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March & April  2015, 2(3&4): 341-361. doi: 10.3934/jdg.2015010

Dynamic club formation with coordination

1. 

Department of Mathematics and LIAAD-INESC, University of Hohenheim, Stuttgart, Germany

2. 

Department of Economics, Vanderbilt University, Nashville, Tennessee, United States

Received  July 2015 Revised  September 2015 Published  November 2015

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.
Citation: Tone Arnold, Myrna Wooders. Dynamic club formation with coordination. Journal of Dynamics & Games, 2015, 2 (3&4) : 341-361. doi: 10.3934/jdg.2015010
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. doi: 10.1016/j.tcs.2008.12.035. Google Scholar

[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. doi: 10.1016/j.jet.2007.07.006. Google Scholar

[3]

T. Arnold and U. Schwalbe, Dynamic coalition formation and the core,, Journal of Economic Behavior and Organization, 49 (2002), 363. Google Scholar

[4]

V. Barham and M. Wooders, First and Second Welfare Theorems for economies with collective goods,, in Topics in Public Finance, (1998), 57. Google Scholar

[5]

E. Bennett and M. Wooders, Income distribution and firm formation,, Journal of Comparative Economics, 3 (1979), 304. Google Scholar

[6]

A. Bogomolnaia and M. O. Jackson, The stability of hedonic coalition structures,, Games and Economic Behavior, 38 (2002), 201. doi: 10.1006/game.2001.0877. Google Scholar

[7]

J. Buchanan, An economic theory of clubs,, Economica, 33 (1965), 1. Google Scholar

[8]

J. P. Conley and H. Konishi, Migration-proof Tiebout equilibrium: Existence and asymptotic efficiency,, Journal of Public Economics, 86 (2000), 243. Google Scholar

[9]

T. Dieckmann, The evolution of conventions with mobile players,, Journal of Economic Behavior & Organization, 38 (1999), 93. Google Scholar

[10]

G. Demange D. Gale and M. Sotomayor, Multi-item auctions,, Journal of Political Economy, 94 (1986), 863. Google Scholar

[11]

A. Fagebaume, D. Gale and M. Sotomayor, A note on the multiple partners assignment game,, Journal of Mathematical Economics, 46 (2010), 388. doi: 10.1016/j.jmateco.2009.06.014. Google Scholar

[12]

G. Hollard, On the existence of a pure strategy Nash equilibrium in group formation games,, Economics Letters, 66 (2000), 283. doi: 10.1016/S0165-1765(99)00193-7. Google Scholar

[13]

R. Holzman and N. Law-Yone, Strong equilibrium in congestion games,, Games and Economic Behavior, 21 (1997), 85. doi: 10.1006/game.1997.0592. Google Scholar

[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. doi: 10.1016/0165-4896(86)90032-6. Google Scholar

[15]

J. G. Kemeny and J. L. Snell, Finite Markov Chains,, Springer-Verlag, (1976). Google Scholar

[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. Google Scholar

[17]

H. Konishi, M. Le Breton and S. Weber, Equilibria in a model with partial rivalry,, Journal of Economic Theory, 72 (1997), 225. doi: 10.1006/jeth.1996.2203. Google Scholar

[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. doi: 10.1006/game.1997.0542. Google Scholar

[19]

H. Konishi, M. Le Breton and S. Weber, Equilibrium in a finite local public goods economy,, Journal of Economic Theory, 79 (1998), 224. doi: 10.1006/jeth.1997.2386. Google Scholar

[20]

A. Kovalenkov and M. Wooders, Approximate cores of games and economies with clubs,, Journal of Economic Theory, 110 (2003), 87. doi: 10.1016/S0022-0531(03)00003-6. Google Scholar

[21]

I. Milchtaich, Congestion games with player-specific payoff functions,, Games and Economic Behavior, 13 (1996), 111. doi: 10.1006/game.1996.0027. Google Scholar

[22]

