Jérôme Renault
email: jerome.renault@tse-fr.eu.

Xavier Venel

A distance for probability spaces, and long-term values in Markov Decision Processes and Repeated Games

Keywords: Markov Decision Process, gambling houses, POMDP, Repeated Games, distance for belief spaces, Kantorovich-Rubinstein duality, limit value, uniform value, general values, characterization of the value

Given a finite set K, we denote by X = ∆(K) the set of probabilities on K and by Z = ∆ f (X) the set of Borel probabilities on X with finite support. Studying a Markov Decision Process with partial information on K naturally leads to a Markov Decision Process with full information on X. We introduce a new metric d * on Z such that the transitions become 1-Lipschitz from (X, . 1 ) to (Z, d * ). In the first part of the article, we define and prove several properties of the metric d * . Especially, d * satisfies a Kantorovich-Rubinstein type duality formula and can be characterized by using disintegrations. In the second part, we characterize the limit values in several classes of "compact non expansive" Markov Decision Processes. In particular we use the metric d * to characterize the limit value in Partial Observation MDP with finitely many states and in Repeated Games with an informed controller with finite sets of states and actions. Moreover in each case we can prove the existence of a generalized notion of uniform value where we consider not only the Cesàro mean when the number of stages is large enough but any evaluation function θ ∈ ∆(IN * ) when the impatience I(θ) = t≥1 |θ t+1θ t | is small enough.

Introduction

The classic model of Markov Decision Processes with finitely many states, particular class of the model of Stochastic Games introduced by [START_REF] Shapley | Stochastic games[END_REF], was explicitly introduced by [START_REF] Bellman | Dynamic programming[END_REF] in the 1950s and has been extensively studied since then. When the set of actions is also finite, [START_REF] Blackwell | Discrete dynamic programming[END_REF] proved the existence of a strategy which is optimal for all discount factors close to 0.This model was generalized later to MDPs with Partial Observations (POMDP), (for references see Araposthatis et al. (1993)). The decision maker observes neither the state nor his payoff. Instead at each stage, he receives a signal which depends on the previous state and his previous action. In order to solve this problem a classic approach is to go back to the classic model of MDPs by introducing an auxiliary problem with full observation and Borel state space : the space of belief on the state as shown in Astrom, K.J. (1965), [START_REF] Sawaragi | Discrete-Time Markovian Decision Processes with Incomplete State Observations[END_REF] and [START_REF] Rhenius | Incomplete information in Markovian decision models : The Annals of Statistics[END_REF]. For optimality criteria like the Cesàro mean and the Abel mean, these two problems are equivalent and the question of the existence of the limit value is the same. Then given some sufficient conditions of ergodicity, one can search for a solution of the Average Cost Optimality Criterion in order to find "the" value of the MDP, for example as in [START_REF] Runggaldier | On the construction of nearly optimal strategies for a general problem of control of partially observed diffusions[END_REF] or as in [START_REF] Borkar | Average cost dynamic programming equations for controlled Markov chains with partial observation[END_REF][START_REF] Borkar | Dynamic programming for ergodic control of markov chains under partial observations : a correction[END_REF]. An introduction to the ACOE in the framework of MDP and the reduction of POMDP can be found in [START_REF] Hernández-Lerma | Adaptive Markov Control Processes[END_REF]. From another point of view, if we know that the limit value exists, the ACOE may be used as a characterization of the value. For finite MDP, for example, [START_REF] Denardo | Multichain Markov renewal programs[END_REF] proved that the limit value is the solution of a linear programming problem deduced from the ACOE. Moreover by standard linear programming results, it is also equal to the solution of a dual problem from which Hordjik and [START_REF] Hordijk | Linear programming and Markov decision chains[END_REF] deduced an optimal strategy. This dual problem focuses on the maximal payoff that the decision maker can guarantee on invariant measures. This approach was extended to different criteria (see [START_REF] Kallenberg | Survey of linear programming for standard and nonstandard Markovian control problems[END_REF] and to a convex analytic approach by Borkar ( for references see Borkar 2002) in order to study problems with a countable state space and a compact action space.

Given an initial POMDP on a finite space K, we will follow the usual approach and introduce a MDP on X = ∆(K) but instead of assuming some ergodicity on the process we will use the structure of ∆(K) and a new metric on Z = ∆ f (∆(K)). We extend and relax the MDP on ∆ f (X) with a uniformly continuous affine payoff function and non-expansive affine transitions. The structure of Z was already used in [START_REF] Rosenberg | Blackwell Optimality in Markov Decision Processes with Partial Observation[END_REF] and in [START_REF] Renault | Uniform value in Dynamic Programming[END_REF]. Under our new metric, we highlight a stronger property since the transitions became 1-Lipschitz on Z and Z is still precompact. We use this property to focus on general evaluations. Given a probability distribution θ on positive integers, we evaluate a sequence of payoffs g = (g t ) t≥1 by γ θ (g) = t θ t g t . In a MDP or a POMDP, the θ-value is then defined as the maximum expected payoff that the player can guarantee with this evaluation. Most of the literature focuses on the n-stage game where we consider the Cesàro mean of length n, and on the λ discounted games, where we consider the Abel mean with parameter λ. The first type of results focuses on the limit when n converges to +∞ and when λ converges to 0 or the relation between them. When there is no player, the relation between them is directly linked to a Hardy-Littlewood theorem (see [START_REF] Filar | Some comments on a theorem of Hardy and Littlewood[END_REF]. One of the limit exists if and only if the other exists and whenever they exist they are equal. [START_REF] Lehrer | A uniform Tauberian Theorem in Dynamic Programming[END_REF] proved that this result extends to the case where there is one player provided we ask for uniform convergence. The other approach focuses on the existence of a good strategy in any long game or for any discount factor close to 0. We say that the MDP has a uniform value. For MDP with finitely many states, Blackwell's result (1962) solved both problems. In POMDPs, [START_REF] Rosenberg | Blackwell Optimality in Markov Decision Processes with Partial Observation[END_REF] proved the existence of the uniform value when the sets of states, actions and signals are finite, and [START_REF] Renault | Uniform value in Dynamic Programming[END_REF] removed the finiteness assumption on signals and actions.

Concerning stochastic games, [START_REF] Mertens | Stochastic games[END_REF] proved the existence of the uniform value when the set of states and the set of actions are finite. The model also generalizes to partial information but the existence of possible private information implies a more complex structure on the auxiliary state space. [START_REF] Mertens | Formulation of Bayesian analysis for games with incomplete information[END_REF] and [START_REF] Mertens | Repeated games[END_REF] introduced the universal belief space which synthesizes all the information for both players in a general repeated game : their beliefs about the state, their beliefs about the beliefs of the other player, etc... So far, the results always concern some subclasses of games where we can explicitly write the auxiliary game in a "small" tractable set. A lot of work has been done on games with one fully informed player and one player with partial information, introduced by Aumann and Maschler (see reference from 1995). A state is chosen at stage 0 and remains fixed for the rest of the game. [START_REF] Renault | The value of Markov chain games with lack of information on one side[END_REF] extended the analysis to a general underlying Markov chain on the state space (see also [START_REF] Neyman | Existence of Optimal Strategies in Markov Games with Incomplete Information[END_REF]. [START_REF] Rosenberg | Stochastic games with a single controller and incomplete information[END_REF] and Renault (2012a) proved the existence of the uniform value when the informed player can additionally control the evolution of the state variable.

The first section is dedicated to the description of the (pseudo)-distance d * on ∆(X) in the general framework when X is a compact subset of a normed vector space. We provide different definitions and show that they all define this pseudodistance. Then we focus on the case where X is a simplex. We prove that d * is a real metric and prove a "Kantorovich-Rubinstein like " duality formula for probabilities with finite support on X. We give new definitions and a characterization by the disintegration mapping. The second section focuses on Gambling Houses and standard Markov Decision Processes. We first introduce the definitions of general limit value and general uniform value. Then we give sufficient conditions for the existence of the general uniform value and a characterization in several "compact" cases of Gambling Houses and Markov Decision Processes, including the finite state case. We study the limit value as a linear function of the initial probability so there are similarities with the convex analytic approach, but we are able to avoid any assumption on the set of actions. Moreover the MDPs that we are considering may not have 0-optimal strategies as shown in [START_REF] Renault | Uniform value in Dynamic Programming[END_REF]. Finally we apply these results to prove the existence of the general uniform value in finite state POMDPs and repeated games with an informed controller.

Example 2.2. Consider the particular case where X = [0, 1] endowed with the usual norm. Then all f in D 1 are linear. As a consequence, d 1 (u, v) = 0 for u = 1/2 δ 0 + 1/2 δ 1 and v = δ 1/2 . We do not have the separation property and d 1 is not a distance in this case. Let us modify the example. X now is the set of probability distributions over 2 elements, viewed as X = {(x, 1x), x ∈ [0, 1]}. We use the norm . 1 to measure the distance between (x, 1x) and (y, 1y), so that

V = IR 2 is endowed with (x 1 , x 2 ) -(y 1 , y 2 ) = |x 1 -y 1 | + |x 2 -y 2 |. Consider f in E such that f ((x, 1-x)) = x(1-x) for all x. f now belongs to D 1 , and d 1 (u, v) ≥ 1/4 > 0 for u = 1/2 δ 0 + 1/2 δ 1 and v = δ 1/2 . One can show that (∆(X), d 1
) is a compact metric space in this case (see proposition 2.15 later), and for applications in this paper d 1 will be a particularly useful distance whenever X is a simplex ∆(K) endowed with xy = k∈K |x ky k |.

Furthermore it is known that the Kantorovitch Rubinstein metric on ∆(X) only depends on the restriction of the norm . on the set X. Especially if for all x, x ′ ∈ X such that x = x ′ , xx ′ = 2, then for all u, v ∈ ∆(X), d KR (u, v) = uv 1 . This is not the case when considering the metric d 1 . Two norms on V giving the same metric on X may leads to different pseudo-metrics on ∆(X). We consider in the next example different norms on the Euclidean space IR K .

Example 2.3. We consider V = IR K , X = {e 1 , .., e K } the set of canonical vectors of V and a norm such that for all k = k ′ , e ke k ′ = 2. We know that d 1 is smaller than the Kantorovitch-Rubinstein metric, so for all u ∈ ∆(X) and v ∈ ∆(X), we have

d 1 (u, v) ≤ u -v 1 .
We first consider the particular case of the norm defined by x-y = 21-1 p xy p where x -

y p = K k=1 |x k -y k | p 1/p
is the usual L p -norm on IR K , with p a fixed positive integer. Given u, v ∈ ∆(X), the function f defined by

∀k ∈ K f (k) = 1 if u(k) ≥ v(k) -1 otherwise, satisfies u(f ) -v(f ) = k∈K |u(k) -v(k)| = u -v 1 . Moreover for all a ≥ 0, b ≥ 0 and k, k ′ ∈ K such that k = k ′ , we have af (k) -bf (k ′ ) ≤ a + b ≤ 2 2 1/p (a p + b p ) 1/p = ae k -be k ′ , and af (k) -bf (k) ≤ |a -b| ≤ |a -b| 2 2 1/p . Therefore f is in D 1 and d 1 (u, v) = u -v 1 , independently 1 of p.
Nevertheless the inequality d 1 (u, v) ≤ u-v 1 may be strict as in the following example. We consider the case K = 3 and given a vector (x 1 , x 2 , x 3 ) ∈ IR 3 , we define the norm (

x 1 , x 2 , x 3 ) = max(|x 1 | + |x 2 |, 2|x 3 |), which satisfies e 1 -e 2 =
e 2e 3 = e 3e 2 = 2. Let f be a function in D 1 , then we have among others the following constraints :

∀a, b ≥ 0 af (e 3 ) -bf (e 1 ) ≤ (-b, 0, a) = max(2a, b) and ∀a ≥ 0 af (e 2 ) ≤ (0, a, 0) = a. Let u = (0, 1/2, 1/2), v = (1, 0, 0) and f ∈ D 1 , then u(f ) -v(f ) = 1 2 f (e 2 ) + 1 2 f (e 3 ) -f (e 1 ) ≤ 1 2 + max(2/2, 1) = 3 2 . By symmetry of D 1 , we deduce that d 1 (u, v) ≤ 3 2 < u -v 1 . In fact one can show that d 1 (u, v) = 3
2 by checking that the function defined by f (e 1 ) = 0, f (e 2 ) = 1 and f (e 3 ) = 2 is in D 1 and satisfies u(f

) -v(f ) = 3 2 .
We now give other expressions for the pseudo-distance d 1 .

Definition 2.4.

d 2 (u, v) = sup (f,g)∈D 2 u(f ) + v(g),
where

D 2 = {(f, g) ∈ E × E, ∀x, y ∈ X, ∀a, b ≥ 0, af (x) + bg(y) ≤ ax -by }.
Definition 2.5.

d + 2 (u, v) = inf ε>0 d ε 2 (u, v), where d ε 2 (u, v) = sup (f,g)∈D ε 2 u(f ) + v(g) and ∀ε > 0, D ε 2 = {(f, g) ∈ E×E, ∀x, y ∈ X, ∀a, b ∈ [0, 1], af (x)+bg(y) ≤ ε+ ax-by }.
Definition 2.6.

d 3 (u, v) = inf γ∈M 3 (u,v) X 2 ×[0,1] 2 λx -µy dγ(x, y, λ, µ),
where

M 3 (u, v) is the set of finite positive measures on X 2 × [0, 1] 2 satisfying for each f in E : (x,y,λ,µ)∈X 2 ×[0,1] 2 λf (x)dγ(x, y, λ, µ) = u(f ), and
(x,y,λ,µ)∈X 2 ×[0,1] 2 µf (y)dγ(x, y, λ, µ) = v(f ).
In the next subsection we will prove the following result.

