Compound voting and the Banzhaf index

Dubey, Pradeep; Einy, Ezra; Haimanko, Ori

doi:10.1016/j.geb.2004.03.002

Cited by 28 publications

(26 citation statements)

References 12 publications

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“…The first axiom is equivalent to the well-known transfer axiom of Dubey (1975), as remarked in Dubey et al (2005). Here we use a different name which reflects its content in a more direct manner.…”

Section: The Axiomsmentioning

confidence: 99%

An axiomatic characterization of the Owen–Shapley spatial power index

Peters

Zarzuelo

2016

Int J Game Theory

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We present an axiomatic characterization of the Owen-Shapley spatial power index for the case where issues are elements of two-dimensional space. This characterization employs a version of the transfer condition, which enables us to unravel a spatial game into spatial games connected to unanimity games. The other axioms include two conditions concerned particularly with the spatial positions of the players, besides spatial versions of anonymity and dummy. The last condition says that dummy players can be left out in a specific way without changing the power of the other players. We show that this condition can be weakened to requiring dummies to have zero power if we add a condition of positional continuity. We also show that the axioms in our characterization(s) are logically independent.

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Section: The Axiomsmentioning

confidence: 99%

An axiomatic characterization of the Owen–Shapley spatial power index

Peters

Zarzuelo

2016

Int J Game Theory

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“…The transfer property says that if going from C ′ to C involves exactly the same increase in control as going from D ′ to D, then the power of each player should change by the same amount when going from C ′ to C as when going from D ′ to D. The transfer property is related to a property with the same name, used to characterize the Shapley value (Shapley, 1953) for (monotonic) simple games (Dubey, 1975). The form in which we present it is closely related to a version of the axiom discussed in Dubey et al (2005) and Einy and Haimanko (2011). The following lemma shows that TP is equivalent to a condition closely related to the original format of the transfer axiom as introduced in Dubey (1975).…”

Section: Constant-sum (Cs)mentioning

confidence: 99%

“…Thus, Bz is the normalized Banzhaf value (Banzhaf, 1965;Dubey et al, 2005). Define the power index ϕ by…”

Section: Constant-summentioning

confidence: 99%

“…For every player, the change in power when extending a mutual control structure C ′ to C should be equal to the change in power when extending a mutual control structure D ′ to D, whenever exactly the same control relations are added going from C ′ to C as when going from D ′ to D. This condition is called transfer property because it is related to the transfer property used by Dubey (1975) to characterize the Shapley value or Shapley-Shubik index (Shapley, 1953;Shapley and Shubik, 1954) for monotonic simple games. See also Dubey et al (2005).…”

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confidence: 99%

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Indirect control and power in mutual control structures

Karos

Peters

2015

Games and Economic Behavior

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In a mutual control structure agents exercise control over each other. Typical examples occur in the area of corporate governance: firms and investment companies exercise mutual control, in particular by owning each others' stocks. In this paper we formulate a general model for such situations. There is a fixed set of agents, and a mutual control structure assigns to each subset (coalition) the subset of agents controlled by that coalition. Such a mutual control structure captures direct control. We propose a procedure in order to incorporate indirect control as well: if S controls T , and S and T jointly control R, then S controls R indirectly. This way, invariant mutual control structures result. Alternatively, mutual control can be described by vectors of simple games, called simple game structures, each simple game describing who controls a certain player, and also those simple games can be updated in order to capture indirect control. We show that both approaches lead to equivalent invariant structures.In the second part of the paper, we axiomatically develop a class of power indices for invariant mutual control structures. We impose four axioms with a plausible interpretation in this framework, which together characterize a broad class of power indices based on dividends resulting both from exercising and from undergoing control. By adding an extra condition a unique power index is singled out. In this index, each player accumulates his Shapley-Shubik power index assignments from controlling other players, diminished by the sum of the Shapley-Shubik power index assignments to other players controlling him.

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“…, g k in Equation 1 depend on disjoint subsets of variables). Under this assumption, Owen [1978] proved that the Penrose-Banzhaf index of a compound game can be recursively computed based on the following principle: the PB-index of a voter ℓ in a two-tiered compound game is equal to (the PB-index of ℓ in the first-tier game g i that he participates in) × (the PB-index of g i in the second-tier game) (see also Dubey, Einy and Haimanko [2005]). …”

Section: Pyramidal Structuresmentioning

confidence: 99%