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On $$\alpha $$ α -constant-sum games

Author

Listed:
  • Wenna Wang

    (Xi’an University of Finance and Economics)

  • René van den Brink

    (VU University)

  • Hao Sun

    (Northwestern Polytechnical University)

  • Genjiu Xu

    (Northwestern Polytechnical University)

  • Zhengxing Zou

    (VU University
    Beijing Jiaotong University)

Abstract

Given any $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] , an $$\alpha $$ α -constant-sum game (abbreviated as $$\alpha $$ α -CS game) on a finite set of players, N, is a function that assigns a real number to any coalition $$S\subseteq N$$ S ⊆ N , such that the sum of the worth of the coalition S and the worth of its complementary coalition $$N\backslash S$$ N \ S is $$\alpha $$ α times the worth of the grand coalition. This class contains the constant-sum games of Khmelnitskaya (Int J Game Theory 32:223–227, 2003) (for $$\alpha = 1$$ α = 1 ) and games of threats of (Kohlberg and Neyman, Games Econ Behav 108:139–145, 2018) (for $$\alpha = 0$$ α = 0 ) as special cases. An $$\alpha $$ α -CS game may not be a classical TU cooperative game as it may fail to satisfy the condition that the worth of the empty set is 0, except when $$\alpha =1$$ α = 1 . In this paper, we (i) extend the $$\alpha $$ α -quasi-Shapley value giving the Shapley value for constant-sum games and quasi-Shapley-value for threat games to any class of $$\alpha $$ α -CS games, (ii) extend the axiomatizations of Khmelnitskaya (2003) and Kohlberg and Neyman (2018) to any class of $$\alpha $$ α -CS games, and (iii) introduce a new efficiency axiom which, together with other classical axioms, characterizes a solution that is defined by exactly the Shapley value formula for any class of $$\alpha $$ α -CS games.

Suggested Citation

  • Wenna Wang & René van den Brink & Hao Sun & Genjiu Xu & Zhengxing Zou, 2022. "On $$\alpha $$ α -constant-sum games," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(2), pages 279-291, June.
  • Handle: RePEc:spr:jogath:v:51:y:2022:i:2:d:10.1007_s00182-021-00792-y
    DOI: 10.1007/s00182-021-00792-y
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    References listed on IDEAS

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    1. van den Brink, Rene, 2007. "Null or nullifying players: The difference between the Shapley value and equal division solutions," Journal of Economic Theory, Elsevier, vol. 136(1), pages 767-775, September.
    2. Kohlberg, Elon & Neyman, Abraham, 2018. "Games of threats," Games and Economic Behavior, Elsevier, vol. 108(C), pages 139-145.
    3. René Brink & Yukihiko Funaki, 2015. "Implementation and axiomatization of discounted Shapley values," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(2), pages 329-344, September.
    4. Anna B. Khmelnitskaya, 2003. "Shapley value for constant-sum games," International Journal of Game Theory, Springer;Game Theory Society, vol. 32(2), pages 223-227, December.
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