Maël Le Treust
email: mael.le-treust@ensea.fr

Tristan Tomala
email: tomala@hec.fr

Information-Theoretic Limits of Strategic Communication

In this article, we investigate strategic information transmission over a noisy channel. This problem has been widely investigated in Economics, when the communication channel is perfect. Unlike in Information Theory, both encoder and decoder have distinct objectives and choose their encoding and decoding strategies accordingly. This approach radically differs from the conventional Communication paradigm, which assumes transmitters are of two types: either they have a common goal, or they act as opponent, e.g. jammer, eavesdropper. We formulate a point-to-point source-channel coding problem with state information, in which the encoder and the decoder choose their respective encoding and decoding strategies in order to maximize their long-run utility functions. This strategic coding problem is at the interplay between Wyner-Ziv's scenario and the Bayesian persuasion game of Kamenica-Gentzkow. We characterize a single-letter solution and we relate it to the previous results by using the concavification method. This confirms the benefit of sending encoded data bits even if the decoding process is not supervised, e.g. when the decoder is an autonomous device. Our solution has two interesting features: it might be optimal not to use all channel resources; the informational content impacts the encoding process, since utility functions capture preferences on source symbols.

I. INTRODUCTION

What are the limits of information transmission between autonomous devices? The internet of things (IoT) considers pervasive presence in the environment of a variety of devices that are able to interact and coordinate with each other in order to create new applications/services and reach common goals.

Unfortunately the goals of wireless devices are not always common, for example adjacent access points in crowded downtown areas, seeking to transmit at the same time, compete for the use of bandwidth.

Such situations require new efficient techniques to coordinate the actions of the devices whose objectives are neither aligned, nor antagonistic. This question differs from the classical paradigm in Information Theory which assumes that devices are of two types: transmitters pursue the common goal of transferring information, while opponents try to mitigate the communication (e.g. the jammer corrupts the information, the eavesdropper infers it, the warden detects the covert transmission). In this work, we seek to provide an information-theoretic look at the intrinsic limits of the strategic communication between interacting autonomous devices having non-aligned objectives.

The problem of "strategic information transmission" has been well studied in the Economics literature since the seminal paper by Crawford-Sobel [START_REF] Crawford | Strategic information transmission[END_REF]. In their model, a better-informed sender transmits a signal to a receiver, who takes an action which impacts both sender/receiver's utility functions. The problem consists in determining the optimal information disclosure policy given that the receiver' bestreply action will impact the sender's utility, see [START_REF] Forges | Non-zero-sum repeated games and information transmission[END_REF] for a survey. In [START_REF] Kamenica | Bayesian persuasion[END_REF], Kamenica-Gentzkow define the "Bayesian persuasion game" where the sender commits to an information disclosure policy before the game starts. This subtle change of rules of the game induces a very different equilibrium solution related to Stackelberg equilibrium [START_REF] Stackelberg | Marketform und Gleichgewicht[END_REF], instead of Nash equilibrium [START_REF] Nash | Non-cooperative games[END_REF]. This problem was later referred to as "information design" in [START_REF] Bergemann | Information design, Bayesian persuasion, and Bayes correlated equilibrium[END_REF], [START_REF] Taneva | Information design[END_REF], [START_REF] Bergemann | Information design: a unified perspective[END_REF]. In most of the articles in the Economics literature, the transmission between the sender and the receiver is noise-free; except the following ones [START_REF] Tsakas | Noisy persuasion[END_REF], [START_REF] Blume | Noisy talk[END_REF], [START_REF] Hernández | Nash codes for noisy channels[END_REF] where the noisy transmission is investigated in a finite block-length regime. Noise-free transmission does not require the use of block coding techniques for compressing the information and mitigating the noise effects. Interestingly, Shannon's mutual information is widely accepted as a cost of information for the problem of "rational inattention" in [START_REF] Sims | Implication of rational inattention[END_REF] and for the problem of "costly persuasion" in [START_REF] Gentzkow | Costly persuasion[END_REF]; in both cases there is no explicit reference to the coding problem.

Entropy and mutual information appear endogenously in repeated games with finite automata and bounded recall [START_REF] Neyman | Strategic entropy and complexity in repeated games[END_REF], [START_REF] Neyman | Repeated games with bounded entropy[END_REF], [START_REF] Neyman | Growth of strategy sets, entropy, and nonstationary bounded recall[END_REF], with private observation [START_REF] Gossner | How to play with a biased coin?[END_REF], or with imperfect monitoring [START_REF] Gossner | Empirical distributions of beliefs under imperfect observation[END_REF], [START_REF] Gossner | Secret correlation in repeated games with imperfect monitoring[END_REF], [START_REF] Gossner | Informationally optimal correlation[END_REF]. In [START_REF] Gossner | Optimal use of communication resources[END_REF], the authors investigate a sender-receiver game with common interests by formulating a coding problem and by using tools from Information Theory. In their model, the sender observes an July 13, 2018 DRAFT infinite source of information and communicates with the receiver through a perfect channel of fixed alphabet. This result was later refined by Cuff in [START_REF] Cuff | Coordination using implicit communication[END_REF] and referred to as the "coordination problem" in several articles [START_REF] Cuff | Coordination capacity[END_REF], [START_REF] Cuff | Hybrid codes needed for coordination over the point-to-point channel[END_REF], [START_REF] Treust | Joint empirical coordination of source and channel[END_REF], [START_REF] Cervia | Strong coordination of signals and actions over noisy channels with two-sided state information[END_REF]. In the literature of Information Theory, the Bayesian persuasion game is investigated in [START_REF] Akyol | Information-theoretic approach to strategic communication as a hierarchical game[END_REF] for Gaussian source and channel with Crawford-Sobel's quadratic cost functions. The authors compute the linear equilibrium's encoding and decoding strategies and relate their results to the literature on "decentralized stochastic control". In [START_REF] Sarıtaş | Nash and stackelberg equilibria for dynamic cheap talk and signaling games[END_REF], [START_REF] Sarıtaş | Quadratic multi-dimensional signaling games and affine equilibria[END_REF], the authors extend the model of Crawford-Sobel to multidimentional sources and noisy channels. They determine whether the optimal encoding policies are linear or based on quantification. Sender-receiver games are also investigated in [START_REF] Akyol | Networked estimation-privacy games[END_REF], [START_REF] Farokhi | Estimation with strategic sensors[END_REF] for the problem of "strategic estimation" involving self-interested sensors; and in [START_REF] Jakub Mareček | Signaling and obfuscation for congestion control[END_REF] for the "strategic congestion control" problem. In [START_REF] Dughmi | Algorithmic bayesian persuasion[END_REF], [START_REF] Dughmi | Algorithmic information structure design: A survey[END_REF], [START_REF] Dughmi | Persuasion with limited communication[END_REF], the authors investigate the computational aspects of the Bayesian persuasion game when the signals are noisy. In [START_REF] Berry | Shannon meets Nash on the interference channel[END_REF], [START_REF] Perlaza | Perfect output feedback in the two-user decentralized interference channel[END_REF] the interference channel coding problem is formulated as a game in which the users, i.e. the pairs of encoder/decoder, are allowed to use any encoding/decoding strategy. The authors compute the set of Nash equilibria for linear deterministic and Gaussian channels.

