Time-Inconsistent Problems for Controlled Markov Chains with Distribution-Dependent Costs: Equilibrium Solutions

TL;DR

This paper introduces a new equilibrium concept for continuous-time controlled Markov chains with distribution-dependent costs, establishing existence, uniqueness, and local optimality, and links the problem to mean-field games with time inconsistency.

Contribution

It proposes a novel definition of equilibrium for time-inconsistent, distribution-dependent Markov control problems and proves its fundamental properties.

Findings

Existence and uniqueness of the equilibrium are established.

The equilibrium is shown to be locally optimal.

The problem is equivalent to an infinite-player mean-field game with time-inconsistent costs.

Abstract

This paper focuses on a class of continuous-time controlled Markov chains with time-inconsistent and distribution-dependent cost functional (in some appropriate sense). A new definition of time-inconsistent distribution-dependent equilibrium in closed-loop sense is given and its existence and uniqueness have been established. Because of the time-inconsistency, it is proved that the equilibrium is locally optimal in an appropriate sense. Moreover, it has been shown that our problem is essentially equivalent to an infinite-player mean-field game with time-inconsistent cost.