On Limit of Value Functions for Various Densities Dmitry V. Khlopin Krasovskii Institute of Mathematics and Mechanics 16, S.Kovalevskaja St., 620990 Yekaterinburg, Russia. Ural Federal University 4, Turgeneva St., 620083, Yekaterinburg, Russia khlopin@imm.uran.ru Abstract The paper is concerned with zero-sum differential games and the asymptotic properties of their value functions. The games with a com- mon dynamics, a running cost, and capabilities of players are consid- ered. Each payoff represents an average of the running cost with respect to the given discount functions (densities); these games differ in densi- ties only. We prove a Tauberian-type Theorem, that is, the fact that the existence of a uniform limit of the value functions for uniform density or for exponential density implies that the value functions uniformly converge to the same limit for arbitrary piecewise continuous density as the time scale parameter tends to zero. 1 Introduction In dynamic optimization, it is not uncommon to normalize the payoff by taking the average over time with respect to a certain probability distribution—for example, when the terminal time is large yet not exactly specified. In this case, for a realization of the process (a function t 7→ z(t)), in addition to the running cost (a function t 7→ g(z(t))), one also considers the payoff in the form of a certain average of the running cost, ∫ ∞ ϱ(t)g(z(t)) dt, 0 with respect to a certain discount function, a probability density function ϱ. Most often, when the problem is considered on infinite horizon, the potential infinity of the interval is emulated by considering the problems where the payoff is taken over increasingly large intervals [0, T ] or in view of increasingly small discounts λ; then, the limits of these problems are studied if such exist. Thus, effectively, for the payoffs ∫ ∞ λϱ(λt)g(z(λt)) dt, (1) 0 Copyright ⃝ c by the paper’s authors. Copying permitted for private and academic purposes. In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamisov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkin (eds.): Proceedings of the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, published at http://ceur-ws.org 328 one considers the asymptotic behavior of the corresponding value functions as the scale parameter λ tends to zero. Usually, the densities of the uniform ϱ(t) = 1[0,1] (t) (Cezaro mean) and exponential ϱ(t) = e−t (Abel mean) distributions are applied. The existence of such a limit of value functions in view of some density means that the value function’s response to a change in the scale parameter λ is very weak when this parameter is sufficiently small. In particular, in the stochastic statement, this value (the asymptotic value) is customarily considered the game value when the planning horizon is infinite [Bewley et al., 1976]. In these statements, one could often obtain, in addition, an asymptotically optimal strategy such that a payoff is close to the optimal one (uniform value)—for sufficiently small values of the scale parameter [Mertens et al., 1981]; however, in this paper, we only consider the value function asymptotics. The existence of uniform limits of value functions for payoffs (1) when averaged with respect to the uniform and/or exponential densities was proved for a broad class of stochastic games [Ziliotto, 2016b], for optimal control problems [Gaitsgori, 1985, Lions et al., 1986, Grüne, 1998, Artstein et al., 2000, Li et al., 2016, Gaitsgori et al., 2013], and for certain classes of differential