Abstract
In this paper, we propose three new axiomatizations of the Owen value, similar as the axiomatizations of the Shapley value of Chun (Int J Game Theory 20(2):183–190, 1991), van den Brink (Int J Game Theory 30(3):309–319, 2002), and Manuel et al. (Math Methods Oper Res 77:1–14, 2013), respectively. Firstly, we show that the additivity and null player property in Owen’s (in: Henn and Moeschlin (eds) Mathematical economics and game theory, Springer-Verlog, Berlin, 1977) axiomatization can be weakened into coalitional strategic equivalence. And then, we prove that the coalitional symmetry (respectively symmetry within union) and additivity in Owen’s (in: Henn and Moeschlin (eds) Mathematical economics and game theory, Springer-Verlog, Berlin, 1977) axiomatization can be weakened into a variation of fairness, named as coalitional fairness (respectively fairness within union). Finally, we show that the two fairness axioms in our second axiomatization can be weakened into two axioms, involving a special relation between players, named as indifference. Besides characterizing the Owen value, we also illustrate that our results can be extended to the Winter value, being a common single-valued solution for cooperative games with a level structure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
For simplicities of notations, we omit the braces for singletons, when no confusion occurs.
Note that E, A, and N are defined over \({\mathcal {CG}}\). As mentioned before, \({\mathcal {G}}\subseteq {\mathcal {CG}}\), they are also well-defined over \({\mathcal {G}}\).
Ba-Sh can be defined for every \((N,v,{\mathcal {C}})\in {\mathcal {CG}}\) and \(i\in N\) by
$$\begin{aligned} \text {Ba-Sh}_i(N,v,{\mathcal {C}}) =\sum _{T\subseteq N:i\in T}\Delta _v(T)\cdot \frac{\sum _{j\in N}\text {Ba-Sh}_i(N,u_T,{\mathcal {C}})}{|{\mathcal {D}}_T|\cdot |C(i)\cap T|}. \end{aligned}$$Following the same line with the Owen value, and considering the fact that Ba-Sh satisfies null players out [implied by A and N (Derks and Haller 1999)], it is easily to verify that Ba-Sh satisfies IPO.
For more details about level games, see Hu and Li (2020) and the references therein.
This definition was first presented by Calvo et al. (1996).
References
Alonso-Meijide JM, Fiestras-Janeiro MG (2002) Modification of the Banzhaf value for games with a coalition structure. Ann Oper Res 109:213–227
Aumann RJ, Drèze JH (1974) Cooperative games with coalition structures. Int J Game Theory 3(4):217–237
Calvo E, Gutiérrez E (2013) The Shapley-solidarity value for games with a coalition structure. Int Game Theory Rev 15(1):1350002
Calvo E, Lasaga JJ, Winter E (1996) The principle of balanced contributions and hierarchies of cooperation. Math Soc Sci 31(3):171–182
Casajus A (2010) Another characterization of the Owen value without the additivity axiom. Theory Decis 69(4):523–536
Casajus A (2011) Differential marginality, van den brink fairness, and the Shapley value. Theory Decis 71(2):163–174
Casajus A (2014) The Shapley value without efficiency and additivity. Math Soc Sci 68:1–4
Chun Y (1989) A new axiomatization of the Shapley value. Games Econ Behav 1(2):119–130
Chun Y (1991) On the symmetric and weighted Shapley values. Int J Game Theory 20(2):183–190
Derks JJM, Haller HH (1999) Null players out? Linear values for games with variable supports. Int Game Theory Rev 1:201–314
Gómez-Rúa M, Vidal-Puga J (2010) The axiomatic approach to three values in games with coalition structure. Eur J Oper Res 207(2):795–806
Hart S, Mas-Colell A (1989) Potential, value, and consistency. Econometrica 57(3):589–614
Hu XF (2020) The weighted Shapley-egalitarian value for cooperative games with a coalition structure. Top 28:193–212
Hu XF, Li DF (2018) A new axiomatization of the Shapley-solidarity value for games with a coalition structure. Oper Res Lett 46(2):163–167
Hu XF, Li DF (2020) The equal surplus division value for cooperative games with a level structure. Group Decis Negotiat. https://doi.org/10.1007/s10726-020-09680-4
Khmelnitskaya AB, Yanovskaya EB (2007) Owen coalitional value without additivity axiom. Math Methods Oper Res 66(2):255–261
Levy A, McLean RP (1989) Weighted coalition structure values. Games Econ Behav 1(3):234–249
Manuel C, González-Arangüena E, van den Brink R (2013) Players indifferent to cooperate and characterizations of the Shapley value. Math Methods Oper Res 77:1–14
Myerson RB (1980) Conference structures and fair allocation rules. Int J Game Theory 9(3):169–182
Owen G (1977) Values of games with a priori unions. In: Henn R, Moeschlin O (eds) Mathematical economics and game theory. Springer-Verlag, Berlin, pp 76–88
Shapley LS (1953) A value for \(n\)-person games. In: Kuhn HW, Tucker AW (eds) Contributions to the theory of games I. Princeton University Press, Princeton, pp 307–317
Shubik M (1962) Incentives, decentralized control, the assignment of joint costs and internal pricing. Manag Sci 8(3):325–343
van den Brink R (2002) An axiomatization of the Shapley value using a fairness property. Int J Game Theory 30(3):309–319
Winter E (1989) A value for cooperative games with levels structure of cooperation. Int J Game Theory 18(2):227–240
Winter E (1992) The consistency and potential for values of games with coalition structure. Games Econ Behav 4:132–144
Young HP (1985) Monotonic solutions of cooperative games. Int J Game Theory 14(2):65–72
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by the National Nature Science Foundation of China (71901076, 71871206) and the National Social Science Fund of China (18ZDA043).
Rights and permissions
About this article
Cite this article
Hu, XF. New axiomatizations of the Owen value. Math Meth Oper Res 93, 585–603 (2021). https://doi.org/10.1007/s00186-021-00743-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-021-00743-z