Dimension of complete simple games with minimum

Freixas, Josep; Puente, María Albina

doi:10.1016/j.ejor.2007.03.050

Cited by 40 publications

(45 citation statements)

References 17 publications

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“…. ∨ v k coincides with a simple game v is the co-dimension of v. Freixas and Puente have shown that there exists a complete simple game with dimension k for every integer k [7]. It is not known yet whether the dimension of a complete simple game is polynomially bounded in the number of its players or can grow exponentially (like for general simple games).…”

Section: Notation and Definitionsmentioning

confidence: 99%

“…Marking this occurrence as an edge in a graph G with vertex set L 23,24 , we can perform a clique search on G. It turns out that G contains 24 452 800 cliques of size 6 but no larger clique. One of the 6-cliques corresponds to L ′ = {1, 4,5,7,8,9,10,11,12,14,15,16,17,18,19,20 This 6-clique is actually the most robust one regarding changes of the relative population distribution in the EU: it is not upset by moves between states, births, or deaths as long as the new relative population vector pop ′ and the old one, pop, based on Table 1, have a · 1 -distance less than 0.0095. This distance could accommodate arbitrary moves of up to 2.5 million EU citizens.…”

Section: Bounds For V Eu28 'S Dimensionmentioning

confidence: 99%

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Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record

Kurz

Napel

2015

Optim Lett

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ABSTRACT. The new voting system of the Council of the European Union cannot be represented as the intersection of six or fewer weighted games, i.e., its dimension is at least 7. This sets a new record for real-world voting bodies. A heuristic combination of different discrete optimization methods yields a representation as the intersection of 13 368 weighted games. Determination of the exact dimension is posed as a challenge to the community. The system's Boolean dimension is proven to be 3.

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Section: Notation and Definitionsmentioning

confidence: 99%

Section: Bounds For V Eu28 'S Dimensionmentioning

confidence: 99%

Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record

Kurz

Napel

2015

Optim Lett

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“…It is interesting that the class of complete games with a unique shiftminimal winning coalition was studied before Freixas and Puente (2008), without noticing that this class is actually the class of conjunctive hierarchical games.…”

Section: Characterizations Of Conjunctive Hierarchical Gamesmentioning

confidence: 99%

“…This allows us to characterize weighted conjunctive hierarchical games and obtain their structural characterization as complete games with a unique shift-minimal winning coalition. We note that the class of complete games with a unique shift-minimal winning coalition was studied in its own right in Freixas and Puente (2008) without any reference to hierarchical games.…”

Section: Introductionmentioning

confidence: 99%

Weightedness and structural characterization of hierarchical simple games

Gvozdeva

Hameed

Slinko

2013

Mathematical Social Sciences

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In this paper we give structural charaterizations for disjunctive and conjunctive hierarchical simple games by characterizing them as complete games with a unique shift-maximal losing and, respectively, shift-minimal winning coalitions. We prove canonical representation theorems for both types of hierarchical games and establish duality between them. We characterize those disjunctive and conjunctive hierarchical games which are weighted majority games. This paper was inspired by Beimel et al. (2008) and Farràs and Padró (2010) characterizations of ideal weighted threshold access structures of secret sharing schemes.

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“…Both disjunctive and conjunctive hierarchical games are seldom weighted (Gvozdeva et al, 2013) or even roughly weighted (Hameed and Slinko, 2015). Freixas and Puente (2008) studied conjunctive hierarchical games (under the name of games with a minimum) and found that their dimension grows linearly in the number of players and asked whether or not in the class of complete games the dimension can grow polynomially or even exponentially. In the present paper we show that the growth of dimension of disjunctive hierarchical games, which are complete, is exponential in the number of players.…”

Section: Introductionmentioning

confidence: 99%

Growth of dimension in complete simple games

O'Dwyer

Slinko

2017

Mathematical Social Sciences

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The concept of dimension in simple games was introduced by Taylor and Zwicker (1993) as a measure of the remoteness of a given game from a weighted game. They demonstrated that the dimension of a simple game can grow exponentially in the number of players. However, the problem of worst-case growth of the dimension in complete games was left open. Freixas and Puente (2008) showed that complete games of arbitrary dimension exist and, in particular, their examples demonstrate that the worst-case growth of dimension in complete games is at least linear. In this paper, using a novel technique of Kurz and Napel (2015), we demonstrate that the growth of dimension in complete games can also be exponential in the number of players.

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Dimension of complete simple games with minimum

Cited by 40 publications

References 17 publications

Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record

Dimension of the Lisbon voting rules in the EU Council: a challenge and new world record

Weightedness and structural characterization of hierarchical simple games

Growth of dimension in complete simple games

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