Z n U n X n Y n V n P T E D φ e (u, v) φ d (u, v)
The non-aligned devices' objectives are captured by utility functions, defined in a similar way as the distortion function for lossy source coding. Coding for several distortion measures is investigated for "multiple descriptions coding" in [START_REF] Gamal | Achievable rates for multiple descriptions[END_REF], for the lossy version of "Steinberg's common reconstruction"

problem in [START_REF] Lapidoth | Constrained source-coding with side information[END_REF], for an alternative measure of "secrecy" in [START_REF] Yamamoto | A rate-distortion problem for a communication system with a secondary decoder to be hindered[END_REF], [START_REF] Yamamoto | Rate-distortion theory for the Shannon cipher system[END_REF], [START_REF] Schieler | Rate-distortion theory for secrecy systems[END_REF], [START_REF] Schieler | The henchman problem: Measuring secrecy by the minimum distortion in a list[END_REF]. In [START_REF] Lapidoth | On the role of mismatch in rate distortion theory[END_REF], Lapidoth investigates the "mismatch source coding problem" in which the sender is constrained by Nature, to encode according to a different distortion than the decoder's one. The coding problems in these works consider two types of transmitters: either they pursue a common goal, or they behave as opponents.

The problem of information transmission has therefore been addressed from two complementary point of views: the economists consider noiseless environment and non-aligned objectives whereas the information theorists consider noisy environment and the objectives are either aligned or opposed. In this work, we propose a novel framework in order to investigate both aspects simultaneously, by considering July 13, 2018 DRAFT noisy environment and non-aligned transmitters' objectives. We formulate the strategic coding problem by considering a joint source-channel scenario in which the decoder observes a state information à la Wyner-Ziv [START_REF] Wyner | The rate-distortion function for source coding with side information at the decoder[END_REF]. This model generalizes the framework we introduced in our previous work, in [START_REF] Treust | Persuasion with limited communication capacity[END_REF]. The encoder and the decoder are endowed with distinct utility functions and they play a multi-dimensional version of the Bayesian persuasion game of Kamenica-Gentkow in [START_REF] Kamenica | Bayesian persuasion[END_REF]. We point out two essential features of the strategic coding problem:

• Each source symbol has a different impact on encoder/decoder's utility functions; so it's optimal to encode each symbol differently.

• In the noiseless version of the Bayesian persuasion game in [START_REF] Kamenica | Bayesian persuasion[END_REF], the optimal information disclosure policy requires a fixed amount of information bits. When the channel capacity is larger than this amount, it is optimal not to use all the channel resource.

The contributions of this article are as follows:

• We characterize the single-letter solution of the strategic coding problem and we relate it to

Wyner-Ziv's rate-distortion function [START_REF] Wyner | The rate-distortion function for source coding with side information at the decoder[END_REF] and to the separation result by . This characterization is based on the set of probability distributions that are achievable for the related coordination problem, under investigation in [51, Theorem IV.2].

• We reformulate our single-letter solution in terms of a concavification as in Kamenica-Gentkow [START_REF] Kamenica | Bayesian persuasion[END_REF], taking into account the information constraint imposed by the noisy channel. This provides an alternative point of view on the problem: given an encoding strategy, the decoder computes its posterior beliefs and chooses a best-reply action accordingly. Knowing this in advance, the encoder discloses the information optimally such as to induce the posterior beliefs corresponding to its optimal actions.

• We reformulate our solution in terms of a concavification of a Lagrangian, so as to relate with the cost of information considered for rational inattention in [START_REF] Sims | Implication of rational inattention[END_REF] and for the costly persuasion in [START_REF] Gentzkow | Costly persuasion[END_REF].

We also provide a bi-variate concavification where the information constraint is integrated along an additional dimension.

• One technical novelty is the characterization of the posterior beliefs induced by Wyner-Ziv's coding.

This confirms the benefit of sending encoded data bits to a autonomous decoder, even if the decoding process is not controlled. In fact, we prove that Wyner-Ziv's coding reveals the exact amount of information needed, no less no more. This property also holds for Shannon's Lossy source coding, as demonstrated in [START_REF] Treust | Persuasion with limited communication capacity[END_REF].

• When the channel is perfect and has a large input alphabet, our coding problem is equivalent to July 13, 2018 DRAFT several i.i.d. copies of the one-shot problem, whose optimal solution is given by our characterization without the information constraint. This noise-free setting is related to the problem of "persuasion with heterogeneous beliefs" under investigation in [START_REF]Bayesian persuasion with heterogeneous priors[END_REF] and [START_REF] Laclau | Public persuasion[END_REF].

• We illustrate our results by considering an example with binary source, states and decoder's actions.

We explain the concavification method and we analyse the impact of the channel noise on the set of achievable posterior beliefs.

• Surprisingly, the decoder's state information has two opposite effects on the optimal encoder's utility:

it enlarges the set of decoder's posterior beliefs, so it may increase encoder's utility; it reveals partial information to the decoder, so it forces some decoder's best-reply actions that might be sub-optimal for the encoder, hence it may decrease encoder's utility. The solution is provided without information constraint in Sec. IV-A and with information constraint in Sec. IV-B. The proofs are stated in App A -D.