Theorem 2.7. For all u and v in ∆(X),

d 1 (u, v) = d 2 (u, v) = d + 2 (u, v) = d 3 (u, v).

Proof of theorem 2.7

The proof is split into several parts.

Proposition 2.8.

d 1 = d 2 = d + 2 . It is plain that d 1 ≤ d 2 ≤ d + 2
, so all we have to prove is d + 2 ≤ d 1 . We start with a lemma. Lemma 2.9. Fix ε > 0, and let f in E be such that :

∀x ∈ X, ∀a ∈ [0, 1], af (x) ≤ ε + a x . Define f by : ∀y ∈ X, f (y) = inf a∈[0,1],b∈(0,1],x∈X 1 b (ε + ax -by -af (x)) .
Then for each y in X,y ≤ f (y) ≤ -f (y) + ε. Moreover f ∈ E 1 , and :

∀x ∈ X, ∀y ∈ X, ∀a ∈ [0, 1], ∀b ∈ [0, 1], a f (x) -b f (y) ≤ aε + by -ax .
Proof of lemma 2.9 : By assumption on f , we have for all y in X, a in

[0, 1], b in (0, 1], x in X : 1 b (ε + ax -by -af (x)) ≥ 1 b (-a x + ax -by ) ≥ -y .
In the definition of f (y), considering a = b = 1 and x = y yields f (y) ≤ -f (y) + ε.

Fix x and y in X, a and b in [0, 1]. We have :

a f (x) -b f (y) = a inf a ′ ,b ′ ,x ′ 1 b ′ (ε + a ′ x ′ -b ′ x -a ′ f (x ′ )) -b inf a ′′ ,b ′′ ,x ′′ 1 b ′′ (ε + a ′′ x ′′ -b ′′ y -a ′′ f (x ′′ )) . If a = 0, then the inequality f (y) ≥ -y leads to -b f (y) ≤ b y . If b = 0, choose a ′ = 0, b ′ = 1 and x ′ = x to get a f (x) ≤ aε + ax . If ab > 0, given η > 0, choose a ′′ , b ′′ , x ′′ η-optimal in the second infimum. We can define x ′ = x ′′ , and choose a ′ ∈ [0, 1] and b ′ ∈ (0, 1] such that a ′ b ′ = b a a ′′ b ′′ . We obtain : a f (x) -b f (y) ≤ bη + ( a b ′ - b b ′′ )ε + ( a ′′ b ′′ bx ′′ -ax - a ′′ b ′′ bx ′′ -by ) ≤ bη + ( a b ′ - b b ′′ )ε + ax -by . If a = b > 0, choose a ′ = a ′′ and b ′ = b ′′ to obtain : f (x) -f (y) ≤ x -y and therefore f is 1-Lipschitz.
Otherwise, we distinguish two cases. Proof of proposition 2.8 : Fix u and v in ∆(X), and consider ε > 0. For each

(f, g) in D ε 2 , we have -f + ε ≥ f ≥ g and (f, f ) in D ε 2 . We also have ( f , f ) ∈ D ε 2 so iterating the construction, we get ( f , f ) ∈ D ε 2 , and -f + ε ≥ f ≥ f . Now, u(f ) + v(g) ≤ u( f ) + v( f) ≤ -u( f ) + ε + v( f).
Hence we have obtained :

d ε 2 (u, v) ≤ ε + sup f ∈C ε(u,v) -u(f ) + v(f ),
where

C ε(u,v) is the set of functions f in E 1 satisfying : ∀x ∈ X, ∀y ∈ X, ∀a ∈ [0, 1], ∀b ∈ [0, 1], af (x)-bf (y) ≤ aε+ ax-by and f (y) ≥ -y .
For each positive k, one can choose f k in E 1 achieving the above supremum for ε = 1/k. Taking a limit point of (f k ) k yields a function f in D 1 such that :

-u(f ) + v(f ) ≥ d + 2 (u, v). The function f * = -f is in D 1 and satisfies u(f * ) - v(f * ) ≥ d + 2 (u, v)
, and the proof of proposition 2.8 is complete.

Proposition 2.10. d + 2 ≥ d 3 . Proof : The proof is based on (a corollary of) Hahn-Banach theorem. Define :

H = C(X 2 × [0, 1] 2 ) and L = {ϕ ∈ H, ∃f, g ∈ C(X) s.t. ∀x, y ∈ X, ∀λ, µ ∈ [0, 1], ϕ(x, y, λ, µ) = λf (x)+µg(y)}.
H is endowed with the uniform norm and L is a linear subspace of H. Note that the unique constant mapping in L is 0. Fix u and v in ∆(X), and let r be the linear form on L defined by r(ϕ) = u(f ) + v(g), where ϕ(x, y, λ, µ) = λf (x) + µg(y) for all x, y, λ, µ.

Fix now ε > 0, and put :

U ε = {ϕ ∈ H, ∀x, y ∈ X, ∀λ, µ ∈ [0, 1], ϕ(x, y, λ, µ) ≤ λx -µy + ε}.
We have :

sup ϕ∈L∩Uε r(ϕ) = d ε 2 (u, v).
U ε is a convex subset of H which is radial at 0, in the sense that : ∀ϕ ∈ H, ∃δ > 0 such that tϕ ∈ U ε as soon as |t| ≤ δ. By a corollary of Hahn-Banach theorem (see theorem 6.2.11 p.202 in Dudley, 2002), r can be extended to a linear form on H such that :

sup ϕ∈Uε r(ϕ) = d ε 2 (u, v). Given ϕ ∈ H, we have εϕ/ ϕ ∞ ∈ U ε , which implies that r(ϕ) ≤ ϕ ∞ d ε 2 (u, v)/ε, so that r belongs to H ′ . And if ϕ ≥ 0, we have tϕ ∈ U ε if t ≤ 0, so that r(ϕ) ≥ d ε 2 (u, v)
/t for all t ≤ 0 and r(ϕ) ≥ 0. By Riesz Theorem, r can be represented by a positive finite measure γ on X 2 × [0, 1] 2 .

Given f in E, one can consider ϕ f ∈ L defined by ϕ f (x, y, λ, µ) = λf (x). r(ϕ = f ) = γ(ϕ f ) gives : u(f ) = (x,y,λ,µ)∈X 2 ×[0,1] 2 λf (x)dγ(x, y, λ, µ), and similarly v(f ) = (x,y,λ,µ)∈X 2 ×[0,1] 2 µf (y)dγ(x, y, λ, µ), and we obtain that γ ∈ M 3 (u, v).

Because γ ≥ 0, sup ϕ∈Uε r(ϕ) = r(ϕ * ) where ϕ * (x, y, λ, µ) = λxµy + ε.

We get d ε 2 (u, v) = X 2 ×[0,1] 2 λx -µy dγ(x, y, λ, µ) + εγ(X 2 × [0, 1] 2 ), so d ε 2 (u, v) ≥ X 2 ×[0,1] 2 λx -µy dγ(x, y, λ, µ) ≥ d 3 (u, v).
Lemma 2.11.

d 3 ≥ d 2 . Proof : Fix (f, g) ∈ D 2 and γ ∈ M 3 (u, v). u(f ) + v(g) = X 2 ×[0,1] 2 λf (x)dγ(x, y, λ, µ) + X 2 ×[0,1] 2 µg(y)dγ(x, y, λ, µ) = X 2 ×[0,1] 2 (λf (x) + µg(y))dγ(x, y, λ, µ) ≤ X 2 ×[0,1] 2
λxµy dγ(x, y, λ, µ).

The case of probabilities over a simplex

We assume here that X = ∆(K), where K is a non empty finite set. We use p = k |p k | for every vector p = (p k ) k∈K in IR K , and view X as the set of vectors in IR K + with norm 1.

X = {p = (p k ) k∈K ∈ IR K + , k∈K p k = 1}.
Recall that for u and v in ∆(X), we have

d 1 (u, v) = sup f ∈D 1 |u(f ) -v(f )|, where D 1 = {f ∈ E, ∀x, y ∈ X, ∀a, b ≥ 0, af (x) -bf (y) ≤ ax -by }.
We now introduce an alternative definition of d 1 using "non revealing game functions". These functions come from the theory of repeated games with incomplete information à la Aumann Maschler (1995), and the interest for the distance d 0 emerged several years ago while doing research on Markov decision processes with partial observation and repeated games with an informed controller (see [START_REF] Renault | Uniform value in Dynamic Programming[END_REF]2012a).

Given a collection of matrices (G k ) k∈K (all of the same finite size I × J) indexed by K and with values in [-1, 1], we define the "non revealing function" f in C(X) by :

∀p ∈ X, f (p) = Val k∈K p k G k , = max x∈∆(I) min y∈∆(J) i∈I,j∈J x(i)y(j) k∈K p k G k (i, j) , = min y∈∆(J) max x∈∆(I) i∈I,j∈J
x(i)y(j)

k∈K p k G k (i, j) .
f (p) is the minmax value of the average matrix k p k G k . The set of all such non revealing functions f , where I, J and (G k ) k∈K vary, is denoted by D 0 .

Clearly, all affine functions from X to [-1, 1] belong to D 0 . It is known that the set of non revealing functions is dense in C(X). However, we only consider here non revealing functions defined by matrices with values in [-1, 1], and D 0 is not dense in the set of continuous functions from X to [-1, 1]. As an example, consider the case where K = {1, 2} and f in E is piecewise-linear with f (1, 0) = f (0, 1) = 0 and f (1/2, 1/2) = 1. If a function g in D 0 is such that g(1/2, 1/2) = 1, then necessarily the values of the two matrix games G 1 and G 2 are also equal to 1 since it is the maximum value. Therefore f is not in

D 0 . In fact f is 1-Lipschitz, however 2f (1/2, 1/2) -f (1, 0) = 2 > 2(1/2, 1/2) -(1, 0) = 1,
so it is not in D 1 which we will see later contains D 0 (see lemma 2.13).

Lemma 2.12. If f , g belong to D 0 and λ ∈ [0, 1], then -f , sup{f, g}, inf{f, g} and λf + (1λ)g are in D 0 . The linear span of D 0 is dense in C(X).

Proof : The proof can be easily deduced from proposition 5.1. page 357 in MSZ, part B. For instance, let f and g in D 0 be respectively defined by the collections of matrices (G k ) k∈K with size I 1 × J 1 and (H k ) k∈K with size I 2 × J 2 .

Defining for each k, i 1 , j 1 : G ′k (i 1 , j 1 ) = -G k (j 1 , i 1 ) yields a family of matrices (G ′k ) k with size J 1 × I 1 inducing -f . So -f ∈ D 0 .

To get that sup{f, g} belongs to D 0 , one can assume w.l.o.g. that

I 1 ∩ I 2 = J 1 ∩ J 2 = ∅. Set I = I 1 ∪ I 2 and J = J 1 × J 2 . Define for each k the matrix game L k in IR I×J by L k (i, (j 1 , j 2 )) = G k (i, j 1 ) if i ∈ I 1 , L k (i, (j 1 , j 2 )) = H k (i, j 2 ) if i ∈ I 2 . Then for each p in X, we have Val( k p k L k ) = sup{f (p), g(p)}, so that sup{f, g} ∈ D 0 .
Lemma 2.13. The closure of D 0 is D 1 .

Proof : We first show that D 0 ⊂ D 1 . Let I and J be finite sets, and (G k ) k∈K be a collection of I × J-matrices with values in [-1, 1]. Consider p and q in X and a and b non negative. Then for all i and j :

k p k aG k (i, j) - k q k bG k (i, j) ≤ k |ap k -bq k | = ap -bq .
As a consequence,

aVal k∈K p k G k -bVal k∈K q k G k = Val k∈K ap k G k -Val k∈K bq k G k ≤ ap -bq
We now show that the closure of D 0 is D 1 . Consider f in D 1 , in particular we have f ∞ ≤ 1. Let p and q be distinct elements in X, and define Y as the linear span of p and q, and define ϕ from Y to IR such that : ϕ(λp + µq) = λf (p) + µf (q) for all reals λ and µ.

If λ ≥ 0 and µ ≥ 0, we have ϕ(λp + µq) ≤ λ + µ = λp + µq . If λ ≥ 0 and µ ≤ 0, we directly use the definition of D 1 to get : ϕ(λp + µq) ≤ λp + µq . As a consequence, ϕ is a linear form with norm at most 1 on Y . By Hahn-Banach theorem, it can be extended to a linear mapping on IR K with the same norm, and we denote by g the restriction of this mapping to X. g is affine with g(p) = ϕ(p) = f (p) and g(q) = ϕ(q) = f (q). Moreover, for each r in X, we have g(r) ≤ r = 1. As a consequence g belongs to D 0 . Because D 0 is stable under the sup and inf operations, we can use Stone-Weierstrass theorem (see for instance lemma A7.2 in Ash p.392) to conclude that f belongs to the closure of D 0 .

Definition 2.14. Given u and v in ∆(X), define :

d 0 (u, v) = sup f ∈D 0 u(f ) -v(f ) Proposition 2.15. d 0 is a distance on ∆(X) metrizing the weak-* topology. Moreover d 0 = d 1 = d 2 = d 3 . Proof : d 0 = d 1 = d 2 = d 3
follows from lemma 2.13 and theorem 2.7. Because the linear span of D 0 is dense in C(X), we obtain the separation property and

d 0 is a distance on ∆(X). Because D 0 ⊂ D 1 ⊂ E 1 , we have d 0 = d 1 ≤ d KR .
Since (∆(X), d KR ) is a compact metric space, the identity map (∆(X), d KR ) to (∆(X), d 0 ) is bicontinuous, and we obtain that (∆(X), d 0 ) is a compact metric space and d 0 and d KR are equivalent. (see for instance proposition 2 page 138 Aubin).

