The unit-level-core for multi-choice games: the replicated core for TU games

This note extends the solution concept of the core for traditional transferable-utility (TU) games to multi-choice TU games, which we name the unit-level-core. It turns out that the unit-level-core of a multi-choice TU game is a “replicated subset” of the core of a corresponding “replicated” TU game. We propose an extension of the theorem of Bondareva (Probl Kybern 10:119–139, 1963) and Shapley (Nav Res Logist Q 14:453–460, 1967) to multi-choice games. Also, we introduce the reduced games for multi-choice TU games and provide an axiomatization of the unit-level-core on multi-choice TU games by means of consistency and its converse.     

The multichoice coalition value

In this paper we define a solution for multichoice games which is a generalization of the Owen coalition value (Lecture Notes in Economics and Mathematical Systems: Essays in Honor of Oskar Morgenstern, Springer, New York, pp. 76–88, 1977) for transferable utility cooperative games and the Egalitarian solution (Peters and Zanks, Ann. Oper. Res. 137, 399–409, 2005) for multichoice games. We also prove that this solution can be seen as a generalization of the configuration value and the dual configuration value (Albizuri et al., Games Econ. Behav. 57, 1–17, 2006) for transferable utility cooperative games.     

Consistent extensions and subsolutions of the core of multi-choice NTU games

We extend the reduced games introduced by Davis and Maschler (Naval Res Log Q 12:223–259, 1965) and Moulin (J Econ Theory 36:120–148, 1985) to multi-choice non-transferable utility games and define two related properties of consistency. We also show that the core proposed by Hwang and Li (Math Methods Oper Res 61:33–40, 2005) violates these two consistency properties. In order to investigate how seriously it violates these two consistency properties, we provide consistent extensions and consistent subsolutions of the core.     

Kalai-Smorodinsky Bargaining Solution Equilibria

Multicriteria games describe strategic interactions in which players, having more than one criterion to take into account, don’t have an a-priori opinion on the relative importance of all these criteria. Roemer (Econ. Bull. 3:1–13, 2005) introduces an organizational interpretation of the concept of equilibrium: each player can be viewed as running a bargaining game among criteria. In this paper, we analyze the bargaining problem within each player by considering the Kalai-Smorodinsky bargaining solution (see Kalai and Smorodinsky in Econometrica 43:513–518, 1975). We provide existence results for the so called Kalai-Smorodinsky bargaining solution equilibria for a general class of disagreement points which properly includes the one considered by Roemer (Econ. Bull. 3:1–13, 2005). Moreover we look at the refinement power of this equilibrium concept and show that it is an effective selection device even when combined with classical refinement concepts based on stability with respect to perturbations; in particular, we consider the extension to multicriteria games of the Selten’s trembling hand perfect equilibrium concept (see Selten in Int. J. Game Theory 4:25–55, 1975) and prove that perfect Kalai-Smorodinsky bargaining solution equilibria exist and properly refine both the perfect equilibria and the Kalai-Smorodinsky bargaining solution equilibria.     

Distributive Lattices with a Generalized Implication: Topological Duality

In this paper we introduce the notion of generalized implication for lattices, as a binary function ⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication as a common abstraction of the notions of annihilator (Mandelker, Duke Math J 37:377–386, 1970), Quasi-modal algebras (Celani, Math Bohem 126:721–736, 2001), and weakly Heyting algebras (Celani and Jansana, Math Log Q 51:219–246, 2005). We introduce the suitable notions of morphisms in order to obtain a category, as well as the corresponding notion of congruence. We develop a Priestley style topological duality for the bounded distributive lattices with a generalized implication. This duality generalizes the duality given in Celani and Jansana (Math Log Q 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (Math Bohem 126:721–736, 2001) for Quasi-modal algebras.     

Multicriteria games and potentials

In this note we study how far the theory of strategic games with potentials, as reported by Monderer and Shapley (Games Econ Behav 14:124–143, 1996), can be extended to strategic games with vector payoffs, as reported by Shapley (Nav Res Logist Q 6:57–61, 1959). The problem of the existence of pure approximate Pareto equilibria for multicriteria potential games is also studied.      

Bidding and sequential coalition formation with externalities

We study coalition formation games with externalities where each agent’s payoff depends on the entire partition. As in Bloch (Games Econ Behav 14:90–123, 1996) and Ray and Vohra (Games Econ Behav 26:268–336, 1999), we assume that coalitions form sequentially and agreements are irreversible. Instead of a fixed order protocol, we use a “bidding mechanism” (see Pérez-Castrillo and Wettstein in Am Econ Rev 92(5):1577–1587, 2002) to determine proposals and transfers among the agents. We show that such a mechanism facilitates the attainment of efficiency; in particular, our game admits a Markov perfect equilibrium with the property of full dynamic efficiency. In addition, the aggregate equilibrium payoffs of our game exhibit monotonicity in time. Nevertheless, inefficient equilibria can also emerge.     

