The Shapley value, the Proper Shapley value, and sharing rules for cooperative ventures

René van den Brink, René Levínský*, Miroslav Zelený

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review


In this note, we discuss two solutions for cooperative transferable utility games, namely the Shapley value and the Proper Shapley value. We characterize positive Proper Shapley values by affine invariance and by an axiom that requires proportional allocation of the surplus according to the individual singleton worths in generalized joint venture games. As a counterpart, we show that affine invariance and an axiom that requires equal allocation of the surplus in generalized joint venture games characterize the Shapley value.

Original languageEnglish
Pages (from-to)55-60
Number of pages6
JournalOperations Research Letters
Issue number1
Early online date28 Nov 2019
Publication statusPublished - 1 Jan 2020



  • Cooperative venture game
  • Equity principle
  • Proper Shapley value
  • Shapley value

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