Characterizations of the Deegan–Packel and Johnston power indices

In this paper, we provide a new characterization of the Deegan–Packel power index, which uses a new monotonicity property, and two characterizations of the Johnston power index by employing some usual properties of the literature and two new transfer properties. Finally, a real-world numerical example taken from the political field illustrates the main differences between these two indices.     

Dimension of complete simple games with minimum

Weighted majority games have the property that players are totally ordered by the desirability relation (introduced by Isbell in [J.R. Isbell, A class of majority games, Quarterly Journal of Mathematics, 7 (1956) 183–187]) because weights induce it. Games for which this relation is total are called complete simple games. Taylor and Zwicker proved in [A.D. Taylor, W.S. Zwicker, Weighted voting, multicameral representation, and power, Games and Economic Behavior 5 (1993) 170–181] that every simple game (or monotonic finite hypergraph) can be represented by an intersection of weighted majority games and consider the dimension of a game as the needed minimum number of them to get it. They provide the existence of non-complete simple games of every dimension and left open the problem for complete simple games.     

A generating functions approach for computing the Public Good index efficiently

In the past years, a combinatorial method based on generating functions was introduced to compute Shapley–Shubik, Banzhaf and other indices for weighted majority games exactly and efficiently. In this paper, taking inspiration from what has already been done, in view of the efficiency of the generating functions method, we define a generating function for computing the Public Good index, maintaining the property of exactness of the resulting algorithm. The main difference with the existing algorithms derives from the fact that the Public Good index takes into account only minimal winning coalitions and counts how many swings of a player involve them. Moreover, we study the computational complexity of the algorithm and we evaluate the Public Good index for the vote share of the Russian Duma in 1995.     

A class of power indices for voting games

A class of power indices is discussed which includes not only the usual power indices but also the class of semivalues and the indices introduced byDeegan/Packel [1978]. Sufficient conditions for members of this class to be symmetric, support-independent, and self-dual are given. We then discuss various conditions under which members of this class coincide with the ShapleyShubik and Banzhaf indices.     

Partitioning sequencing situations and games

The interaction between sequencing situations and cooperative games starting from the paper of Curiel et al. [Curiel, I., Pederzoli, G., Tijs S., 1989. Sequencing games. European Journal of Operational Research 40, 344–351], has become an established line of research within the theory of operation research games.     

On the ordinal equivalence of the Johnston, Banzhaf and Shapley power indices

In this paper, we characterize the games in which Johnston, Shapley-Shubik and Penrose-Banzhaf-Coleman indices are ordinally equivalent, meaning that they rank players in the same way. We prove that these three indices are ordinally equivalent in semicomplete simple games, which is a newly defined class that contains complete games and includes most of the real-world examples of binary voting systems. This result constitutes a twofold extension of Diffo Lambo and Moulen’s result (Diffo Lambo and Moulen, 2002) in the sense that ordinal equivalence emerges for three power indices (not just for the Shapley-Shubik and Penrose-Banzhaf-Coleman indices), and it holds for a class of games strictly larger than the class of complete games.     

A fuzzy approach to cooperative n-person games

The object of this paper is to provide a systematic treatment of bargaining procedures as a basis for negotiation. An innovative fuzzy logic approach to analyze n-person cooperative games is developed. A couple of indices, the Good Deal Index and the Counterpart Convenience Index are proposed to characterize the heuristic of bargaining and to provide a solution concept. The indices are examined theoretically and experimentally by analyzing three case studies. The results verify the validity of the approach.     

The Shapley–Shubik index for multi-criteria simple games

In this paper we address multi-criteria simple games which constitute an extension of the basic framework of voting systems and related social-choice situations. For these games, we propose the extended Shapley–Shubik index as the natural generalization of the Shapley–Shubik index in conventional simple games, and establish an axiomatic characterization of this power index.     

Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces

Viscosity approximation methods for a family of finite nonexpansive mappings are established in Banach spaces. The main theorems extend the main results of Moudafi [Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55] and Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] to the case of finite mappings. Our results also improve and unify the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159], Browder [Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Archiv. Ration. Mech. Anal. 24 (1967) 82–90], Cho et al. [Some control conditions on iterative methods, Commun. Appl. Nonlinear Anal. 12 (2) (2005) 27–34], Ha and Jung [Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990) 330–339], Halpern [Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961], Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], Jung et al. [Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl. 2005 (2) (2005) 125–135], Jung and Kim [Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc. 34 (1) (1997) 93–102], Lions [Approximation de points fixes de contractions, C.R. Acad. Sci. Ser. A-B, Paris 284 (1977) 1357–1359], O’Hara et al. [Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], Reich [Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (12) (1997) 3641–3645], Takahashi and Ueda [On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546–553], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992) 486–491], Xu [Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2) (2002) 240–256], and Zhou et al. [Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press] among others.     

Generalized coalitional semivalues

In this paper we change some axioms in the axiom system which defines coalitional semivalues [Albizuri, M.J., Zarzuelo J.M., 2004. On coalitional semivalues. Games and Economic Behavior 49, 221–243] and we define generalized coalitional semivalues. Generalized coalitional semivalues, like coalitional semivalues, are “compositions” of semivalues, but they form a broader set of “compositions”. Like coalitional semivalues, generalized coalitional semivalues are extensions to the coalitional context of semivalues for transferable utility games [Dubey, P., Neyman, A., Weber, R.J., 1981. Value theory without efficiency. Mathematics of Operations Research 6, 122–128].     

Balancedness of multiple machine sequencing games revisited

We study m-sequencing games, which were introduced by [Hamers, H., Klijn, F., Suijs, J., (1999). On the balancedness of multiple machine sequencing games. European Journal of Operational Research 119, 678–691]. We answer the open question whether all these games are balanced in the negative. We do so, by an example of a 3-sequencing situation with 5 jobs, whose associated 3-sequencing game has an empty core. The counterexample finds its basis in an inconsistency in [Hamers et al., ibid], which was probably overlooked by the authors. This observation demands for a detailed reconsideration of their proofs.¹     

The minimum sum representation as an index of voting power

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.     

A stochastic and asymmetric-information framework for a dominant-manufacturer supply chain

Consider a dominant manufacturer wholesaling a product to a retailer, who in turn retails it to the consumers at $p/unit. The retail-market demand volume varies with p according to a given demand curve. This basic system is commonly modeled as a manufacturer-Stackelberg ([mS]) game under a “deterministic and symmetric-information” (“det-sym-i”) framework. We first explain the logical flaws of this framework, which are (i) the dominant manufacturer-leader will have a lower profit than the retailer under an iso-elastic demand curve; (ii) in some situations the system’s “correct solution” can be hyper-sensitive to minute changes in the demand curve; (iii) applying volume discounting while keeping the original [mS] profit-maximizing objective leads to an implausible degenerate solution in which the manufacturer has dictatorial power over the channel. We then present an extension of the “stochastic and asymmetric-information” (“sto-asy-i”) framework proposed in Lau and Lau [Lau, A., Lau, H.-S., 2005. Some two-echelon supply-chain games: Improving from deterministic–symmetric-information to stochastic-asymmetric-information models. European Journal of Operational Research 161 (1), 203–223], coupled with the notion that a profit-maximizing dominant manufacturer may implement not only [mS] but also “[pm]”—i.e., using a manufacturer-imposed maximum retail price. We show that this new framework resolves all the logical flaws stated above. Along the way, we also present a procedure for the dominant manufacturer to design a profit-maximizing volume-discount scheme using stochastic and asymmetric demand information.     

A remark on the law of the logarithm for weighted sums of random variables with multidimensional indices

In this work, a sharp upper bound on the law of the logarithm for the weighted sums of random variables with multidimensional indices is obtained. The main result improves the result in [Li, Rao and Wang, 1995. On strong law of large numbers and the law of the logarithm for weighted sums of independent random variables with multidimensional indices. J. Multivariate Anal. 52, 181–198], partly.     

Inferior players in simple games

Power indices like those of Shapley and Shubik (1954) or Banzhaf (1965) measure the distribution of power in simple games. This paper points at a deficiency shared by all established indices: players who are inferior in the sense of having to accept (almost) no share of the spoils in return for being part of a winning coalition are assigned substantial amounts of power. A strengthened version of the dummy axiom based on a formalized notion of inferior players is a possible remedy. The axiom is illustrated first in a deterministic and then a probabilistic setting. With three axioms from the Banzhaf index, it uniquely characterizes the Strict Power Index (SPI). The SPI is shown to be a special instance of a more general family of power indices based on the inferior player axiom. Received: December 1999/Final version: June 2001     

On general best proximity pairs and equilibrium pairs in free abstract economies

In this paper, using Lassonde’s fixed point theorem for Kakutani factorizable multifunctions and Park’s fixed point theorem for acyclic factorizable multifunctions, we will prove new existence theorems for general best proximity pairs and equilibrium pairs for free abstract economies, which generalize the previous best proximity theorems and equilibrium existence theorems due to Srinivasan and Veeramani [P.S. Srinivasan, P. Veeramani, On best approximation pair theorems and fixed point theorems, Abstr. Appl. Anal. 2003 (1) (2003) 33–47; P.S. Srinivasan, P. Veeramani, On existence of equilibrium pair for constrained generalized games, Fixed Point Theory Appl. 2004 (1) (2004) 21–29], and Kim and Lee [W.K. Kim, K.H. Lee, Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl. 316 (2006) 433–446] in several aspects.     

A new index of power for simplen-person games

Measures of (a priori) power play a useful role in assessing the character of interpersonal interaction found in collective decision making bodies. We propose and axiomatically characterize an alternative power index to the familiarShapley/Shubik andBanzhaf indices which can be used for such purposes. The index presented is shown to be unique for the class of simplen-person games. By subsequent generalization of the index and its axioms to the class ofn-person games in characteristic function form we obtain an analog to theShapley value.     

A general characterization for non-balanced games in terms of U-cycles

In a paper by Cesco [Cesco, J.C., 2003. Fundamental cycles of pre-imputations in non-balanced TU-games. International Journal of Game Theory 32, 211–222], it was proven that the existence of a certain type of cycles of pre-imputations, fundamental cycles, is equivalent to the non-balancedness of a TU-game, i.e., the emptiness of the core of the game. There are two characteristic sub-classes related to fundamental cycles: U-cycles and maximal U-cycles. In this note we show that it is enough to consider U-cycles in obtaining a similar characterization for non-balanced TU-games.     

Point-based sufficient conditions for metric regularity of implicit multifunctions

We obtain some point-based sufficient conditions for the metric regularity in Robinson’s sense of implicit multifunctions in a finite-dimensional setting. The new implicit function theorem (which is very different from the preceding results of Ledyaev and Zhu [Yu.S. Ledyaev, Q.J. Zhu, Implicit multifunctions theorems, Set-Valued Anal. 7 (1999) 209–238], Ngai and Théra [H.V. Ngai, M. Théra, Error bounds and implicit multifunction theorem in smooth Banach spaces and applications to optimization, Set-Valued Anal. 12 (2004) 195–223], Lee, Tam and Yen [G.M. Lee, N.N. Tam, N.D. Yen, Normal coderivative for multifunctions and implicit function theorems, J. Math. Anal. Appl. 338 (2008) 11–22]) can be used for analyzing parametric constraint systems as well as parametric variational systems. Our main tools are the concept of normal coderivative due to Mordukhovich and the corresponding theory of generalized differentiation.