Two equivalence results for time inconsistent problems due to non-decomposable discounting

Abstract

In the context of stochastic optimal control problems with a general discount function, we introduce the notions of decomposability and of super- and sub-decomposability of the discount function. We clarify that failure to decomposability leads to time inconsistency. A popular way to attack a time inconsistent problem with a non-decomposable discount function is the game theoretical approach. For such a problem, inspired by [1], we prove two equivalence results. In both results, we prove the existence of a time-consistent optimization problem that is equivalent, in the Nash equilibrium sense, to the original time-inconsistent one. Specifically, in the first result, discounting disappears, as the original discount function is incorporated into the new running utility. On the other hand, in the second result the equivalent time consistent problem is characterized by any given decomposable discount function. Notably, we prove that the same equivalence results – where equivalence is now meant in the dynamic programming sense – hold also with an original time-consistent problem with a decomposable discount function.