games [Buckdahn et al., 2011, Cannarsa et al., 2015] in the so-called nonexpansive-like case. It turns out that, for stochastic games with a finite number of states and actions [Mertens et al., 1981], for discrete-time control problems [Lehrer et al., 1992], for general control problems [Oliu-Barton et al., 2013], for differential games [Khlopin, 2016], and a broad class of stochastic games [Ziliotto, 2016a, Ziliotto, 2016b], there holds the following Tauberian theorem: the uniform convergence of value functions under payoffs (1) with one of these densities (uniform or exponential) guarantees that the value functions in view of the other payoff also converge uniformly—to the same limit. The general approach, which deduces these Tauberian theorems from the Dynamic Programming Principle, is considered in [Khlopin, in print]. Such Tauberian theorems guarantee that if there is a uniform asymptotics for one of these densities (uniform or exponential) then, in addition to the value function’s insensitivity to the choice of the discount parameter λ for payoff (1), this asymptotics is also insensitive to the choice between these two densities. Often, it is also possible to prove insensitivity to the choice of the density ϱ from quite a broad class. Thus, we can find a sufficient asymptotic condition on the densities ϱλ , λ > 0, under which the uniform convergence of value functions as λ → 0 for the payoffs with uniform or/and exponential densities (for Cesaro and/or Abel means) implies the uniform convergence to the same limit (as λ → 0) of the value functions for the payoffs ∫ ∞ ϱλ (t)g(z(t)) dt. (2) 0 For example, to this end, for discrete-time control processes, in paper [Monderer et al., 1993], the following sufficient asymptotic condition on a family of ϱλ , λ > 0 was proposed: all densities ϱλ are nonincreasing and ∫ T lim ϱλ (t) dt = 0 ∀T > 0. (3) λ→0 0 For Markov decision processes, the sufficiency of the asymptotic condition lim V0∞ [ϱλ ] = 0 (4) λ→0 △ was refined in [Ziliotto, 2016c]; here, V0∞ [µ] is the total variation of a real-valued function µ on R+ = [0, ∞). For zero-sum differential games with the Isaacs condition, from the uniform convergence of the value functions for Cesaro means, it follows (see [Khlopin, 2015]) that the value functions with payoffs (2) converge to the same limit as λ → 0 for a family of densities ϱλ , λ > 0 if this family enjoys (3) and q[ϱλ ](r) lim sup V0 [ln ϱλ ] < +∞ ∀r ∈ (0, 1); (5) λ→0 ∫ q[ϱ ](r) here, for each r ∈ (0, 1), the quantile q[ϱλ ](r) is the minimal solution of the equation 0 λ ϱλ (t)dt = r. The main aim of this paper to prove that in zero-sum differential games with the Isaacs condition the existence of a uniform limit of the value functions for uniform density or for exponential density implies the uniform convergence (to the same limit) for the values in view of payoffs (1) for every piecewise continuous density ϱ. To this end, first, we will improve condition (5) (see (12)), and, then, apply this improved sufficient asymptotic condition for payoffs (2). The cornerstone of this proof is the sufficiency of (3)&(5), proved in [Khlopin, 2015] for zero-sum differential games with Isaacs condition. 