley-Shubik and Banzhaf index are very similar, it may happen in particular instances (games) that they the rank voters differently; see also the book by Shubik (1982).…”

Section: Contribution Of Lloyd Shapley To Game Theorymentioning

confidence: 99%

Modification of Shapley Value and its Implementation in Decision Making

Zaremba¹,

Zaremba²,

Suchenek

2017

Foundations of Management

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Abstract:The article presents a solution of a problem that is critical from a practical point of view: how to share a higher than usual discount of $10 million among 5 importers. The discount is a result of forming a coalition by 5 current, formerly competing, importers. The use of Shapley value as a concept for co-operative games yielded a solution that was satisfactory for 4 lesser importers and not satisfactory for the biggest importer. Appropriate modification of Shapley value presented in this article allowed to identify appropriate distribution of the saved purchase amount, which according to each player accurately reflects their actual strength and position on the importer market. A computer program was used in order to make appropriate calculations for 325 permutations of all possible coalitions. In the last chapter of this paper, we recognize the lasting contributions of Lloyd Shapley to the cooperative game theory, commemorating his recent (March 12, 2016) descent from this world.

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Section: Contribution Of Lloyd Shapley To Game Theorymentioning

confidence: 99%

Modification of Shapley Value and its Implementation in Decision Making

Zaremba¹,

Zaremba²,

Suchenek

2017

Foundations of Management

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“…On the other hand, the Banzhaf value (introduced by Banzhaf (1965) to measure voting power in voting games and generalized by Owen (1975) and Dubey and Shapley (1979) to general TU-games) is the solution Ba: G N → IR N that assigns to every player its expected marginal contribution given that every combination of the other players has equal probability of being the coalition that is already present when that player enters. Thus, it assigns to every player in a game its average marginal contribution, i.e.…”

Section: Preliminariesmentioning

confidence: 99%

Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games

Brink

2009

SSRN Journal

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Three well-known solutions for cooperative TU-games are the Shapley value, the Banzhaf value and the equal division solution. In the literature various axiomatizations of these solutions can be found. Axiomatizations of the Shapley value often use efficiency which is not satisfied by the Banzhaf value. On the other hand, the Banzhaf value satisfies collusion neutrality which is not satisfied by the Shapley value. Both properties seem desirable. However, neither the Shapley value nor the Banzhaf value satisfy both. The equal division solution does satisfy both axioms and, moreover, together with symmetry these axioms characterize the equal division solution. Further, we show that there is no solution that satisfies efficiency, collusion neutrality and the null player property. Finally, we show that a solution satisfies efficiency, collusion neutrality and linearity if and only if there exist exogenous weights for the players such that in any game the worth of the 'grand coalition' is distributed proportional to these weights.

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“…19 When the number of voters increases, while holding the bloc sizes constant, we can show that the substantive results for the powers of the blocs do not change. Under our assumptions the global voting body, G, can be closely approximated by an "oceanic game" for which we have analytical results from Dubey and Shapley (1979). An oceanic game is a limiting case of a legislature in which the number of voters is considered to increase without limit, while each voter has a progressively smaller weight, in the limit infinitesimal, such that the bloc sizes remain fixed.…”

Section: Generalisationsmentioning

confidence: 99%

Voting Power and Voting Blocs

Leech

2004

SSRN Journal

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Abstract. We investigate the method of power indices to study voting power of members of a legislature that has voting blocs. Our analysis is theoretical, intended to contribute to a theory of positive political science in which social actors are motivated by the pursuit of power as measured by objective power indices. Our starting points are the papers by Riker (Behavioural Science, 1959, "A test of the adequacy of the power index") and Coleman (American Sociological Review, 1973, "Loss of Power"). We argue against the Shapley-Shubik index and show that anyway the Shapley-Shubik index per head is inappropriate for voting blocs. We apply the Penrose index (the absolute Banzhaf index) to a hypothetical voting body with 100 members. We show how the power indices of individual bloc members can be used to study the implications of the formation of blocs and how voting power varies as bloc size varies. We briefly consider incentives to migrate between blocs. This technique of analysis has many real world applications to legislatures and international bodies. It can be generalised in many ways: our analysis is a priori (assuming formal voting and ignoring actual voting behaviour) but can be made empirical with voting data reflecting behaviour; it examines the consequences of two blocs but can easily be extended to more.

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Mathematical Properties of the Banzhaf Power Index

Cited by 597 publications

References 55 publications

Modification of Shapley Value and its Implementation in Decision Making

Modification of Shapley Value and its Implementation in Decision Making

Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games

Voting Power and Voting Blocs

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