Anna Borisovna Khmelnitskaya

Social choice with independent subgroup utility scales

Abstract. In this article, the kinds of utility comparisons that can be made may differ in distinct population subgroups. Within each subgroup, utility is either ordinally or cardinally measurable. Levels and differences of utility may or may not be interpersonally comparable within a subgroup. No utility comparisons are possible between subgroups. Given these informational assumptions, it is shown that any continuous social welfare ordering that satisfies the weak Pareto principle only depends on the utilities of one of the subgroups. The class of social welfare orderings consistent with these assumptions is determined by the scale type of the dictatorial subgroup.

Values for games with two-level communication structures

We consider a new model of a TU game endowed with both coalition and two-level communication structures that applies to various network situations. The approach to the value is close to that of both Myerson (1977) and Aumann and Dreze (1974): it is based on ideas of component efficiency and of one or another deletion link property, and it treats an a priori union as a self-contained unit; moreover, our approach incorporates also the idea of the Owens quotient game property (1977). The axiomatically introduced values possess an explicit formula representation and in many cases can be quite simply computed. The results obtained are applied to the problem of sharing an international river, possibly with a delta or multiple sources, among multiple users without international firms.

Marginalist and efficient values for TU games

Abstract We derive an explicit formula for a marginalist and efficient value for TU game which possesses the null-player property and is either continuous or monotonic. We show that every such value has to be additive and covariant as well. It follows that the set of all marginalist, efficient, and monotonic values possessing the null-player property coincides with the set of random-order values, and, thereby, the last statement provides an axiomatization without the linearity axiom for the latter which is similar to that of Young for the Shapley value. Another axiomatization without linearity for random-order values is provided by marginalism, efficiency, monotonicity and covariance.

Semiproportional values for TU games

Abstract. The goal of the paper is to introduce a family of values for transferable utility cooperative games that are proportional for two-person games and as well satisfying some combinatorial structure composed by contributions of complementary coalitions or, to less extent, marginal contributions by players.

The Average Covering Tree Value for Directed Graph Games

We introduce a single-valued solution concept, the so-called average covering tree value, for the class of transferable utility games with limited communication structure represented by a directed graph. The solution is the average of the marginal contribution vectors corresponding to all covering trees of the directed graph. The covering trees of a directed graph are those (rooted) trees on the set of players that preserve the dominance relations between the players prescribed by the directed graph. The average covering tree value is component efficient and under a particular convexity-type condition is stable. For transferable utility games with complete communication structure the average covering tree value equals to the Shapley value of the game. If the graph is the directed analog of an undirected graph the average covering tree value coincides with the gravity center solution.

A social capital index

In this paper we propose a social capital measure for individuals belonging to a social network. To do this, we use a game theoretical approach and so we suppose that these individuals are also involved in a cooperative TU-game modelling the economic or social interests that motivate their interactions. We propose as a measure of individual social capital the difference between the Myerson and the Shapley values of actors in the social network and explore the properties of such a measure. This definition is close to our previous measure of centrality (Gomez et al., 2003) and so in this paper we also study the relation between social capital and centrality, finding that this social capital measure can be considered as a vector magnitude with two additive components: centrality and positional externalities. Finally, several real political examples are used to show the agreement of our conclusions with the reality in these situations.

An Owen-type value for games with two-level communication structure

We introduce an Owen-type value for games with two-level communication structure, which is a structure where the players are partitioned into a coalition structure such that there exists restricted communication between as well as within the a priori unions of the coalition structure. Both types of communication restrictions are modeled by an undirected communication graph. We provide an axiomatic characterization of the new value using an efficiency, two types of fairness (one for each level of the communication structure), and a new type of axiom, called fair distribution of the surplus within unions, which compares the effect of replacing a union in the coalition structure by one of its maximal connected components on the payoffs of these components. The relevance of the new value is illustrated by an example. We also show that for particular two-level communication structures the Owen value and the Aumann–Drèze value for games with coalition structure, the Myerson value for communication graph games, and the equal surplus division solution appear as special cases of this new value.

Shapley value for constant-sum games

It is proved that Young’s [4] axiomatization for the Shapley value by marginalism, efficiency, and symmetry is still valid for the Shapley value defined on the class of nonnegative constant-sum games with nonzero worth of grand coalition and on the entire class of constant-sum games as well.

Social welfare functions for different subgroup utility scales

This paper characterizes social welfare functions for different scales of individual utility measurement in distinct population subgroups. Different combinations of ordinal, interval, ratio, and translation scales are studied. We consider situations when utility comparisons among subgroups of individuals by unit and/or zeropoint can or cannot be made, that is when subgroup utility scales are dependent or independent. We show that for combinations of independent subgroup utility scales every corresponding social welfare function is completely determined by the opinions of only one subgroup of individuals when there is not more than one subgroup with ratio or translation scale measurable utilities. Otherwise it also might be determined by the union of subgroups corresponding to ratio and translation scales, and is in agreement with the utility scales of the decisive coalition. We also investigate social welfare functions admissible given various combinations of interval scales with a common unit that combine individual utilities from different subgroups, or with a common zero that lead to the existence of a dictatorial subgroup.

The prenucleolus and the prekernel for games with communication structures

It is well-known that the prekernel on the class of TU games is uniquely determined by non-emptiness, Pareto efficiency (EFF), covariance under strategic equivalence (COV), the equal treatment property, the reduced game property (RGP), and its converse. We show that the prekernel on the class of TU games restricted to the connected coalitions with respect to communication structures may be axiomatized by suitably generalized axioms. Moreover, it is shown that the prenucleolus, the unique solution concept on the class of TU games that satisfies singlevaluedness, COV, anonymity, and RGP, may be characterized by suitably generalized versions of these axioms together with a property that is called “independence of irrelevant connections”. This property requires that any element of the solution to a game with communication structure is an element of the solution to the game that allows unrestricted cooperation in all connected components, provided that each newly connected coalition is sufficiently charged, i.e., receives a sufficiently small worth. Both characterization results may be extended to games with conference structures.