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Conditions for the uniqueness of the Gately point for cooperative games

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  • Jochen Staudacher
  • Johannes Anwander

Abstract

We are studying the Gately point, an established solution concept for cooperative games. We point out that there are superadditive games for which the Gately point is not unique, i.e. in general the concept is rather set-valued than an actual point. We derive conditions under which the Gately point is guaranteed to be a unique imputation and provide a geometric interpretation. The Gately point can be understood as the intersection of a line defined by two points with the set of imputations. Our uniqueness conditions guarantee that these two points do not coincide. We provide demonstrative interpretations for negative propensities to disrupt. We briefly show that our uniqueness conditions for the Gately point include quasibalanced games and discuss the relation of the Gately point to the $\tau$-value in this context. Finally, we point out relations to cost games and the ACA method and end upon a few remarks on the implementation of the Gately point and an upcoming software package for cooperative game theory.

Suggested Citation

  • Jochen Staudacher & Johannes Anwander, 2019. "Conditions for the uniqueness of the Gately point for cooperative games," Papers 1901.01485, arXiv.org.
    • Handle: RePEc:arx:papers:1901.01485
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    References listed on IDEAS

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    1. Gately, Dermot, 1974. "Sharing the Gains from Regional Cooperation: A Game Theoretic Application to Planning Investment in Electric Power," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 15(1), pages 195-208, February.
    2. Rodica Branzei & Dinko Dimitrov & Stef Tijs, 2008. "Models in Cooperative Game Theory," Springer Books, Springer, edition 0, number 978-3-540-77954-4, March.
    3. R.J. Aumann & S. Hart (ed.), 2002. "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3.
    4. Chakravarty,Satya R. & Mitra,Manipushpak & Sarkar,Palash, 2014. "A Course on Cooperative Game Theory," Cambridge Books, Cambridge University Press, number 9781107691322, December.
    5. Otten, G.J.M., 1993. "Characterizations of a Game Theoretical Cost Allocation Method," Discussion Paper 1993-37, Tilburg University, Center for Economic Research.
    6. Chakravarty,Satya R. & Mitra,Manipushpak & Sarkar,Palash, 2015. "A Course on Cooperative Game Theory," Cambridge Books, Cambridge University Press, number 9781107058798, December.
    7. Sandler, Todd & Tschirhart, John T, 1980. "The Economic Theory of Clubs: An Evaluative Survey," Journal of Economic Literature, American Economic Association, vol. 18(4), pages 1481-1521, December.
    8. Otten, G.J.M., 1993. "Characterizations of a Game Theoretical Cost Allocation Method," Other publications TiSEM 18a0262e-a6d3-4bd9-bdb0-6, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Robert P. Gilles & Lina Mallozzi, 2022. "Gately Values of Cooperative Games," Papers 2208.10189, arXiv.org, revised Jul 2023.
    2. Robert P. Gilles & Lina Mallozzi, 2023. "Game theoretic foundations of the Gately power measure for directed networks," Papers 2308.02274, arXiv.org.
    3. Robert P. Gilles & Lina Mallozzi, 2023. "Game Theoretic Foundations of the Gately Power Measure for Directed Networks," Games, MDPI, vol. 14(5), pages 1-19, September.
    4. Gilles, Robert P. & Mallozzi, Lina, 2022. "Generalised Gately Values of Cooperative Games," QBS Working Paper Series 2022/06, Queen's University Belfast, Queen's Business School.

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