In this paper, we investigate the equal surplus division value for cooperative games with a level structure, which is a sequence of coalition structures becoming coarser and coarser. We propose three axiomatizations of the value. Among them, the first two use different variations of the recent population solidarity axiom, and the third one invokes a special reduced game consistency axiom. Due to the existence of a level structure, our axioms impose special restrictions on the players we focus. We show that our value can be characterized with these axioms and other variations of well-known axioms, such as efficiency, standardness, and quotient game property. Besides characterizing the value, we also connect it to the recent field of ordered tree cooperative games, wherein we find that the iterative value can be viewed as a special case of our value.
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Note that van den Brink et al. (2016) focus on convex combinations of the equal division value and the ESD-value, thus axiomatizations of the two values are special cases of their axiomatizations. In this paper, for illustration, we just focus on the ESD-value.
van den Brink et al. (2016) use a different population solidarity proposed by Chun and Park (2012). Nonetheless, as indicated by Chun and Park (2012), “population fairness implies both population solidarity and population fair-ranking”. It is therefore the population solidarity in van den Brink et al. (2016) can be replaced with population fairness, which is stronger than our population solidarity.
Albizuri MJ, Aurrecoechea J, Zarzuelo JM (2006) Configuration values: extensions of the coalitional owen value. Games Econ Behav 57(1):1–17
Alonso-Meijide J, Álvarez-Mozos M, Fiestras-Janeiro M (2012) Notes on a comment on 2-efficiency and the Banzhaf value. Appl Math Lett 25(7):1098–1100
Álvarez-Mozos M, Tejada O (2011) Parallel characterizations of a generalized Shapley value and a generalized Banzhaf value for cooperative games with level structure of cooperation. Decis Support Syst 52(1):21–27
Álvarez-Mozos M, van den Brink R, van der Laan G, Tejada O (2013) Share functions for cooperative games with levels structure of cooperation. Eur J Oper Res 224(1):167–179
Álvarez-Mozos M, van den Brink R, van der Laan G, Tejada O (2017) From hierarchies to levels: New solutions for games with hierarchical structure. Int J Game Theory 46(4):1–25
Aumann RJ, Drèze JH (1974) Cooperative games with coalition structures. Int J Game Theory 3(4):217–237
Banzhaf JF (1965) Weighted voting doesn’t work: a mathematical analysis. Rutgers Law Rev 19:317–343
Béal S, Férrières S, Rémila E, Solal P (2018) Axiomatization of an allocation rule for ordered tree TU-games. Math Soc Sci 93:132–140
Béal S, Rémila E, Solal P (2015) Preserving or removing special players: what keeps your payoff unchanged in TU-games? Math Soc Sci 73:23–31
Calvo E, Gutiérrez E (2010) Solidarity in games with a coalition structure. Math Soc Sci 60(3):196–203
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, Huettner F (2014) Null, nullifying, or dummifying players: the difference between the Shapley value, the equal division value, and the equal surplus division value. Econ Lett 122(2):167–169
Chun Y, Park B (2012) Population solidarity, population fair-ranking, and the egalitarian value. Int J Game Theory 41(2):255–270
Driessen TSH, Funaki Y (1991) Coincidence of and collinearity between game theoretic solutions. Oper Res Spektrum 13(1):15–30
Ferrières S (2017) Nullified equal loss property and equal division values. Theory Decis 83(3):385–406
Funaki Y, Yamato T (2001) The core and consistency properties: a general characterisation. Int Game Theory Rev 3:175–187
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
Gómez-Rúa M, Vidal-Puga J (2011) Balanced per capita contributions and level structure of cooperation. Top 19(1):167–176
Hu XF (2019) Coalitional surplus desirability and the equal surplus division value. Econ Lett 179:1–4
