# American Institute of Mathematical Sciences

• Previous Article
For claims problems, another compromise between the proportional and constrained equal awards rules
• JDG Home
• This Issue
• Next 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
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 and 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), 1552-1563. 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.

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), 1552-1563. 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.
 [1] Marco Tosato, Jianhong Wu. An application of PART to the Football Manager data for players clusters analyses to inform club team formation. Big Data & Information Analytics, 2018  doi: 10.3934/bdia.2018002 [2] Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics and Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 [3] Jian Hou, Liwei Zhang. A barrier function method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1091-1108. doi: 10.3934/jimo.2014.10.1091 [4] Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A penalty method for generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2012, 8 (1) : 51-65. doi: 10.3934/jimo.2012.8.51 [5] Elvio Accinelli, Bruno Bazzano, Franco Robledo, Pablo Romero. Nash Equilibrium in evolutionary competitive models of firms and workers under external regulation. Journal of Dynamics and Games, 2015, 2 (1) : 1-32. doi: 10.3934/jdg.2015.2.1 [6] Dean A. Carlson. Finding open-loop Nash equilibrium for variational games. Conference Publications, 2005, 2005 (Special) : 153-163. doi: 10.3934/proc.2005.2005.153 [7] Shunfu Jin, Haixing Wu, Wuyi Yue, Yutaka Takahashi. Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2407-2424. doi: 10.3934/jimo.2019060 [8] Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control and Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 [9] Xiaona Fan, Li Jiang, Mengsi Li. Homotopy method for solving generalized Nash equilibrium problem with equality and inequality constraints. Journal of Industrial and Management Optimization, 2019, 15 (4) : 1795-1807. doi: 10.3934/jimo.2018123 [10] Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics and Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537 [11] Xiaolin Xu, Xiaoqiang Cai. Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium. Journal of Industrial and Management Optimization, 2008, 4 (4) : 843-859. doi: 10.3934/jimo.2008.4.843 [12] Rui Mu, Zhen Wu. Nash equilibrium points of recursive nonzero-sum stochastic differential games with unbounded coefficients and related multiple\\ dimensional BSDEs. Mathematical Control and Related Fields, 2017, 7 (2) : 289-304. doi: 10.3934/mcrf.2017010 [13] Mei Ju Luo, Yi Zeng Chen. Smoothing and sample average approximation methods for solving stochastic generalized Nash equilibrium problems. Journal of Industrial and Management Optimization, 2016, 12 (1) : 1-15. doi: 10.3934/jimo.2016.12.1 [14] Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1 [15] Bin Zhou, Hailin Sun. Two-stage stochastic variational inequalities for Cournot-Nash equilibrium with risk-averse players under uncertainty. Numerical Algebra, Control and Optimization, 2020, 10 (4) : 521-535. doi: 10.3934/naco.2020049 [16] Janos Kollar. The Nash conjecture for threefolds. Electronic Research Announcements, 1998, 4: 63-73. [17] William Geller, Bruce Kitchens, Michał Misiurewicz. Microdynamics for Nash maps. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1007-1024. doi: 10.3934/dcds.2010.27.1007 [18] Matt Barker. From mean field games to the best reply strategy in a stochastic framework. Journal of Dynamics and Games, 2019, 6 (4) : 291-314. doi: 10.3934/jdg.2019020 [19] Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260 [20] Sheri M. Markose. Complex type 4 structure changing dynamics of digital agents: Nash equilibria of a game with arms race in innovations. Journal of Dynamics and Games, 2017, 4 (3) : 255-284. doi: 10.3934/jdg.2017015

Impact Factor: