A local interaction dynamic for the matching problem

In the matching problem agents with partially overlapping interests must be matched pairwise. It has inspired many physicists working on complex systems who studied the properties of the stable state and the ground state by employing the tools of statistical mechanics. Here, we examine the matching problem from a different perspective by studying a dynamic evolution of a matching system. We propose a model where agents interact locally and selfishly to maximize their benefit. We investigate the dynamic and steady-state properties of our model in two different cases: when mutual benefits between agents are symmetrical and when they are not. In particular, we show analytically that the global benefit of the society in the stationary state is far from the ground state in both cases, and this distance increases with the number of agents. However, a society with symmetrical interests performs better than one with asymmetrical interests. Possible practical implications of our findings are discussed.