Core extensions for non-balanced TU-games

A family of core extensions for cooperative TU-games is introduced. These solution concepts are non-empty when applied to non-balanced games yet coincide with the core whenever the core is non-empty. The extensions suggest how an exogenous regulator can sustain a stable and efficient outcome, financing a subsidy via individual taxes. Economic and geometric properties of the solution concepts are studied. When taxes are proportional, the proportional prenucleolus is proposed as a single-valued selection device. An application of these concepts to the decentralization of a public goods economy is discussed. We wish to thank the editor, a referee, an anonymous reviewer, Beth Allen, Marc Dudey, Yakar Kannai, Herve Moulin, Marcel Richter, Luis Sánchez-Mier, and the participants in the Microeconomic Theory Workshop at Rice University for their helpful comments.     

Prosperity properties of TU-games

An important open problem in the theory of TU-games is to determine whether a game has a stable core (Von Neumann-Morgenstern solution (1944)). This seems to be a rather difficult combinatorial problem. There are many sufficient conditions for core-stability. Convexity is probably the best known of these properties. Other properties implying stability of the core are subconvexity and largeness of the core (two properties introduced by Sharkey (1982)) and a property that we have baptized extendability and is introduced by Kikuta and Shapley (1986). These last three properties have a feature in common: if we start with an arbitrary TU-game and increase only the value of the grand coalition, these properties arise at some moment and are kept if we go on with increasing the value of the grand coalition. We call such properties prosperity properties. In this paper we investigate the relations between several prosperity properties and their relation with core-stability. By counter examples we show that all the prosperity properties we consider are different. Received: June 1998/Revised version: December 1998     

Farsighted coalitional stability in TU-games

We study farsighted coalitional stability in the context of TU-games. We show that every TU-game has a nonempty largest consistent set and that each TU-game has a von Neumann–Morgenstern farsighted stable set. We characterize the collection of von Neumann–Morgenstern farsighted stable sets. We also show that the farsighted core is either empty or equal to the set of imputations of the game. In the last section, we explore the stability of the Shapley value. The Shapley value of a superadditive game is a stable imputation: it is a core imputation or it constitutes a von Neumann–Morgenstern farsighted stable set. A necessary and sufficient condition for a superadditive game to have the Shapley value in the largest consistent set is given.     

Axioms for infinite matroids

We give axiomatic foundations for infinite matroids with duality, in terms of independent sets, bases, circuits, closure and rank. Continuing work of Higgs and Oxley, this completes the solution to a problem of Rado of 1966.     

Axioms for Sequential Convergence

It is of general knowledge that those (ultra)filter convergence relations coming from a topology can be characterized by two natural axioms. However, the situation changes considerable when moving to sequential spaces. In case of unique limit points Kisyński (Colloq Math 7:205–211, 1959/1960) obtained a result for sequential convergence similar to the one for ultrafilters, but the general case seems more difficult to deal with. Finally, the problem was solved by Koutnik (Closure and topological sequential convergence. In: Convergence Structures 1984 (Bechyně, 1984). Math. Res., vol. 24, pp. 199–204. Akademie-Verlag, Berlin, 1985). In this paper we present an alternative approach to this problem. Our goal is to find a characterization more closely related to the case of ultrafilter convergence. We extend then the result to characterize sequential convergence relations corresponding to Fréchet topologies, as well to those corresponding to pretopological spaces.      

Fundamental cycles of pre-imputations in non-balanced TU-games

In this paper we prove a characterization for the subclass of non-balanced T U-games. The result is stated in terms of certain class of cycles of pre-imputations. A cycle is a finite sequence of pre-imputations, where each pair of neighbouring elements are interrelated to each other through a transfer of some amount of utility from members of a certain coalition to the members of the complementary coalition, with the understanding that individual gains or losses within any coalition are proportional to the number of members of the coalition. These cycles are strongly connected with a transfer scheme designed to reach a point in the core of a T U-game provided this set is non-empty.The main result of this paper provides an alternative characterization of balanced TU-games to Shapley-Bondarevas theorem.I would like to thank A. Cali for her helpful observations, and two anonymous referees for their detailed comments and suggestions to improve substantially the presentation of the paper.     

Note On linear consistency of anonymous values for TU-games

In the framework of values for TU-games, it is shown that a particular type of consistency, called linear consistency, together with some kind of standardness for two-person games, imply efficiency, anonymity, linearity, as well as uniqueness of the value. Among others, this uniform treatment generalizes Sobolev's axiomatization of the Shapley value. Revised version: December 2001     

An axiomatic approach to egalitarianism in TU-games

A core concept is a solution concept on the class of balanced games that exclusively selects core allocations. We show that every continuous core concept that satisfies both the equal treatment property and a new property called independence of irrelevant core allocations (IIC) necessarily selects egalitarian allocations. IIC requires that, if the core concept selects a certain core allocation for a given game, and this allocation is still a core allocation for a new game with a core that is contained in the core of the first game, then the core concept also chooses this allocation as the solution to the new game. When we replace the continuity requirement by a weak version of additivity we obtain an axiomatization of the egalitarian solution concept that assigns to each balanced game the core allocation minimizing the Euclidean distance to the equal share allocation.     

Axioms for expected utility inn-person games

International Journal of Game Theory - The von Neumann and Morgenstern utility axioms apply to an individual's preferences on a set of probability distributions that is closed under convex...     

Axioms forS-pregroups

AnS-pregroup is a subset of a group such that no nontrivial reduced word (a string of length 2 of elements of the set such that the product of no adjacent pair is in the set) represents the identity when placed in the group. To complete the definition we also require anS-pregroup to contain the identity, be closed under inverses and generate the parent group. In [1] John Stallings, coining the term S-pregroups, asked for an internal axiomatic characterization ofS-pregroups. In this paper we describe such a system by modifying the axioms for a pregroup.Presented by J. Mycielski.     

Global Rank Axioms for Poset Matroids

An excellent introduction to the topic of poset matroids is due to Barnabei, Nicoletti and Pezzoli. In this paper, we investigate the rank axioms for poset matroids; thereby we can characterize poset matroids in a “global” version and a “pseudo-global” version. Some corresponding properties of combinatorial schemes are also obtained.     

Axioms for a theory of semantic equivalence

Gödel-type semantic completeness theorems are established for a formal theory of semantic equivalence based on L.A. Zadeh's concept of a linguistic variable. The linguistics that is employed allows for the expression of propositions such as “it is not the case that ‘young’ is semantically equivalent with ‘not old’”, or, in symbols (young(x) ≅ ∼old(x)).The result is a two-leveled semantics which, at the lower level, is a multivalent interpretation of fuzzy assertions (e.g., ∼old(x)) in terms of fuzzy subsets of a given universe and, at the upper level, is a two-valued (classical) interpretation based on the fact that two closed fuzzy assertions either do or do not have the same truth value. The main proof is of the Henkin variety, employing adaptations of the algebraic methods found in Rasiowa [9] and Rasiowa and Sikorski [10].     

Axioms for a fuzzy measure of inequality

It has been argued that the concept of inequality is inherently imprecise. A difficulty with standard inequality measures is that they generally make no allowances for this, and when they do, it is by dropping the ‘completeness’ axiom in ranking social states (e.g. the Lorenz criterion). It is argued here that the erring axiom is not ‘completeness’ but ‘exactness’ which, being implicit, tends to escape notice. A fuzzy measure of inequality, along with a set of necessary and sufficient axioms, is established. The new measure has several attractive properttes: It allows for tentative judgements and doubts. It is easy to interpret and compute, and the Gini ranking turns out to be a nearest exact approximation of it.