Tail optimality and preferences consistency for intertemporal optimization problems

Given an intertemporal optimization problem over a time interval $[t_0,T]$ and a control plan associated to it, we introduce the four notions of local and global tail optimality of the control plan, and local and global preferences consistency of the agent. While the notion of tail optimality of a control plan is not new, the main innovation of this paper is the definition of preferences consistency of an agent, that is a novel concept. We prove that, in the case of a emph{linear} time-consistent problem where dynamic programming can be applied, the optimal control plan is globally tail-optimal and the agent is globally preferences-consistent. Opposite, in the case of a emph{non-linear} problem that gives rise to time inconsistency, we find that global tail optimality and global preferences consistency do not coexist. We analyze three common ways to attack a time-inconsistent problem: (i) precommitment approach, (ii) dynamically optimal approach, (iii) consistent planning approach. We find that none of the three approaches keeps simultaneously the desirable properties of global tail optimality and global preferences consistency: the existing approaches to time inconsistency are flawed in various ways. We also prove that if the performance criterion includes a convex function of expected final wealth and a globally tail-optimal plan exists, then the three approaches coincide and the problem is linear. The contribution of the paper is to disentangle the notion of time consistency into the two notions of tail optimality and preferences consistency. The analysis should shed light on the price to be paid in terms of tail optimality and preferences consistency with each of the three approaches currently available for time inconsistency.