A constructive study of Markov equilibria in stochastic games with strategic complementarities

We study a class of discounted infinite horizon stochastic games with strategic complementarities. Using monotone operators on the space of values and strategies, we prove existence of a Stationary Markov Nash equilibrium under different set of assumptions than Curtat (1996), Amir (2002, 2005) or Nowak (2007) via constructive methods. In addition, we provide monotone comparative statics results for ordered perturbations of the space of stochastic games. Under slightly stronger assumptions, we prove the stationary Markov Nash equilibrium values form a complete lattice of Lipschitz stochastic processes, with least and greatest equilibrium value functions being the uniform limit of successive approximations iterations from pointwise lower and upper bounds. Finally, we discuss the relationship between our function-based monotone methods (in pointwise orders), and correspondence-based monotone methods (in set inclusion orders) that have been proposed in the recent literature stemming from the seminal work of Kydland and Prescott (1977, 1980) and Abreu, Pearce, and Stacchetti (1990).