Abstract
I propose a model of coalitional bargaining with claims in order to find solutions for games with non-transferable utility and externalities. I show that, for each such game, payoff configurations exist which will not be renegotiated. In the ordinal game derived from these payoff configurations, a core stable partition can be found, i.e. a partition in which no group of players has an incentive to jointly change their coalitions.
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Notes
For \(x,y\in \mathbb R^S\) I write \(x\ge y\) if \(x_i\ge y_i\) for all \(i\in S\), I write \(x>y\) if \(x\ge y\) and \(x\ne y\), and I write \(x\gg y\) if \(x_i>y_i\) for all \(i\in S\).
For a vector \(d=(d_i)_{i\in N}\in \mathbb R^N\) I write \(d_S\) for \((d_i)_{i\in S}\in \mathbb R^S\).
Note that in the original definition of Chun and Thomson (1992) it is required that \(c\notin X\). I do not impose this condition.
See Karos (2015) for details.
A proper monotonic simple game is a function \(v:\mathcal P\rightarrow \left\{ 0,1\right\} \) such that \(v\left( N\right) =1, v\left( S\right) \le v\left( T\right) \) if \(S\subseteq T\), and \(v\left( S\right) + v\left( N{\setminus } S\right) \le 1\) for all nonempty \(S\subsetneq N\).
That is \(V(S)=\left\{ x\in \mathbb R^S:\;\sum _{i\in S}x_i\le v(S)\right\} \) for all \(S\in \mathcal P\).
In order to avoid brackets I use the notation jk for coalition \(\{j,k\}\) and accordingly \(\{i,jk\}\) for the partition \(\{\{i\},\{j,k\}\}\) here and in the remainder of the paper.
This idea is based on Shenoy (1979).
A deeper investigation of this solution in the context of proper monotonic simple games, including a example where uniqueness fails for \(\mu =0\), can be found in Karos (2013).
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References
Aumann R (1967) A survey of cooperative games without side payments. In: Shubik M (ed) Essays in mathematical economics in honor of Oskar Morgenstern. Princeton University Press, Princeton, pp 3–27
Aumann R, Maschler M (1985) Game theoretic analysis of a bankruptcy problem from the Talmud. J Econ Theory 36:195–213
Banerjee S, Konishi H, Sönmez T (2001) Core in a simple coalition formation game. Soc Choice Welf 19:135–153
Bogomolnaia A, Jackson M (2002) The stability of hedonic coalition structures. Games Econ Behav 38:201–230
Chun Y, Thomson W (1992) Bargaining problems with claims. Math Soc Sci 24:19–33
Curiel I, Maschler M, Tijs S (1987) Bankruptcy games. Z Oper Res 31:143–159
Davis M, Maschler M (1963) Existence of stable payoff configurations for cooperative games. Bull Am Math Soc 69:106–108
de Clippel G, Serrano S (2008) Marginal contributions and externalities in the value. Econometrica 76:1413–1436
Dimitrov D, Haake C (2008) Stable governments and the semistrict core. Games Econ Behav 62:460–475
Drèze J, Greenberg J (1980) Hedonic coalitions: optimalilty and stability. Econometrica 48:987–1003
Gillies D (1959) Solutions to general non-zero-sum games. In: Kuhn A, Luce R (eds) Contributions to the theory of games. Princeton University Press, Princeton, pp 47–85
Harsanyi J (1963) A simplified bargaining model for the n-person cooperative game. Int Econ Rev 4:194–220
Hougaard J, Moreno-Ternero J, Østerdaal (2012) A unifying framework for the problem of adjudicating conflicting claims. J Math Econ 48:107–114
Hougaard J, Moreno-Ternero J, Østerdaal (2013) Rationing in the presence of baseline. Soc Choice Welf 40:1047–1066
Iehlé V (2007) The core-partition of hedonic games. Math Soc Sci 54:176–185
Kalai E (1977) Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45:1623–1630
Kalai E, Smorodinsky M (1975) Other solutions to Nash’s bargaining problem. Econometrica 43:513–518
Karos D (2013) Bargaining and power. FEEM Working Paper No. 63
Karos D (2014) Coalition formation in general apex games under monotonic power indices. Games Econ Behav 87:239–252
Karos D (2015) Stable partitions for games with non-tranferable utilities and externalities. Oxford Working Paper Series No. 741
Luce R, Raiffa H (1957) Games and decisions. Wiley, New York
Moulin H (2000) Priority rules and other assymmetric rationing methods. Econometrica 68:643–684
Nash J (1950) The bargaining problem. Econometrica 18:155–162
Nydegger R, Owen G (1974) Two-person bargaining: an experimental test of the Nash axioms. Int J Game Theory 3:239–249
Oosterbeek H, Sloof R, van de Kuilen G (2004) Cultural differences in ultimatum game experiments: evidence from a meta-analysis. Exp Econ 6:171–188
Otten G, Borm P, Peleg B, Tijs S (1998) The MC-value for monotonic NTU-games. Int J Game Theory 27:37–47
Peleg B (1963) Bargaining sets for cooperative games without side payments. Isr J Math 1:197–200
Peters H (1992) Axiomatic bargaining theory. Kluwer Academic, Dordrecht
Ray D, Vohra R (1997) Equilibrium binding agreements. J Econ Theory 73:30–78
Shenoy P (1979) On coalition formation: a game-theoretical approach. Int J Game Theory 8:133–164
Thrall R, Lucas W (1963) \(n\)-person games in partition function form. Naval Res Logist Q 10:281–298
von Neumann J, Morgenstern O (1944) Theory of games and economic behaviour. Princeton University Press, Princeton
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I thank Hans Peters, Hervé Moulin, and Geoffroy de Clippel for their useful comments on a previous version of this paper. Also thanks to the two anonymous referees and the editor in charge for their insightful suggestions.
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Karos, D. Stable partitions for games with non-transferable utility and externalities. Int J Game Theory 45, 817–838 (2016). https://doi.org/10.1007/s00182-015-0487-5
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DOI: https://doi.org/10.1007/s00182-015-0487-5
Keywords
- Games with non-transferable utility in partition function form
- Bargaining with claims
- Ordinal games
- Core stable partitions