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Shapley-like values without symmetry

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  • Jacob North Clark
  • Stephen Montgomery-Smith

Abstract

Following the work of Lloyd Shapley on the Shapley value, and tangentially the work of Guillermo Owen, we offer an alternative non-probabilistic formulation of part of the work of Robert J. Weber in his 1978 paper "Probabilistic values for games." Specifically, we focus upon efficient but not symmetric allocations of value for cooperative games. We retain standard efficiency and linearity, and offer an alternative condition, "reasonableness," to replace the other usual axioms. In the pursuit of the result, we discover properties of the linear maps that describe the allocations. This culminates in a special class of games for which any other map that is "reasonable, efficient" can be written as a convex combination of members of this special class of allocations, via an application of the Krein-Milman theorem.

Suggested Citation

  • Jacob North Clark & Stephen Montgomery-Smith, 2018. "Shapley-like values without symmetry," Papers 1809.07747, arXiv.org, revised May 2019.
    • Handle: RePEc:arx:papers:1809.07747
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    References listed on IDEAS

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    1. Monderer, Dov & Samet, Dov, 2002. "Variations on the shapley value," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 54, pages 2055-2076, Elsevier.
    2. Roger B. Myerson, 1977. "Graphs and Cooperation in Games," Mathematics of Operations Research, INFORMS, vol. 2(3), pages 225-229, August.
    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. Sergiu Hart, 2006. "Shapley Value," Discussion Paper Series dp421, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    5. Gomez, Daniel & Gonzalez-Aranguena, Enrique & Manuel, Conrado & Owen, Guillermo & del Pozo, Monica & Tejada, Juan, 2003. "Centrality and power in social networks: a game theoretic approach," Mathematical Social Sciences, Elsevier, vol. 46(1), pages 27-54, August.
    6. Mertens, Jean-Francois, 2002. "Some other economic applications of the value," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 3, chapter 58, pages 2185-2201, Elsevier.
    7. Charalambos D. Aliprantis & Kim C. Border, 2006. "Infinite Dimensional Analysis," Springer Books, Springer, edition 0, number 978-3-540-29587-7, May.
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