Uniform Tauberian theorem in differential games

Khlopin, D. V.

doi:10.1134/s0005117916040172

Cited by 12 publications

(10 citation statements)

References 22 publications

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“…We would like to note that Theorem 6 and Theorem 4 for more strong assumptions were also proved in [31] and [32], respectively.…”

mentioning

confidence: 86%

“…A variant of the following theorem was also announced in [29,30] and proved in [31] with the aid of nonanticipating operators [17], [18].…”

Section: Remarkmentioning

confidence: 99%

“…However, now, the situation has changed. Firstly, a Tauberian theorem was proposed for differential games in nonergodic case in [31]. In [50], a very general approach to proof of Tauberian theorem was proposed for games with two players with opposite goals in discrete setting.…”

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confidence: 99%

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A uniform Tauberian theorem in dynamic games

Khlopin¹

2018

Sb. Math.

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The paper is concerned with two-person dynamic zero-sum games in continuous setting. 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 λdiscounted games as the discount tends to zero. Under quite weak assumptions on the game, we prove the Uniform Tauberian Theorem: existence of a uniform limit for one of the value functions implies the uniform convergence of the other one to the same limit.We also prove the analogs of the One-sided Tauberian Theorem, i.e., the inequalities on asymptotics for the lower and upper games. Also, a variant of the theorem for discretetime games is treated separately. Special attention is devoted to the case of differential games. The key roles in the proof were played by Bellman's optimality principle and the closedness of strategies under concatenation.

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“…We would like to note that Theorem 6 and Theorem 4 for more strong assumptions were also proved in [31] and [32], respectively.…”

mentioning

confidence: 86%

“…A variant of the following theorem was also announced in [29,30] and proved in [31] with the aid of nonanticipating operators [17], [18].…”

Section: Remarkmentioning

confidence: 99%

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A uniform Tauberian theorem in dynamic games

Khlopin¹

2018

Sb. Math.

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“…for some constant c. However, we stress again the fact that due to the lack of coercivity and due to the lack of small time controllability of our model, we do not expect the existence of a continuous viscosity solutions of the ergodic equation. This problem has been overcome in several other frameworks: we can quote for instance [34,19,37,14,7,5,3,6,9,24,23], for related problems see also [2,28] and the references therein. Following techniques developed in [6] we prove in the first part of Theorem 2.2 that the limit of u T /T exists and is equal to a constant.…”

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confidence: 99%

Ergodic behavior of control and mean field games problems depending on acceleration

Cardaliaguet,

Mendico

2020

Preprint

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The goal of this paper is to study the long time behavior of solutions of the first-order mean field game (MFG) systems with a control on the acceleration. The main issue for this is the lack of small time controllability of the problem, which prevents to define the associated ergodic mean field game problem in the standard way. To overcome this issue, we first study the long-time average of optimal control problems with control on the acceleration: we prove that the time average of the value function converges to an ergodic constant and represent this ergodic constant as a minimum of a Lagrangian over a suitable class of closed probability measure. This characterization leads us to define the ergodic MFG problem as a fixed-point problem on the set of closed probability measures. Then we also show that this MFG ergodic problem has at least one solution, that the associated ergodic constant is unique under the standard monotonicity assumption and that the time-average of the value function of the time-dependent MFG problem with control of acceleration converges to this ergodic constant.

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“…Paper [49] exhibits a more subtle approach. In stochastic games, the value is a fixed point of [26,27,49]) is required, although a reduction to the typical formalization will apparently require some of these.…”

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confidence: 99%

Tauberian Theorem for Value Functions

Khlopin

2017

Dyn Games Appl

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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.

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Uniform Tauberian theorem in differential games

Cited by 12 publications

References 22 publications

A uniform Tauberian theorem in dynamic games

A uniform Tauberian theorem in dynamic games

Ergodic behavior of control and mean field games problems depending on acceleration

Tauberian Theorem for Value Functions

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