Proper cores and dominance cores of fuzzy games

Many kinds of proper cores and dominance cores of fuzzy games are proposed in this paper. We also consider two similar concepts of payoff of a fuzzy coalition that are called the true payoff and quasi-payoff. The different concepts of proper cores and dominance cores will be proposed based on the true payoff and quasi-payoff of a fuzzy coalition. Some mild sufficient conditions are provided in this paper to guarantee the equalities of different proper cores and dominance cores.     

Stable cores of large games

We give general conditions, based on the largeness of the core, under which cores of exact TU games are their unique von Neumann-Morgenstern stable sets. We show that this condition is satisfied by convex games and by nonatomic exact market games. In this way, we extend and unify earlier results existing in literature. Under some additional conditions we also prove the equivalence between the core and the Mas-Colell bargaining set.We thank Jean-Francois Mertens, Enrico Minelli, William Thomson, and two anonymous referees for helpful comments. We also thank seminar audiences at CORE, Cornell, Pescara, and Rochester. We gratefully acknowledge the financial support of the Ministero dell’Istruzione, dell’Universitá e della Ricerca.     

Cooperative games with large cores

A payoff vector in ann-person cooperative game is said to be acceptable if no coalition can improve upon it. The core of a game consists of all acceptable vectors which are feasible for the grand coalition. The core is said to be large if for every acceptable vectory there is a vectorx in the core withx?y. This paper examines the class of games with large cores.     

Characterization of cores of assignment games

We consider the assignment game of Shapley and Shubik (1972). We prove that the class of possible cores of such games (expressed in terms of payoffs for players on one side of the market) is exactly the same as a special class of polytopes, called 45-lattices. These results parallel similar work done by Conway (in Knuth, 1976) and Blair (1984) for marriage markets.Research supported by the Office of Naval Technology.     

Approximate cores of replica games and economies. Part I: Replica games,externalities, and approximate cores

Sufficient conditions are demonstrated for the non-emptiness of approximate cores of sequences of replica games, i.e. for all sufficiently large replications the games have non-empty approximate cores and the approximation can be made arbitrarily ‘good’. The conditions are simply that the games are superadditive and satisfy a non-restrictive ‘per-capita’ boundedness assumption (these properties are satisfied by games derived from well-known models of replica economies). It is argued that the results can be applied to a broad class of games derived from economic models, including ones with external economies and diseconomies, indivisibilities, and non-convexities. To support this claim, in Part I applications to an economy with local public goods are provided, and in Part II, to a general model of a coalition production economy with few restrictions on production technology sets and with (possibly) indivisibilities in consumption. Additional examples in Part I illustrate the generality of the result.     

Nonemptiness conditions for cores of discrete cooperative games

Translated from Programmnoe Oborudovanie i Voprosy Prinyatiya Reshenii, pp. 210–219, 1989.     

Competitive outcomes in the cores of market games

The competitive outcomes of an economic system are known, under quite general conditions, always to lie in the core of the associated cooperative game. It is shown here that every “market game” (i.e., one that arises from an exchange economy with money) can be represented by a “direct market” whose competitive outcomes completely fill up the core. It is also shown that it can be represented by a market having any given core outcome as itsunique competitive outcome, or, more generally, having any given compact convex subset of the core as its full set of competitive outcomes.     

Symmetric solutions for symmetric games with large cores

International Journal of Game Theory - Symmetric solutions (symmetric stable sets) and their uniqueness are investigated for symmetric games when the cores are large enough to have intersections...     

Values and cores of fuzzy games with infinitely many players

The problem of the existence ofvalues (FA-valued, linear, positive, symmetric and efficient operators) on symmetric spaces of “fuzzy games” (that is, ideal set functions of bounded variation) arises naturally from [8], [18], [23] and [2], [3], [4] where it is implicitely approached for technical purposes. In our present work, this problem is approached in itself for the main reason that it is essentially related with the problem of the existence of significant countable additive measures lying in the cores of the “market games”. In fact, it is shown here that there exists a continuous value on the closed subspacebv′ICA ofIBV spanned by thebv′ functions of “fuzzy probability measures” ([9]), this values is “diagonal” onpICA, the closed subspace ofbv′ICA spanned by the natural powers of the fuzzy measures and this is used to prove the main result stating that the cooperative markets contained inpICA have unique fuzzy measures in their cores which are exactly the corresponding diagonal values. This result is of interest because it is providing a tool of determiningCA measures lying in the cores of large classes of games which are not necessarily “non-atomic” and, specially, because it is opening a way toward a new approach of the “Value Equivalence Principle” for differentiable markets with a continuum of traders which are not “perfectly competitive”.     

Aubin cores and bargaining sets for convex cooperative fuzzy games

In this paper, we deal with Aubin cores and bargaining sets in convex cooperative fuzzy games. We first give a simple and direct proof to the well-known result (proved by Branzei et al. (Fuzzy Sets Syst 139:267–281, 2003)) that for a convex cooperative fuzzy game v, its Aubin core C(v) coincides with its crisp core C ^cr (v). We then introduce the concept of bargaining sets for cooperative fuzzy games and prove that for a continuous convex cooperative fuzzy game v, its bargaining set coincides with its Aubin core, which extends a well-known result by Maschler et al. for classical cooperative games to cooperative fuzzy games. We also show that some results proved by Shapley (Int J Game Theory 1:11–26, 1971) for classical decomposable convex cooperative games can be extended to convex cooperative fuzzy games.     

