On the Computation of Value Correspondences for Dynamic Games

Abstract

Recursive game theory provides theoretic procedures for computing the equilibrium payoff or value sets of repeated games and the equilibrium payoff or value correspondences of dynamic games. In this paper, we propose and implement outer and inner approximation methods for equilibrium value correspondences that naturally occur in the analysis of dynamic games. The procedure utilizes set-valued step functions. We provide an application to a bilateral insurance game with storage.

Repeated moral hazard and recursive Lagrangeans

“…While there are various algorithms that perform this task in an efficient way (e.g. Sleet and Yeltekin (2007)), the problem is a nightmare even for low N and m. It is easy to see that for the simple model in section 2 the correspondence U is actually one interval. However, in the more general framework presented in section 3 with N agents with endogenous states x ∈ X, there are N different intervals for each point of the natural state space X, i.e.…”

“…These techniques are well understood and widely used for full information problems of optimal policy and enforcement frictions, but MM do not analyze their applicability to environments with private information. Sleet and Yeltekin (2008) make a crucial contribution in applying recursive Lagrangean techniques to dynamic models with privately observed idiosyncratic preference shocks. This paper instead focuses on a particular class of dynamic models with hidden actions, i.e.…”

“…While in principle the Lagrangean approach can deal with models with tens of state variables and agents 6 , APS would require an unreasonable amount of CPU time in such cases, even with the efficient techniques developed by Sleet and Yeltekin (2007). Moreover, for models that have been solved with APS, the Lagrangean approach would save computational and coding time and therefore it might be a more suitable choice for calibration purposes.…”

“…2 An illustration with a simple dynamic agency model Sleet and Yeltekin (2008) has shown how to extend MM's ideas to dynamic economies with hidden shocks 10 . However, many economic situations can be better represented as dynamic principal-agent relationship: unemployment insurance, worker's compensation, bank lending, etc.…”

Preface: Special Issue on Dynamic Games in Macroeconomics

“…1 It is well known that computing the equilibrium value set can be a serious complication of implementing strategic dynamic programming methods. In the paper of Sleet of Yeltekin [28], the authors discuss how APS methods can be applied to interesting classes of dynamic games and then provide a new method for computing equilibrium value correspondences. The authors new method is based on outer and inner approximations of the equilibrium value correspondence via step-valued set functions.…”