Dynamic club formation with coordination

Tone  Arnold; Myrna  Wooders

doi:10.3934/jdg.2015010

Article Contents

2015, Volume 2, Issue 3&4: 341-361. Doi: 10.3934/jdg.2015010

This issue Previous Article Why do stable clearinghouses work so well? - Small sets of stable matchings in typical environments, and the limits-on-manipulation theorem of Demange, Gale and Sotomayor Next Article For claims problems, another compromise between the proportional and constrained equal awards rules

Dynamic club formation with coordination

Tone Arnold¹ and
Myrna Wooders^2,

1.
Department of Mathematics and LIAAD-INESC, University of Hohenheim, Stuttgart
2.
Department of Economics, Vanderbilt University, Nashville, Tennessee

Received: June 30, 2015

Revised: August 31, 2015

Published: June 2015

Abstract Related Papers Cited by

Abstract

We present a dynamic model of club formation in a society of identicalpeople. Coalitions consisting of members of the same club can form for oneperiod and coalition members can jointly deviate. The dynamic process isdescribed by a Markov chain defined by myopic optimization on the part ofcoalitions. We define a Nash club equilibrium (NCE) as a strategy profilethat is immune to such coalitional deviations. For single-peakedpreferences, we show that, if one exists, the process will converge to a NCEprofile with probability one. NCE is unique up to a renaming of players andlocations. Further, NCE corresponds to strong Nash equilibrium in the clubformation game. Finally, we deal with the case where NCE fails to exist.When the population size is not an integer multiple of an optimal club size,there may be `left over' players who prevent the process from `settlingdown'. To treat this case, we define the concept of $k$-remainderNCE, which requires that all but $k$ players are playing a Nash clubequilibrium, where $k$ is defined by the minimal number of left overplayers. We show that the process converges to an ergodic NCE, that is, aset of states consisting only of $k$-remainder NCE and provide somecharacterization of the set of ergodic NCE.

Keywords:

Mathematics Subject Classification: 62P20, 37N40.

Citation:

Related Papers

Cited by

References

[1]	H. Ackermann, R. Röglin and B. Vöcking, Pure Nash equilibria in player-specific and weighted congestion games, Theoretical Computer Science, 410 (2009), 1552-1563.doi: 10.1016/j.tcs.2008.12.035.
[2]	N. Allouch and M. Wooders, Price taking equilibrium in economies with multiple memberships in clubs and unbounded club sizes, Journal of Economic Theory, 140 (2008), 246-278.doi: 10.1016/j.jet.2007.07.006.
[3]	T. Arnold and U. Schwalbe, Dynamic coalition formation and the core, Journal of Economic Behavior and Organization, 49 (2002), 363-380.
[4]	V. Barham and M. Wooders, First and Second Welfare Theorems for economies with collective goods, in Topics in Public Finance, D. Pines, E. Sadka and I. Zilcha, (eds.), Cambridge University Press, (1998), 57-88.
[5]	E. Bennett and M. Wooders, Income distribution and firm formation, Journal of Comparative Economics, 3 (1979), 304-317.
[6]	A. Bogomolnaia and M. O. Jackson, The stability of hedonic coalition structures, Games and Economic Behavior, 38 (2002), 201-230.doi: 10.1006/game.2001.0877.
[7]	J. Buchanan, An economic theory of clubs, Economica, 33 (1965), 1-14.
[8]	J. P. Conley and H. Konishi, Migration-proof Tiebout equilibrium: Existence and asymptotic efficiency, Journal of Public Economics, 86 (2000), 243-262.
[9]	T. Dieckmann, The evolution of conventions with mobile players, Journal of Economic Behavior & Organization, 38 (1999), 93-111.
[10]	G. Demange D. Gale and M. Sotomayor, Multi-item auctions, Journal of Political Economy, 94 (1986), 863-872.
[11]	A. Fagebaume, D. Gale and M. Sotomayor, A note on the multiple partners assignment game, Journal of Mathematical Economics, 46 (2010), 388-392.doi: 10.1016/j.jmateco.2009.06.014.
[12]	G. Hollard, On the existence of a pure strategy Nash equilibrium in group formation games, Economics Letters, 66 (2000), 283-287.doi: 10.1016/S0165-1765(99)00193-7.
[13]	R. Holzman and N. Law-Yone, Strong equilibrium in congestion games, Games and Economic Behavior, 21 (1997), 85-101.doi: 10.1006/game.1997.0592.
[14]	M. Kaneko and M. Wooders, The core of a game with a continuum of players and finite coalitions: The model and some results, Mathematical Social Sciences, 12 (1986), 105-137.doi: 10.1016/0165-4896(86)90032-6.
[15]	J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer-Verlag, New York, 1976.
[16]	H. Konishi, S. Weber and M. Le Breton, Free mobility equilibrium in a local public goods economy with congestion, Research in Economics, 51 (1997), 19-30.
[17]	H. Konishi, M. Le Breton and S. Weber, Equilibria in a model with partial rivalry, Journal of Economic Theory, 72 (1997), 225-327.doi: 10.1006/jeth.1996.2203.
[18]	H. Konishi, M. Le Breton and S. Weber, Pure strategy Nash equilibria in a group formation game with positive externalities, Games and Economic Behavior, 21 (1997), 161-182.doi: 10.1006/game.1997.0542.
[19]	H. Konishi, M. Le Breton and S. Weber, Equilibrium in a finite local public goods economy, Journal of Economic Theory, 79 (1998), 224-244.doi: 10.1006/jeth.1997.2386.
[20]	A. Kovalenkov and M. Wooders, Approximate cores of games and economies with clubs, Journal of Economic Theory, 110 (2003), 87-120.doi: 10.1016/S0022-0531(03)00003-6.
[21]	I. Milchtaich, Congestion games with player-specific payoff functions, Games and Economic Behavior, 13 (1996), 111-124.doi: 10.1006/game.1996.0027.
[22]	I. Milchtaich and and E. Winter, Stability and segregation in group formation, Games and Economic Behavior, 38 (2002), 318-346.doi: 10.1006/game.2001.0878.
[23]	D. Monderer and L. S. Shapley, Potential games, Games and Economic Behavior, 14 (1996), 124-143.doi: 10.1006/game.1996.0044.
[24]	F. H. Page Jr. and M. Wooders, Networks and clubs, Journal of Economic Behavior & Organization, 64 (2007), 406-425.
[25]	D. Ray and R. Vohra, Equilibrium binding agreements, Journal of Economic Theory, 73 (1997), 30-78.doi: 10.1006/jeth.1996.2236.
[26]	R. W. Rosenthal, A class of games possessing pure-strategy Nash equilibria, International Journal of Game Theory, 2 (1973), 65-67.doi: 10.1007/BF01737559.
[27]	A. Roth and M. Sotomayor, Two-sided Matching: A Study in Game-theoretic Modeling and Analysis, Cambridge University Press, Cambridge, 1990.doi: 10.1017/CCOL052139015X.
[28]	M. Shubik, Bridge Game Economy: An example of Invisibilities, Journal of Political Economy, 79 (1971), 909-912.
[29]	M. Shubik and M. Wooders, Approximate cores of replica games and economies: Part I. Replica games, externalities, and approximate Cores, Mathematical Social Sciences, 6 (1983), 27-48.doi: 10.1016/0165-4896(83)90044-6.
[30]	M. Sotomayor, Three remarks on the stability of the many-to-many matching, Mathematical Social Sciences, 38 (1999), 55-70.doi: 10.1016/S0165-4896(98)00048-1.
[31]	C. Tiebout, A pure theory of local expenditures, Journal of Political Economy, 64 (1956), 416-424.
[32]	M. Wooders, Equilibria, the core, and jurisdiction structures in economies with a local public good, Journal of Economic Theory, 18 (1978), 328-348.doi: 10.1016/0022-0531(78)90087-X.
[33]	M. Wooders, The Tiebout Hypothesis: Near optimality in local public good economies, Econometrica, 48 (1980), 1467-1486.doi: 10.2307/1912819.
[34]	M. Wooders, Multijurisdictional economies, the Tiebout Hypothesis, and sorting, Proceedings of the National Academy of Sciences, 96 (1999), 10585-10587.