Tauberian Theorem for Value Functions

Abstract

For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long run average cost as the time horizon tends to infinity and the limit of value functions of λ -discounted games as the discount tends to zero. We prove that the Dynamic Programming Principle for value functions directly leads to the Tauberian Theorem-that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub-and super-optimality principles (the Dynamic Programming Principle).With their aid, we obtain certain inequalities on asymptotics of sub-and super-solutions, which lead to the Tauberian Theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point; a very simple stochastic game model is also considered.

LP Formulations of Discrete Time Long-Run Average Optimal Control Problems: The NonErgodic Case

“…The ergodic case, when these limits are constants (that is, when they do not depend on the initial condition y 0 ), was studied, for example, in [3,5,7,17] (see also references therein). Results for the non-ergodic case were obtained in [12,22,23,28,31,32,33]. In particular, it was results of [12] that were instrumental for obtaining the IDLP representation for the aforementioned limits for systems evolving in continuous time in [10].…”

LP Based Bounds for Cesàro and Abel Limits of the Optimal Values in Non-ergodic Stochastic Systems

“…If a function ϕ is even, then 3 4 ϕ(yu) + 1 4 ϕ(−yu) − ϕ(y) ≡ 0 (since u is either equal to 1 or to −1). Therefore, (29) is satisfied for all γ ∈ P(G), while (30) Thus, the constraints (30) ensure that the occupational measures γ generated by the state-control trajectories satisfy the property γ(Y \ Y y0 ) = 0. This is consistent with the system's dynamics (see (24)), according to which the only states attended by the state trajectories are y 0 and −y 0 .…”

“…In this paper, we study asymptotic properties of problems of control of stochastic discrete time systems with time averaging and time discounting optimality criteria, and we establish that the Cesàro and Abel limits of the optimal values in such problems can be evaluated with the help of a certain infinite-dimensional (ID) linear programming (LP) problem and its dual. Note that matters related to the existence and the equality of such limits have been investigated by many; see, e.g., [1], [2], [6], [8], [14], [18], [21], [22], [30], [33], [34], [37], [41]. A distinct feature of the present paper is that the Cesàro and Abel limits of the optimal values are evaluated with the help of LP tools.…”

Linear Programming Estimates for Cesaro and Abel Limits of Optimal Values in Optimal Control Problems

“…(called Cesàro and Abel limits, respectively). Matters related to the existence and the equality of Cesàro and Abel limits of the optimal values have been addressed by many authors (see, e.g., [1,5,6,11,16,17,22,24,25,26]). A special feature and the novelty of our consideration is that we are making use of occupational measure reformulations of problems (3) and ( 4) and utilize the LP duality theory.…”