Remark : one can show that allowing for infinite sets I, J in the definition of D 0 (still assuming that all games k p k G k have a value) would not change the value of d 0 .

From now on, we just write d * (u, v) for the distance

d 0 = d 1 = d 2 = d 3 on ∆(X).
Elements of X can be viewed as elements of ∆(X) (using Dirac measures), and it is well known that for p, q in X, we have : d KR (δ p , δ q ) = pq . We have the same result with d * .

Lemma 2.16. For p, q in X, we have d * (δ p , δ q ) = pq .

Proof :

Define K 1 = {k ∈ K, p k ≥ q k }, and K 2 = K\K 1 . Consider f affine on X such that f (k) = +1 if k ∈ K 1 , and f (k) = -1 if k ∈ K 2 . Then f ∈ D 1 , and d * (δ p , δ q ) ≥ |f (p) -f (q)| = p -q .
The other inequality is clear.

We now present a dual formulation for our distance, in the spirit of Kantorovich duality formula from optimal transport. For any u, v in ∆(X), we denote by Π(u, v) the set of transference plans, or couplings, of u and v, that is the set of probability distributions over X × X with first marginal u and second marginal v. Recall (see for instance Villani 2003, p.207) :

d KR (u, v) = sup f ∈E 1 |u(f ) -v(f )| = min γ∈Π(u,v) (x,y)∈X×X
xy dγ(x, y)

We will concentrate on probabilities on X with finite support. We denote by Z = ∆ f (X) the set of such probabilities.

Definition 2.17. Let u and v be in Z with respective supports U and V . We define M 4 (u, v) as the set

(α, β) ∈ (IR + U ×V ) 2 , s.t.∀x ∈ U, ∀y ∈ V, y ′ ∈V α(x, y ′ ) = u(x) and x ′ ∈U β(x ′ , y) = v(y) . And d 4 (u, v) = inf (α,β)∈M 4 (u,v) (x,y)∈U ×V xα(x, y) -yβ(x, y)
Notice that diagonal elements in M 4 (u, v), i.e. measures α such that (α, α) ∈ M 4 (u, v), coincide with elements of Π(u, v). M 4 (u, v) is a polytope in the Euclidean space (IR U ×V ) 2 , so the infimum in the definition of d 4 (u, v) is achieved.

Theorem 2.18. (Duality formula) Let u and v be in Z with respective supports U and V .

d * (u, v) = sup f ∈D 1 |u(f ) -v(f )| = min (α,β)∈M 4 (u,v) (x,y)∈U ×V xα(x, y) -yβ(x, y)
where

D 1 = {f ∈ E, ∀x, y ∈ X, ∀a, b ≥ 0, af (x) -bf (y) ≤ ax -by }, and M 4 (u, v) = (α, β) ∈ IR + U ×V × IR + U ×V , s.t. ∀(x, y) ∈ U × V, y ′ ∈V α(x, y ′ ) = u(x) and x ′ ∈U β(x ′ , y) = v(y) .
The proof is postponed to the next subsection. We conclude this part by a simple but fundamental property of the distance d * .

Definition 2.19. Given a finite set S, we define the posterior mapping ψ S from ∆(K × S) to ∆(X) by :

ψ S (π) = s∈S π(s)δ p(s)
where for each s, π(s

) = k π(k, s) and p(s) = (p k (s)) k∈K ∈ X is the posterior on K given s (defined arbitrarily if π(s) = 0) : for each k in K, p k (s) = π(k,s) π(s) .
ψ S (π) is a probability with finite support over X. Intuitively, think of a joint variable (k, s) being selected according to π, and an agent just observes s. His knowledge on K is then represented by p(s). And ψ S (π) represents the ex-ante information that the agent will know about the variable k. ∆(K × S) is endowed as usual with the . 1 norm. One can show that ψ S is continuous whenever X is endowed with the weak-* topology. Intuitively, ψ S (π) has less information than π, because the agent does not care about s itself but just on the information about k given by s. So one may hope that the mapping ψ S is 1-Lipschitz (non expansive) for a well chosen distance on ∆(X). This is not the case if one uses the Kantorovich-Rubinstein distance d KR , as shown by the example below :

Example 2.20. Consider the case where K = {a, b, c} and S = {α, β}. We denote by π and π ′ the following laws on ∆(K × S) :

S S K   1 4 0 0 1 2 1 4 0   and   1 4 0 0 1 2 0 1 4   . π π ′ Their disintegrations are respectively ψ S (π) = 1 2   1 2 0 1 2   + 1 2   0 1 0   and ψ S (π ′ ) = 1 4   1 0 0   + 3 4   0 2 3 1 3   . We define the test function f : ∆(K) → [-1, 1] by f   0 1 0   = 1 3 , f   1 2 0 1 2   = - 1 3 , f   0 2 3 1 3   = 1 and f   1 0 0   = 2 3 . We have π-π ′ = 1 2 and since f is 1-Lipschitz, d KR (ψ S (π), ψ S (π ′ )) ≥ ψ S (π ′ )(f )- ψ S (π)(f ) = 11 12 -0 > 1 2 . The posterior mapping ψ S is not 1-Lipschitz from (∆(K × S), . 1 ) to (∆(X), d KR ) .
However, the next proposition shows that the distance d * has the desirable property.

Proposition 2.21. For each finite set S, the mapping

ψ S is 1-Lipschitz from (∆(K × S), . 1 ) to (∆ f (X), d * ).
Moreover, d * is the largest distance on Z having this property : given u and v in Z, we have

d * (u, v) = inf{ π -π ′ 1 , s.t. S finite, ψ S (π) = u, ψ S (π ′ ) = v}.
Proof : First fix S and π, π ′ in ∆(K × S). Write u = ψ S (π), u ′ = ψ S (π ′ ). For any f in D 1 , we have :

u(f ) -u ′ (f ) = s∈S (π(s)f (p(s)) -π ′ (s)f (p ′ (s)) ≤ s∈S π(s)p(s) -π ′ (s)p ′ (s) ≤ s∈S (π(k, s)) k -(π ′ (k, s)) k ≤ s∈S k∈K |π(k, s) -π ′ (k, s)| = π -π ′ 1 . So d * (u, u ′ ) ≤ π -π ′ 1 , and ψ S is 1-Lipschitz.
Let now u and v be in Z. There exists (α, , y). By definition of M 4 (u, v), π and π ′ are probabilities and

β) ∈ M 4 (u, v) such that d * (u, v) = (x,y)∈U ×V α(x, y)x -β(x, y)y . Define S = U × V and π, π ′ ∈ ∆(K × S) by π(k, (x, y)) = x(k)α(x, y) and π ′ (k, (x, y)) = y(k)β(x
π -π ′ 1,K×S = k∈K,(x,y)∈U ×V |x(k)α(x, y) -y(k)β(x, y)| = (x,y)∈U ×V α(x, y)x -β(x, y)y .

Proof of the duality formula

Let u and v be in ∆(X), and denote by U and V the respective supports of u and v. We write S = X 2 × [0, 1] 2 , and we start with a lemma, where no finiteness assumption on U or V is needed.

Lemma 2.22. For each γ ∈ M 3 (u, v), we have :

X 2 ×[0,1] 2 λx-µy dγ(x, y, λ, µ) = 2+ U ×V ×[0,1] 2 ( λx -µy -λ -µ) dγ(x, y, λ, µ). Proof : Write A(γ) = S λx -µy dγ(x, y, λ, µ). By definition of M 3 (u, v), we have : S λ1 x /
∈U dγ = 0, and

S µ1 y / ∈V dγ = 0.
So that λ1 x / ∈U = µ1 y / ∈V = 0 γ. a.e. We can write :

A(γ) = S 1 x∈U,y∈V λx -µy dγ(x, y, λ, µ) + S 1 x∈U,y /
∈V λxµy dγ(x, y, λ, µ)

+ S 1 x / ∈U,y∈V λx -µy dγ(x, y, λ, µ) + S 1 x / ∈U,y / ∈V λx -µy dγ(x, y, λ, µ) = S 1 x∈U,y∈V λx-µy dγ(x, y, λ, µ)+ S 1 x∈U,y / ∈V λdγ(x, y, λ, µ)+ S 1 x / ∈U,y∈V µdγ(x, y, λ, µ)+0.
We also have by definition of M 3 (u, v) that 1 = S 1 x∈U λdγ, so that :

1 = S 1 x∈U,y∈V λdγ + S 1 x∈U,y / ∈V λdγ.
And similarly 1 = S 1 x∈U,y∈V µdγ + S 1 x / ∈U,y∈V µdγ. We obtain :

A(γ) = 2 + S 1 x∈U,y∈V λx -µy dγ(x, y, λ, µ) - S 1 x∈U,y∈V λdγ - S 1 x∈U,y∈V µdγ.
We assume in the sequel that U and V are finite, and define d 5 (u, v) as follows :

Definition 2.23. Define

M 5 (u, v) = (α, β) = (α(x, y), β(x, y)) (x,y)∈U ×V ∈ (IR U ×V ) 2 , s.t.∀x ∈ U, ∀y ∈ V, α(x, y) ≥ 0, β(x, y) ≥ 0, y ′ ∈V α(x, y ′ ) ≤ u(x) and x ′ ∈U β(x ′ , y) ≤ v(y) . And d 5 (u, v) = inf (α,β)∈M 5 (u,v) 2+ (x,y)∈U ×V ( xα(x, y) -yβ(x, y) -α(x, y) -β(x, y)) . M 5 (u, v) is a polytope in the Euclidean space (IR U ×V ) 2 , so the infimum in the definition of d 5 (u, v) is achieved. Lemma 2.24. d 3 (u, v) ≥ d 5 (u, v).
Proof : Let γ be in M 3 (u, v). Fix for a while (x, y) in U × V , and assume that γ(x, y) > 0. We define γ(.|x, y) the conditional probability on [0, 1] 2 given (x, y) by : for all

ϕ ∈ C([0, 1] 2 ), [0,1] 2 ϕ(λ, µ)dγ(λ, µ|x, y) = 1 γ(x, y) (x ′ ,y ′ ,λ,µ)∈S 1 x ′ =x,y ′ =y ϕ(λ, µ)dγ(x ′ , y ′ , λ, µ). So that γ(x, y) [0,1] 2 ( λx-µy -λ-µ)dγ(λ, µ|x, y) = (λ,µ)∈[0,1] 2 ( λx-µy -λ-µ)dγ(x, y, λ, µ).
The mapping Ψ : (λ, µ) → λx-µy -λ-µ is convex so by Jensen's inequality we get :

(λ,µ)∈[0,1] 2 ( λx -µy -λ -µ)dγ(λ, µ|x, y) ≥ x (λ,µ)∈[0,1] 2 λdγ(λ, µ|x, y) -y (λ,µ)∈[0,1] 2 µdγ(λ, µ|x, y) - (λ,µ)∈[0,1] 2 λdγ(λ, µ|x, y) - (λ,µ)∈[0,1] 2 µdγ(λ, µ|x, y).
We write :

P (x, y) = (λ,µ)∈[0,1] 2 λdγ(λ, µ|x, y) and Q(x, y) = (λ,µ)∈[0,1] 2 µdγ(λ, µ|x, y), so that (λ,µ)∈[0,1] 2 ( λx-µy -λ-µ)dγ(λ, µ|x, y) ≥ xP (x, y)-yQ(x, y) -P (x, y)-Q(x, y).

Now, by lemma 2.22

A(γ) = 2 + x∈U,y∈V (λ,µ)∈[0,1] 2 ( λx -µy -λ -µ) dγ(x, y, λ, µ) = 2 + x∈U,y∈V,γ(x,y)>0 (λ,µ)∈[0,1] 2 ( λx -µy -λ -µ) dγ(x, y, λ, µ) ≥ 2 + x∈U,y∈V,γ(x,y)>0 γ(x, y) ( xP (x, y) -yQ(x, y) -P (x, y) -Q(x, y)) . For (x, y) in U × V , define α(x, y) = γ(x, y)P (x, y) ≥ 0 and β(x, y) = γ(x, y)Q(x, y) ≥ 0 (with α(x, y) = β(x, y) = 0 if γ(x, y) = 0). We get : A(γ) ≥ 2 + x∈U,y∈V ( xα(x, y) -yβ(x, y) -α(x, y) -β(x, y)) .
And we have, for each x in U :

y∈V α(x, y) = y∈V,γ(x,y)>0 (λ,µ)∈[0,1] 2 λdγ(x, y, λ, µ) ≤ (y,λ,µ)∈X×[0,1] 2 λdγ(x, y, λ, µ) = u(x).
where the last equality comes from the definition of M 3 (u, v). Similarly, for each y in V we can show that x∈U β(x, y) ≤ v(y), and lemma 2.24 is proved.

