Elena Yanovskaya

Note On linear consistency of anonymous values for TU-games

Abstract. 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 Sobolevs axiomatization of the Shapley value.

The prenucleolus for games with restricted cooperation

A game with restricted cooperation is a triple (N,v,Ω), where N is a finite set of players, Ω⊂2N is a nonempty collection of feasible coalitions such that N∈Ω, and v:Ω→R is a characteristic function. The definition implies that if Ω=2N, then the game (N,v,Ω)=(N,v) is the classical transferable utility (TU) cooperative game.

Competitive Division of a Mixed Manna

A mixed manna contains goods (that everyone likes), bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others. If all items are goods and utility functions are homothetic, concave (and monotone), the Competitive Equilibrium with Equal Incomes maximizes the Nash product of utilities: hence it is welfarist (determined utility-wise by the feasible set of pro les), single-valued and easy to compute. We generalize the Gale-Eisenberg Theorem to a mixed manna. The Competitive division is still welfarist and related to the product of utilities or disutilities. If the zero utility pro le (before any manna) is Pareto dominated, the competitive pro le is unique and still maximizes the product of utilities. If the zero pro le is unfeasible, the competitive pro les are the critical points of the product of disutilities on the eciency frontier, and multiplicity is pervasive. In particular the task of dividing a mixed manna is either good news for everyone, or bad news for everyone. We re ne our results in the practically important case of linear preferences, where the axiomatic comparison between the division of goods and that of bads is especially sharp. When we divide goods and the manna improves, everyone weakly bene ts under the competitive rule; but no reasonable rule to divide bads can be similarly Resource Monotonic. Also, the much larger set of Non Envious and Ecient divisions of bads can be disconnected so that it will admit no continuous selection.

Nash social welfare orderings

Abstract The paper considers the problem of description of social welfare orderings (SWOs) on the entire utility (Euclidean) space R n satisfying Scale Independence. These orderings and the functions representing them are called the Nash social welfare orderings and the Nash social welfare functions (SWFs), respectively. The more properties of the SWOs into consideration are Strong Pareto and two variants of weakening for continuity: the ‘orthant’ continuity, meaning continuity of a SWO on each separate orthant of the utility space, and ‘upper preserving in the limit’ (UPL) that is equivalent to existence of maximal elements for a SWO on each compact set. The complete characterization of the Nash SWOs satisfying these continuity conditions is given. For a fixed arbitrary orthant of the utility space each Strong Pareto Nash SWO is representable by a Nash (Cobb–Douglas) type SWFs (if the SWO is orthantly continuous) or by a lexicographical ordering defined by a collection of such functions (if the SWO satisfies only UPL). Vectors from the different orthants are compared either by the orthant rules (if they coincide for the orthants), or by a linear ordering on the set of orthants.

Consistent and covariant solutions for TU games

Abstract.One of the important properties characterizing cooperative game solutions is consistency. This notion establishes connections between the solution vectors of a cooperative game and those of its reduced game. The last one is obtained from the initial game by removing one or more players and by giving them the payoffs according to a specific principle (e.g. a proposed payoff vector). Consistency of a solution means that the restriction of a solution payoff vector of the initial game to any coalition belongs to the solution set of the corresponding reduced game. There are several definitions of the reduced games (cf., e.g., the survey of T. Driessen [2]) based on some intuitively acceptable characteristics. In the paper some natural properties of reduced games are formulated, and general forms of the reduced games possessing some of them are given. The efficient, anonymous, covariant TU cooperative game solutions satisfying the consistency property with respect to any reduced game are described.

Consistency For Proportional Solutions

One of the properties characterizing cooperative game solutions is consistency connecting solution vectors of a cooperative game with finite set of players and its reduced game defined by removing one or more players and by assigning them the payoffs according to some specific principle (e.g., a proposed payoff vector). Consistency of a solution means that any part (defined by a coalition of the original game) of a solution payoff vector belongs to the solution set of the corresponding reduced game. In the paper the proportional solutions for TU-games are defined as those depending only on the proportional excess vectors in the same manner as translation covariant solutions depend on the usual Davis–Maschler excess vectors. The general form of the reduced games defining consistent proportional solutions is given. The efficient, anonymous, proportional TU cooperative game solutions meeting the consistency property with respect to any reduced game are described.

Monotonicity properties of interval solutions and the Dutta-Ray solution for convex interval games

This paper examines several monotonicity properties of value-type interval solutions on the class of convex interval games and focuses on the Dutta-Ray (DR) solution for such games. Well known properties for the classical DR solution are extended to the interval setting. In particular, it is proved that the interval DR solution of a convex interval game belongs to the interval core of that game and Lorenz dominates each other interval core element. Consistency properties of the interval DR solution in the sense of Davis-Maschler and of Hart-Mas-Colell are verified. An axiomatic characterization of the interval DR solution on the class of convex interval games with the help of bilateral Hart-Mas-Colell consistency and the constrained egalitarianism for two-person interval games is given.

STRONGLY CONSISTENT SOLUTIONS TO BALANCED TU GAMES

Consistency properties of game solutions connect between themselves the solution sets of games with different sets of players. In the paper, the strongly consistent solutions with respect to the Davis–Maschler definition of the reduced games to the class of balanced cooperative TU games with finite sets of players are considered. A cooperative game solution σ to a class of a TU cooperative game is called strongly consistent if for any and , where is the reduced game of Γ on the player set S and with respect to x. Evidently, all consistent single-valued solutions are strongly consistent. In the paper, we characterise anonymous, covariant bounded and strongly consistent to the class of balanced games. The core, its relative interior and the prenucleolus are among them. However, they are not unique solutions satisfying these axioms. Thus, more axioms are necessary in order to characterise these solutions with strong consistency. One of such axioms is the definition of a solution for the class of balanced two-person games. It is sufficient for the axiomatisation of the prenucleolus without the single-valuedness axiom. If we add the closed graph property of the solution correspondence to the given axioms, then the system characterises only the core. The two axiomatisations are the main result of the paper. An example of a strongly consistent solution different from the prenucleolus, the core and its relative interior is given.

Correspondence between social choice functions and solutions of cooperative games

Abstract Ordinal solutions of cooperative transferable utility (TU) games are defined as ones that can be represented as social choice functions in a space of the utilities of players and coalitions. We provide an axiomatic characterization of a social choice function of this type. The corresponding solution of a cooperative TU game turns out to be an extended (pre)nucleolus or a set-valued analogue of a (pre)nucleolus.

The bounded core for games with restricted cooperation

A game with restricted (incomplete) cooperation is a triple (N, v, Ω), where N represents a finite set of players, Ω ⊂ 2N is a set of feasible coalitions such that N ∈ Ω, and v: Ω → R denotes a characteristic function. Unlike the classical TU games, the core of a game with restricted cooperation can be unbounded. Recently Grabisch and Sudhölter [9] proposed a new solution concept—the bounded core—that associates a game (N, v,Ω) with the union of all bounded faces of the core. The bounded core can be empty even if the core is nonempty. This paper gives two axiomatizations of the bounded core. The first axiomatization characterizes the bounded core for the class Gr of all games with restricted cooperation, whereas the second one for the subclass Gbcr ⊂ Gr of the games with nonempty bounded cores.