329 2 Differential Game Consider a system in Rm controlled by two players, ẋ = f (x, p, q), x(0) ∈ Rm , t ≥ 0, p(t) ∈ P, q(t) ∈ Q; (6) here, P and Q are non-empty compact subsets of finite-dimensional Euclidean spaces. Assume that the functions f : Rm × P × Q → Rm and g : Rm × P × Q → [0, 1] are continuous, and let these functions be Lipschitz continuous in the state variable; namely, there exists a constant L > 0 such that, for all x, y ∈ Rm , p ∈ P, and q ∈ Q, f (x, p, q) − f (y, p, q) + g(x, p, q) − g(y, p, q) ≤ L x − y . Denote by P and Q the sets of all Borel measurable functions R+ ∋ t 7→ p(t) ∈ P and R+ ∋ t 7→ q(t) ∈ Q, respectively. So, for each pair (p, q) ∈ P × Q, for every initial condition x(0) = x∗ ∈ Rm , system (6) generates the unique solution x(·) = y(·; x∗ , p, q) defined for the whole R+ . We will essentially refer to the results proved in [Khlopin, 2015]. As a consequence, we need to admit all assumptions on differential games from [Khlopin, 2015]. To make it happen, we also impose the Isaacs condition (“the saddle point condition in a small game”) [Krasovskii et al., 1988] [ ] [ ] max min ⟨s, f (x, p, q)⟩ + g(x, p, q) = min max ⟨s, f (x, p, q)⟩ + g(x, p, q) ∀x, s ∈ Rm . p∈P q∈Q q∈Q p∈P It is easy to see that, for each nonnegative function ϱ : R+ → R+ , it implies that, for all t ∈ R+ , x, s ∈ Rm , [ ] [ ] max min ⟨s, f (x, p, q)⟩ + ϱ(t)g(x, p, q) = min max ⟨s, f (x, p, q)⟩ + ϱ(t)g(x, p, q) . (7) p∈P q∈Q q∈Q p∈P Let D be the set of all probability density functions having their support in R+ . For a density ϱ ∈ D and a number r ∈ (0, 1), let the quantile q[ϱ](r) be the minimum number such that ∫ q[ϱ](r) ϱ(t) dt = r. 0 For a given density ϱ ∈ D and an initial position x∗ ∈ Rm , let the goal of the first player be to maximize the payoff function ∫ ∞ △ c[ϱ](x∗ , p, q) = ϱ(t)g(y(t; x∗ , p, q), p(t), q(t)) dt, (8) 0 and let the task of the second one be to minimize it. There are many ways to define a game and the sets of strategies for each player; for a very well made review encompassing a large number of formalizations, refer to [Subbotin, 1995, Subsect.14,15]. We will consider the nonanticipating strategies (see [Eliott et al., 1972]). Definition 1 A map α : Q → P is called a nonanticipating strategy of the first player if, for all t > 0 and q, q ′ ∈ Q, from q|[0,t] = q ′ |[0,t] it follows that α[q]|[0,t] = α[q ′ ]|[0,t] . A map β : P → Q is called a nonanticipating strategy of the second player if, for all t > 0 and p, p′ ∈ P, from p|[0,t] = p′ |[0,t] it follows that β[p]|[0,t] = β[p′ ]|[0,t] . We denote by A and B the sets of all nonanticipating strategies of the first player and of the second player, respectively. For each density ϱ ∈ D, define the corresponding value function by the following rule: ∫ ∞ △ ( ( ) ) V[ϱ](x∗ ) = sup inf ϱ(t)g y t; x∗ , α[q], q , α[q](t), q(t) dt ∀x∗ ∈ Rm ; (9) α∈A q∈Q 0 also, define ∫ ∞ △ ( ( ) ) V+ [ϱ](x∗ ) = inf sup ϱ(t)g y t; x∗ , p, β[p] , p(t), β[p](t) dt ∀x∗ ∈ Rm . β∈B p∈P 0 330 For each density ϱ ∈ D with bounded supp ϱ, condition (7) guarantees ([Krasovskii et al., 1988], [Subbotin, 1995], [Cardaliaguet et al., 2000]) the equality V+ [ϱ] ≡ V[ϱ]. (10) △ In the general case, for each density ϱ ∈ D, define the sequence of densities ϱn = n+1 n ϱ · 1[0,q[ϱ]( n+1 )] ∈ D. Since n supp ϱn is compact, passing to the limit as n → ∞ in (8) and (10), we see that the payoffs c[ϱn ] converge to c[ϱ] and the value functions V[ϱn ] = V+ [ϱn ] converge to V[ϱ] = V+ [ϱ] as n → ∞. Now, we have proved (10) for all ϱ ∈ D. 3 The Main Result For each density ϱ ∈ D and an arbitrary λ > 0, it is also possible to introduce the density ϱλscale by the rule ϱλscale (t) = λϱ(λt) ∀t ≥ 0. Set ϖλ (t) = λ · 1[0,1/λ] , πλ (t) = λ · e−λt , ∀λ > 0, t ≥ 0. Thus, we define the uniform and exponential density families. For an interval [a, b) ⊂ R and a function y : [a, b) → R ∪ {∞}, denote by Vab [y] the