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

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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: July 2015

Revised: September 2015

Published: July 2015

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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.

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Mathematics Subject Classification: 62P20, 37N40.

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