II. STRATEGIC CODING PROBLEM

A. Coding Strategies and Utility Functions

We consider the i.i.d. distribution of information source/state P(u, z) and the memoryless channel distribution T (y|x) depicted in Fig. 1. Uppercase letters U denote the random variables, lowercase letters u denote the realizations and calligraphic fonts U denote the alphabets. Notations U n , X n , Y n , Z n , V n stand for sequences of random variables of information source u n = (u 1 , . . . , u n ) ∈ U n , decoder's state information z n ∈ Z n , channel inputs x n ∈ X n , channel outputs y n ∈ Y n and decoder's actions v n ∈ V n , respectively. The sets U , Z, X , Y, V have finite cardinality and the notation ∆(X ) stands for the set of probability distributions over X , i.e. the probability simplex. With a slight abuse of notation, we denote by Q(x) ∈ ∆(X ) the probability distribution, as in [54, pp. 14]. For example the joint

probability distribution Q(x, v) ∈ ∆(X ×V) decomposes as: Q(x, v) = Q(v)×Q(x|v) = Q(x)×Q(v|x).
The distance between two probability distributions Q(x) and P(x) is based on L 1 norm, denoted by: • The encoder E chooses an encoding strategy σ and the decoder D chooses a decoding strategy τ , defined by:

||Q -P|| 1 = x∈X |Q(x) -P(x)|. Notation U --X --Y
σ : U n -→ ∆(X n ), (1)
τ : Y n × Z n -→ ∆(V n ). (2)
Both strategies (σ, τ ) are stochastic.

• The strategies (σ, τ ) induces a joint probability distribution

P σ,τ ∈ ∆(U n × Z n × X n × Y n × V n ) over
the n-sequences of symbols, defined by:

n i=1 P u i , z i × σ x n u n × n i=1 T y i x i × τ v n y n , z n . (3)
The encoding and decoding strategies (σ, τ ) correspond to the problem of joint source-channel coding with decoder's state information studied in [START_REF] Merhav | On joint source-channel coding for the Wyner-Ziv source and the Gel'fand-Pinsker channel[END_REF], based on Wyner-Ziv's setting in [START_REF] Wyner | The rate-distortion function for source coding with side information at the decoder[END_REF]. Unlike these previous works, we investigate the case where the encoder and the decoder are autonomous decisionmakers, who choose their own encoding σ and decoding τ strategies. In this work, the encoder and the decoder have non-aligned objectives captured by distincts utility functions, depending on the source symbol U , the decoder's state information Z and on decoder's action V .

Definition II.2 (Utility Functions) • The single-letter utility functions of E and D are defined by:

φ e : U × Z × V -→ R, (4)
φ d : U × Z × V -→ R. (5)
• The long-run utility functions Φ n e (σ, τ ) and Φ n d (σ, τ ) are evaluated with respect to the probability distribution P σ,τ induced by the strategies (σ, τ ):

Φ n e (σ, τ ) =E σ,τ 1 n n i=1 φ e (U i , Z i , V i ) = u n ,z n ,v n P σ,τ u n , z n , v n • 1 n n i=1 φ e (u i , z i , v i ) , (6)
Φ n d (σ, τ ) = u n ,z n ,v n P σ,τ u n , z n , v n • 1 n n i=1 φ d (u i , z i , v i ) . (7)

B. Bayesian Persuasion Game

We investigate the strategic communication between autonomous devices who choose the encoding σ and decoding τ strategies in order to maximize their own long-run utility functions Φ n e (σ, τ ) and Φ n d (σ, τ ). We assume that the encoding strategy σ is designed and observed by the decoder in advance, i.e. before the transmission starts; then the decoder is free to choose any decoding strategy τ . This framework corresponds to the Bayesian persuasion game [START_REF] Kamenica | Bayesian persuasion[END_REF], in which the encoder commits to its strategy σ and announces it to the decoder, who chooses strategy τ accordingly. We assume that the strategic communication takes place as follows:

• The encoder E chooses and announces an encoding strategy σ to the decoder D.

• The sequences (U n , Z n , X n , Y n ) are drawn according to the probability distribution:

n i=1 P(u i , z i ) × σ(x n |u n ) × n i=1 T (y i |x i ). • The decoder D knows σ, observes the sequences of symbols (Y n , Z n ) and draws a sequence of actions V n according to τ (v n |y n , z n ).
By knowing σ in advance, the decoder D can compute the set of best-reply decoding strategies.

Definition II.3 (Decoder's Best-Replies) For any encoding strategy σ, the set of best-replies decoding strategies BR d (σ), is defined by:

BR d (σ) = τ, s.t. Φ n d (σ, τ ) ≥ Φ n d (σ, τ ), ∀ τ = τ . (8)
In case there are several best-reply strategies, we assume that the decoder chooses the one that minimizes encoder's utility: min τ ∈BR d (σ) Φ n e (σ, τ ), so that encoder's utility is robust to the exact specification of decoder's strategy.

The coding problem under investigation consists in maximizing the encoder's long-run utility:

sup σ min τ ∈BR d (σ) Φ n e (σ, τ ). (9)
Problem ( 9) raises the following interesting question: is it optimal for an autonomous decoder to extract the encoded information? We provide a positive answer to this question by showing that the actions induced by Wyner-Ziv's decoding τ wz coincide with those induced by any best-reply τ BR ∈ BR d (σ wz )

to Wyner-Ziv's encoding σ wz , for a large fraction of stages. We characterize a single-letter solution to [START_REF] Bergemann | Information design, Bayesian persuasion, and Bayes correlated equilibrium[END_REF], by refining Wyner-Ziv's result for source coding with decoder's state information [START_REF] Wyner | The rate-distortion function for source coding with side information at the decoder[END_REF].

