Francesc Llerena

The vector lattice structure of the n-person TU games

Abstract We show that any cooperative TU game is the maximum of a finite collection of a specific class of the convex games: the almost positive games. These games have non-negative dividends for all coalitions of at least two players. As a consequence of the above result we show that the class of modular games is a set of generators of the distributive lattice of all cooperative TU games. Finally, we characterize zero-monotonic games using a strong max-convex decomposition.

An axiomatization of the nucleolus of assignment markets

On the domain of two-sided assignment markets with agents’ reservation values, the nucleolus is axiomatized as the unique solution that satisfies consistency with respect to Owen’s reduced game and symmetry of maximum complaints of the two sides. As an adjunt, we obtain a geometric characterization of the nucleolus by means of a strong form of the bisection property that characterizes the intersection between the core and the kernel of a coalitional game in (Math Opr Res 4:303–338, 1979).

Convex decomposition of games and axiomatizations of the core and the D-core

This paper provides an axiomatic framework to compare the D-core (the set of undominated imputations) and the core of a cooperative game with transferable utility. Theorem 1 states that the D-core is the only solution satisfying projection consistency, reasonableness (from above), (*)-antimonotonicity, and modularity. Theorem 2 characterizes the core replacing (*)-antimonotonicity by antimonotonicity. Moreover, these axioms also characterize the core on the domain of convex games, totally balanced games, balanced games, and superadditive games.

The pairwise egalitarian solution for the assignment game

Abstract In this note we consider the pairwise egalitarian solution (Sanchez-Soriano, 2003) on the domain of assignment games and study its relation with the core. Strengthening the dominant diagonal condition (Solymosi and Raghavan, 2001), we introduce k -dominant diagonal assignment games ( k ≥ 1 ) , analyzing for which values of k the pairwise egalitarian solution fulfills the standards of fairness represented by the Lorenz domination and the kernel. We also characterize the Thompson’s fair division point (Thompson, 1981) for arbitrary assignment games.

Sequentially compatible payoffs and the core in TU-games

Abstract In the context of cooperative TU-games, we introduce a recursive procedure to distribute the surplus of cooperation when there is an exogenous ordering among the set of players N. In each step of the process, using a given notion of reduced games, an upper and a lower bound for the payoff to the player at issue are required. Sequentially compatible payoffs are defined as those allocation vectors that meet these recursive bounds. For a family of reduction operations, the behavior of this new solution concept is analyzed. For any ordering of N, the core of the game turns out to be the set of sequentially compatible payoffs when the Davis–Maschler reduced games are used. Finally, we study which reduction operations give an advantage to the first player in the ordering.

Rationality, aggregate monotonicity and consistency in cooperative games: some (im)possibility results

On the domain of cooperative games with transferable utility, we investigate if there are single-valued solutions that reconcile individual rationality, core selection, consistency and monotonicity (with respect to the worth of the grand coalition). This paper states some impossibility results for the combination of core selection with either complement consistency (Moulin, J Econ Theory 36:120–148, 1985) or projected consistency (Funaki, Dual axiomatizations of solutions of cooperative games. Mimeo, Tokyo, 1998), and core selection, max consistency (Davis and Maschler, Naval Res Logist Q 12:223–259, 1965) and monotonicity. By contrast, possibility results are manifest when combining individual rationality, projected consistency and monotonicity.

On the existence of the Dutta–Ray’s egalitarian solution

A class of balanced games, called exact partition games, is introduced. Within this class, it is shown that the egalitarian solution of Dutta and Ray (1989) behaves as in the class of convex games. Moreover, we provide two axiomatic characterizations by means of suitable properties such as consistency, rationality and Lorenz-fairness. As a by-product, alternative characterizations of the egalitarian solution over the class of convex games are obtained.

Reduced games and egalitarian solutions

For a class of reduced games satisfying a monotonicity property, we introduce a family of set-valued solution concepts based on egalitarian considerations and consistency principles, and study its relation with the core. Regardless of the reduction operation we consider, the intersection between both sets is either empty or a singleton containing the lexmax solution Arin et al. (Math Soc Sci 46:327–345, 2003). This result induces a procedure for computing the lexmax solution for a class of games that contains games with large core Sharkey (Int J Game Theory 11:175–182, 1982).