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

Molinero, Xavier; Riquelme, Fabián; Serna, María

doi:10.1016/j.mathsocsci.2015.04.008

Cited by 12 publications

(7 citation statements)

References 79 publications

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“…As a result, a width = 24 and a length = 14 was obtained. For the width, the following three optimal combinations were found (players are ordered according to Appendix A): Solution 1: {2, 5, 6, 7, 8, 9, 10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26, 27} Solution 2: {3, 5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27} Solution 3: {4,5,6,7,8,9,10,11,12,13,14,…”

Section: Methodsmentioning

confidence: 99%

“…Simple games are one of the most fundamental models for decision-making [3], and have been used to solve problems arising voting systems, social choice theory, logic and threshold logic, circuit complexity, artificial intelligence, geometry, linear programming, Sperner theory, order theory, among other disciplines [3][4][5]. Besides the traditional forms of representation of simple games [6], since the 2010s, different formulations based on graphs have emerged, with the aim of applying the knowledge acquired in cooperative game theory in network science [7]. In this context, influence games arise as simple games defined by influence graphs (i.e., weighted, labeled graphs) on which an influence spread phenomenon is exerted.…”

Section: Introductionmentioning

confidence: 99%

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Extremal Coalitions for Influence Games Through Swarm Intelligence-Based Methods

Riquelme¹,

Olivares²,

Mu駉z³

et al. 2022

Computers, Materials &Amp; Continua

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An influence game is a simple game represented over an influence graph (i.e., a labeled, weighted graph) on which the influence spread phenomenon is exerted. Influence games allow applying different properties and parameters coming from cooperative game theory to the contexts of social network analysis, decision-systems, voting systems, and collective behavior. The exact calculation of several of these properties and parameters is computationally hard, even for a small number of players. Two examples of these parameters are the length and the width of a game. The length of a game is the size of its smaller winning coalition, while the width of a game is the size of its larger losing coalition. Both parameters are relevant to know the levels of difficulty in reaching agreements in collective decision-making systems. Despite the above, new bio-inspired metaheuristic algorithms have recently been developed to solve the NP-hard influence maximization problem in an efficient and approximate way, being able to find small winning coalitions that maximize the influence spread within an influence graph. In this article, we apply some variations of this solution to find extreme winning and losing coalitions, and thus efficient approximate solutions for the length and the width of influence games. As a case study, we consider two real social networks, one formed by the 58 members of the European Union Council under nice voting rules, and the other formed by the 705 members of the European Parliament, connected by political affinity. Results are promising and show that it is feasible to generate approximate solutions for the length and width parameters of influence games, in reduced solving time.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extremal Coalitions for Influence Games Through Swarm Intelligence-Based Methods

Riquelme¹,

Olivares²,

Mu駉z³

et al. 2022

Computers, Materials &Amp; Continua

Self Cite

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“…Any of those set families determine uniquely the game and constitute one of the usual forms of representation for simple games [33]. Observe that the size of such representation is not, in general, polynomial in the number of players [27]. Example 2.…”

Section: Simple and Influence Gamesmentioning

confidence: 99%

Measuring satisfaction and power in influence based decision systems

Molinero

Riquelme

Serna

2019

Knowledge-Based Systems

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“…The subsets of N are called coalitions, the set N is the grand coalition and each X ∈ W is a winning coalition. The complement of the family of winning coalitions is the family of losing coalitions L, i.e., L = P(N ) \ W. Any of those set families determine uniquely the game Γ and constitute one of the usual forms of representation for simple games [20], although the size of the representation is not, in general, polynomial in the number of players (see also [16]).…”

Section: Simple and Influence Gamesmentioning

confidence: 99%

Measuring satisfaction in societies with opinion leaders and mediators

Molinero¹,

Riquelme²,

Serna³

2016

Preprint

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An opinion leader-follower model (OLF) is a two-action collective decision-making model for societies, in which three kinds of actors are considered: opinion leaders, followers, and independent actors. In OLF the initial decision of the followers can be modified by the influence of the leaders. Once the final decision is set, a collective decision is taken applying the simple majority rule [22]. We consider a generalization of OLF, the gOLF models which allow collective decision taken by rules different from the single majority rule. Inspired in this model we define two new families of collective decision-making models associated with influence games [15]. We define the oblivious and non-oblivious influence models. We show that gOLF models are nonoblivious influence models played on a two layered bipartite influence graph.Together with OLF models, the satisfaction measure was introduced and studied. We analyze the computational complexity of the satisfaction measure for gOLF models and the other collective decision-making models introduced in the paper. We show that computing the satisfaction measure is #P-hard in all the considered models. On the other hand, we provide two subfamilies of decision models in which the satisfaction measure can be computed in polynomial time.Exploiting the relationship with influence games, we can relate the satisfaction measure with the Rae index of an associated simple game. The Rae index is closely related to the Banzhaf. Thus, our results also extend the families of simple games for which computing the Rae index and the Banzhaf value is computationally hard.

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

Cited by 12 publications

References 79 publications

Extremal Coalitions for Influence Games Through Swarm Intelligence-Based Methods

Extremal Coalitions for Influence Games Through Swarm Intelligence-Based Methods

Measuring satisfaction and power in influence based decision systems

Measuring satisfaction in societies with opinion leaders and mediators

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