Remark II.4 (Stackelberg v.s. Nash Equilibrium) The optimization problem of ( 9) is a Bayesian persuasion game [START_REF] Kamenica | Bayesian persuasion[END_REF], [START_REF] Gentzkow | Costly persuasion[END_REF] also referred to as Information Design problem [START_REF] Bergemann | Information design, Bayesian persuasion, and Bayes correlated equilibrium[END_REF], [START_REF] Taneva | Information design[END_REF], [START_REF] Bergemann | Information design: a unified perspective[END_REF]. This corresponds July 13, 2018 DRAFT to a Stackelberg equilibrium [START_REF] Stackelberg | Marketform und Gleichgewicht[END_REF] in which the encoder is the leader and the decoder is the follower, unlike the Nash equilibrium [START_REF] Nash | Non-cooperative games[END_REF] in which the two devices choose their strategy simultaneously.

Remark II.5 (Equal Utility Functions) When assuming that the encoder and decoder have a common objective, i.e. have equal utility function φ e = φ d , our problem boils down to the classical approach of Wyner-Ziv [START_REF] Wyner | The rate-distortion function for source coding with side information at the decoder[END_REF] and Merhav-Shamai [START_REF] Merhav | On joint source-channel coding for the Wyner-Ziv source and the Gel'fand-Pinsker channel[END_REF], in which both strategies (σ, τ ) are chosen jointly, in order to maximize the utility function:

sup σ min τ ∈BR d (σ) Φ n e (σ, τ ) = max (σ,τ )
Φ n e (σ, τ ), [START_REF] Taneva | Information design[END_REF] or to minimize a distortion function.

III. CHARACTERIZATIONS

A. Linear Program with Information Constraint

Before stating our main result, we define the encoder's optimal utility Φ ⋆ e .

Definition III.1 (Target Distributions) We consider an auxiliary random variable

W ∈ W with |W| = min |U | + 1, |V| |Z| .
The set Q 0 of target probability distributions is defined by:

Q 0 = P(u, z) × Q(w|u), s.t., max P(x) I(X; Y ) -I(U ; W |Z) ≥ 0 . ( 11
)
We define the set Q 2 Q(u, z, w) of single-letter best-replies of the decoder:

Q 2 Q(u, z, w) = argmax Q(v|z,w) E Q(u,z,w) ×Q(v|z,w) φ d (U, Z, V ) . ( 12
)
The encoder's optimal utility Φ ⋆ e is given by:

Φ ⋆ e = sup Q(u,z,w)∈Q0 min Q(v|z,w)∈ Q 2 (Q(u,z,w)) E Q(u,z,w) ×Q(v|z,w) φ e (U, Z, V ) . (13)
We discuss the above definitions.

• The information constraint [START_REF] Bergemann | Information design: a unified perspective[END_REF] of the set Q 0 involves the channel capacity max P(x) I(X; Y ) and the Wyner-Ziv's information rate I(U ; W |Z) = I(U ; W ) -I(W ; Z), stated in [START_REF] Wyner | The rate-distortion function for source coding with side information at the decoder[END_REF]. It corresponds to the separation result by Shannon [START_REF] Shannon | A mathematical theory of communication[END_REF], extended to the Wyner-Ziv setting by Merhav-Shamai in [START_REF] Merhav | On joint source-channel coding for the Wyner-Ziv source and the Gel'fand-Pinsker channel[END_REF].

• For the clarity of the presentation, the set Q 2 Q(u, z, w) contains stochastic functions Q(v|z, w), even if for the linear problem [START_REF] Tsakas | Noisy persuasion[END_REF], some optimal Q(v|z, w) are deterministic. If there are several optimal Q(v|z, w), we assume the decoder chooses the one that minimize encoder's utility:

min Q(v|z,w)∈ Q 2 (Q(u,z,w))
E φ e (U, Z, V ) , so that encoder's utility is robust to the exact specification of Q(v|z, w).

• The supremum over Q(u, z, w) ∈ Q 0 is not a maximum since the function min Q(v|z,w)∈ Q 2 (Q(u,z,w))
E φ e (U, Z, V )

July 13, 2018 DRAFT is not continuous with respect to Q(u, z, w).

• In [51, Theorem IV.2], the author shows that the sets Q 0 and Q 2 correspond to the target probability distributions Q(u, z, w) × Q(v|z, w) that are achievable for the problem of empirical coordination, see also [START_REF] Cuff | Coordination capacity[END_REF], [START_REF] Treust | Joint empirical coordination of source and channel[END_REF]. As noticed in [START_REF] Treust | Empirical coordination, state masking and state amplification: Core of the decoder's knowledge[END_REF] and [START_REF] Treust | State leakage and coordination of actions: Core of decoder's knowledge[END_REF], the tool of Empirical Coordination allows us to characterize the "core of the decoder's knowledge", that captures what the decoder can infer about all the random variables of the problem.

• The value Φ ⋆ e corresponds to the Stackelberg equilibrium of an auxiliary one-shot game in which the decoder chooses Q(v|z, w), knowing in advance that the encoder has chosen Q(w|u) ∈ Q 0 and the utility functions are: E φ e (U, Z, V ) and E φ d (U, Z, V ) .

Remark III.2 (Equal Utility Functions) When assuming that the decoder's utility function is equal to the encoder's utility function:

φ d (u, z, v) = φ e (u, z, v), then the set Q 2 Q(u, z, w) is equal to argmax Q(v|z,w) E φ e (U, Z, V )
. Thus, we have:

min Q(v|z,w)∈ Q 2 (Q(u,z,w)) E Q(u,z,w) ×Q(v|z,w) φ e (U, Z, V ) = max Q(v|z,w) E Q(u,z,w) ×Q(v|z,w) φ e (U, Z, V ) . (14)
Hence, the encoder's optimal utility Φ ⋆ e is equal to:

Φ ⋆ e = sup Q(u,z,w)∈Q0 min Q(v|z,w)∈ Q 2 (Q(u,z,w)) E Q(u,z,w) ×Q(v|z,w) φ e (U, Z, V ) (15)
= sup

Q(u,z,w)∈Q0 max Q(v|z,w) E Q(u,z,w) ×Q(v|z,w) φ e (U, Z, V ) (16)
= max

Q(u,z,w)∈Q 0 , Q(v|z,w) E Q(u,z,w) ×Q(v|z,w) φ e (U, Z, V ) . ( 17
)
The supremum in ( 16) is replaced by a maximum in [START_REF] Neyman | Strategic entropy and complexity in repeated games[END_REF] due to the compacity of Q 0 and the continuity of function max Q(v|z,w) E φ e (U, Z, V ) with respect to Q(u, z, w).