On a class of secant-like methods for solving nonlinear equations

We provide a semilocal convergence analysis for a certain class of secant-like methods considered also in Argyros (J Math Anal Appl 298:374–397, 2004, 2007), Potra (Libertas Mathematica 5:71–84, 1985), in order to approximate a locally unique solution of an equation in a Banach space. Using a combination of Lipschitz and center-Lipschitz conditions for the computation of the upper bounds on the inverses of the linear operators involved, instead of only Lipschitz conditions (Potra, Libertas Mathematica 5:71–84, 1985), we provide an analysis with the following advantages over the work in Potra (Libertas Mathematica 5:71–84, 1985) which improved the works in Bosarge and Falb (J Optim Theory Appl 4:156–166, 1969, Numer Math 14:264–286, 1970), Dennis (SIAM J Numer Anal 6(3):493–507, 1969, 1971), Kornstaedt (1975), Larsonen (Ann Acad Sci Fenn, A 450:1–10, 1969), Potra (L’Analyse Numérique et la Théorie de l’Approximation 8(2):203–214, 1979, Aplikace Mathematiky 26:111–120, 1981, 1982, Libertas Mathematica 5:71–84, 1985), Potra and Pták (Math Scand 46:236–250, 1980, Numer Func Anal Optim 2(1):107–120, 1980), Schmidt (Period Math Hung 9(3):241–247, 1978), Schmidt and Schwetlick (Computing 3:215–226, 1968), Traub (1964), Wolfe (Numer Math 31:153–174, 1978): larger convergence domain; weaker sufficient convergence conditions, finer error bounds on the distances involved, and a more precise information on the location of the solution. Numerical examples further validating the results are also provided.     

Order-Compactifications of Totally Ordered Spaces: Revisited

Order-compactifications of totally ordered spaces were described by Blatter (J Approx Theory 13:56–65, 1975) and by Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Their results generalize a similar characterization of order-compactifications of linearly ordered spaces, obtained independently by Fedorčuk (Soviet Math Dokl 7:1011–1014, 1966; Sib Math J 10:124–132, 1969) and Kaufman (Colloq Math 17:35–39, 1967). In this note we give a simple characterization of the topology of a totally ordered space, as well as give a new simplified proof of the main results of Blatter (J Approx Theory 13:56–65, 1975) and Kent and Richmond (J Math Math Sci 11(4):683–694, 1988). Our main tool will be an order-topological modification of the Dedekind-MacNeille completion. In addition, for a zero-dimensional totally ordered space X, we determine which order-compactifications of X are Priestley order-compactifications.     

A characterization of convex TU games by means of the Mas-Colell bargaining set (à la Shimomura)

Within the class of zero-monotonic games, we prove that a cooperative game with transferable utility is convex if and only if the core of the game coincides with the Mas-Colell bargaining set (à la Shimomura, in Int J Game Theory 26:283–302, 1997).     

Consistent cost sharing

A new concept of consistency for cost sharing solutions is discussed, analyzed, and related to the homonymous and natural property within the rationing context. Main result is that the isomorphism in Moulin and Shenker (J Econ Theory 64:178–201, 1994) pairs each additive and consistent single-valued mechanism with a corresponding monotonic and consistent rationing method. Then this answers the open question in Moulin (Econometrica 68:643–684, 2000; Handbook of social choice and welfare. Handbooks in economics, pp 289–357, 2002) whether such notion for cost sharing exists. The conclusion is that renown solutions like the average and serial cost sharing mechanisms are consistent, whereas the Shapley–Shubik mechanism is not. Average cost sharing is the only strongly consistent element in this class. The two subclasses of incremental and parametric cost sharing mechanisms are further analyzed as refinement of the main result.     

Revisiting Generalized Nash Games and Variational Inequalities

Generalized Nash games with shared constraints represent an extension of Nash games in which strategy sets are coupled across players through a shared or common constraint. The equilibrium conditions of such a game can be compactly stated as a quasi-variational inequality (QVI), an extension of the variational inequality (VI). In (Eur. J. Oper. Res. 54(1):81–94, 1991), Harker proved that for any QVI, under certain conditions, a solution to an appropriately defined VI solves the QVI. This is a particularly important result, given that VIs are generally far more tractable than QVIs. However Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007) suggested that the hypotheses of this result are difficult to satisfy in practice for QVIs arising from generalized Nash games with shared constraints. We investigate the applicability of Harker’s result for these games with the aim of formally establishing its reach. Specifically, we show that if Harker’s result is applied in a natural manner, its hypotheses are impossible to satisfy in most settings, thereby supporting the observations of Facchinei et al. But we also show that an indirect application of the result extends the realm of applicability of Harker’s result to all shared-constraint games. In particular, this avenue allows us to recover as a special case of Harker’s result, a result provided by Facchinei et al. (Oper. Res. Lett. 35(2):159–164, 2007), in which it is shown that a suitably defined VI provides a solution to the QVI of a shared-constraint game.     

