Evolutionary Game of Coalition Building Under External Pressure

We study the fragmentation-coagulation, or merging and splitting, model as introduced in Kolokoltsov (Math Oper Res, 2016, in press, doi:10.1287/moor.2016.0838), where N small players can form coalitions to resist to the pressure exerted by the principal. It is a Markov chain in continuous time, and the players have a common reward to optimize. We study the behavior as N grows and show that the problem converges to a (one player) deterministic optimization problem in continuous time, in the infinite dimensional state space `1. We apply the method developed in Gast et al. (IEEE Trans Autom Control 57:2266–2280, 2012), adapting it to our different framework. We use tools involving dynamics in `1, generators of Markov processes, martingale problems, and coupling of Markov chains.