I. Milchtaich and and E. Winter, Stability and segregation in group formation,, Games and Economic Behavior, 38 (2002), 318. doi: 10.1006/game.2001.0878. Google Scholar

[23]

D. Monderer and L. S. Shapley, Potential games,, Games and Economic Behavior, 14 (1996), 124. doi: 10.1006/game.1996.0044. Google Scholar

[24]

F. H. Page Jr. and M. Wooders, Networks and clubs,, Journal of Economic Behavior & Organization, 64 (2007), 406. Google Scholar

[25]

D. Ray and R. Vohra, Equilibrium binding agreements,, Journal of Economic Theory, 73 (1997), 30. doi: 10.1006/jeth.1996.2236. Google Scholar

[26]

R. W. Rosenthal, A class of games possessing pure-strategy Nash equilibria,, International Journal of Game Theory, 2 (1973), 65. doi: 10.1007/BF01737559. Google Scholar

[27]

A. Roth and M. Sotomayor, Two-sided Matching: A Study in Game-theoretic Modeling and Analysis,, Cambridge University Press, (1990). doi: 10.1017/CCOL052139015X. Google Scholar

[28]

M. Shubik, Bridge Game Economy: An example of Invisibilities,, Journal of Political Economy, 79 (1971), 909. Google Scholar

[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. doi: 10.1016/0165-4896(83)90044-6. Google Scholar

[30]

M. Sotomayor, Three remarks on the stability of the many-to-many matching,, Mathematical Social Sciences, 38 (1999), 55. doi: 10.1016/S0165-4896(98)00048-1. Google Scholar

[31]

C. Tiebout, A pure theory of local expenditures,, Journal of Political Economy, 64 (1956), 416. Google Scholar

[32]

M. Wooders, Equilibria, the core, and jurisdiction structures in economies with a local public good,, Journal of Economic Theory, 18 (1978), 328. doi: 10.1016/0022-0531(78)90087-X. Google Scholar

[33]

M. Wooders, The Tiebout Hypothesis: Near optimality in local public good economies,, Econometrica, 48 (1980), 1467. doi: 10.2307/1912819. Google Scholar

[34]

M. Wooders, Multijurisdictional economies, the Tiebout Hypothesis, and sorting,, Proceedings of the National Academy of Sciences, 96 (1999), 10585. Google Scholar

show all references

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. doi: 10.1016/j.tcs.2008.12.035. Google Scholar

[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. doi: 10.1016/j.jet.2007.07.006. Google Scholar

[3]

T. Arnold and U. Schwalbe, Dynamic coalition formation and the core,, Journal of Economic Behavior and Organization, 49 (2002), 363. Google Scholar

[4]

V. Barham and M. Wooders, First and Second Welfare Theorems for economies with collective goods,, in Topics in Public Finance, (1998), 57. Google Scholar

[5]

E. Bennett and M. Wooders, Income distribution and firm formation,, Journal of Comparative Economics, 3 (1979), 304. Google Scholar

[6]

A. Bogomolnaia and M. O. Jackson, The stability of hedonic coalition structures,, Games and Economic Behavior, 38 (2002), 201. doi: 10.1006/game.2001.0877. Google Scholar

[7]

J. Buchanan, An economic theory of clubs,, Economica, 33 (1965), 1. Google Scholar

[8]

J. P. Conley and H. Konishi, Migration-proof Tiebout equilibrium: Existence and asymptotic efficiency,, Journal of Public Economics, 86 (2000), 243. Google Scholar

[9]

T. Dieckmann, The evolution of conventions with mobile players,, Journal of Economic Behavior & Organization, 38 (1999), 93. Google Scholar

[10]

G. Demange D. Gale and M. Sotomayor, Multi-item auctions,, Journal of Political Economy, 94 (1986), 863. Google Scholar

[11]

A. Fagebaume, D. Gale and M. Sotomayor, A note on the multiple partners assignment game,, Journal of Mathematical Economics, 46 (2010), 388. doi: 10.1016/j.jmateco.2009.06.014. Google Scholar