Hyperpolygon spaces and their cores

Given an -tuple of positive real numbers , Konno (2000) defines the hyperpolygon space , a hyperkähler analogue of the Kähler variety parametrizing polygons in with edge lengths . The polygon space can be interpreted as the moduli space of stable representations of a certain quiver with fixed dimension vector; from this point of view, is the hyperkähler quiver variety defined by Nakajima. A quiver variety admits a natural -action, and the union of the precompact orbits is called the core. We study the components of the core of , interpreting each one as a moduli space of pairs of polygons in with certain properties. Konno gives a presentation of the cohomology ring of ; we extend this result by computing the -equivariant cohomology ring, as well as the ordinary and equivariant cohomology rings of the core components.

     

Computing the cores of strategic games with punishment–dominance relations

In this paper, we discuss the computational complexity of the strategic cores of a class of n-person games defined by Masuzawa (Int J Game Theory 32:479–483, 2003), which includes economic situations with monotone externality. We propose an algorithm for finding an α-core strategy of any game in this class which, counting the evaluation of a payoff for a strategy profile as one step, terminates after O(n ³· M) operations, where M is the maximum size of a strategy set of any of the n players. The idea underlying this method is based on the property of reduced games. This paper is based on a part of the doctoral dissertation of the author. The author thanks Mikio Nakayama, Masashi Umezawa, William Thomson, an associate editor, and the anonymous referee for their helpful comments, suggestions, and advice. Thanks are also due to Yukihiko Funaki for a comment that led the author to this subject. The author is responsible for errors and inadvertencies.     

Loops, their cores and symmetric spaces

This paper is devoted to the relations among affine symmetric spaces, smooth Bol and Moufang loops, smooth left distributive quasigroups and differentiable 3-nets. The results are used to prove the analyticity of smooth Moufang loops and left distributive quasigroups with involutive left translations as well as to show the Lie nature of transformation groups naturally related to some classes of smooth binary systems and 3-nets. In the last section we establish power series expansion for local loops with weak associativity conditions and apply the methods of the previous sections in order to describe geodesic loops having euclidean lines either as their geodesic lines or as geodesic lines of their core. The first author was partly supported by the Deutsche Forschungsgemeinschaft and by OTKA Grant no. T020545.     

Multi-choice clan games and their core

In this paper, we consider market situations with two corners. One corner consists of a group of powerful agents with yes-or-no choices and clan behavior. The other corner consists of non-powerful agents with multi-choices regarding the extent at which cooperation with the clan can be achieved. Multi-choice clan games arise from such market situations. The focus is on the analysis of the core of multi-choice clan games. Several characterizations of multi-choice clan games by the shape of the core are given, and the connection between the convexity of a multi-choice clan game and the stability of its core is studied.      

On the restricted cores and the bounded core of games on distributive lattices

A game with precedence constraints is a TU game with restricted cooperation, where the set of feasible coalitions is a distributive lattice, hence generated by a partial order on the set of players. Its core may be unbounded, and the bounded core, which is the union of all bounded faces of the core, proves to be a useful solution concept in the framework of games with precedence constraints. Replacing the inequalities that define the core by equations for a collection of coalitions results in a face of the core. A collection of coalitions is called normal if its resulting face is bounded. The bounded core is the union of all faces corresponding to minimal normal collections. We show that two faces corresponding to distinct normal collections may be distinct. Moreover, we prove that for superadditive games and convex games only intersecting and nested minimal collection, respectively, are necessary. Finally, it is shown that the faces corresponding to pairwise distinct nested normal collections may be pairwise distinct, and we provide a means to generate all such collections.     

Winning strategies in club games and their applications

We present results concerning winning strategies and tactics in club games on ??λ. We show that there is generally no winning tactic for the player trying to get inside the club. The bound‐countable game turns out to be rather fruitful and adds to some previous results about the construction of elementary substructures and their localization in certain intervals. We show that Player II has a winning strategy in the bound‐countable game, thus establishing a new ZFC result. The applications given are new proofs for two cardinal diamonds and the impossibility of collapsing cardinals to ?₂ under certain conditions (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cooperative games under interval uncertainty: on the convexity of the interval undominated cores

Uncertainty is a daily presence in the real world. It affects our decision making and may have influence on cooperation. Often uncertainty is so severe that we can only predict some upper and lower bounds for the outcome of our actions, i.e., payoffs lie in some intervals. A suitable game theoretic model to support decision making in collaborative situations with interval data is that of cooperative interval games. Solution concepts that associate with each cooperative interval game sets of interval allocations with appealing properties provide a natural way to capture the uncertainty of coalition values into the players’ payoffs. This paper extends interval-type core solutions for cooperative interval games by discussing the set of undominated core solutions which consists of the interval nondominated core, the square interval dominance core, and the interval dominance core. The interval nondominated core is introduced and it is shown that it coincides with the interval core. A straightforward consequence of this result is the convexity of the interval nondominated core of any cooperative interval game. A necessary and sufficient condition for the convexity of the square interval dominance core of a cooperative interval game is also provided.     

A conjecture of Shapley and Shubik on competitive outcomes in the cores of NTU market games

It is shown that for every NTUmarket game, there is amarket thatrepresents the game whosecompetitive payoff vectors completely fill up theinner core of the game. It is also shown that for every NTU market game and for any point in its inner core, there is a market that represents the game and further has the given inner core point as itsunique competitive payoff vector. These results prove a conjecture of Shapley and Shubik.Journal of Economic Literature Classification Numbers: C71, D51.