Analytic solution for the nucleolus of a three‐player cooperative game

Abstract

The nucleolus solution for cooperative games in characteristic function form is usually computed numerically by solving a sequence of linear programming (LP) problems, or by solving a single, but very large-scale, LP problem. This paper proposes an algebraic method to compute the nucleolus solution analytically (i.e., in closed-form) for a three-player cooperative game in characteristic function form. We …rst consider cooperative games with empty core and derive a formula to compute the nucleolus solution. Next, we examine cooperative games with non-empty core and calculate the nucleolus solution analytically for …ve possible cases arising from the relationship among the value functions of di¤erent coalitions.

Common mistakes in computing the nucleolus

“…A number of approaches have been developed in order to compute it, as reviewed by Leng andParlar (2010) andÇ etiner (2013). Although linear programming and duality have been correctly used in several approaches (e.g.…”

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

“…This game has two minimum-sum integer representations, where the quota is 99 and the weight vectors are w 1 = (38, 31,31,28,23,12,11,8,6,5,3,1) and, respectively, w 2 = (37, 31,31,28,23,12,11,8,7,5,3,1). The weight vectors differ with respect to two non-symmetric players (i.e., players 1 and 9).…”

“…A complementary practical question is how to compute ν(Γ). In general, ν(Γ) can be obtained by solving a sequence of integer linear programs, analogous to a suggestion by Maschler [2] (p. 615) that is implemented in most of the studies compiled by Leng and Parlar [11] (p. 669):…”

Finding the nucleolus of any n-person cooperative game by a single linear program

“…Since then, one can find some improvements on the computation of the nucleolus in particular classes of games, but not much has been done on the general case. In this regard, it is worth underlying the paper by Leng and Parlar [10], which develops an algebraic method for finding the nucleolus of any 3-player game with non-empty core. This method is based on a division of different cases depending on the values of the characteristic function.…”