Simultaneous penalization and subsidization for stabilizing grand cooperation

Lindong Liu, Xiangtong Qi, Zhou Xu

Research output: Journal article publicationJournal articleAcademic researchpeer-review

9 Citations (Scopus)


In this paper we propose a new instrument, a simultaneous penalization and subsidization, for stabilizing the grand coalition and enabling cooperation among all players of an unbalanced cooperative game. The basic idea is to charge a penalty z from players who leave the grand coalition, and at the same time provide a subsidy ω to players who stay in the grand coalition. To formalize this idea, we establish a penalty-subsidy function ω(z) based on a linear programming model, which allows a decision maker to quantify the trade-off between the levels of penalty and subsidy. By studying function ω(z), we identify certain properties of the trade-off. To implement the new instrument, we design two algorithms to construct function ω(z) and its approximation. Both algorithms rely on solving the value of ω(z) for any given z, for which we propose two effective solution approaches. We apply the new instrument to a class of machine scheduling games, showing its wide applicability.

Original languageEnglish
Pages (from-to)1362-1375
Number of pages14
JournalOperations Research
Issue number5
Publication statusPublished - Sept 2018


  • Cooperative game
  • Grand coalition stability
  • Parallel machine scheduling game
  • Simultaneous penalization and subsidization

ASJC Scopus subject areas

  • Computer Science Applications
  • Management Science and Operations Research


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