Łukasz Balbus

Existence of perfect equilibria in a class of multigenerational stochastic games of capital accumulation

In this paper we introduce a model of multigenerational stochastic games of capital accumulation where each generation consists of m different players. The main objective is to prove the existence of a perfect stationary equilibrium in an infinite horizon game. A suitable change in the terminology used in this paper provides (in the case of perfect altruism between generations) a new Nash equilibrium theorem for standard stochastic games with uncountable state space.

A Constructive Study of Markov Equilibria in Stochastic Games with Strategic Complementarities

We study a class of infinite horizon, discounted stochastic games with strategic complementarities. In our class of games, we prove the existence of a stationary Markov Nash equilibrium, as well as provide methods for constructing this least and greatest equilibrium via a simple successive approximation schemes. We also provide results on computable equilibrium comparative statics relative to ordered perturbations of the space of games. Under stronger assumptions, we prove the stationary Markov Nash equilibrium values form a complete lattice, with least and greatest equilibrium value functions being the uniform limit of approximations starting from pointwise lower and upper bounds.

Construction of Nash equilibria in symmetric stochastic games of capital accumulation

Abstract.A large class of nonzero-sum symmetric stochastic games of capital accumulation/resource extraction is considered. An iterative method leading to a Nash equilibrium in the infinite horizon game with the discounted evaluation is studied.

Time consistent Markov policies in dynamic economies with quasi-hyperbolic consumers

We study the question of existence and computation of time-consistent Markov policies of quasi-hyperbolic consumers under a stochastic transition technology in a general class of economies with multidimensional action spaces and uncountable state spaces. Under standard complementarity assumptions on preferences, as well as a mild geometric condition on transition probabilities, we prove existence of time-consistent solutions in Markovian policies, and provide conditions for the existence of continuous and monotone equilibria. We present applications of our methods to habit formation models, environmental policies, and models of consumption under borrowing constraints, and hence show how our methods extend the results obtained by Harris and Laibson (Econometrica 69:935–957, 2001) to a broad class of dynamic economies. We also present a simple successive approximation scheme for computing extremal equilibrium, and provide some results on the existence of monotone equilibrium comparative statics in the model’s deep parameters.

Robust Markov Perfect Equilibria in a Dynamic Choice Model with Quasi-hyperbolic Discounting

A stochastic dynamic choice model with the transition probability depending on an unknown parameter is specified and analysed in this chapter. The main feature in our model is an application of the quasi-hyperbolic discounting concept to describe the situation in which agent’s preferences may hinge on time. This requirement, in turn, leads to a non-cooperative infinite horizon stochastic game played by a countably many selves representing him during the play. As a result, we provide two existence theorems for a robust Markov perfect equilibrium (RMPE) and discuss its properties.

Markov Stationary Equilibria in Stochastic Supermodular Games with Imperfect Private and Public Information

We study a class of discounted, infinite horizon stochastic games with public and private signals and strategic complementarities. Using monotone operators defined on the function space of values and strategies (equipped with a product order), we prove existence of a stationary Markov–Nash equilibrium via constructive methods. In addition, we provide monotone comparative statics results for ordered perturbations of our space of games. We present examples from industrial organization literature and discuss possible extensions of our techniques for studying principal-agent models.

A Strategic Dynamic Programming Method for Studying Short Memory Equilibria of Stochastic Games with Uncountable Number of States

We study a class of infinite horizon stochastic games with uncountable number of states. We first characterize the set of all (nonstationary) short-term (Markovian) equilibrium values by developing a new (Abreu et al. in Econometrica 58(5):1041–1063, 1990)-type procedure operating in function spaces. This (among others) proves Markov perfect Nash equilibrium (MPNE) existence. Moreover, we present techniques of MPNE value set approximation by a sequence of sets of discretized functions iterated on our approximated APS-type operator. This method is new and has some advantages as compared to Judd et al. (Econometrica 71(4):1239–1254, 2003), Feng et al. (Int Econ Rev 55(1):83–110, 2014), or Sleet and Yeltekin (Dyn Games Appl doi:10.1007/s13235-015-0139-1, 2015). We show applications of our approach to hyperbolic discounting games and dynamic games with strategic complementarities.

Stochastic bequest games

In this paper, we prove the existence of a stationary Markov perfect equilibrium for a stochastic version of the bequest game. A novel feature in our approach is the fact that the transition probability need not be non-atomic and therefore, the deterministic production function is not excluded from consideration. Moreover, in addition to the common expected utility we also deal with a utility that takes into account an attitude of the generation towards risk.

NASH EQUILIBRIA IN UNCONSTRAINED STOCHASTIC GAMES OF RESOURCE EXTRACTION

A class of nonzero-sum symmetric stochastic games of capital accumulation/resource extraction is considered. It is shown that Nash equilibria in the games with some natural constraints are also equilibrium solutions in unconstrained games and dominate in the Pareto sense an equilibrium leading to exhausting the entire resource stock in the first period of the game.