If we consider that the utility function is equal to minus the distortion function:

φ e (u, z, v) = -d(u, v)
as in [50, Definition 1], then we recover the distortion-rate function corresponding to [50, Theorem 1]:

Φ ⋆ e = -min Q(u,z,w)∈Q 0 , Q(v|z,w) E Q(u,z,w) ×Q(v|z,w) d(U, V ) . (18)

B. Main Result

We introduce the notation N ⋆ = N \ {0} and we characterize the encoder's long-run optimal utility (9) by using Φ ⋆ e .

July 13, 2018 DRAFT Theorem III.3 (Main Result) The long-run optimal utility of the encoder satisfies:

∀ε > 0, ∃n ∈ N ⋆ , ∀n ≥ n, sup σ min τ ∈BR d (σ) Φ n e (σ, τ ) ≥ Φ ⋆ e -ε, (19)
∀n ∈ N, sup

σ min τ ∈BR d (σ) Φ n e (σ, τ ) ≤ Φ ⋆ e . (20)
The proofs of the achievability [START_REF] Neyman | Growth of strategy sets, entropy, and nonstationary bounded recall[END_REF] and the converse [START_REF] Gossner | How to play with a biased coin?[END_REF] results are given in App. B and C. When removing decoder's state information Z = ∅, we recover our previous result in [START_REF] Treust | Persuasion with limited communication capacity[END_REF]Theorem 4.3]. As a consequence, Theorem III.3 characterizes the limit behaviour of long-run optimal utility of the encoder:

lim n→+∞ sup σ min τ ∈BR d (σ) Φ n e (σ, τ ) = Φ ⋆ e . (21)

C. Concavification

The concavification of a function f is the smallest concave function cav f : X → R ∪ {-∞} that majorizes f on X. In this section, we reformulate the encoder's optimal utility Φ ⋆ e in terms of a concavification, similarly to [6, Corollary 1] and [START_REF] Treust | Persuasion with limited communication capacity[END_REF]Definition 4.2]. This alternative approach simplifies the optimization problem in [START_REF] Blume | Noisy talk[END_REF], by plugging the decoder's posterior beliefs and best-reply actions into the encoder's utility function. This also provides a nice interpretation: the goal of the strategic communication is to control the posterior beliefs of the decoder knowing it will take a best-reply action afterwards.

Before the transmission, the decoder holds a prior belief corresponding to the source's statistics P(u) ∈ ∆(U ). After observing the pair of symbols (w, z) ∈ W × Z, the decoder updates its posterior belief

Q(•|z, w) ∈ ∆(U ) according to Bayes rule: Q(u|z, w) = P(u,z)Q(w|u) u ′ P(u ′ ,z)Q(w|u ′ )
, for all (u, z, w) ∈ U ×W ×Z.

Definition III.4 (Best-Reply Action) For each symbol z ∈ Z and belief p ∈ ∆(U ), the decoder chooses the best-reply action v ⋆ (z, p) that belongs to the set V ⋆ (z, p), defined by:

V ⋆ (z, p) = argmin v∈argmax Ep φ d (U,z,v) E p φ e (U, z, v) . (22)
If several actions are best-replies to symbol z ∈ Z and belief p ∈ ∆(U ), the decoder chooses one of the worst action for encoder's utility. This is a reformulation of the minimum in [START_REF] Blume | Noisy talk[END_REF].

Definition III.5 (Robust Utility Function) For each symbol z ∈ Z and belief p ∈ ∆(U ), the encoder's robust utility function is defined by:

ψ e (z, p) = E p φ e (U, z, v ⋆ (z, p)) . (23)
Definition III.6 (Average Utility and Average Entropy) For each belief p ∈ ∆(U ), we define the average encoder's utility function Ψ e (p) and average entropy function h(p):

Ψ e (p) = u,z p(u) • P(z|u) • ψ e z, p(u) • P(z|u) u ′ p(u ′ ) • P(z|u ′ ) , (24)
h(p) = u,z p(u) • P(z|u) • log 2 u ′ p(u ′ ) • P(z|u ′ ) p(u) • P(z|u) . ( 25
)
The conditional probability distribution P(z|u) is given by the information source.

Lemma 1 (Concavity) The average entropy h(p) is concave in p ∈ ∆(U ).

Proof. [Lemma 1] The average entropy h(p) in ( 25) is a reformulation of the conditional entropy H(U |Z)

as a function of the probability distribution p ∈ ∆(U ), for a fixed P(z|u). The mutual information I(U ; Z) [58, pp. 23]), and the entropy H(U ) is concave in p ∈ ∆(U ). Hence the

is convex in p ∈ ∆(U ) (see
conditional entropy h(p) = H(U |Z) = H(U ) -I(U ; Z) is concave in p ∈ ∆(U ).
Theorem III.7 (Concavification) The solution Φ ⋆ e of (13) is the concavification of Ψ e (p) evaluated at the prior distribution P(u), under an information constraint:

Φ ⋆ e = sup w λ w • Ψ e (p w ) s.t. w λ w • p w = P(u) ∈ ∆(U ),
and

w λ w • h(p w ) ≥ H(U |Z) -max P(x) I(X; Y ) , (26)
where the supremum is taken over λ w ∈ [0, 1] summing up to 1 and p w ∈ ∆(U ), for each w ∈ W with

|W| = min |U | + 1, |V| |Z| .
The proof of Theorem III.7 is stated in App. A. The "splitting Lemma" by Aumann and Maschler [START_REF] Aumann | Repeated Games with Incomplete Information[END_REF], also called "Bayes plausibility" in [START_REF] Kamenica | Bayesian persuasion[END_REF], ensures that there is a one-to-one correspondance between the conditional distribution Q(w|u) = λw•pw(u)

P(u)
and the parameters (λ w , p w ) w∈W , also referred to as the "splitting of the prior belief". Formulation [START_REF] Cuff | Coordination capacity[END_REF] provides an alternative point of view on the encoder's optimal utility (13).