[12]

G. Hollard, On the existence of a pure strategy Nash equilibrium in group formation games,, Economics Letters, 66 (2000), 283. doi: 10.1016/S0165-1765(99)00193-7. Google Scholar

[13]

R. Holzman and N. Law-Yone, Strong equilibrium in congestion games,, Games and Economic Behavior, 21 (1997), 85. doi: 10.1006/game.1997.0592. Google Scholar

[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. doi: 10.1016/0165-4896(86)90032-6. Google Scholar

[15]

J. G. Kemeny and J. L. Snell, Finite Markov Chains,, Springer-Verlag, (1976). Google Scholar

[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. Google Scholar

[17]

H. Konishi, M. Le Breton and S. Weber, Equilibria in a model with partial rivalry,, Journal of Economic Theory, 72 (1997), 225. doi: 10.1006/jeth.1996.2203. Google Scholar

[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. doi: 10.1006/game.1997.0542. Google Scholar

[19]

H. Konishi, M. Le Breton and S. Weber, Equilibrium in a finite local public goods economy,, Journal of Economic Theory, 79 (1998), 224. doi: 10.1006/jeth.1997.2386. Google Scholar

[20]

A. Kovalenkov and M. Wooders, Approximate cores of games and economies with clubs,, Journal of Economic Theory, 110 (2003), 87. doi: 10.1016/S0022-0531(03)00003-6. Google Scholar

[21]

I. Milchtaich, Congestion games with player-specific payoff functions,, Games and Economic Behavior, 13 (1996), 111. doi: 10.1006/game.1996.0027. Google Scholar

[22]

I. Milchtaich and and E. Winter, Stability and segregation in group formation,, Games and Economic Behavior, 38 (2002), 318. doi: 10.1006/game.2001.0878. Google Scholar

[23]

D. Monderer and L. S. Shapley, Potential games,, Games and Economic Behavior, 14 (1996), 124. doi: 10.1006/game.1996.0044. Google Scholar

[24]

F. H. Page Jr. and M. Wooders, Networks and clubs,, Journal of Economic Behavior & Organization, 64 (2007), 406. Google Scholar

[25]

D. Ray and R. Vohra, Equilibrium binding agreements,, Journal of Economic Theory, 73 (1997), 30. doi: 10.1006/jeth.1996.2236. Google Scholar

[26]

R. W. Rosenthal, A class of games possessing pure-strategy Nash equilibria,, International Journal of Game Theory, 2 (1973), 65. doi: 10.1007/BF01737559. Google Scholar

[27]

A. Roth and M. Sotomayor, Two-sided Matching: A Study in Game-theoretic Modeling and Analysis,, Cambridge University Press, (1990). doi: 10.1017/CCOL052139015X. Google Scholar

[28]

M. Shubik, Bridge Game Economy: An example of Invisibilities,, Journal of Political Economy, 79 (1971), 909. Google Scholar

[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. doi: 10.1016/0165-4896(83)90044-6. Google Scholar

[30]

M. Sotomayor, Three remarks on the stability of the many-to-many matching,, Mathematical Social Sciences, 38 (1999), 55. doi: 10.1016/S0165-4896(98)00048-1. Google Scholar

[31]

C. Tiebout, A pure theory of local expenditures,, Journal of Political Economy, 64 (1956), 416. Google Scholar

[32]

M. Wooders, Equilibria, the core, and jurisdiction structures in economies with a local public good,, Journal of Economic Theory, 18 (1978), 328. doi: 10.1016/0022-0531(78)90087-X. Google Scholar

[33]

M. Wooders, The Tiebout Hypothesis: Near optimality in local public good economies,, Econometrica, 48 (1980), 1467. doi: 10.2307/1912819. Google Scholar

[34]

M. Wooders, Multijurisdictional economies, the Tiebout Hypothesis, and sorting,, Proceedings of the National Academy of Sciences, 96 (1999), 10585. Google Scholar

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