Lemma 2.25. d 5 (u, v) ≥ d 4 (u, v).
Proof : Consider (α * , β * ) achieving the minimum in the definition of d 5 (u, v). Assume that there exists x * such that y∈V α(x * , y) < u(x * ). For any x in X and z in IR K + , one can check that the mapping l : (α → xαzα) is nonincreasing from IR + to IR (as the sum of the mappings l k : (α → |αx kz k |αx k ), each l k being non increasing in α). As a consequence, one can choose any y * in V and increase α(x * , y * ) in order to saturate the constraint without increasing the objective. So we can assume without loss of generality that y∈V α(x * , y) = u(x * ) for all x * and similarly x∈U β(x, y * ) = v(y * ) for all y * . Consequently,

d 5 (u, v) = 2 + (x,y)∈U ×V ( xα * (x, y) -yβ * (x, y) -α * (x, y) -β * (x, y)) = (x,y)∈U ×V xα * (x, y) -yβ * (x, y) ≥ d 4 (u, v). Lemma 2.26. d 4 (u, v) ≥ d 2 (u, v). Proof : Fix (f, g) ∈ D 2 and (α, β) ∈ M 4 (u, v). u(f ) + v(g) = x∈U f (x)u(x) + y∈Y g(y)v(y) = (x,y)∈U ×V f (x)α(x, y) + g(y)β(x, y) ≤ (x,y)∈U ×V α(x, y)x -β(x, y)y ≤ d 4 (u, v).
We have shown that

d 3 (u, v) ≥ d 5 (u, v) ≥ d 4 (u, v) ≥ d 2 (u, v) = d 3 (u, v) = d 1 (u, v).
This ends the proof of theorem 2.18.

Long-term values for compact non expansive Markov Decision Processes

In this section we consider Markov Decision Processes, or Controlled Markov Chains, with bounded payoffs and transitions with finite support. We will consider two closely related models of MDP and prove in each case the existence and a characterization for a general notion of long-term value. The first model deals with MDP without any explicit action set (hence, payoffs only depend on the current state), such MDP will be called gambling houses using the terminology of gambling theory (see [START_REF] Maitra | Discrete gambling and stochastic games[END_REF]. We will assume in this setup that the set of states X is metric compact and that the transitions are non expansive with respect to the KR-distance on ∆(X). Since we only use the KRdistance here, the theorem for the first model, namely theorem 3.9, does not use the distance for belief spaces studied in section 2. The second model is the standard model of Markov Decision Processes with states, actions, transitions and payoffs, and we will assume that the state space X is a compact subset of a simplex ∆(K). We will need for this second case an assumption of non expansiveness for the transitions which is closely related to the distance d * introduced in section 2, see theorem 3.19 later. The applications in sections 4.1 and 4.2 will be based on the second model.

Long-term values for Gambling Houses

In this section we consider Markov Decision Processes of the following form. There is a non empty set of states X, a transition given by a multi-valued mapping F : X ⇉ ∆ f (X) with non empty values, and a payoff (or reward) function r : X → [0, 1]. The idea is that given an initial state x 0 in X, a decision-maker (or player) can choose a probability with finite support u 1 in F (x 0 ), then x 1 is selected according to u 1 and there is a payoff r(x 1 ). Then the player has to select u 2 in F (x 1 ), x 2 is selected according to u 1 and the player receives the payoff r(x 2 ), etc... Note that there is no explicit action set here, and that the transitions take values in ∆ f (X) and hence all have finite support.

We say that Γ = (X, F, r) is a Gambling House. We assimilate the elements in X with their Dirac measures in ∆(X), and in case the values of F only consist of Dirac measures on X, we view F as a correspondence from X to X and say that Γ is a deterministic Gambling House (or a Dynamic Programming problem). In general we write Z = ∆ f (X), and an element in Z is written u = x∈X u(x)δ x . The set of stages is IN * = {1, ..., t, ....}, and a probability distribution over stages is called an evaluation. Given an evaluation θ = (θ t ) t≥1 and an initial stage x 0 in X, the θ-problem Γ θ (x 0 ) is the problem induced by a decision-maker starting from x 0 and maximizing the expectation of t≥1 θ t r(x t ).

Formally, we first linearly extend r and F to ∆ f (X) by defining for each u = x∈X u(x)δ x in Z, the payoff r(u) = x∈X r(x)u(x) and the transition

F (u) = { x∈X u(x)f (x), s.t. f : X → Z and f (x) ∈ F (x)∀x ∈ X}.
We also define the mixed extension of F as the correspondence from Z to itself which associates to every u = x∈X u(x)δ x in ∆ f (X) the image :

F (u) = x∈X u(x)f (x), s.t. f : X → Z and f (x) ∈ convF (x) ∀x ∈ X .
The graph of F is the convex hull of the graph of F . Moreover F is an affine correspondence, as shown by the lemma below.

Lemma 3.1. ∀u, u ′ ∈ Z, ∀α ∈ [0, 1], F (αu + (1 -α)u ′ ) = α F (u) + (1 -α) F (u ′ ). Proof : The ⊂ part is clear. To see the reverse inclusion, let v = α x∈X u(x)f (x)+ (1 -α) x∈X u ′ (x)f ′ (x) be in α F (u) + (1 -α) F (u ′ ), with transparent notations. Define g(x) = αu(x)f (x) + (1 -α)u ′ (x)f ′ (x) αu(x) + (1 -α)u ′ (x) ,
for each x such that the denominator is positive. Then g(x) ∈ convF (x), and

v = x∈X (αu(x) + (1 -α)u ′ (x))g(x) ∈ F (αu + (1 -α)u ′ ).
Definition 3.2. A pure play, or deterministic play, at x 0 is a sequence σ = (u 1 , ..., u t , ...) ∈ Z ∞ such that u 1 ∈ F (x 0 ) and u t+1 ∈ F (u t ) for each t ≥ 1.

A play, or mixed play, at x 0 is a sequence σ = (u 1 , ..., u t , ...) ∈ Z ∞ such that u 1 ∈ convF (x 0 ) and u t+1 ∈ F (u t ) for each t ≥ 1. We denote by Σ(x 0 ) the set of mixed plays at x 0 .

A pure play is a particular case of a mixed play. Mixed plays corresponds to situations where the decision-maker can select, at every stage t and state x t-1 , randomly the law u t of the new state. A mixed play at x 0 naturally induces a probability distribution over the set (X × ∆ f (X)) ∞ of sequences (x 0 , u 0 , x 1 , u 1 , ...), where X and Z are endowed with the discrete σ-algebra and (X × ∆ f (X)) ∞ is endowed with the product σ-algebra.

Definition 3.3. Given an evaluation θ, the θ-payoff of a play σ = (u 1 , ..., u t , ...) is defined as : γ θ (σ) = t≥1 θ t r(u t ), and the θ-value at x 0 is :

v θ (x 0 ) = sup σ∈Σ(x 0 ) γ θ (σ).
It is easy to see that the supremum in the definition of v θ can be taken over the set of pure plays at x 0 . We have the following recursive formula. For each evaluation θ = (θ t ) t≥1 such that θ 1 < 1, we denote by θ + the "shifted" evaluation ( θ t+1 1-θ 1 ) t≥1 . We extend linearly v θ to Z, so that the recursive formula can be written :

∀θ ∈ ∆(IN * ), ∀x ∈ X, v θ (x) = sup u∈convF (x) (θ 1 r(u) + (1 -θ 1 )v θ + (u)) .
And by linearity the supremum can be taken over F (x). It is also easy to see that for all evaluation θ and initial state x, we have the inequality :

|v θ (x) -sup u∈F (x) v θ (u)| ≤ θ 1 + t≥2 |θ t -θ t-1 |.
(1)

In this paper, we are interested in the limit behavior when the decision-maker is very patient. Given an evaluation θ, we define :

I(θ) = t≥1 |θ t+1 -θ t |
The decision-maker is considered as patient whenever I(θ) is small, so I(θ) may be seen as the impatience of θ (see Sorin, 2002 p. 105 and[START_REF] Renault | General long-term values in Dynamic Programming[END_REF]. When θ = (θ t ) t≥1 is non increasing, then I(θ) is just θ 1 . A classic example is when θ = n t=1 1 n δ t , the value v θ is just denoted v n and the evaluation corresponds to the average payoff from stage 1 to stage n. In this case I(θ) = 1/n -→ n→∞ 0. We also have I(θ) = 1/n if θ = m+n t=m 1 n δ t for some non-negative m. Another example is the case of discounted payoffs, when θ = (λ(1λ) t-1 ) t≥1 for some discount factor λ ∈ (0, 1], and in this case I(θ) = λ -→ λ→0 0.

Definition 3.4. The Gambling House Γ = (X, F, r) has a general limit value v * if (v θ ) uniformly converges to v * when I(θ) goes to zero, i.e. :

∀ε > 0, ∃α > 0, ∀θ, ( I(θ) ≤ α =⇒ (∀x ∈ X, |v θ (x) -v * (x)| ≤ ε) ) .
The existence of the general limit value implies in particular that (v n ) n and (v λ ) λ converge to the same limit when n goes to +∞ and λ goes to 0. This is coherent with the result of [START_REF] Lehrer | A uniform Tauberian Theorem in Dynamic Programming[END_REF], which states that the uniform convergence of (v n ) n and (v λ ) λ are equivalent.

In the definition of the general limit value, we require all value functions to be close to v * when the patience is high, but the plays used may depend on the precise expression of θ. In the following definition, we require the same play to be simultaneously optimal for all θ patient enough.

Definition 3.5. The Gambling House Γ = (X, F, r) has a general uniform value if it has a general limit value v * and moreover for each ε > 0 one can find α > 0 and for each initial state x a mixed play σ(x) at x satisfying :

∀θ, (I(θ) ≤ α =⇒ (∀x ∈ X, γ θ (σ(x)) ≥ v * (x) -ε) ) .
Up to now, the literature in repeated games has focused on the evaluations θ = n t=1 1 n δ t and θ = (λ(1λ) t-1 ) t≥1 . The standard (Cesàro)-uniform value can be defined by restricting the evaluations to be Cesàro means : for each ε > 0 one can find n 0 and for each initial state x a mixed play σ(x) at x satisfying : 2011) considered deterministic Gambling Houses and characterized the uniform convergence of the value functions (v n ) n . He also proved the existence of the standard Cesàro-uniform value under some assumptions, including the case where the set of states X is metric precompact, the transitions are non expansive and the payoff function is uniformly continuous. As a corollary, he proved the existence of the uniform value in Partial Observation Markov Decision Processes with finite set of states (after each stage the decision-maker just observes a stochastic signal more or less correlated to the new state).

∀n ≥ n 0 , ∀x ∈ X, γ n (σ(x)) ≥ v * (x) -ε. Recently, Renault (
We now present our main theorem for Gambling Houses. Equation (1) implies that the general limit value v * necessarily has to satisfy some rigidity property. The function v * (or more precisely its linear extension to Z) can only be an "excessive function" in the terminology of potential theory [START_REF] Choquet | Existence et unicité des représentations intégrales au moyen des points extrémaux dans les cônes convexes[END_REF]) and gambling houses [START_REF] Dubins | How to gamble if you must : Inequalities for stochastic processes[END_REF]Savage 1965, Maitra and[START_REF] Maitra | Discrete gambling and stochastic games[END_REF].

Definition 3.6. An affine function w defined on Z (or ∆(X)) is said to be excessive if for all x in X, w(x) ≥ sup u∈F (x) w(u).

Example 3.7. Let us consider the splitting transition given by K a finite set, X = ∆(K) and ∀x ∈ X, F (x) = {u ∈ ∆(X), p∈X u(p)p = x}. Then the function w from Z = ∆(X) to [0, 1] is excessive if and only if the restriction of w to X is concave. Moreover given u, u ′ ∈ ∆(X), u ′ ∈ F (u) if and only u ′ is the sweeping of u as defined by [START_REF] Choquet | Existence et unicité des représentations intégrales au moyen des points extrémaux dans les cônes convexes[END_REF] : for all continuous concave functions

f from X to [0, 1], u ′ (f ) ≤ u(f ).
Assume now that X is a compact metric space and r is continuous. r is naturally extended to an affine continuous function on ∆(X) by r(u) = p∈X r(p)du(p) for all Borel probabilities on X. In the following definition, we consider the closure of the graph of F within the (compact) set ∆(X × X). Definition 3.8. An element u in ∆(X) is said to be an invariant measure of the Gambling House Γ = (X, F, r) if (u, u) ∈ cl(Graph F ). The set of invariant measures of Γ is denoted by R, so that :

R = {u ∈ ∆(X), (u, u) ∈ cl(Graph F )}.
R is a convex compact subset of ∆(X). Recall that for u and u ′ in ∆(X), the Kantorovich-Rubinstein distance between u and u ′ is denoted by

d KR (u, u ′ ) = sup f ∈E 1 |u(f ) -u ′ f )|.
Theorem 3.9. Consider a Gambling House Γ = (X, F, r) such that X is a compact metric space, r is continuous and F is non expansive with respect to the KR distance :

∀x ∈ X, ∀x ′ ∈ X, ∀u ∈ F (x), ∃u ′ ∈ F (x ′ )s.t. d KR (u, u ′ ) ≤ d(x, x ′ ).
Then the Gambling House has a general uniform value v * characterized by :

∀x ∈ X, v * (x) = inf w(x), w : ∆(X) → [0, 1] affine C 0 s.t.
(1) ∀y ∈ X, w(y) ≥ sup

u∈F (y) w(u) and (2)∀u ∈ R, w(u) ≥ r(u) .
That is, v * is the smallest continuous affine function on X which is 1) excessive and 2) above the running payoff r on invariant measures.

Notice that : 1) when Γ = (X, F, r) is deterministic, the hypotheses are satisfied as soon as X is metric compact for some metric d, r is continuous and F is non expansive for d.

2) when X is finite, one can use the distance d(x, x ′ ) = 2 for all x = x ′ in X, so that for u and u

′ in ∆(X), d KR (u, u ′ ) = u -u ′ 1 = x∈X |u(x) -u ′ (x)|
, and the hypotheses are automatically satisfied. We will prove later a more general result for a model of MDP with finite state space, allowing for explicit actions influencing transitions and payoffs (see corollary 3.20).

