Dynamic Club Formation with Coordination

Abstract

We present a dynamic model of jurisdiction formation in a society of identical people. The process is described by a Markov chain that is de¯ned by myopic optimization on the part of the players. We show that the process will converge to a Nash equilibrium club structure. Next, we allow for coordination between members of the same club, i. e. club members can form coalitions for one period and deviate jointly. We de¯ne a Nash club equilibrium (NCE) as a strategy con¯guration that is immune to such coalitional deviations. We show that, if one exists, this modi¯ed process will converge to a NCE con¯guration with probability one. Finally, we deal with the case where a NCE fails to exist due to indivisibility problems. When the population size is not an integer multiple of the optimal club size, there will be left over players who prevent the process from settling down. We de¯ne the concept of an approximate Nash club equilibrium (ANCE), which means that all but k players are playing a Nash club equilibrium, where k is de¯ned by the minimal number of left over players. We show that the modified process converges to an ergodic set of states each of which is ANCE.