Tail optimality and preferences consistency for intertemporal optimization problems

When an intertemporal optimization problem over a time interval [t0, T ] is linear and can be solved via dynamic programming, the Bellman’s principle holds, and the optimal control map has the desirable feature of being tail-optimal in the right queue; moreover, the optimizer keeps solving the same problem at any time time t with renovated conditions: we will say that he is preferences-consistent. Opposite, when an intertemporal optimization problem is non-linear and cannot be tackled via dynamic programming, the Bellman’s principle does not hold and, according to existing literature, the problem gives raise to time inconsistency. Currently, there are three different ways to attack a time-inconsistent problem: (i) precommitment approach, (ii) dynamically optimal approach, (iii) game theoretical approach. The three approaches coincide when the problem is linear and can be solved via dynamic programming. However, for non-linear time-inconsistent problems none of the three approaches presents simultaneously the two features of tail optimality and preferences consistency that hold for linear problems. In this paper, given an optimization problem and the control map associated to it, we formulate the four notions of local and global tail optimality of the control map, and local and global preferences consistency of the optimizer. While the notion of tail optimality of a control map is not new in optimization theory, to the best of our knowledge the notion of preferences consistency of an optimizer is novel. We prove that, due to the validity of the Bellman’s principle, in the case of a linear problem the optimal control map is globally tail-optimal and the optimizer is globally preferences-consistent. Opposite, in the case of a non-linear problem global tail optimality and global preferences consistency do not coexist. For the precommitment approach, there is local tail optimality and local preferences consistency at initial time t0. For the dynamically optimal approach, there is global preferences consistency, but not even local tail optimality. For the game theoretical approach, there is neither local tail optimality nor local preferences consistency with respect to the original non-linear problem, but there is global tail optimality and global preferences consistency with respect to a different linear problem. This 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.