Remark 3.10. The formula also holds when there is no decision maker, i.e. when F is single-valued, and there are some similarities with the Von Neumann ergodic theorem (1932). Let Z be a Hilbert space and Q be a linear isometry on Z, this theorem states that for all z ∈ Z, the sequence z n = 1 n n t=1 Q t (z) converges to the projection z * of z on the set R of fixed points of Q. Using the linearity and the non expansiveness leads to a characterization by the set of fixed points. In particular, having in mind linear payoff functions of the form (z →< l, z >), we have that the projection z * of z on R is characterized by : ∀l ∈ Z, < l, z * >=< l * , z >= inf{< l ′ , z >, l ′ ∈ R and < l ′ , r >≥< l, r > ∀r ∈ R}.

Example 3.11. We consider here a basic periodic sequence of 0 and 1. Let X = {0, 1} and for all x ∈ X, F (x) = {1 -x} and r(x) = x. There is a unique invariant measure u = 1/2δ 0 + 1/2δ 1 , and the general uniform value exists and satisifes v * (x) = 1 2 for all states x. Notice that considering evaluations θ = (θ t ) t such that θ t is small for each t without requiring I(θ) small, would not necessarily lead to v * . Consider for instance θ n = n t=1 1 n δ 2t for each n, we have v θ n (x) = x for all x in X.

Example 3.12. The state space is the unit circle, let X = {x ∈ C, |x| = 1} and F (e iα ) = e i(α+1) for all real α. If we denote by µ the uniform distribution (Haar probability measure) on the circle, the mapping F is µ-ergodic and µ is Finvariant. By Birkhoff's theorem (1931), we know that the time average converges to the space average µ-almost surely. Here µ is the unique invariant measure, and we obtain that the general uniform value is the constant :

∀x ∈ X, v * (x) = 1 2π 2π 0
r(e iα )dα.

Notice that the value v θ (x) converges to v * (x) for all x in X, and not only for µ-almost all x in X.

Example 3.13. Let Γ = (X, F, r) be a MDP satisfying the hypotheses of the theorem 3.9 such that for all x ∈ X, δ x ∈ F (x). Therefore the set R is equal to ∆(X). In the terminology of Gambling Theory (see [START_REF] Maitra | Discrete gambling and stochastic games[END_REF], Γ is called a leavable gambling house since at each stage the player can stay at the current state. The limit value v * is here characterized by :

v * = inf{v : X → [0, 1] C 0 , v is excessive and v ≥ r}.
In the above formula, v excessive means : ∀x ∈ X, v(x) ≥ sup u∈F (x) IE u (v). This is a variant of the fundamental theorem of gambling theory (see section 3.1 in Maitra Sudderth 1996).

Example 3.14. The following deterministic Gambling House, which is an extension of example 1.4.4. in [START_REF] Sorin | A first course on Zero-Sum repeated games[END_REF] and of example 5.2 of [START_REF] Renault | Uniform value in Dynamic Programming[END_REF], shows that the assumptions of theorem 3.9 allow for many speeds of convergence to the limit value v * . Here l > 1 is a fixed parameter, X is the simplex {x = (p a , p b , p c ) ∈ IR 3 + , p a + p b + p c = 1} and the initial state is x 0 = (1, 0, 0). The payoff is r(p a , p b , p c ) = p bp c , and the transition is defined by : F (p a , p b , p

c ) = {((1 -α -α l )p a , p b + αp a , p c + α l p a ), α ∈ [0, 1/2]}.
The probabilistic interpretation is the following : there are 3 points a, b and c, and the initial point is a. The payoff is 0 at a, it is +1 at b, and -1 at c. At point a, the decision maker has to choose α ∈ [0, 1/2] : then b is reached with probability α, c is reached with probability α l , and the play stays in a with the remaining probability 1αα l . When b (resp. c) is reached, the play stays at b (resp. c) forever. So the decision maker starting at point a wants to reach b and to avoid c. By playing at each stage α > 0 small enough, he can get as close to b as he wants.

Back to our deterministic setup, we use norm . 1 and obtain that X is compact, F is non expansive and r is continuous, so that theorem 3.9 applies. The limit value is given by v * (p a , p b , p c ) = p a + p b , and if we denote by x λ the value v λ (x 0 ), we have for all λ ∈ (0, 1] : x λ = φ(x λ ), where for all x ∈ IR,

φ(x) = max α∈[0,1/2] (1 -λ)(1 -α -α l )x + α.
Since x λ ∈ (0, 1), the first order condition gives (1λ)x λ (-1lα l-1 ) + 1 = 0 and we can obtain :

x λ = 1 (1 -λ) l λ (1 -λ)(l -1) l-1 l + 1 -1
.

Finally we can compute an equivalent of x λ as λ goes to 0. We have

λ (1 -λ)(l -1) l-1 l = ( 1 l -1 ) l-1 l λ l-1 l (1 + o(λ l-1 l )) so that v λ (x 0 ) = (1 -λ) 1 l ( 1 l-1 ) l-1 l λ l-1 l + o(λ 2l-2 l ) + 1 v λ (x 0 ) = 1 -Cλ l-1 l + o(λ l-1 l ) with C = l (l -1) (l-1) l .

Long-term values for standard MDPs

A standard Markov Decision Problem Ψ is given by a non empty set of states X, a non empty set of actions A, a mapping q : X × A → ∆ f (X) and a payoff function g : X × A → [0, 1]. At each stage, the player learns the current state x and chooses an action a. He then receives the payoff g(k, a), a new state is drawn accordingly to q(k, a) and the game proceeds to the next stage. Definition 3.15. A pure, or deterministic, strategy is a sequence of mappings σ = (σ t ) t≥1 where σ t : (X × A) t-1 → A for each t. A strategy (or behavioral strategy) is a sequence of mappings σ = (σ t ) t≥1 where σ t : (X × A) t-1 → ∆ f (A) for each t. We denote by Σ the set of strategies.

A pure strategy is a particular case of strategy. An initial state x 1 in X and a strategy σ naturally induce a probability distribution with finite support over the set of finite histories (X × A) n for all n, which can be uniquely extended to a probability over the set (X × A) ∞ of infinite histories. Definition 3.16. Given an evaluation θ and an initial state x 1 in X, the θ-payoff of a strategy σ at x 1 is defined as γ θ (x 1 , σ) = IE x 1 ,σ t≥1 θ t g(x t , a t ) , and the θ-value at x 1 is :

v θ (x 1 ) = sup σ∈Σ γ θ (x 1 , σ).
As for gambling houses, it is easy to see that the supremum can be taken over the smaller set of pure strategies, and one can derive a recursive formula linking the value functions. General limit and uniform values are defined as in the previous subsection 3.1. Definition 3.17. Let Ψ = (X, A, q, g) be a standard MDP.

Ψ has a general limit value v * if (v θ ) uniformly converges to v * when I(θ) goes to zero, i.e. for each ε > 0 one can find α > 0 such that :

∀θ, ( I(θ) ≤ α =⇒ (∀x ∈ X, |v θ (x) -v * (x)| ≤ ε) ) .
Ψ has a general uniform value if it has a general limit value v * and moreover for each ε > 0 one can find α > 0 and a behavior strategy σ(x) for each initial state x satisfying :

∀θ, (I(θ) ≤ α =⇒ (∀x ∈ X, γ θ (x, σ(x)) ≥ v * (x) -ε) ) .
We now present a notion of invariance for the MDP Ψ. The next definition will be similar to definition 3.8, however one needs to be slightly more sophisticated here to incorporate the payoff component. Assume now that X is a compact metric space, and define for each (u, y)

in ∆ f (X) × [0, 1], F (u, y) = x∈X u(x)q(x, a(x)), x∈X u(x)g(x, a(x)) , where a : X → ∆ f (A) .
where q(x, .) and g(x, .) have been linearly extended for all x. We have defined a correspondence F from ∆ f (X) × [0, 1] to itself. It is easy to see that F always is an affine correspondence (see lemma 3.26 later). In the following definition we consider the closure of the graph of F within the compact set (∆(X) × [0, 1]) 2 , with the weak topology.

Definition 3.18. An element (u, y) in ∆(X) × [0, 1] is said to be an invariant couple for the MDP Ψ if ((u, y), (u, y)) ∈ cl(Graph( F )). The set of invariant couples of Ψ is denoted by RR.

Our main result for standard MDPs is the following, where X is assumed to be a compact subset of a simplex ∆(K). Recall that D 1 = {f ∈ C(∆(K)), ∀x, y ∈ ∆(K), ∀a, b ≥ 0, af (x)bf (y) ≤ axby 1 }, and any f in D 1 is linearly extended to ∆(∆(K)).

Theorem 3.19. Let Ψ = (X, A, q, g) be a standard MDP where X is a compact subset of a simplex ∆(K), such that :

∀x ∈ X, ∀y ∈ X, ∀a ∈ A, ∀f ∈ D 1 , ∀α ≥ 0, ∀β ≥ 0, |αf (q(x, a)) -βf (q(y, a))| ≤ αx -βy 1 and |αg(x, a) -βg(y, a)| ≤ αx -βy 1 .
then Ψ has a general uniform value v * characterized by : for all x in X, v * (x) = inf w(x), w : ∆(X) → [0, 1] affine C 0 s.t.

(1) ∀x ′ ∈ X, w(x ′ ) ≥ sup a∈A w(q(x ′ , a)) and ( 2) ∀(u, y) ∈ RR, w(u) ≥ y .

The proof of theorem 3.19 will be in section 3.4. An immediate corollary is when the state space is finite.

Corollary 3.20. Consider a standard MDP (K, A, q, g) with a finite set of states K. Then it has a general uniform value v * , and for each state k :

v * (k) = inf w(k), w : ∆(K) → [0, 1] affine s.t.
(1) ∀k ′ ∈ K, w(k ′ ) ≥ sup a∈A w(q(k ′ , a)) and ( 2)∀(p, y) ∈ RR, w(p) ≥ y .

with

RR = {(p, y) ∈ ∆(K) × [0, 1], ((p, y), (p, y)) ∈ cl(conv(Graph(F )))} and F (k, y) = {(q(k, a), g(k, a)), a ∈ A}.
Proof : K is viewed as a subset of the simplex ∆(K), endowed with the L 1 -norm. Fix k, k ′ in K, a in A, α ≥ 0 and β ≥ 0. We have

αk -βk ′ = |α -β| if k = k ′ , α + β otherwise. First, |αg(k, a) -βg(k ′ , a)| ≤ |α -β|g(k, a) if k = k ′ α + β otherwise , so in all cases |αg(k, a) -βg(k ′ , a)| ≤ αk -βk ′ . Secondly, consider f ∈ D 1 . f takes values in [-1, 1], so similarly we have : |αf (q(k, a)) -βf (q(k ′ , a))| ≤ αk -βk ′ .
So we can apply theorem 3.19, and the graph of F is the convex hull of the graph of F . Remark 3.21. When the set of actions is finite, we are in the setting of [START_REF] Blackwell | Discrete dynamic programming[END_REF] and the value is characterized by the Average Cost Optimality Equation In fact in this setting, our characterization leads to a dual formulation of a result of [START_REF] Denardo | Multichain Markov renewal programs[END_REF]. [START_REF] Denardo | Multichain Markov renewal programs[END_REF] showed that the value v * is the smallest (pointwise) excessive function for which there exists a vector h ∈ IR K such that (v * , h) is superharmonic in the sense of Hordjik and [START_REF] Hordijk | Linear programming and Markov decision chains[END_REF] , i.e.

∀k ∈ K, a ∈ A v * (k) + h(k) ≥ r(k, a) + k ′ q(k, a)(k ′ )h(k ′ ).
(2)

Given a function w the existence of a vector h such that (w, h) is superharmonic is a linear programming problem with K ×A inequalities. By Farkas' lemma it has a solution if and only if a dual problem has no solution, and the dual programming problem is to find a solution π ∈ IR K×A of the following system :

∀k ∈ K a ′ ∈A π(k, a ′ ) = k ′ ∈K,a ′ ∈A π(k ′ , a ′ )q(k ′ , a ′ )(k) ∀(k, a) ∈ K × A π(k, a) ≥ 0 ∀k ∈ K a ′ ∈A π(k, a)g(k, a ′ ) > v(k).
If we denote by p the marginal of π on K and define for all k such that p(k) > 0, σ(k) = π(k,a) p(k) and set σ(k) to any probability otherwise, then σ is a strategy in the MDP. Moreover p is invariant under σ and the stage payoff y is greater than v(p), thus the couple (p, y) is in RR and the condition (2) in corollary 3.20 is not satisfied. Reciprocally since the action state is compact, given (p, y) ∈ RR, there exists a strategy σ such that p is invariant under σ and the payoff is y. Therefore if the condition (2) is not true then there exists h ∈ IR k such that (w, h) is superharmonic. Note that Denardo and Fox state a dual of the minimization problem and obtain an explicit dual maximization problem whose solution is the value. Hordjik and Kallenberg exhibit from the solutions of this dual problem an optimal strategy.

Proof of theorem 3.9

In this section we consider a compact metric space (X, d), and we use the Kantorovich-Rubinstein distance d = d KR on ∆(X). We write Z = ∆ f (X), Z = ∆(X). We start with a lemma.

