Universal characterization sets for the nucleolus in balanced games

We provide a new mo dus op erandi for the computation of the nucleolus in co op- erative games with transferable utility. Using the concept of dual game we extend the theory of characterization sets. Dually essential and dually saturated coalitions determine b oth the core and the nucleolus in monotonic games whenever the core is non-empty. We show how these two sets are related with the existing charac- terization sets. In particular we prove that if the grand coalition is vital then the intersection of essential and dually essential coalitions forms a characterization set itself. We conclude with a sample computation of the nucleolus of bankruptcy games - the shortest of its kind

On the Single-Valuedness of the Pre-Kernel

Based on results given in the recent book by Meinhardt (2013), which presents a dual characterization of the pre-kernel by a finite union of solution sets of a family of quadratic and convex objective functions, we could derive some results related to the uniqueness of the pre-kernel. Rather than extending the knowledge of game classes for which the pre-kernel consists of a single point, we apply a different approach. We select a game from an arbitrary game class with a single pre-kernel element satisfying the non-empty interior condition of a payoff equivalence class, and then establish that the set of related and linear independent games which are derived from this pre-kernel point of the default game replicates this point also as its sole pre-kernel element. In the proof we apply results and techniques employed in the above work. Namely, we prove in a first step that the linear mapping of a pre-kernel element into a specific vector subspace of balanced excesses is a singleton. Secondly, that there cannot exist a different and non-transversal vector subspace of balanced excesses in which a linear transformation of a pre-kernel element can be mapped. Furthermore, we establish that on the restricted subset on the game space that is constituted by the convex hull of the default and the set of related games, the pre-kernel correspondence is single-valued, and therefore continuous. Finally, we provide sufficient conditions that preserve the pre-nucleolus property for related games even when the default game has not a single pre-kernel point.Comment: 24 pages, 2 table

Consistency, converse consistency, and aspirations in TU-games

In problems of choosing ‘aspirations’ for TU-games, we study two axioms, ‘MW-consistency’ and ‘converse MW-consistency.’ In particular, we study which subsolutions of the aspiration correspondence satisfy MW-consistency and/or converse MW-consistency. We also provide axiomatic characterizations of the aspiration kernel and the aspiration nucleolus

The Monoclus of a Coalitional Game

The analysis of single-valued solution concepts for coalitional games with transferable utilities has a long tradition. Opposed to most of this literature we will not deal with solution concepts that provide payoffs to the players for the grand coalition only, but we will analyze allocation scheme rules, which assign payoffs to all players in all coalitions. We introduce four closely related allocation scheme rules for coalitional games. Each of these rules results in a population monotonic allocation scheme (PMAS) whenever the underlying coalitional game allows for a PMAS. The driving force behind these rules are monotonicities, which measure the payoff difference for a player between two nested coalitions. From a functional point of view these monotonicities can best be compared with the excesses in the definition of the (pre-)nucleolus. Two different domains and two different collections of monotonicities result in four allocation scheme rules. For each of the rules we deal with nonemptiness, uniqueness, and continuity, followed by an analysis of conditions for (some of) the rules to coincide. We then focus on characterizing the rules in terms of subbalanced weights. Finally, we deal with computational issues by providing a sequence of linear programs.cooperative game theory;population monotonic allocation schemes;allocation scheme rules

Strongly Essential Coalitions and the Nucleolus of Peer Group Games

Most of the known efficient algorithms designed to compute the nucleolus for special classes of balanced games are based on two facts: (i) in any balanced game, the coalitions which actually determine the nucleolus are essential; and (ii) all essential coalitions in any of the games in the class belong to a prespeci ed collection of size polynomial in the number of players.We consider a subclass of essential coalitions, called strongly essential coalitions, and show that in any game, the collection of strongly essential coalitions contains all the coalitions which actually determine the core, and in case the core is not empty, the nucleolus and the kernelcore.As an application, we consider peer group games, and show that they admit at most 2n - 1 strongly essential coalitions, whereas the number of essential coalitions could be as much as 2n-1. We propose an algorithm that computes the nucleolus of an n-player peer group game in O(n2) time directly from the data of the underlying peer group situation.game theory;algorithm;cooperative games;kernel estimation;peer games

The SD-prenucleolus for TU games

We introduce and characterize a new solution concept for TU games. The new soluction is called SD-prenucleolus and is a lexicographic value although is not a weighted prenucleolus. The SD-prenucleolus satisfies several desirable poperties and is the only known solution that satisfies core stability, strong aggegate monotonicity and null player out property in the class of balanced games. The SD-prenucleolus is the only known solution that satisfies core stability continuity and is monotonic in the class of veto balanced games.TU games, prenucleolus, per capita prenucleolus