The Integer Nucleolus of Directed Simple Games: A Characterization and an Algorithm

Reiner Wolff

doi:10.3390/g8010016

Abstract

We study the class of directed simple games, assuming that only integer solutions are admitted; i.e., the players share a resource that comes in discrete units. We show that the integer nucleolus—if nonempty—of such a game is composed of the images of a particular payoff vector under all symmetries of the game. This payoff vector belongs to the set of integer imputations that weakly preserve the desirability relation between the players. We propose an algorithm for finding the integer nucleolus of any directed simple game with a nonempty integer imputation set. The algorithm supports the parallel execution of multiple threads in a computer application. We also consider the integer prenucleolus and the class of directed generalized simple games.

Highlights

The nucleolus is a popular solution concept for cooperative transferable-utility (TU) games because it is a set of game outcomes that are in a sense most acceptable “... as a compromise between the players” (Schmeidler [1], p. 1163)
We have been concerned with the class of directed simple games, under the assumption that only integer solutions are feasible
The players of such a game face an integer allocation problem, in that they share a resource that comes in discrete units

Summary

Introduction

The nucleolus is a popular solution concept for cooperative transferable-utility (TU) games because it is a set of game outcomes that are in a sense most acceptable “... as a compromise between the players” (Schmeidler [1], p. 1163). Given a directed simple game Γt , denote by θW the subvector of θ associated with the coalitions in W (Γt ) \ N, and by P(Γt ) := x ∈ I (Γt ) : x1 ≥ · · · ≥ xn ∧ | xi − x j | ∈ {0, 1} for all i 6= j, i ∼ D j the subset of integer imputations in anti-lexicographic order that satisfy Lemmas 1 and 2. If Γt is a directed simple game and I (Γt ) 6= ∅, we may search P(Γt ) for y without reference to the coalitions not contained in W (Γt ) In view of this observation, we suggest an algorithm to compute the integer nucleolus of any such game.

Performance Analysis

Extension to the Integer Prenucleolus

Conclusions