Lemma 3.22. Let F : X ⇉ ∆ f (X) be non expansive for d KR . Then the mixed extension of F is 1-Lipschitz from ∆ f (X) to ∆ f (X) for d KR .
Proof of lemma 3.22. We first show that the mapping (p → convF (p)) is non expansive from X to Z. Indeed, consider p and p ′ in X, and u = i∈I α i u i , with I finite, α i ≥ 0, u i ∈ F (p) for each i, and i∈I α i = 1. By assumption for each i one can find

u ′ i in F (p ′ ) such that d KR (u i , u ′ i ) ≤ d(p, p ′ ). Define u ′ = i∈I α i u ′ i in convF (p ′ ).
We have :

d KR (u, u ′ ) = sup f ∈E 1 i α i u i (f ) - i α i u ′ i (f ) , = sup f ∈E 1 i∈I α i (u i (f ) -u ′ i (f )), ≤ i∈I α i d KR (u i , u ′ i ), ≤ d(p, p ′ ).
We now prove that F is 1-Lipschitz from Z to Z. Let u 1 , u 2 be in Z and v 1 = p∈X u 1 (p)f 1 (p), where f 1 (p) ∈ convF (p) for each p. By the Kantorovich duality formula, there exists a coupling γ = (γ(p, q)) (p,q)∈X×X in ∆ f (X × X) with first marginal u 1 and second marginal u 2 satisfying :

d KR (u 1 , u 2 ) = (p,q)∈X×X γ(p, q)d(p, q).
For each p, q in X by the first part of this proof there exists f p (q) ∈ convF (q) such that d KR (f p (q), f 1 (p)) ≤ d(p, q). We define :

f 2 (q) = p∈X γ(p, q) u 2 (q)
f p (q) ∈ convF (q), and v 2 = q∈X u 2 (q)f 2 (q) ∈ F (u 2 ).

We now conclude.

d KR (v 1 , v 2 ) = d KR p∈X u 1 (p)f 1 (p), q∈X u 2 (q)f 2 (q) = d KR p,q γ(p, q)f 1 (p), q,p γ(p, q)f p (q) ≤ p,q γ(p, q)d KR (f 1 (p), f p (q)) ≤ p,q γ(p, q)d(p, q) = d KR (u 1 , u 2 ).
The mixed extension of F is 1-lipschitz.

We now consider a Gambling House Γ = (X, F, r) and assume the hypotheses of theorem 3.9 are satisfied. We will work2 with the deterministic Gambling House Γ = (∆ f (X), F , r). Recall that r is extended to an affine and continuous mapping on ∆(X) whereas F is an affine non expansive correspondence from Z to Z.

For p in X, the pure plays in Γ at the initial state δ p coincide with the mixed plays in Γ at the initial state p. As a consequence, the θ-value for Γ at p coincides with the θ-value for Γ at δ p , which is written v θ (p) = v θ (δ p ). Because F and r are affine on Z, the θ-value for Γ, as a function defined on Z, is the affine extension of the original v θ defined on X. So we have a unique value function v θ which is defined on Z and is affine. Because F is 1-Lipschitz and r is uniformly continuous, all the value functions v θ have the same modulus of continuity as r, so (v θ ) θ is an equicontinuous family of mappings from Z to [0, 1]. Consequently, we extend v θ to an affine mapping on Z with the same modulous of continuity, and the family (v θ ) θ now is an equicontinuous 3 family of mappings from Z to [0, 1].

We define R and v * as in the statements of theorem 3.9, so that for all x in For each k, let σ k = (u k 1 , ..., u k t , ...) ∈ Z ∞ be a play at δ p for Γ which is almost optimal for the θ k -value, in the sense that t≥1 θ k

t r(u k t ) ≥ v θ k (p) -ε. Define : u(k) = ∞ t=1 θ k t u k t ∈ Z, and u ′ (k) = ∞ t=1 θ k t u k t+1 ∈ Z.
u(k) and u ′ (k) are well-defined limits of normal convergent series in the Banach space C(X) ′ . Because F is affine, its graph is a convex set and (u(k),

u ′ (k)) ∈ cl(Graph( F )) for each k. Moreover, we have d(u(k), u ′ (k)) ≤ diam(X)(θ k 1 + ∞ t=2 |θ k t -θ k t-1 |), where diam(X) is the diameter of X. Consequently, t≥1 |θ k t+1 -θ k t | -→ k→∞ 0 implies d(u(k), u ′ (k)) -→ k→∞ 0. Considering a limit point of the sequence (u(k), u ′ (k)) k ,
we obtain some u in R. By assumption on w, w(u) ≥ r(u). Moreover, for each k

we have r(u(k)) = t≥1 θ k t r(u k t ) ≥ v θ k (p) -ε, so r(u) ≥ v(p) -ε. Because w is excessive, we obtain that for each k the sequence (w(u k t )) t is non increasing, so w(u(k)) = t≥1 θ k t w(u k t ) ≤ w(p).
So we obtain :

w(p) ≥ w(u) ≥ r(u) ≥ v(p) -ε.
This is true for all ε, so w ≥ v.

Proposition 3.25. Γ has a general uniform value.

Proof of proposition 3.25 : First we can extend the notion of mixed play to Z. A mixed play at u 0 ∈ Z, is a sequence σ = (u 1 , ..., u t , ...) ∈ Z ∞ such that u t+1 ∈ F (u t ) for each t ≥ 0, and we denote by Σ(u 0 ) the set of mixed play at u 0 . Given t, T in IN, n ∈ IN * and u 0 ∈ Z, we define for each mixed play σ = (u t ) t≥1 ∈ Σ(u 0 ) the auxiliary payoff :

γ t,n (σ) = 1 n t+n l=t+1
r(u l ), and β T,n (σ) = inf t∈{0,...,T } γ t,n (σ).

And we also define the auxiliary value function : for all u in Z,

h T,n (u 0 ) = sup σ∈Σ(u 0 ) β T,n (σ).
Clearly, β T,n (σ) ≤ γ 0,n (σ) and h T,n (u 0 ) ≤ v n (u 0 ). We can write :

h T,n (u 0 ) = sup σ∈Σ(u 0 ) inf θ∈∆({0,...,T }) 1 n T t=0 θ t t+n l=t+1 r(u l ) = sup σ∈Σ(u 0 ) inf θ∈∆({0,...,T }) T +n l=1 β l (θ, n)r(u l ).
j ≥ 1, denote by θ j the maximum of θ on the block B j = {(j -1)n 0 + 1, ..., jn 0 }. For all t ∈ B j , we have :

θ j ≥ θ t ≥ θ j - t ′ ∈{(j-1)n 0 +1,...jn 0 -1} |θ t ′ +1 -θ t ′ |.
As a consequence, for all j we have :

jn 0 t=(j-1)n 0 +1 θ t r(z ′ t ) ≥ θ j jn 0 t=(j-1)n 0 +1 r(z ′ t ) -n 0 t ′ ∈{(j-1)n 0 +1,...,jn 0 -1} |θ t ′ +1 -θ t ′ | ≥ jn 0 t=(j-1)n 0 +1 θ t (v * (p) -2ε) -n 0 t ′ ∈{(j-1)n 0 +1,...,jn 0 -1} |θ t ′ +1 -θ t ′ |
and by summing over j, we get :

γ θ (x 0 , σ ′ ) ≥ v * (p) -2ǫ -n 0 I(θ) ≥ v * (p) -3ǫ
as soon as I(θ) is small enough.

Proof of theorem 3.19

Assume that X is a compact subset of a simplex ∆(K), and let Ψ = (X, A, q, g) be a standard MDP such that : ∀x ∈ X, ∀y ∈ X, ∀a ∈ A, ∀f ∈ D 1 , ∀α ≥ 0, ∀β ≥ 0, |αf (q(x, a))βf (q(y, a))| ≤ αxβy 1 and |αg(x, a)βg(y, a)| ≤ αxβy 1 .

We write Z = ∆ f (X) × [0, 1], and Z = ∆(X) × [0, 1]. We will use the metric d * = d 0 = d 1 = d 2 = d 3 on ∆(∆(K)) introduced in section 2.3 and its restriction to ∆(X), so that Z is a compact metric space. For all (u, y), (u ′ , y ′ ) ∈ ∆ f (X) ×[0, 1], we put d((u, y), (u ′ , y ′ )) = max(d * (u, u ′ ), |yy ′ |) so that (Z, d) is a precompact metric space. Recall we have defined the correspondence F from Z to itself such that for all (u, y) in Z,

F (u, y) = {(Q(u, σ), G(u, σ)) s.t. σ : X → ∆ f (A)} , with the notations Q(u, σ) = x∈X u(x)q(x, σ(x)) and G(u, σ) = x∈X u(x)g(x, σ(x)).
And we simply define the payoff function r from Z to [0, 1] by r(u, y) = y for all (u, y) in Z. We start with a crucial lemma, which shows the importance of the duality formula of theorem 2.18. Lemma 3.26. F is an affine and non expansive correspondence from Z to itself.

Proof of lemma 3.26. We first show that : ∀u,

u ′ ∈ ∆ f (X), ∀α ∈ [0, 1], ∀y, y ′ ∈ [0, 1], F (αu + (1 -α)u ′ , αy + (1 -α)y ′ ) = α F (u, y) + (1 -α) F (u ′ , y ′ ).
First the transition does not depend on the second coordinate so we can forget it for the rest of the proof. The ⊂ part is clear. To see the reverse inclusion, consider

σ : X → ∆ f (A), σ ′ : X → ∆ f (A) and v = α x∈X u(x)q(x, σ(x)) + (1 - α) x∈X u ′ (x)q(x, σ ′ (x)) in α F (u) + (1 -α) F (u ′ ). Define σ * (x) = αu(x)σ(x) + (1 -α)u ′ (x)σ ′ (x) αu(x) + (1 -α)u ′ (x) ,
for each x such that the denominator is positive. Then v = x∈X (αu + (1α)u ′ (x))q(x, σ * (x)), and F is affine.

We now prove that F is non expansive. Let z = (u, y) and z ′ = (u ′ , y ′ ) be in Z. We have d((u, y), (u ′ , y ′ )) ≥ d * (u, u ′ ) and denote by U and U ′ the respective supports of u and u ′ . By the duality formula of theorem 2.18, there exists α

= (α(p, p ′ )) (p,p ′ )∈U ×U ′ and β = (β(p, p ′ )) (p,p ′ )∈U ×U ′ with non-negative coordinates satisfying : p ′ ∈U ′ α(p, p ′ ) = u(p) for all p ∈ U, p∈U β(p, p ′ ) = u ′ (p ′ ) for all p ′ ∈ U ′ , and d * (u, u ′ ) = (p,p ′ )∈U ×U ′ p α(p, p ′ ) -p ′ β(p, p ′ ) 1 . Consider now v = Q(u, σ) = p∈U u(p)q(p, σ(p)) for some σ : X → ∆ f (A). We define for all p ′ in U ′ : σ ′ (p ′ ) = p∈U β(p, p ′ ) u ′ (p ′ ) σ(p),
and v ′ = Q(u ′ , σ ′ ) = p ′ ∈U ′ u ′ (p ′ )q(p ′ , σ ′ (p ′ ))
. Then v ′ ∈ F (u ′ , y ′ ), and for each test function ϕ in D 1 we have :

|ϕ(v) -ϕ(v ′ )| = | p,p ′ α(p, p ′ )ϕ(q(p, σ(p))) -β(p, p ′ )ϕ(q(p ′ , σ(p)))| = | p,p ′ ,a α(p, p ′ )σ(p)(a)ϕ(q(p, a)) -β(p, p ′ )σ(p)(a)ϕ(q(p ′ , a))| ≤ p,p ′ α(p, p ′ )p -β(p, p ′ )p ′ 1 = d * (u, u ′ ),
and therefore

d * (v, v ′ ) ≤ d * (u, u ′ ).
In addition we have a similar result on the payoff,

|G(u, σ) -G(u ′ , σ ′ )| = | p,p ′ α(p, p ′ )g(p, σ(p)) -β(p, p ′ )g(p ′ , σ(p))| ≤ p,p ′ α(p, p ′ )p -β(p, p ′ )p ′ 1 ≤ d * (u, u ′ ). Thus we have d((Q(u, σ), R(u, σ)), (Q(u ′ , σ ′ ), R(u ′ , σ ′ ))) ≤ d * (u, u ′ ) ≤ d(z, z ′ ).
Recall that the set of invariant couples of the MDP Ψ is :

RR = {(u, y) ∈ Z, ((u, y), (u, y)) ∈ cl(Graph( F ))},
and the function v * : X -→ IR is defined by :

v * (x) = inf w(x), w : ∆(X) → [0, 1] affine C 0 s.t.
(1) ∀y ∈ X, w(y) ≥ sup a∈A w(q(y, a)) and ( 2) ∀(u, y) ∈ RR, w(u) ≥ y .

We now consider the deterministic Gambling House Γ = (Z, F , r). Z is precompact metric, F is affine non expansive and r is obviously affine and uniformly continuous. Given an evaluation θ, the θ-value of Γ at z 0 = (u, y) is denoted by vθ (u, y) = vθ (u) and does not depend on y. The recursive formula of section 3.1 yields :

∀(u, y) ∈ Z, vθ (u) = sup (u ′ ,y ′ )∈ F (u) θ 1 y ′ + (1 -θ 1 )v θ + (u ′ ) = sup σ∈X→∆ f (A) (θ 1 G(u, σ) + (1 -θ 1 )v θ + (Q(u, σ))) .
Because F and r are affine, vθ is affine in u and the supremum in the above expression can be taken over the function from X to A. Because F is non expansive and r is 1-Lipschitz, each vθ is 1-Lipschitz. We denote by v θ the θ-value of the MDP Ψ and linearly extend it to ∆ f (X). It turns out that the recursive formula satisfied by v θ is similar to the above recursive formula for vθ , so that v θ (u) = vθ (u, y) for all u in ∆ f (X) and y in [0, 1]. As a consequence, the existence of the general limit value in both problems Γ and Ψ is equivalent. Moreover, a deterministic play in Γ induces a strategy in Ψ, so that the existence of a general uniform value in Γ will imply the existence of the general uniform value in Ψ (note that deterministic and mixed plays in Γ are equivalent since F has convex values).