• The optimal solution Φ ⋆ e can be found by the concavification method [START_REF] Aumann | Repeated Games with Incomplete Information[END_REF]. In Sec IV, we provide an example that illustrates the optimal splitting and the corresponding expected utility.

• When the channel is perfect and has a large input alphabet |X | ≥ min(|U |, |V| |Z| ), the strategic coding problem is equivalent to several i.i.d. copies of the one-shot problem, whose optimal solution is given by our characterization without the information constraint [START_REF] Cuff | Coordination capacity[END_REF]. This noise-free setting is related to the problem of persuasion with heterogeneous beliefs under investigation in [START_REF]Bayesian persuasion with heterogeneous priors[END_REF] and [START_REF] Laclau | Public persuasion[END_REF].

July 13, 2018 DRAFT

• The information constraint w λ w • h(p w ) ≥ H(U |Z) -max P(x) I(X; Y ) in ( 26) is a reformulation of I(U ; W |Z) ≤ max P(x) I(X; Y ) in [START_REF] Bergemann | Information design: a unified perspective[END_REF]:

w λ w • h(p w ) = w λ w • H(U |Z, W = w) (27) =H(U |Z, W ). (28)
• The dimension of the problem ( 26 • The cardinality of W is also restricted to the vector of recommended actions |W| = |V| |Z| , telling to the decoder which action to play in each state. Otherwise assume that two posteriors p w1 and p w2 induce the same vectors of actions

v 1 = (v 1 1 , . . . , v 1 |Z| ) = v 2 = (v 2 1 , . . . , v 2
|Z| ). Then, both posteriors p w1 and p w2 can be replaced by their average:

p = λ w1 • p w1 + λ w2 • p w2 λ w1 + λ w2 , (29)
without changing the utility and still satisfying the information constraint:

h( p) ≥ λ w1 • h(p w1 ) + λ w2 • h(p w2 ) λ w1 + λ w2 (30)
=⇒

w =w 1 , w =w 2 λ w • h(p w ) + (λ w1 + λ w2 ) • h( p) ≥ H(U |Z) -max P(x) I(X; Y ). (31)
Inequality [START_REF] Akyol | Information-theoretic approach to strategic communication as a hierarchical game[END_REF] comes from the concavity of h(p), stated in Lemma 1.

Following the arguments of [49, Theorem 5.1], the splitting under information constraint of Theorem III.7 can be reformulated in terms of Lagrangian and in terms of a general concavification Ψe (p, ν) defined by:

Ψe (p, ν) =      Ψ e (p), if ν ≤ h(p), -∞, otherwise, (32)
Theorem III.8 The optimal solution Φ ⋆ e reformulates as:

Φ ⋆ e = inf t≥0 cav Ψ e + t • h P(u) -t • H(U |Z) -max P(x) I(X; Y ) (33)
= cav Ψe P(u), H(U |Z)max

P(x) I(X; Y ) . (34)
Equation ( 33) is the concavification of a Lagrangian that integrates the information constraint.

u 2 u 1 w 2 w 1 z 2 z 1 1 -α 1 -β 1 -p 0 p 0 α β 1 -δ 1 δ 1 1 -δ 2 δ 2 Fig. 2. Joint probability distribution P(u, z) × Q(w|u) depending on parameters p0 ∈ [0, 1], δ1 ∈ [0, 1], δ2 ∈ [0, 1], α ∈ [0, 1] and β ∈ [0, 1].
Fig. 3, 4 and do not depend on the state z. In this example, the goal of the decoder is to choose the action v that matches the source symbol u, whereas the goal of encoder is to persuade the decoder to take the action v 2 . After receiving the pair of symbols (w, z), the decoder updates its posterior belief

u 2 u 1 v 1 v 2 0 0 1 1
it plays v ⋆ 1 if p(u2) ∈ [0, 0.6] and v ⋆ 2 if p(u2) ∈ ]0.6, 1].
utility function of the encoder (see Definition III.5), given by:

ψ e (p) = min v∈argmax p(u 1 )•φ d (u 1 ,v)+p(u 2 )•φ d (u 2 ,v) p(u 1 ) • φ e (u 1 , v) + p(u 2 ) • φ e (u 2 , v), (35)
= 1 p(u 2 ) ∈]0.6, 1] . (36)

A. Concavification without Information Constraint

In this section, we assume that the channel capacity is large enough max P(x) I(X; Y ) ≥ log 2 |U |, so we investigate the concavification of Ψ e (p) (see Definition III.6), without information constraint:

Φ • e = sup w λ w • Ψ e (p w ) s.t. w λ w • p w = P(u) ∈ ∆(U ) . (37)
The correlation between random variables (U, Z) is fixed whereas the correlation between random variables (U, W ) is chosen strategically by the encoder. This imposes a strong relationship between the three different kinds of posterior beliefs: Q(u|z), Q(u|w), Q(u|w, z). We denote by p 01 ∈ [0, 1], 1] the belief parameters after observing the state information Z only:

p 02 ∈ [0,
p 01 = Q(u 2 |z 1 ) = p 0 • δ 2 p 0 • δ 2 + (1 -p 0 ) • (1 -δ 1 ) , ( 38
)
p 02 = Q(u 2 |z 2 ) = p 0 • (1 -δ 2 ) p 0 • (1 -δ 2 ) + (1 -p 0 ) • δ 1 . ( 39
)
The beliefs parameters p 01 , p 02 are given by the source probability distribution P(u, z) ∈ ∆(U × Z)

and correspond to the horizontal dotted lines in Fig. 8, for p 0 = 0.5, δ 1 = 0.7, δ 2 = 0.9. We denote by July 13, 2018 DRAFT q 1 ∈ [0, 1], q 2 ∈ [0, 1] the belief parameters after observing only the symbol W sent by the encoder:

q 1 = Q(u 2 |w 1 ) = p 0 • β p 0 • β + (1 -p 0 ) • (1 -α) , (40)
q 2 = Q(u 2 |w 2 ) = p 0 • (1 -β) p 0 • (1 -β) + (1 -p 0 ) • α . ( 41
)
By inverting the system of equations ( 40) -( 41), we express the cross-over probabilities α(q 1 , q 2 ) and β(q 1 , q 2 ) as functions of the target belief parameters (q 1 , q 2 ):

     α(q 1 , q 2 ) = (1-q2)•(q1-p0) (1-p0)•(q1-q2) β(q 1 , q 2 ) = q1•(p0-q2) p0•(q1-q2) (42)
Lemma 2 (Feasible Posteriors) The parameters α(q 1 , q 2 ) and β(q 1 , q 2 ) belong to the interval [0, 1] if and only if q 1 ≤ p 0 ≤ q 2 or q 2 ≤ p 0 ≤ q 1 .

