arXiv subjects

Josep Freixas

Publications and source records attributed to Josep Freixas.

10 recordsLinked to original sources

arXiv2023

Nash's bargaining problem and the scale-invariant Hirsch citation index

A number of citation indices have been proposed for measuring and ranking the research publication records of scholars. Some of the best known indices, such as those proposed by Hirsch and Woeginger, are designed to reward most highly those records that strike some balance between productivity (number of papers published), and impact (frequency with which those papers are cited). A large number of rarely cited publications will not score well, nor will a very small number of heavily cited papers. We discuss three new citation indices, one of which was independently proposed in \cite{FHLB}. Each rests on the notion of scale invariance, fundamental to John Nash's solution of the two-person bargaining problem. Our main focus is on one of these -- a scale invariant version of the Hirsch index. We argue that it has advantages over the original; it produces fairer rankings within subdisciplines, is more decisive (discriminates more finely, yielding fewer ties) and more dynamic (growing over time via more frequent, smaller increments), and exhibits enhanced centrality and tail balancedness. Simulations suggest that scale invariance improves robustness under Poisson noise, with increased decisiveness having no cost in terms of the number of ``accidental" reversals, wherein random irregularities cause researcher A to receive a lower index value than B, although A's productivity and impact are both slightly higher than B's. Moreover, we provide an axiomatic characterization of the scale invariant Hirsch index, via axioms that bear a close relationship, in discrete analogue, to those used by Nash in \cite{Nas50}. This argues for the mathematical naturality of the new index. An earlier version was presented at the 5th World Congress of the Game Theory Society, Maastricht, Netherlands in 2016.

math.CO↗

arXiv2018

Bounds for the Nakamura number

The Nakamura number is an appropriate invariant of a simple game to study the existence of social equilibria and the possibility of cycles. For symmetric quota games its number can be obtained by an easy formula. For some subclasses of simple games the corresponding Nakamura number has also been characterized. However, in general, not much is known about lower and upper bounds depending of invariants on simple, complete or weighted games. Here, we survey such results and highlight connections with other game theoretic concepts.

math.CO↗

arXiv2017

Characterization of threshold functions: state of the art, some new contributions and open problems

This paper has a twofold scope. The first one is to clarify and put in evidence the isomorphic character of two theories developed in quite different fields: on one side, threshold logic, on the other side, simple games. One of the main purposes in both theories is to determine when a simple game is representable as a weighted game, which allows a very compact and easily comprehensible representation. Deep results were found in threshold logic in the sixties and seventies for this problem. However, game theory has taken the lead and some new results have been obtained for the problem in the last two decades. The second and main goal of this paper is to provide some new results on this problem and propose several open questions and conjectures for future research. The results we obtain depend on two significant parameters of the game: the number of types of equivalent players and the number of types of shift-minimal winning coalitions.

cs.GT↗

arXiv2014

The cost of getting local monotonicity

Manfred Holler introduced the Public Good index as a proposal to divide a public good among players. In its unnormalized version, i.e., the raw measure, it counts the number of times that a player belongs to a minimal winning coalition. Unlike the Banzhaf index, it does not count the remaining winning coalitions in which the player is crucial. Holler noticed that his index does not satisfy local monotonicity, a fact that can be seen either as a major drawback or as an advantage.

cs.GT↗

arXiv2014

The golden number and Fibonacci sequences in the design of voting structures

Some distinguished types of voters, as vetoers, passers or nulls, as well as some others, play a significant role in voting systems because they are either the most powerful or the least powerful voters in the game independently of the measure used to evaluate power. In this paper we are concerned with the design of voting systems with at least one type of these extreme voters and with few types of equivalent voters, with this purpose in mind we enumerate these special classes of games and find out that its number always follows a Fibonacci sequence with smooth polynomial variations. As a consequence we find several families of games with the same asymptotic exponential behavior excepting of a multiplicative factor which is the golden number or its square. From a more general point of view, our studies are related with the design of voting structures with a predetermined importance ranking.

math.CO↗

arXiv2013

On minimum integer representations of weighted games

We study minimum integer representations of weighted games, i.e., representations where the weights are integers and every other integer representation is at least as large in each component. Those minimum integer representations, if the exist at all, are linked with some solution concepts in game theory. Closing existing gaps in the literature, we prove that each weighted game with two types of voters admits a (unique) minimum integer representation, and give new examples for more than two types of voters without a minimum integer representation. We characterize the possible weights in minimum integer representations and give examples for $t\ge 4$ types of voters without a minimum integer representation preserving types, i.e., where we additionally require that the weights are equal within equivalence classes of voters.

math.CO↗

arXiv2013

Enumeration of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum

This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players of type 1.

math.CO↗

arXiv2012

On $α$-roughly weighted games

Gvozdeva, Hemaspaandra, and Slinko (2011) have introduced three hierarchies for simple games in order to measure the distance of a given simple game to the class of (roughly) weighted voting games. Their third class $\mathcal{C}_α$ consists of all simple games permitting a weighted representation such that each winning coalition has a weight of at least 1 and each losing coalition a weight of at most $α$. For a given game the minimal possible value of $α$ is called its critical threshold value. We continue the work on the critical threshold value, initiated by Gvozdeva et al., and contribute some new results on the possible values for a given number of voters as well as some general bounds for restricted subclasses of games. A strong relation beween this concept and the cost of stability, i.e. the minimum amount of external payment to ensure stability in a coalitional game, is uncovered.

math.CO↗

arXiv2009

A Fibonacci sequence for linear structures with two types of components

We investigate binary voting systems with two types of voters and a hierarchy among the members in each type, so that members in one class have more influence or importance than members in the other class. The purpose of this paper is to count, up to isomorphism, the number of these voting systems for an arbitrary number of voters. We obtain a closed formula for the number of these systems, this formula follows a Fibonacci sequence with a smooth polynomial variation on the number of voters.

math.CO↗

arXiv2008

The Complexity of Testing Properties of Simple Games

Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes-no voting system is a set of rules that specifies exactly which collections of ``yea'' votes yield passage of the issue at hand. A collection of ``yea'' voters forms a winning coalition. We are interested on performing a complexity analysis of problems on such games depending on the game representation. We consider four natural explicit representations, winning, loosing, minimal winning, and maximal loosing. We first analyze the computational complexity of obtaining a particular representation of a simple game from a different one. We show that some cases this transformation can be done in polynomial time while the others require exponential time. The second question is classifying the complexity for testing whether a game is simple or weighted. We show that for the four types of representation both problem can be solved in polynomial time. Finally, we provide results on the complexity of testing whether a simple game or a weighted game is of a special type. In this way, we analyze strongness, properness, decisiveness and homogeneity, which are desirable properties to be fulfilled for a simple game.

cs.GT↗