A minimum dimensional class of simple games

Freixas, Josep; Marciniak, Dorota

doi:10.1007/s11750-009-0115-2

Cited by 13 publications

(19 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We surmise that several alternative ways of accomplishing this might be available, but at present none of them seems simple and convincing. Perhaps some notion of dimension for simple voting games, such as those introduced in Freixas and Marciniak (2009) The adjective 'raw' refers to an index prior to normalization by the sum of voting weights. Non-numbered games have quota q = w(N ) − q + 1, where q is the quota of the preceding game.…”

Section: Discussionmentioning

confidence: 99%

The minimum sum representation as an index of voting power

Freixas

Kaniovski

2014

European Journal of Operational Research

Self Cite

View full text Add to dashboard Cite

We propose a new power index based on the minimum sum representation (MSR) of a weighted voting game. The MSR offers a redesign of a voting game, such that voting power as measured by the MSR index becomes proportional to voting weight. The MSR index is a coherent measure of power that is ordinally equivalent to the Banzhaf, Shapley-Shubik and Johnston indices. We provide a characterization for a bicameral meet as a weighted game or a complete game, and show that the MSR index is immune to the bicameral meet paradox. We discuss the computation of the MSR index using a linear integer program and the inverse MSR problem of designing a weighted voting game with a given distribution of power.

show abstract

Section: Discussionmentioning

confidence: 99%

The minimum sum representation as an index of voting power

Freixas

Kaniovski

2014

European Journal of Operational Research

Self Cite

View full text Add to dashboard Cite

show abstract

“…The result for intersection (dimension concept) was first shown in [38] for hypergraphs, and then expressed for simple games in [52]. The result for union (codimension concept) was introduced for simple games in [25]. A simple game is said to be of dimension (codimension) k if and only if it can be represented as the intersection (union) of exactly k weighted games, but not as the intersection (union) of (k − 1) weighted games.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…Thus, as any simple game can be represented as an intersection or union of a finite number of weighted games, we have an alternative way to show the completeness of the family of weighted influence games with respect to the class of simple games (Theorem 1). However, as the dimension, the codimension, and the representation as a boolean weighted game of a simple game might be exponential in the number of players (but bounded by the number of maximal losing, minimal winning coalitions, or both, respectively) [28,25,22], we cannot conclude that any simple game can be represented by a weighted influence game whose number of agents is polynomial in the number of players. For the particular case of unweighted influence game we know the following.…”

Section: Expressivenessmentioning

confidence: 99%

Cooperation through social influence

Molinero

Riquelme

Serna

2015

European Journal of Operational Research

View full text Add to dashboard Cite

We consider a simple and altruistic multiagent system in which the agents are eager to perform a collective task but where their real engagement depends on the willingness to perform the task of other influential agents. We model this scenario by an influence game, a cooperative simple game in which a team (or coalition) of players succeeds if it is able to convince enough agents to participate in the task (to vote in favor of a decision). We take the linear threshold model as the influence model. We show first the expressiveness of influence games showing that they capture the class of simple games. Then we characterize the computational complexity of various problems on influence games, including measures (length and width), values (Shapley-Shubik and Banzhaf) and properties (of teams and players). Finally, we analyze those problems for some particular extremal cases, with respect to the propagation of influence, showing tighter complexity characterizations.

show abstract

“…In fact the relationship might be complex as it is known that, given two realizations, determine whether both represent the same weighted game is an NP-hard problem (Matsui and Matsui, 2000). For some studies related to the minimal realization of a weighted game, see Freixas and Kurz, 2014;Freixas andMarciniak, 2009 andFreixas andMolinero, 2010. Despite the fact that weighted games are a strict subclass of simple games, a well known result says that every simple game can be expressed as the intersection of a finite number of weighted…”

Section: Weighted Representationmentioning

confidence: 99%

“…We refer to such representation as co-vector-weighted representation form (co-VWRF). It is known that any simple game can be expressed as the union of a finite family of weighted games (Freixas and Marciniak, 2009). Therefore, co-VWRF is a valid representation form for simple games.…”

Section: Weighted Representationmentioning

confidence: 99%

Forms of representation for simple games: Sizes, conversions and equivalences

Molinero

Riquelme

Serna

2015

Mathematical Social Sciences

View full text Add to dashboard Cite

h i g h l i g h t s• We survey the main forms of representation for simple games, regular games and weighted games in literature.• We study the complexity of the conversion problem, i.e., the problem of computing a representation of a simple game given another representation.• We prove that several changes of representation that require exponential time can be solved with polynomial-delay. a b s t r a c tSimple games are cooperative games in which the benefit that a coalition may have is always binary, i.e., a coalition may either win or loose. This paper surveys different forms of representation of simple games, and those for some of their subfamilies like regular games and weighted games. We analyze the forms of representations that have been proposed in the literature based on different data structures for sets of sets. We provide bounds on the computational resources needed to transform a game from one form of representation to another one. This includes the study of the problem of enumerating the fundamental families of coalitions of a simple game. In particular we prove that several changes of representation that require exponential time can be solved with polynomial-delay and highlight some open problems.

show abstract

A minimum dimensional class of simple games

Abstract: Simple games, Hypergraphs, Boolean algebra, Dimension, Codimension, 05C65, 91A12, 94C10,

Cited by 13 publications

References 10 publications

The minimum sum representation as an index of voting power

The minimum sum representation as an index of voting power

Cooperation through social influence

Forms of representation for simple games: Sizes, conversions and equivalences

Contact Info

Product

Resources

About