It is thus sufficient to show that Γ has a general uniform value given by v * , and we can mimic the end of the proof of theorem 3.9. Lemma 3.23 applies word for word. Finally, one can proceed almost exactly as in propositions 3.24 and 3.25 to show that Γ, hence Ψ, has a general uniform value given by v * .

4 Applications to partial observation and games

POMDP with finitely many states

We now consider a more general model of MDP with actions where after each stage, the decision maker does not perfectly observe the state. A MDP with partial observation, or POMDP, Γ = (K, A, S, q, g) is given by a finite set of states K, a non empty set of actions A and a non empty set of signals S. The transition q now goes from K × A to ∆ f (S × K) (by assumption the support of the signals at each state is finite) and the payoff function g still goes from K × A to [0, 1]. Given an initial probability p 1 on K, the POMDP Γ(p 1 ) is played as following. An initial state k 1 in K is selected according to p 1 and is not told to the decision maker. At every stage t he selects an action a t ∈ A. He has a (unobserved) payoff g(k t , a t ) and a pair (s t , k t+1 ) is drawn according to q(k t , a t ). The player learns s t , and the play proceeds to stage t + 1 with the new state k t+1 . A behavioral strategy is now a sequence (σ t ) t≥1 of applications with for each t, σ t : (A × S) t-1 → ∆ f (A). As usual, an initial probability on K and a behavior strategy σ induce a probability distribution over (K × A × S) ∞ and we can define the θ-values and the notions of general limit and uniform values accordingly.

Theorem 4.1. A POMDP with finitely many states has a general uniform value, i.e. there exists v * : ∆(K) → IR with the following property : for each ε > 0 one can find α > 0 and for each initial probability p a behavior strategy σ(p) such that for each evaluation θ with I(θ) ≤ α,

∀p ∈ ∆(K), |v θ (p) -v * (p)| ≤ ε and γ θ (σ(p)) ≥ v * (p) -ε.
Proof : We introduce Ψ an auxiliary MDP on X = ∆(K) with the same set of actions A and the following payoff and transition functions :

• r : X × A -→ [0, 1] such that r(p, a) = k∈K p(k)g(k, a) for all p in X and a ∈ A, • q : X × A → ∆ f (X) such that q(p, a) = s∈S k
p k q(k, a)(s) δ q(p,a,s) , where q(p, a, s) ∈ ∆(K) is the belief on the new state after playing a at p and observing the signal s :

∀k ′ ∈ K, q(p, a, s)(k ′ ) = q(p, a)(k ′ , s) q(p, a)(s) = k p k q(k, a)(k ′ , s) k p k q(k, a) (s)
.

The POMDP Γ(p 1 ) and the standard MDP Ψ(p 1 ) have the same value for all θ-evaluations. And for each strategy σ in Ψ(p 1 ), the player can guarantee the same payoff in the original game Γ(p 1 ) by mimicking the strategy σ. So if we prove that Ψ has a general uniform value it will imply that the POMDP Γ has a general uniform value.

To conclude the proof, we will simply apply theorem 3.19 to the MDP Ψ. We need to check the assumptions on the payoff and on the transition.

Consider any p, p ′ in X, a ∈ A, α ≥ 0 and β ≥ 0. We have :

|αr(p, a) -βr(p ′ , a)| = | k (αp(k) -βp ′ (k))g(k, a)| ≤ αp -βp ′
Moreover for any f ∈ D 1 , we have :

|αq(p, a)(f ) -β q(p ′ , a)(f )| = | s∈S (αq(p, a)(s)f (q(p, a, s)) -βq(p ′ , a)(s)f (q(p ′ , a, s))) | ≤ s αq(p, a)(., s) -βq(p ′ , a)(., s) ≤ s,k,k ′ |αp(k ′ )q(k ′ , a)(k, s) -βp ′ (k ′ )q(k ′ , a)(k, s)| ≤ s,k,k ′ q(k ′ , a)(k, s)|αp(k ′ ) -βp ′ (k ′ )| = αp -βp ′ .
where the first inequality comes from the definition of D 1 . By theorem 3.19, the MDP Ψ has a general uniform value and we deduce that the POMDP Γ has a general uniform value.

Example 4.2. Let Γ = (K, A, S, q, g, p 1 ) be a POMDP where K = {k 1 , k 2 }, A = {a, b}, S = {s} and p 1 = δ k 1 . The initial state is k 1 and since there is only one signal, the decision maker will obtain no additional information on the state. We say that he is in the dark. The payoff is given by g(0, a) = g(0, b) = g(1, b) = 0 and g(1, a) = 1, and the transition by q(1, a) = q(1, b) = δ 1,s , q(0, a) = δ 0,s and q(0, b) = 1 2 δ 0,s + 1 2 δ 1,s . On one hand if the decision maker plays a then the state stays the same and he receives a payoff of 1 if and only if the state is 1, on the other hand if he plays b then he receives a payoff of 0 but the probability to be in state 1 increases.

We define the function r from X = ∆(K) to [0, 1] by r((p, 1p), a) = 1p and r((p, 1p), b) = 0 for all p ∈ [0, 1], and the function q from X to ∆ f (X) by q((p, 1p), a) = δ (p,1-p) and q((p, 1p), b) = δ (p/2,1-p/2) .

Then the standard MDP Ψ = (∆(K), A, r, q) is the MDP associated in the previous proof to Γ. This MDP is deterministic since the decision maker is in the dark.

In this example, the existence of a general uniform value is immediate. If we fix n ∈ IN, the strategy σ = b n a ∞ which plays n times b and then a for the rest of the game, guarantees a stage payoff of (1 -1 2 n ) from stage n + 1 on, so the game has a general uniform value equal to 1. Finally if we consider the discounted evaluations, one can show that the speed of convergence of v λ is slower than λ :

v λ (p 1 ) = 1 - ln(λ) ln(2) λ + O(λ).
All the spaces are finite but the partial observation implies that the speed of convergence is slower than λ contrary to the perfect observation case where it is well known that the convergence is in O(λ).

Remark 4.3. It is unknown if the uniform value exists in pure strategies, i.e. if the behavior strategies σ(p) of theorem 4.1 can be chosen with values in A. This was already an open problem for the Cesàro-uniform value (see [START_REF] Rosenberg | Blackwell Optimality in Markov Decision Processes with Partial Observation[END_REF][START_REF] Renault | Uniform value in Dynamic Programming[END_REF] for different proofs requiring the use of behavioral strategies). In our proof, there are two related places where the use of lotteries on actions is important. First in the proof of the convergence of the function h T,n (within the proof of theorem 3.9), we used Sion's theorem in order to inverse a supremum and an infimum so we need the convexity of the set of strategies. Secondly when we prove that the extended transition is 1-Lipschitz (see lemma 3.26), the coupling between the two distributions u and u ′ introduces some randomization.

Zero-sum repeated games with an informed controller

We finally consider zero-sum repeated games with an informed controller. We start with a general model Γ = (K, I, J, C, D, q, g) of zero-sum repeated game, where we have 5 non empty finite sets : a set of states K, two sets of actions I and J and two sets of signals C and D, and we also have a transition mapping q from K ×I ×J to ∆(K ×C ×D) and a payoff function g from K ×I ×J to [0, 1]. Given an initial probability π on ∆(K ×C ×D), the game Γ(π) = Γ(K, I, J, C, D, q, g, π) is played as follows : at stage 1, a triple (k 1 , c 1 , d 1 ) is drawn according to π, player 1 learns c 1 and player 2 learns d 1 . Then simultaneously player 1 chooses an action i 1 in I and player 2 chooses an action j 1 in J. Player 1 gets a (unobserved) payoff r(k 1 , i 1 , j 1 ) and player 2 the opposite. Then a new triple (k 2 , c 2 , d 2 ) is drawn accordingly to q(k 1 , i 1 , j 1 ). Player 1 observes c 2 , player 2 observes d 2 and the game proceeds to the next stage, etc... A (behavioral) strategy for player 1 is a sequence σ = (σ t ) t≥1 where for each t ≥ 1, σ t is a mapping from (C ×I) t-1 ×C to ∆(I). Similarly a strategy for player 2 is a sequence of mappings τ = (τ t ) t≥1 where for each t ≥ 1 ,τ t is a mapping from (D × J) t-1 × D to ∆(J). We denote respectively by Σ and τ the set of strategies of player 1 and player 2. An initial distribution π and a couple of strategies (σ, τ ) defines for each t a probability on the possible histories up to stage t. And by Kolmogorov extension theorem, it can be uniquely extended to a probability on the set of infinite histories (K × C × D × I × J) +∞ .

Given θ an evaluation function, we define the θ-payoff of (σ, τ ) in Γ(π) as the expectation under IP π,σ,τ of the payoff function,

γ θ (π, σ, τ ) = IE π,σ,τ t θ t r(k t , i t , j t ) .
By Sion's theorem the game γ θ (π) has a value :

v θ (π) = max σ∈Σ min τ ∈τ γ θ (π, σ, τ ) = min τ ∈τ max σ∈Σ γ θ (π, σ, τ ),
and we can define the general limit value as in the MDP framework. Note that we do not ask the convergence to be uniform for all π in ∆(K × C × D), because we will later make some assumptions, in particular on the initial distribution.

Definition 4.4. The repeated game Γ(π) = (K, I, J, C, D, q, g, π) has a general limit value v * (π) if v θ (π) converges to v * (π) when I(θ) goes to zero, i.e. :

∀ε > 0, ∃α > 0, ∀θ, ( I(θ) ≤ α =⇒ (|v θ (π) -v * (π)| ≤ ε) ) .
And we can define a general uniform value by symmetrizing the definition for MDP.

Definition 4.5. The repeated game Γ(π) has a general uniform value if it has a general limit value v * and for each ε > 0 one can find α > 0 and a couple of strategies σ * and τ * such that for all evaluations θ with I(θ) ≤ α :

∀τ ∈ τ , γ θ (π, σ * , τ ) ≥ v * (π) -ε and ∀σ ∈ Σ, γ θ (π, σ, τ * ) ≤ v * (π) + ε.
We now focus on the case of a repeated game with an informed controller. We follow the definitions introduced in Renault (2012a). The first one concerns the information of the first player. We assume that he is always fully informed of the state and of the signal of the second player : Assumption 4.6. There exist two mappings k :

C → K and d : C → D such that, if E denotes {(k, c, d) ∈ K × C × D, k(c) = k, d(c) = d}, we have : ∀(k, i, j) ∈ K × I × J, q(k, i, j)(E) = 1, and π(E) = 1.
Moreover we will assume that only player 1 has a meaningful influence on the transitions, in the following sense.

Assumption 4.7. The marginal of the transition on K × D is not influenced by player 2's action. For k in K, i in I and j in J, we denote by q(k, i) the marginal of q(k, i, j) on K × D.

The second player may influence the signal of the first player but he can not prevent him neither to learn the state nor to learn his own signal. Moreover he can not influence his own information, thus he has no influence on his beliefs about the state or about the beliefs of player 1 about his beliefs. A repeated game satisfying assumptions 4.6 and 4.7 is called a repeated game with an informed controller. It was proved in Renault (2012a) that for such games the Cesàro-uniform value exists and we will generalize it here to the general uniform value.

Example 4.8. We consider Γ a zero-sum repeated game with incomplete information as studied by Aumann and Maschler (see reference from 1995). It is defined by a finite family (G k ) k∈K of payoff matrices in [0, 1] I×J and p ∈ ∆(K) an initial probability. At the first stage, some state k is selected according to p and told to player 1 only. The second player knows the initial distribution p but not the realization. Then the matrix game G k is repeated over and over. At each stage the players observe past actions but not their payoff. Formally it is a zero-sum repeated game Γ = (K, I, J, C, D, q, g) as defined previously, with C = K ×I ×J and D = I ×J, and for all (k, i, j) ∈ K ×I ×J, g(k, i, j) = G k (i, j) and q(k, i, j) = δ k,(k,i,j),(i,j) . For all p ∈ ∆(K), we denote by Γ(p) the game where the initial probability π ∈ ∆(K ×C ×D) is given by π = k∈K p(k)δ k,(k,i 0 ,j 0 ),(i 0 ,j 0 ) with (i 0 , j 0 ) ∈ I × J fixed.

For each n, we denote by v n (p) the value of the n-stage game with initial probability p, where the payoff is the expected mean average of the n first stages. It is known that it satisfies the following dynamic programming formula :

v n (p) = sup a∈∆(I) K 1 n r(p, a) + n -1 n k∈K,i∈I p k a k (i)v n-1 (q(p, a, i)) .
where p ∈ ∆(K), r(p, a) = min j ( k p k G k (a k , j)) and q(p, a, i) is the conditional belief on ∆(K) given p, a, i :

q(p, a, i)(k ′ ) = k p(k)a k (i)q(k, i)(k ′ ) k p(k)a k (i)
.

Starting from a belief p about the state, if player 2 observes action i and knows that the distribution of actions of player 1 is a, then he updates his beliefs to q(p, a, i). Aumann and Maschler have proved that the limit value exists and is characterized by

v * = cavf * = inf{v : ∆(K) → [0, 1], v concave v ≥ f * }, where f * (p) = V al k p k G k for all p ∈ ∆(K).
The function f * is the value of the game, called the non-revealing game, where player 1 is forbidden to use his information.