total variation of the function y in [a, b). Theorem 1 Let a non-empty subset Ω ⊂ Rm be strongly invariant with respect to system (6). For a given map U∗ : Ω → [0, 1], the following conditions are equivalent: 1) for each piecewise continuous on (0, ∞) density µ ∈ D, there holds lim sup V[µλscale ](x∗ ) − U∗ (x∗ ) = 0; λ→0 x∗ ∈Ω 2) the value functions V[ϖλ ] converge to U∗ uniformly in Ω as λ → 0, i.e., lim sup V[ϖλ ](x∗ ) − U∗ (x∗ ) = 0; (11) λ→0 x∗ ∈Ω 3) the value functions V[πλ ] converge to U∗ uniformly in Ω as λ → 0, i.e., lim sup V[πλ ](x∗ ) − U∗ (x∗ ) = 0; λ→0 x∗ ∈Ω 4) for every family of densities µλ ∈ D, λ > 0, it follows from (3) and q[µλ ](r) lim sup V0 [µ] · q[µλ ](r) < +∞ ∀r ∈ (0, 1) (12) λ→0 that the value functions V[µλ ] converge to U∗ uniformly in Ω as λ → 0, i.e., lim sup V[µλ ](x∗ ) − U∗ (x∗ ) = 0. (13) λ→0 x∗ ∈Ω 4 The Proof of Theorem 1 The implications 2) ⇒ 3), 3) ⇒ 2) were proved in [Khlopin, 2016]. From (ϖ1 )λscale = ϖλ for all positive λ, it follows that 1) ⇒ 2). It remains to verify 2) ⇒ 4) ⇒ 1). To do this, we need the following proposition proved in [Khlopin, 2015]: Proposition 1 Assume that the value functions V[ϖλ ], λ > 0 converge to a function U∗ uniformly in Ω as λ → 0, i.e., (11) holds. Let a family of µλ ∈ D, λ > 0, satisfy (3) and (5). Then, for all positive δ < 1, there exists a positive λδ such that, for all positive λ < λδ , 1 V[µλ ](x∗ ) > U∗ (x∗ ) − 8δ ln ∀x∗ ∈ Ω. δ 331 4.1 The Proof of 2) ⇒ 4). The proof is by reductio ad absurdum. Assume the converse. Then, for a positive ε < 1/20, there exists a family of densities µ̂λ ∈ D, λ > 0 such that (3), (12), and lim sup sup |V[µ̂λ ](x∗ ) − U∗ (x∗ )| ≥ 3ε (14) λ→0 x∗ ∈Ω hold. Choose positive δ < 1 and M such that 1 q[µ ](1−ε) 8δ ln < ε, lim sup V0 λ [µ] · q[µλ ](1 − ε) < M. δ λ→0 Now, for all positive λ, define the mapping µλ : R+ → R+ by the following rule: ε µλ (t) = µ̂λ (t) + ∀t ∈ [0, q[µ̂λ ](1 − ε)] q[µ̂λ ](1 − ε) and µλ (t) = 0 otherwise. Then, ∫ ∞ ∫ q[µ̂λ ](1−ε) ∫ q[µ̂λ ](1−ε) µλ (t)dt = µλ (t)dt = µ̂λ (t)dt + ε = 1, (15) 0 0 0 that is, µλ ∈ D. Now, since µ̂λ enjoys (3), we see that q[µ̂λ ](1 − ε) tends to ∞ as λ → 0, and ∫ T ∫ T ∫ min{T,q[µ̂λ ](1−ε)} ε lim sup µλ (t) dt ≤ lim µ̂λ (t) dt + lim dt = 0 λ→0 0 λ→0 0 λ→0 0 q[µ̂λ ](1 − ε) holds for all positive T ; thus, the family of µλ enjoys (3). Next, note that, for all x, y > 0, there holds max{x, y} max{x, y} |x − y| | ln x − ln y| = ln ≤ −1= ; min{x, y} min{x, y} min{x, y} moreover, by (15), q[µλ ](r) < q[µ̂λ ](1 − ε) for all λ > 0 and r ∈ (0, 1). Then, we obtain q[µ̂λ ](1−ε) q[µ̂λ ](1−ε) q[µλ ](r) q[µ̂λ ](1−ε) V0 [µλ ] V [µ̂λ ] · q[µ̂λ ](1 − ε) M V0 [ln µλ ] ≤ V0 [ln µλ ] ≤ ≤ 0 < inf t∈[0,q[µ̂λ ](1−ε)) µλ (t) ε ε for all r ∈ (0, 1), λ > 0. Thus, the family of µλ enjoys (5). Since the family of µλ enjoys all assumptions of Proposition 1, we can find a positive λδ such that 1 V[µλ ](x∗ ) > U∗ (x∗ ) − 8δ ln ∀x∗ ∈ Ω, λ ∈ (0, λδ ). (16) δ △ △ △ Consider a new differential game. Define the sets P− = Q, Q− = P and the maps f − (x, p− , q − ) = f (x, q − , p− ), △ g − (x, p− , q − ) = 1 − g(x, q − , p− ) for all x ∈ Rm , p− ∈ P− , q − ∈ Q− . By (7) with s = −s− , we have [ ] [ ] max min ⟨s− , f − (x, p− , q − )⟩ + ϱ(t)g − (x, p− , q − ) = min max ⟨s− , f − (x, p− , q − )⟩ + ϱ(t)g − (x, p− , q − ) p− ∈P− q − ∈Q− q − ∈Q− p− ∈P− for all s− , x ∈ Rm , t ≥ 0, ϱ ∈ D. Thus, the Isaacs condition also holds. △ In addition, by P − = Q and Q− = P, we obtain y − (x∗ , p− , q − ) = y(x∗ , q − , p− ) for all x∗ ∈ Rm , p− ∈ P − , q − ∈ Q− . Then, Ω is a strongly invariant