Abstract
The approximate core and the aspiration core are two non-empty solutions for cooperative games that have emerged in order to give an answer to cooperative games with an empty core. Although the approximate core and the aspiration core come from two different ideas, we show that both solutions are related in a very interesting way in partitioning games (or superadditive games). In fact, we prove that the approximate core converges to the aspiration core in partitioning games (or superadditive games).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
\((N,\pi ,\bar{v})\) is called a game with restricted cooperation in Pulido and Sánchez (2006). In that paper the grand coalition is always feasible (\(N\in \pi \)) and the players are not reorganized in partitions taken from \(\pi .\)
The 1981 version of the paper is the Cowles Foundation Discussion Paper No. 612 that was published in 1983.
As usual, \(2^{N}\) denotes the set of all the coalitions (subsets) of \(N.\)
There could be other pairs \((N,\pi )\) such that \((N,v)\) is associated to \( (N,\pi )\) for some \(\pi \subset 2^{N}\).
\(N\) is identified with \(N_{1}\). If \(S\subset N, \) we have that \(S\) is identified with \(\{(i,1):i\in S\}.\) Then, we can consider that \(S\) is a subset of \(N_{r}.\)
This replication is due to Kaneko and Wooders (1982).
The payoffs in the \(\epsilon \)-core may not have the equal-treatment property.
On the class of normalized games in \(GS(N,\pi )\), (games \((N,v)\) such that \(v(i)\ge 0\) for all \(i\in N\) and \(v(N)\le \left| N\right| \)) how large \(r\) is only depends on \(\pi ,\) and it is independent of the function \(v.\) In the class of all games \(GS(N,\pi ),\) how large \(r\) is depends on \(\pi \) and \(v.\)
As usual, \((x-\epsilon )=(x^{iq}-\epsilon )_{(i,q)\in N\times \{1,\ldots ,r\}.}\)
The game \((N,\tilde{v})\) is called the balanced cover of \((N,v).\)
The limit notion is the classical one used in set theory. Given a set \(X\) and an indexed collection of subsets \((A_{\epsilon })_{\epsilon \in (0,\infty )}\) of \(X\) such that \(A_{\epsilon }\subset A_{\epsilon ^{\prime }}\) if \(\epsilon <\epsilon ^{\prime },\) the limit of \(A_{\epsilon }\) when \(\epsilon \) tends to zero is,
$$\begin{aligned} \lim \limits _{\epsilon \rightarrow 0}A_{\epsilon }=\bigcap \limits _{\epsilon >0}A_{\epsilon }. \end{aligned}$$See footnote 7.
If \(A\subset \mathbf {R}^{n},\) then an element \(x\) is in \(A-\{(\epsilon ,\ldots ,\epsilon )\}\) if and only if \(x\) \(=\) \((y_{1}-\epsilon ,\ldots ,y_{n}-\epsilon )\) with \(y\) \(\in A\).
References
Aguilera N, Escalante M (2010) A polyhedral approach to the stability of a family of coalitions. Discret Appl Math 158:379–396
Albers W (1979) Core and kernel variants based on imputations and demand profiles. In: Moeschlin O, Pallaschke D (eds) Game theory and related topics. North Holland, Amsterdam
Auriol I, Marchi E (2009) Configuration values: extensions of the Owen value. Int Game Theory Rev 11(01):99–109
Arribillaga R. (2013) “ Axiomatizing core extensions on NTU games” (mimeo, Universidad Nacional de San Luis).
Bejan C, Gomez JC (2012) Axiomatizing core extensions. Int J Game Theory 41(4):885–898
Bennett E (1983) The aspiration approach to predicting coalition formation and payoff distribution in sidepayment games. Int J Game Theory 12:1–28
Cesco J (2012) Non-empty core-type solutions over balanced coalitions in TU-games. Int Game Theory Rev 14(3):14–18
Cross J (1967) Some theoretic characteristics of economic and political coalition. J Confl Resol 11:184–195
Gillies D (1959) Solutions to general non-zero-sum games. In: Tucker TW, Luce RD (eds) Contributions to the Theory of Games IV. Princeton University Press, Princeton, pp 47–85
Kaneko M, Wooders M (1982) Core of partitioning games. Math Soc Sci 3:313–327
Kovalenkov A, Wooders M (2003) Approximate cores of games and economies with clubs. J Econ Theory 110(1):87–120
Pulido MA, Sánchez-Soriano J (2006) Characterization of the core in games with restricted cooperation. Eur J Oper Res 175:860–869
Shapley L, Shubik M (1966) Quasi-cores in a monetary economy with non-convex preferences. Econometrica 34:805–827
Shapley L, Shubik M (1972) The assignment game I: the core. Int J Game Theory 11:111–130
Shubik M (1971) The Bridge Game economy: An example of indivisibilities. J Polit Econ 79:909–912
Solymosi T (2008) Bargaining sets and the core in partitioning games. Cent Eur J Oper Res 16(4):425–440
Wooders M (1983) The epsilon core of a large replica game. J Math Econ 11:277–300
Wooders M (2008) Small group effectiveness, per capita boundedness and nonemptiness of approximate cores. J Math Econ 44:888–906
Acknowledgments
I acknowledge financial support from Universidad Nacional de San Luis, through grant 319502, and from Consejo Nacional de Investigaciones Cient íficas y Técnicas (CONICET), through grant PIP 112-200801-00655. I thanks anonymous referees for valuable comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arribillaga, R.P. Convergence of the approximate cores to the aspiration core in partitioning games. TOP 23, 521–534 (2015). https://doi.org/10.1007/s11750-014-0351-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-014-0351-y