On the regular variation of ratios of jointly Fréchet random variables

We provide a necessary and sufficient condition for the ratio of two jointly α-Fréchet random variables to be regularly varying. This condition is based on the spectral representation of the joint distribution and is easy to check in practice. Our result motivates the notion of the ratio tail index, which quantifies dependence features that are not characterized by the tail dependence index. As an application, we derive the asymptotic behavior of the quotient correlation coefficient proposed in Zhang (Ann Stat 36(2):1007–1030, 2008) in the dependent case. Our result also serves as an example of a new type of regular variation of products, different from the ones investigated by Maulik et al (J Appl Probab 39(4):671–699, 2002).     

On Borell-Brascamp-Lieb Inequalities on Metric Measure Spaces

In this paper we introduce the notion of a Borell-Brascamp-Lieb inequality for metric measure spaces (M,d,m) denoted by BBL(K,N) for two numbers K,N ∈ ℝ with N ≥ 1. In the first part we prove that BBL(K,N) holds true on metric measure spaces satisfying a curvature-dimension condition CD(K,N) developed and studied by Lott and Villani in (Ann Math 169:903–991, 2007) as well as by Sturm in (Acta Math 196(1):133–177, 2006). The aim of the second part is to show that BBL(K,N) is stable under convergence of metric measure spaces with respect to the L ₂-transportation distance.     

<Emphasis Type="Italic">π</Emphasis>-formulas implied by Dougall’s summation theorem for <Subscript>5</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>-series

The general summation theorem for well-poised ₅ F ₄-series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132, 1907) is shown to imply several infinite series of Ramanujan-type for 1/π and 1/π ², including those due to Bauer (J. Reine Angew. Math. 56:101–121, 1859) and Glaisher (Q. J. Math. 37:173–198, 1905) as well as some recent ones by Levrie (Ramanujan J. 22:221–230, 2010).     

<Emphasis Type="Italic">p</Emphasis>-additive games: a class of totally balanced games arising from inventory situations with temporary discounts

We introduce a new class of totally balanced cooperative TU games, namely p-additive games. It is inspired by the class of inventory games that arises from inventory situations with temporary discounts (Toledo Ph.D. thesis, Universidad Miguel Hernández de Elche, 2002) and contains the class of inventory cost games (Meca et al. Math. Methods Oper. Res. 57:481–493, 2003). It is shown that every p-additive game and its corresponding subgames have a nonempty core. We also focus on studying the character of concave or convex and monotone p-additive games. In addition, the modified SOC-rule is proposed as a solution for p-additive games. This solution is suitable for p-additive games, since it is a core-allocation which can be reached through a population monotonic allocation scheme. Moreover, two characterizations of the modified SOC-rule are provided. This work was partially supported by the Spanish Ministry of Education and Science and Generalitat Valenciana (grants MTM2005-09184-C02-02, ACOMP06/040, CSD2006-00032). Authors acknowledge valuable comments made by the Editor and the referee.     

On Ulm’s method using divided differences of order one

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Tzv Akad Nauk Est SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones in Burmeister (Z Angew Math Mech 52:101–110, 1972), Kornstaedt (Aequ Math 13:21–45, 1975), Moser (1973), and Potra and Pták (Cas Pest Mat 108:333–341, 1983) do not. We also show that under the same hypotheses and computational cost, finer error bounds can be obtained. Some error bounds are also shown to be sharp. Numerical examples are also provided further validating the results.     

Yoneda Structures from 2-toposes

A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the Yoneda structure (Street and Walters, J. Algebra, 50:350–379, 1978) it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a Yoneda structure are presented. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2-category (Street, Lecture Notes in Math., 420:104–133, 1974) and provides a self-contained development of the necessary background material on Yoneda structures.      

Convexity properties for interior operator games

Interior operator games arose by abstracting some properties of several types of cooperative games (for instance: peer group games, big boss games, clan games and information market games). This reason allow us to focus on different problems in the same way. We introduced these games in Bilbao et al. (Ann. Oper. Res. 137:141–160, 2005) by a set system with structure of antimatroid, that determines the feasible coalitions, and a non-negative vector, that represents a payoff distribution over the players. These games, in general, are not convex games. The main goal of this paper is to study under which conditions an interior operator game verifies other convexity properties: 1-convexity, k-convexity (k≥2 ) or semiconvexity. But, we will study these properties over structures more general than antimatroids: the interior operator structures. In every case, several characterizations in terms of the gap function and the initial vector are obtained. We also find the family of interior operator structures (particularly antimatroids) where every interior operator game satisfies one of these properties.