The proof of Lemma 2 is stated in App. D. Thanks to the Markov chain property Z --U --W the posterior beliefs Q(u|w, z) reformulate in terms of Q(u|w): We define the belief parameters

Q(u|w, z) = Q(u, z, w) Q(z, w) = Q(u, z|w) u ′ Q(u ′ , z|w) = Q(u|w)P(z|u) u ′ Q(u ′ |w)P(z|u ′ ) , ∀(u, z, w) ∈ U × Z × W. (43
p 1 ∈ [0, 1], p 2 ∈ [0, 1], p 3 ∈ [0, 1], p 4 ∈ [0, 1] after observing (W, Z)
and we express them as functions of q 1 , q 2 :

p 1 = Q(u 2 |w 1 , z 1 ) = q 1 • δ 2 (1 -q 1 ) • (1 -δ 1 ) + q 1 • δ 2 , ( 44
)
p 2 = Q(u 2 |w 1 , z 2 ) = q 1 • (1 -δ 2 ) (1 -q 1 ) • δ 1 + q 1 • (1 -δ 2 ) , ( 45
)
p 3 = Q(u 2 |w 2 , z 1 ) = q 2 • δ 2 (1 -q 2 ) • (1 -δ 1 ) + q 2 • δ 2 , ( 46
)
p 4 = Q(u 2 |w 2 , z 2 ) = q 2 • (1 -δ 2 ) (1 -q 2 ) • δ 1 + q 2 • (1 -δ 2 ) . ( 47
)
Fig. 6 represents (p 1 , p 2 , p 3 , p 4 ) as a functions of q 1 ∈ [0, p 0 ] and q 2 ∈ [p 0 , 1]. In fact, the beliefs q 1 and q 2 , are the key parameters since they control the decoder's best-reply action v ⋆ (p) through the beliefs (p 1 , p 2 , p 3 , p 4 ). We define the two following functions of q ∈ [0, 1]:

p 1 (q) = q • δ 2 (1 -q) • (1 -δ 1 ) + q • δ 2 , ( 48
)
p 2 (q) = q • (1 -δ 2 ) (1 -q) • δ 1 + q • (1 -δ 2 ) . ( 49
)
Given the belief threshold γ = 0.6 at which the decoder changes its action, we define the parameters ν 1

and ν 2 such that p 1 (ν 1 ) = γ and p 2 (ν 2 ) = γ.

γ = p 1 (ν 1 ) ⇐⇒ ν 1 = γ • (1 -δ 1 ) δ 2 • (1 -γ) + γ • (1 -δ 1 ) , ( 50
)
γ = p 2 (ν 2 ) ⇐⇒ ν 2 = γ • δ 1 γ • δ 1 + (1 -δ 2 ) • (1 -γ) . ( 51
)
This belief parameters ν 1 and ν 2 allow to reformulate the utility function of the encoder as a function of belief Q(u|w) (see Fig. 7), instead of belief Q(u|w, z) (see Fig. 8).

The solution Φ • e corresponds to the concavification of the function Ψ e , defined over q ∈ [0, 1]:

Ψ e (q) = (1 -q) • (1 -δ 1 ) + q • δ 2 • ψ e p 1 (q) + (1 -q) • δ 1 + q • (1 -δ 2 )
• ψ e p 2 (q) , (52)

Φ • e = cav Ψ e (p 0 ) = sup λ, q,q ′ λ • Ψ e (q) + (1 -λ) • Ψ e (q ′ ) s.t. λ • q + (1 -λ) • q ′ = p 0 . ( 53
)
Fig. 7 represents the utility function Ψ e (q) of the encoder, depending on the belief q ∈ [0, 1]. When the belief q ∈ [ν 1 , ν 2 ], then ψ e p 1 (q) = 1 whereas ψ e p 2 (q) = 0. The optimal splitting is represented by the square and circle. This indicates that the optimal posterior beliefs are (p 1 , p 2 ) = (γ, p 2 (ν 1 )) and

(p 3 , p 4 ) = (p 1 (ν 2 ), γ), as in Fig. 8.

When the decoder has no state information, the optimal solution by Kamenica-Gentzkow [START_REF] Kamenica | Bayesian persuasion[END_REF] is the concavification of the function ψ e (p) = 1 p ∈]0.6, 1] , corresponding to Φ e in Fig. 8. In this example, the decoder's state information Z decreases the optimal utility of the encoder Φ e ≥ Φ • e .

0 1 1 p(u 2 ) p 0 = 0 . 5 Φ e = 0.833333 v ⋆ 1 v ⋆ 2 p 0 1 = 0 . 7 5 0 0 0 0 p 0 2 = 0 . 1 2 5 0 0 0 p 2 = p 2 ( ν 1 ) = 0 . 0 6 6 6 6 7
p 3 = p 1 ( ν 2 ) = 0 . 9 6 9 2 3 1 p 1 = p 4 = γ = 0 . 6 Φ • e = 0.

643750

Fig. 8. Optimal encoder's utility depending on the belief p(u2) ∈ [0, 1] after observing (W, Z), for parameters p0 = 0.5, δ1 = 0.7, δ2 = 0.9 and γ = 0.6.

