Abstract
The Shapley value, one of the most widespread concepts in operations Research applications of cooperative game theory, was defined and axiomatically characterized in different game-theoretic models. In this paper we provide a new axiomatization of the Shapley value which probably is the most important one-point solution concept for the cooperative games using a differential marginality axiom. This axiom states that two players’ payoff differential is completely determined by the differences of their marginal contributions. Efficiency means that the worth generated by the grand coalition is fully allocated to the players. Null player property means that the marginal contributions of null players in a game are zero payoffs. In this study we show that the Shapley value is redefined by efficiency axiom, the null player property and differential marginality axiom.
Keywords
Game theory Cooperative games Shapley value Axiomatic characterization Fairness axiom Differential marginality axiom.
References
- A. Casajus, “Differential marginality, van den Brink fairness, and the Shapley value,” Theory Decis., 71 (2), 163-174, 2011.
- A. Casajus, “The Shapley value without efficiency and additivity,” Math. Social Sci., 68, 1-4, 2014.
- D. B. Gillies, “Solutions to general non-zero-sum games,” in Contributions to the Theory of Games IV, vol. 40, A. W. Tucker, R. D. Luce, Ed. Princeton: Princeton UP, 1959, pp 47-85.
- H. P. Young, “Monotonic solutions of cooperative games,” Int. J. Game Theory, 14, 65-72, 1985.
- J. C. Harsanyi, “A Bargaining Model for Cooperative n-Person Games,” in Contributions to the Theory of Games IV, vol. 40, A. W. Tucker, R. D. Luce, Ed. Princeton: Princeton UP, 1959, pp 625-355.
- J. von Neumann, “Zur theorie der gesellschaftsspiele,” Math. Ann., 100 (1), 295-320, 1928.
- J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944.
- L. S. Shapley, “A value for n-person games,” Ann. Math. Stud., 28, 307-317, 1953
- O. Palancı, M. Ekici and S. Z. A. Gök, “On the equal surplus sharing interval solutions and an application,” J. Dyn. Games., 8 (2), 139-150, 2021.
- R. van den Brink, “An axiomatization of the Shapley value using a fairness property,” Int. J. Game Theory, 30, 309-319, 2001.
- S. H. Tijs, “Bounds for the core and the τ-value,” in Game Theory and Mathematical Economics, O. Moeschlin, D. Pallaschke, Ed. Amsterdam: North-Holland Publishing Company, 1981, pp 123-132.
- S. H. Tijs, Introduction to Game Theory, Hindustan Book Agency, India, 2003.
- U. A. Yılmaz, S. Z. A. Gök, M. Ekici and O. Palancı, “On the grey equal surplus sharing solutions,” Int. J. Supply Oper. Manag., 5 (1), 1-10, 2018.
- Y. Chun, “On the symmetric and weighted Shapley values,” Int. J. Game Theory, 20, 183-190, 1991.
Abstract
İşbirlikçi oyun teorisinin yöneylem araştırması uygulamalarındaki en yaygın kullanılan çözümlerinden biri olan Shapley değeri farklı oyun teorisi modellerinde tanımlanmış ve aksiyomatik olarak karakterize edilmiştir. Bu makalede diferansiyel marjinaliti aksiyomu kullanılarak işbirlikçi oyunlardaki en önemli çözüm kavramlarından biri olan Shapley değeri için yeni bir karakterizasyon verilecektir. Bu aksiyom, iki oyuncunun ödeme farklılıklarının sadece marjinal katkılarının farklılıklarına bağlı olduğunu göstermektedir. Verimlilik aksiyomu, oyuncuların ödemelerinin toplamının büyük koalisyonun ödemesine eşit olması olarak tanımlanmaktadır. Null oyuncu özelliği, bir oyunda null oyuncu varsa bu oyuncunun oyuna herhangi bir katkısı olmaması demektir. Bu çalışmada, Shapley değeri verimlilik aksiyomu, null oyuncu özelliği ve diferansiyel marjinaliti aksiyomları ile yeniden tanımlanacaktır.
Keywords
Oyun teorisi İşbirlikçi oyunlar Shapley değeri Aksiyomatik karakterizasyon Adalet aksiyomu Diferansiyel marjinaliti aksiyomu.
References
- A. Casajus, “Differential marginality, van den Brink fairness, and the Shapley value,” Theory Decis., 71 (2), 163-174, 2011.
- A. Casajus, “The Shapley value without efficiency and additivity,” Math. Social Sci., 68, 1-4, 2014.
- D. B. Gillies, “Solutions to general non-zero-sum games,” in Contributions to the Theory of Games IV, vol. 40, A. W. Tucker, R. D. Luce, Ed. Princeton: Princeton UP, 1959, pp 47-85.
- H. P. Young, “Monotonic solutions of cooperative games,” Int. J. Game Theory, 14, 65-72, 1985.
- J. C. Harsanyi, “A Bargaining Model for Cooperative n-Person Games,” in Contributions to the Theory of Games IV, vol. 40, A. W. Tucker, R. D. Luce, Ed. Princeton: Princeton UP, 1959, pp 625-355.
- J. von Neumann, “Zur theorie der gesellschaftsspiele,” Math. Ann., 100 (1), 295-320, 1928.
- J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944.
- L. S. Shapley, “A value for n-person games,” Ann. Math. Stud., 28, 307-317, 1953
- O. Palancı, M. Ekici and S. Z. A. Gök, “On the equal surplus sharing interval solutions and an application,” J. Dyn. Games., 8 (2), 139-150, 2021.
- R. van den Brink, “An axiomatization of the Shapley value using a fairness property,” Int. J. Game Theory, 30, 309-319, 2001.
- S. H. Tijs, “Bounds for the core and the τ-value,” in Game Theory and Mathematical Economics, O. Moeschlin, D. Pallaschke, Ed. Amsterdam: North-Holland Publishing Company, 1981, pp 123-132.
- S. H. Tijs, Introduction to Game Theory, Hindustan Book Agency, India, 2003.
- U. A. Yılmaz, S. Z. A. Gök, M. Ekici and O. Palancı, “On the grey equal surplus sharing solutions,” Int. J. Supply Oper. Manag., 5 (1), 1-10, 2018.
- Y. Chun, “On the symmetric and weighted Shapley values,” Int. J. Game Theory, 20, 183-190, 1991.
Details
| Primary Language | Turkish |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Makaleler |
| Authors | |
| Publication Date | November 25, 2022 |
| Published in Issue | Year 2022 |