Layered networks, equilibrium dynamics, and stable coalitions

An important aspect of network dynamics that has been missing from our understanding of network dynamics in various applied settings is the influence of strategic behavior in determining equilibrium network dynamics. Our main objective hear to say what we can regarding the emergence of stable club networks - and therefore, stable coalition structures - based on the stability properties of strategically determined equilibrium network formation dynamics. Because club networks are layered networks, our work here can be thought of as a first work on the dynamics of layered networks. In addition to constructing a discounted stochastic game model (i.e., a DSG model) of club network formation, we show that (1) our DSG of network formation possesses a stationary Markov perfect equilibrium in players' membership action strategies and (2) we identify the assumptions on primitives which ensure that the induced equilibrium Markov process of layered club network formation satisfies the Tweedie Stability Conditions (2001) and that (3) as a consequence, the equilibrium Markov network formation processes generates a unique decomposition of the set of state-network pairs into a transient set together with finitely many basins of attraction. Moreover, we show that if there is a basin containing a vio set (a visited infinitely often set) of club networks sufficiently close together, then the coalition structures across club networks in the vio set will be the same (i.e., closeness across networks in a vio set leads to invariance in coalition structure across networks in a vio set).