Theorem 4.9. A zero-sum repeated game with an informed controller has a general uniform value.

Proof of theorem 4.9 : Assume that Γ(π) = (K, I, J, C, D, q, g, π) is a repeated game with an informed controller. The proof will consist of 5 steps. First we introduce an auxiliary standard Markov Decision Process Ψ(π) on the state space X = ∆(K). Then we show that for all evaluations θ, the repeated game Γ(π) and the MDP Ψ(π) have the same θ-value. In step 3 we check that the MDP satisfies the assumption of theorem 3.19 so it has a general limit value and a general uniform value v * . As a consequence the repeated game has a general limit value v * (π). Then we prove that player 1 can use an ǫ-optimal strategy of the auxilliary MDP in order to guarantee v * (π)ǫ in the original game. Finally we prove that Player 2 can play by blocks in the repeated game in order to guarantee v * (π) + ǫ. And we obtain that v * (π) can be guaranteed by both players in the repeated game, so it is the general uniform value of Γ(π).

For every P ∈ ∆(K × C × D), we denote by P the marginal of P on K × D and we put P = ψ D (P ) where ψ D is the disintegration with respect to D (recall proposition 2.21) : for all µ ∈ ∆(K × D), ψ D (µ) = d∈D µ(d)δ µ(.|d) .

Step 1 : We put X = ∆(K) and A = ∆(I) K and for every p in X, a in A and b in ∆(J), we define : p k a k (i)q(k, i) ∈ ∆(K × D), q(p, a) = ψ D (q(p, a)) = d∈D q(p, a)(d)δ q(p,a,d) ∈ ∆ f (X).

Here q(p, a, d) ∈ ∆(K) is the belief of the second player on the new state after observing the signal d and knowing that player 1 has played a at p :

∀k ′ ∈ K, q(p, a, d)(k ′ ) = q(p, a)(k ′ , d) q(p, a)(d) = k p k q(k, a(k))(k ′ , d) k p k q(k, a(k))(d)
.

We define the auxiliary MDP Ψ = (X, A, q, r), and denote the θ-value in the MDP by vθ . The MDP with initial state π has strong links with the repeated game Γ(π).

Step 2 : By proposition 4.23, part b) in Renault 2012a), we have for all evaluations θ with finite support : v θ (π) = vθ (π).

The proof relies on the same recursive formula satisfied by v and v, and the equality can be easily extended to any evaluation θ. where v θ + is naturally linearly extended to ∆ f (X). As a consequence if Ψ(π) has a general limit value so does the repeated game Γ(π).

Step 3 : Let us check that Ψ satisfies the assumption of 3.19. Consider p, p ′ in X, a in A, and α ≥ 0 and β ≥ 0. We have : So Ψ = (X, A, q, r) has a general limit value and a general uniform value that we denote by v * . As a consequence, Γ(π) has a general limit value v * (π).

Step 4 : Given ε > 0, there exists α > 0 and a strategy σ in the MDP Ψ(π) such that the θ-payoff in the MDP is large : γθ (π, σ) ≥ v * (π)ε whenever I(θ) ≤ α. Moreover if we look at the end of the proof of theorem 3.19 we can choose σ to be induced by a deterministic play in the Gambling House Γ with state space Z = ∆ f (X) × [0, 1]. As a consequence one can mimic σ to construct a strategy σ * in the original repeated game Γ(π) such that : ∀τ ∈ τ , γ θ (π, σ * , τ ) ≥ v * (π)ε whenever I(θ) ≤ α.

Step 5 : Finally we show that player 2 can also guarantee the value v * in the repeated game Γ. Note that in the repeated game he can not compute the state variable in ∆(K) without knowing the strategy of player 1. Nevertheless he has no influence on the transition function so playing independently by large blocks will be sufficient for him in order to guarantee v * (π). We use the following characterization of the value proved in Renault (2012a) :

v * (π) = inf n sup m v m,n (π).
where v m,n is the value of the game with payoff function the Cesàro mean of the stage payoffs between stages m and m + n. We proceed as in proposition 4.22 of Renault 2012a. Fix n 0 ≥ 1, then we consider the strategy τ * which for each j ∈ IN, plays optimally in the game with the evaluation the Cesàro mean for the payoffs on the block of stages B j = {n 0 (j -1)+1, ..., n 0 j}. Since player 2 does not influence the state it is well defined and this strategy guarantees sup t≥0 v t,n 0 (z) for the overall Cesàro mean.

Let θ ∈ ∆(IN * ) and σ be a strategy of player 1. For each j ≥ 1, denote by θ j the minimum of θ on the block B j = {(j -1)n 0 + 1, ..., jn 0 }. We have Given ǫ, there exists n 0 such that sup t≥0 v t,n 0 (π) ≤ v * (π) + ǫ. Fix α = ǫ n 0 and τ * defined as before then for all θ such that I(θ) ≤ α, we have sup σ∈Σ γ θ (π, σ, τ * ) ≤ v * (π) + 2ǫ, and this concludes the proof of theorem 4.9.

Example 4.10. The computation of the value in two-player repeated game with incomplete information is a difficult problem as shown in the next example introduced in [START_REF] Renault | The value of Markov chain games with lack of information on one side[END_REF] and partially solved by [START_REF] Hörner | Markov Games with One-Sided Information[END_REF]. The value exists by a theorem in [START_REF] Renault | The value of Markov chain games with lack of information on one side[END_REF] but the value has been computed only for some values of the parameters. The set of states is K = {k 1 , k 2 }, the set of actions of player 1 is I = {T, B}, the set of actions of player 2 is J = {L, R}, and the payoff is given by L R L R T B 1 0 0 0 and T B 0 0 0 1 .

k 1 k 2
The evolution of the state does not depend on the actions : at each stage the state stays the same with probability p and changes to the other state with probability 1p. At each stage, both players observe the past actions played but only player 1 is informed of the current state (with previous notation C = K × I × J and D = I × J). For each p ∈ [0, 1], it defines a repeated game Γ p . In the case p = 1, the matrix is fixed for all the game thus it is a repeated game with incomplete information on one side à la Aumann [START_REF] Aumann | Repeated games with incomplete information[END_REF]. For all other positive values of p, the process is ergodic so the limit value is constant, and it is sufficient to study the case p ∈ [1/2, 1) by symmetry of the problem. [START_REF] Hörner | Markov Games with One-Sided Information[END_REF] proved that if p ∈ [1/2, 2/3), then the value is v p = p 4p-1 . If p ≥ 2/3, we do not know the value except for p * , the solution of 9p 3 -12p 2 + 6p -1 = 0, where one has v p = p * 1-3p * +6(p * ) 2 .

  If a b b ′′ ≤ 1, we define b ′ = a b b ′′ and a ′ = a ′′ and we get a f (x)b f (y) ≤ bη + axby . If a b b ′′ > 1, we define b ′ = 1 and a ′ = a ′′ b b ′′ a ∈ [0, 1] and obtain a f (x)b f (y) ≤ bη + aε + axby . Thus for all η > 0, we have a f (x)b f (y) ≤ bη + aε + axby , and therefore a f (x)b f (y) ≤ aε + axby .

  r(p, a, b) = (k,i,j)∈K×I×J p k a k (i)b(j)g(k, i, j) ∈ [0, 1],

  ∀θ ∈ ∆(IN * ), ∀p ∈ X, v θ (p) = sup a∈A inf b∈B ( θ 1 r(p, a, b) + (1θ 1 )v θ + (q(p, a)) ) .

  |αr(p, a)βr(p ′ , a)| ≤ sup b∈∆(J) |αr(p, a, b)βr(p ′ , a, b)| ≤ sup b∈∆(J)| k∈K αp k g(k, a k , b)βp ′k g(k, a k , b)| ≤ sup b∈∆(J) k∈K |αp kβp ′k | = αpβp ′ 1 . Moreover, let ϕ : ∆(K) -→ IR be in D 1 . |αϕ(q(p, a))βϕ(q(p ′ , a))| = d∈D (αq(p, a)(d)ϕ(q(p, a, d))β q(p ′ , a)(d)ϕ(q(p ′ , a, d))) ≤ d∈D α q(p, a)(d) q(p, a, d)β q(p ′ , a)(d) q(p ′ , a, d) 1 ≤ d∈D α (q(p, a)(k ′ , d)) k ′β (q(p ′ , a)(k ′ , d)) k ′ 1 ≤ d∈D k∈K αp k (q(k, a)(k ′ , d)) k ′βp ′k (q(k, a)(k ′ , d)) k ′ 1 ≤ d∈D k ′ ∈K k∈K q(k, a)(k ′ , d)|αp kβp ′k | = αpβp ′ 1 .

  n 0 (π) + n 0 I(θ).

Similarly, the same result holds for the case p = +∞, i.e. where xy = 2 xy ∞ .

A variant of the proof would be to consider the Gambling House on ∆(X) where the transition correspondence is defined so that its graph is the closure of the graph of F . Part 1) of lemma

3.23 shows this correspondence is also non expansive. 3. Z being precompact, this is enough to obtain the existence of a general limit value, see Renault 2012b. Here we will moreover obtain a characterization of this value and the existence of the general uniform value.

Acknowledgements. The authors gratefully acknowledge the support of the Agence Nationale de la Recherche, under grant ANR JEUDY, ANR-10-BLAN 0112,

X, v * (x) = inf w(x), w : Z → [0, 1] affine C 0 s.t.

(1) ∀y ∈ X, w(y) ≥ sup u∈F (y)

w(u) and (2)∀u ∈ R, w(u) ≥ r(u) .

We start with a technical lemma using the non-expansiveness of F .

Lemma 3.23. 1) Given (u, u ′ ) in cl(Graph( F )), v in Z and ε > 0, there exists

2) Given a sequence (z t ) t≥0 of elements of Z such that (z t , z t+1 ) ∈ cl(Graph( F )) for all t ≥ 1, for each ε one can find a sequence (z ′ t ) t≥0 of elements of Z such that (z ′ t ) t≥1 is a play at z ′ 0 , and d(z t , z ′ t ) ≤ ε for each t ≥ 0.

Proof of lemma 3.23 : 1) For all ε > 0 there exists (z,

for each t. Proposition 3.24. Γ has a general limit value given by v * .

Proof of proposition 3.24 : By Ascoli's theorem, it is enough to show that any limit point of (v θ ) θ (for the uniform convergence) coincides with v * . We thus assume that (v θ k ) k uniformly converges to v on Z when k goes to ∞, for a family of evaluations satisfying :

And we need to show that

It is plain that v can be extended to an affine function on Z and has the same modulus of continuity of r. Because t≥1 |θ k t+1θ k t | -→ k→∞ 0, we have by equation (1) of section 3.1 that : ∀y ∈ X, v(y) = sup u∈F (y) v(u).

Let now u be in R. By lemma 3.23 for each ε one can find u 0 in Z and a play (u 1 , u 2 , ..., u t , ...) such that u t ∈ F (u t-1 ) and d(u, u t ) ≤ ε for all t ≥ 0. Because r is uniformly continuous, we get v(u) ≥ r(u).

By definition of v * as an infimum, we obtain : v * ≤ v.

B)

We show that v * ≥ v. Let w be a continous affine mapping from Z to [0, 1] satisfying (1) and (2) of the definition of v * . It is enough to show that w(p) ≥ v(p) for each p in X. Fix p in X and ε > 0.

where for each l in 1, ..., T + n,

By construction, F is affine, so Σ(u 0 ) is a convex subset of Z ∞ . ∆({0, ..., T }) is convex compact and the payoff T +n l=1 β l (θ, n)r(u l ) is affine both in θ and in σ. We can apply a standard minmax theorem to get :

We write θ t = 0 for t > T and for each l ≥ 0 :

The impatience of β(θ, n) goes to zero as n goes to infinity, uniformly in θ. So we can use the previous proposition 3.24 to get :

and the convergence is uniform over Z. Consequently, if we fix ε > 0 there exists n 0 such that for all u 0 in Z, for all T ≥ 0, there exists a play σ T = (u T t ) t≥1 in Σ(u 0 ) such that the average payoff is good on every interval of n 0 stages starting before T + 1 : for all t = 0, ..., T , γ t,n 0 (σ T ) ≥ v * (u 0 )ε.

We fix u 0 in Z and consider, for each T , the play σ T = (u T t ) t≥1 in Σ(u) as above. By a diagonal argument we can construct for each t ≥ 1 a limit point z t in Z of the sequence (u T t ) T ≥0 such that for each t we have (z t , z t+1 ) ∈ cl(Graph( F )), with z 0 = u 0 . For each m ≥ 0, we have 1 n 0 m+1+n 0 t=m+1 r(u T t ) ≥ v * (u 0 )ε for T large enough, so at the limit we get :

r being uniformly continuous, there exists α such that |r(z)-r(z ′ )| ≤ ε as soon as d(z, z ′ ) ≤ α. By lemma 3.23, one can find a σ ′ = (z ′ 1 , ...., z ′ t , ...) at Σ(z 0 ) such that for each t, d(z t , z ′ t ) ≤ α. We obtain that for each m ≥ 0, 1

Consequently we have proved : ∀ε > 0, there exists n 0 such that for all initial state p in X, there exists a mixed play σ ′ = (z ′ t ) t at p such that : ∀m ≥ 0,

) be an evaluation, it is now easy to conclude. First if v * (p) -2ǫ < 0, then any play is 2ǫ-optimal. Otherwise, for each