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We show that all discounted stochastic games DSGs satisfying the usual assumptions have Nash payoff selection correspondences having fixed points. Our fixed point result is surprising because it is well known that Nash payoff selection correspondences are badly behaved, being in general neither convex valued nor closed valued in the appropriate topologies (in this case the weak star topologies). Here we show that because all DSGs satisfying the usual assumptions have upper Caratheodory (uC) Nash (equilibrium) correspondences containing uC Nash sub-correspondences having the 3M property (defined here), these uC Nash sub-correspondences are continuum valued and therefore induce interval-valued uC player payoff sub-correspondences - and therefore, Caratheodory approximable uC player payoff sub-correspondences. Finally, because these uC player payoff sub-correspondences are Caratheodory approximable, their induced Nash payoff selection sub-correspondences have fixed points - implying that the DSGs to which they belong have stationary Markov perfect equilibria.

Under mild assumptions on primitives, we show that all parameterized state-contingent games (PSGs) have upper Caratheodory (uC) Nash (equilibrium) correspondences which contain minimal uC Nash correspondences having the 3M property (defined here). This implies that all PSGs have Nash correspondences made up of minimal uC Nash correspondences taking closed, connected, and essential Nash equilibrium values (essential in the sense of Fort, 1950). It then follows from Fu and Page (2022b), that because all PSGs have continuum valued minimal Nash correspondences, all PSGs have Caratheodory approximable Nash payoff correspondences - which in turn implies that all PSGs have approximable Nash payoff selection correspondences, and therefore have Nash payoff selection correspondences with fixed points.

We show that any measurable selection valued correspondence induced by the composition of an m-tuple of real-valued Caratheodory functions with an upper Caratheodory (uC) correspondence has fixed points if the underlying uC correspondence in the composition contains a continuum valued uC sub-correspondence. Moreover, this composition of the m-tuple of real-valued Caratheodory functions with the continuum valued uC sub-correspondence induces a measurable selection valued sub-correspondence that is weak star upper semicontinuous.

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).