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

Authors

  • Mingming Leng,

    1. Department of Computing and Decision Sciences, Faculty of Business, Lingnan University, 8 Castle Peak Road, Tuen Mun, Hong Kong
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  • Mahmut Parlar

    Corresponding author
    1. DeGroote School of Business, McMaster University, Hamilton, Ontario L8S 4M4, Canada
    • DeGroote School of Business, McMaster University, Hamilton, Ontario L8S 4M4, Canada
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  • The authors wish to thank the associate editor and two anonymous referees for their insightful comments that helped improve the paper.

Abstract

The nucleolus solution for cooperative games in characteristic function form is usually computed numerically by solving a sequence of linear programing (LP) problems, or by solving a single, but very large-scale, LP problem. This article 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 first consider cooperative games with empty core and derive a formula to compute the nucleolus solution. Next, we examine cooperative games with nonempty core and calculate the nucleolus solution analytically for five possible cases arising from the relationship among the value functions of different coalitions. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010

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