Hu XF, Li DF (2015) Analytic relationship between Shapley and Winter values. Oper Res Trans 19:114–120 (In Chinese)
Hu XF, Li DF (2016) The multi-step Shapley value of transferable utility cooperative games with a level structure. Syst Eng Theory Pract 36(7):1863–1870 (In Chinese)
Hu XF, Li DF (2017) The collective value of transferable utility cooperative games with a level structure. J Syst Sci Math Sci 37(1):172–185 (In Chinese)
Hu XF, Li DF (2018) A new axiomatization of a class of equal surplus division values for TU games. RAIRO Oper Res 52:935–942
Hu XF, Li DF, Xu GJ (2018) Fair distribution of surplus and efficient extensions of the Myerson value. Econ Lett 165:1–5
Hu XF, Xu GJ, Li DF (2019) The egalitarian efficient extension of the Aumann–Drèze value. J Optim Theory Appl 181:1033–1052
Kamijo Y (2009) A two-step Shapley value for cooperative games with coalition structures. Int Game Theory Rev 11(2):207–214
Kamijo Y (2013) The collective value: a new solution for games with coalition structures. Top 21(3):572–589
Kamijo Y, Kongo T (2010) Axiomatization of the Shapley value using the balanced cycle contributions property. Int J Game Theory 39(4):563–571
Khmelnitskaya AB, Yanovskaya EB (2007) Owen coalitional value without additivity axiom. Math Methods Oper Res 66(2):255–261
Kongo T (2018) Effects of players’ nullification and equal (surplus) division values. Int Game Theory Rev 20(1):1750029
Lehrer E (1988) An axiomatization of the Banzhaf value. Int J Game Theory 17(2):89–99
Li DF, Hu XF (2017) The \(\tau \)-value of transferable utility cooperative games with level structures. J Syst Eng 32(2):177–187 (In Chinese)
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) Math Econ Game Theory. Springer, 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, vol II. Princeton University Press, Princeton, pp 307–317
Tejada O, Álvarez-Mozos M (2018) Graphs and (levels of) cooperation in games: two ways how to allocate the surplus. Math Soc Sci 93:114–122
Thomson W (1983) Problems of fair division and the egalitarian solution. J Econ Theory 31(2):211–226
Thomson W (1995) Population monotonic allocation rules. In: Barnett WA, Moulin H, Salles M, J, S. N., (eds) Social choice, welfare, and ethics, pp 79–124. Cambridge University Press, Cambridge
van den Brink R (2007) Null or nullifying players: the difference between the Shapley value and equal division solutions. J Econ Theory 136(1):767–775
van den Brink R, Chun Y, Funaki Y, Park B (2016) Consistency, population solidarity, and egalitarian solutions for TU-games. Theory Dec 81(3):427–447
van den Brink R, Funaki Y (2009) Axiomatizations of a class of equal surplus sharing solutions for TU-games. Theory Decis 67(3):303–340
van den Brink R, van der Laan G, Moes N (2015) Values for transferable utility games with coalition and graph structure. Top 23(1):1–23
Vidal-Puga JJ (2005) Implementation of the levels structure value. Ann Oper Res 137(1):191–209
Winter E (1989) A value for cooperative games with levels structure of cooperation. Int J Game Theory 18(2):227–240
Xu GJ, van den Brink R, van den Laan G, Sun H (2015) Associated consistency characterization of two linear values for TU games by matrix approach. Linear Algebra Its Appl 471:224–240
Yokote K, Kongo T, Funaki Y (2019) Relationally equal treatment of equals and affine combinations of values for TU games. Soc Choice Welf 53:197–212
This work was supported by the National Nature Science Foundation of China (71901076) and the National Social Science Fund of China (18ZDA043).
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Communicated by Edith van der Wal.
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Hu, XF., Li, DF. The Equal Surplus Division Value for Cooperative Games with a Level Structure. Group Decis Negot (2020). https://doi.org/10.1007/s10726-020-09680-4
- Cooperative game
- Level structure
- Equal surplus division value
- Ordered tree cooperative game
Mathematics Subject Classification