set for this dynamics. Moreover, thanks to Q− = P and A− = B, for each density ϱ ∈ D, the map Ω ∈ x∗ 7→ 1 − V[ϱ](x∗ ) = 1 − V+ [ϱ](x∗ ) ∫ ∞ ( ( ( ) )) = sup inf ϱ(t) 1 − g y t; x∗ , p, β[p] , p(t), β[p](t) dt β∈B p∈P 0 ∫ ∞ ( ( ( ) )) = sup − inf − ϱ(t) 1 − g y − t; x∗ , α− [q − ], q − , q − (t), α− [q − ](t) dt α− ∈A− q ∈Q 0 ∫ ∞ ( ( ) )) = sup − inf − ϱ(t)g − y − t; x∗ , α− [q − ], q − , α− [q − ](t), q − (t) dt α− ∈A− q ∈Q 0 332 is the value function (9) of the new game. In particular, (11) holds for this game with asymptotics U∗− ≡ 1 − U∗ . Applying Proposition 1 for this game, we can choose a positive λ− δ such that 1 1 − V[µλ ](x∗ ) > 1 − U∗ (x∗ ) − 8δ ln ∀x∗ ∈ Ω δ holds for all positive λ < λ− δ . Together with (16), it implies 1 lim sup sup V[µλ ](x∗ ) − U∗ (x∗ ) ≤ 8δ ln < ε. λ→0 x∗ ∈Ω δ Then, thanks to (14), we obtain lim sup sup V[µ̂λ ](x∗ ) − V[µλ ](x∗ ) > 2ε. (17) λ→0 x∗ ∈Ω However, ∫ ∞ ∫ ∞ |µλ (t) − µ̂λ (t)|dt = ε + µ̂λ (t)dt = 2ε, 0 q[µ̂λ ](1−ε) therefore, by 0 ≤ g ≤ 1, for all x∗ ∈ Ω, α ∈ A, q ∈ Q, λ > 0, we have ∫ ∞ |µλ (t) − µ̂λ (t)|g(y(t; x∗ , α[q](t), q(t)), α[q](t), q(t))dt ≤ 2ε, ∫ 0∞ inf µλ (t)g(y(t; x∗ , α[q](t), q(t)), α[q](t), q(t))dt − q∈Q 0 ∫ ∞ − inf µ̂λ (t)g(y(t; x∗ , α[q](t), q(t)), α[q](t), q(t))dt ≤ 2ε q∈Q 0 V[µλ ](x∗ ) − V[µ̂λ ](x∗ ) ≤ 2ε. We obtain a contradiction with (17). This contradiction proves the implication 2) ⇒ 4). 4.2 The Proof of 4) ⇒ 1) Consider a piecewise continuous in (0, ∞) density ϱ. Fix a number ε > 0; now, there exists a sufficiently large natural n > 3 such that n5 < ε. For all n ∈ N, n > 3, set △ △ rn = q[ϱ](1/n), sn = q[ϱ](1 − 1/n). Since the piecewise continuous function ϱ ∫is Riemann integrable ∫s on [rn , sn∫], there exists a staircase function s sn µn : R+ → R, supp µn ⊂ [rn , sn ] such that rnn µn (t) dt = rnn ϱ(t) dt = n−2 n , rn |µn (t) − ϱ(t)| dt < 1/n hold. In ∫∞ particular, its total variation in [0, ∞) is finite. Since 0 µn (t)dt = n−2 n , put △ n △ nsn ∞ µ̄n = µn ∈ D, M = sn V0∞ [µ̄n ] = V [µn ] ∈ R. n−2 n−2 0 Now, for all λ > 0, we have [ ] [ ] q[µ̄n ](1 − ε) V0∞ (µ̄n )λscale · q (µ̄n )λscale (1 − ε) = λV0∞ [µ̄n ] = V0∞ [µ̄n ]q[µ̄n ](1 − ε) ≤ sn V0∞ [µ̄n ] = M, (18) ∫ ∞ ∫ ∞ λ ∫ sn ( ) 3 n 5 ϱλscale (t) − (µ̄n )λscale (t) dt = ϱ(t) − µ̄n (t) dt < + − 1 ϱ(t) dt = < ε. 0 0 n rn n−2 n Thus, we have sup V[ϱλscale ](x∗ ) − V[(µ̄n )λscale ](x∗ ) ≤ ε ∀λ > 0. (19) x∗ ∈Ω 333 Consider some positive T . For all positive λ < rn /T , we have µ̄n |[0,λT ] ≡ 0 and ∫ T ∫ T (µ̄n )λscale (t) dt = λ µ̄n (λt) dt = 0. 0 0 Thus, (3) holds for densities (µ̄n )λscale , λ > 0. Thanks to (18), the densities (µ̄n )λscale , λ > 0 also satisfy (12). Applying condition 4) for this family, we give lim sup |V[(µ̄n )λscale ](x∗ ) − U∗ (x∗ )| = 0. λ→0 x∗ ∈Ω Accounting for (19), we obtain lim sup sup |V[ϱλscale ](x∗ ) − U∗ (x∗ )| ≤ ε. λ→0 x∗ ∈Ω Since the choice of a positive number ε was arbitrary, the implication 4) ⇒ 1) is proved.  5 Conclusion Based on the results from [Khlopin, 2015], for differential games with the Isaacs condition, we managed to prove that their value functions’ insensitivity to the choice of the scale parameter for the exponential or uniform distribution family implies the same result with respect to all densities of a relatively general form. Apparently, it can be proved for all dynamic games by the game value map method (similar [Khlopin, in print]), however, this should be investigated further. In addition, we propose a new condition (3)&(12), which is sufficient for (13). 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