Z is investigated in [START_REF] Treust | Persuasion with limited communication capacity[END_REF] and the corresponding information constraint is given by:

p 0 -q 2 q 1 -q 2 • H b (q 1 ) + q 1 -p 0 q 1 -q 2 • H b (q 2 ) ≥ H(U ) -max P(x) I(X; Y ) (58)
⇐⇒Q(w 1 ) • H(U |W = w 1 ) + Q(w 2 ) • H(U |W = w 2 ) ≥ H(U ) -max P(x) I(X; Y ) (59)
⇐⇒I(U ; W ) ≤ max P(x) I(X; Y ). (60)
Fig. 9 shows that the decoder's state information Z enlarges the set of posterior beliefs Q(u|w) compatible with the information constraint of Q 0 .

The optimal posterior beliefs are represented on Fig. 10, by the square and the circle. Due to the restriction imposed by the channel, the optimal posterior beliefs are (ν 3 , ν 2 ) instead of (ν 1 , ν 2 ), and this reduces the encoder's optimal utility to Φ ⋆ e ≃ 0.63 instead of Φ • e ≃ 0.64. In fact, the posterior beliefs (ν 1 , ν 2 ) do not satisfy the information constraint: λh(ν 1 ) + (1λ)h(ν 2 ) < H(U |Z)max P(x) I(X; Y ), whereas the pair of posterior beliefs (ν 3 , ν 2 ) lies at the boundary of the blue region in Fig. 9. Posterior beliefs (ν 3 , ν 2 ) determine the posterior beliefs (p 1 , p 2 , p 3 , p 4 ), represented in Fig. 11, that satisfy the July 13, 2018 DRAFT C in green, for p0 = 0.5, δ1 = 0.7, δ2 = 0.9 and channel capacity C = 0.1.

p 0 = 0 . 5 p 0 = 0.5 0 1 q 2 1 q 1 ν 3 = 0 . 3 8 3 ν 2 = 0.913
0 1 1 q p 0 = 0 . 5 ν 1 = 0 . 3 3 3 3 3 3 ν 2 = 0 . 9 1 3 0 4 3 Φ ⋆ e = 0.6337 ν 3 = 0 . 3 8 3 ( v ⋆ 1 , v ⋆ 1 ) ( v ⋆ 2 , v ⋆ 2 ) ( v ⋆ 2 , v ⋆ 1 )
H(U |Z) -C Fig. 10. Optimal encoder's utility depending on the belief parameter q ∈ [0, 1] after observing W , for parameters p0 = 0.5, δ1 = 0.7, δ2 = 0.9, γ = 0.6 and C = 0.1. The curve represents the average entropy h(q) = H(U |Z) defined in [START_REF] Cover | Elements of Information Theory[END_REF], as a function of q ∈ [0, 1].

July 13, 2018 DRAFT

0 1 1 p(u 2 ) p 0 = 0 . 5 H(U |Z) -C v ⋆ 1 v ⋆ 2 p 0 1 = 0 . 7 5 0 0 0 0 p 0 2 = 0 . 1 2 5 0 0 0 p 2 = p 2 ( ν 1 ) = 0 . 0 8 1 5
p 3 = p 1 ( ν 2 ) =
0 . 9 6 9 2 3 1

p 4 = γ = 0 . 6 p 1 = 0 . 6 5 0 6
Φ ⋆ e = 0.6337 reformulation of the information constraint:

λ w1,z1 H b (p 1 ) + λ w1,z2 H b (p 2 ) + λ w2,z1 H b (p 3 ) + λ w2,z2 H b (p 4 ) = H(U |Z) -max P(x) I(X; Y ), (61)
and provide the corresponding expected utility Φ ⋆ e ≃ 0.63.

Remark IV.1 (Impact of the State Information Z) The state information Z at the decoder has two effects:

• When the communication is restricted (i.e. max P(x) I(X; Y ) < log 2 |U |), it enlarges the set of posterior beliefs Q(u|w), so it may increase the encoder's utility.

• Since it reveals partial information to the decoder, it forces the decoder to choose a best-reply actions that might be sub-optimal for the encoder, so it may decrease the encoder's utility.

Depending on the problem, the state information Z may increase or decrease the encoder's optimal utility.

July 13, 2018 DRAFT Ziv's source coding converge to the target posterior beliefs.

A. Zero Capacity

We first investigate the case of zero capacity.

Lemma 3 If the channel has zero capacity: max P(x) I(X; Y ) = 0, then we have:

∀n ∈ N, ∀σ, min τ ∈BR d (σ) Φ n e (σ, τ ) = Φ ⋆ e . (68)
Proof. [Lemma 3] The zero capacity max P(x) I(X; Y ) = 0 implies that any probability distribution P(u, z)×Q(w|u) ∈ Q 0 satisfies I(U ; W |Z) = 0, corresponding to the Markov chain property U --Z--W , i.e. Q(u|z, w) = P(u|z) for all (u, z, w) ∈ U × Z × W.

Φ ⋆ e = sup Q(u,z,w)∈Q0 min Q(v|z,w)∈ Q 2 (Q(u,z,w)) E Q(u,z,w) ×Q(v|z,w) φ e (U, Z, V ) (69)
= sup

Q(u,z,w)∈Q0 E Q(u,z,w) φ e (U, Z, V ⋆ (z, Q(u|w, z))) (70) = sup Q(u,z,w)∈Q0 E Q(u,z,w) φ e (U, Z, V ⋆ (z, P(u|z))) (71)
=E P(u,z) φ e (U, Z, V ⋆ (z, P(u|z))) . (72)
Equation ( 70) is a reformulation by using the best-reply action v ⋆ z, p of Definition III.4 for symbol z ∈ Z and the belief Q(u|w, z).

Equation (71) comes from Markov chain property U --Z --W that allows to replace the belief Q(u|w, z) by P(u|z).

Equation (72) comes from removing the random variable W since it has no impact on the function φ e (u, z, v ⋆ (z, P(u|z))).

For any n and for any encoding strategy σ, the encoder's long-run utility is given by:

min τ ∈BR d (σ)
Φ n e (σ, τ ) = min

τ ∈BR d (σ) u n ,z n ,x n , y n ,v n n i=1 P u i , z i × σ x n u n × n i=1 T y i × τ v n y n , z n • 1 n n i=1 φ e (u i , z i , v i ) (73) = min τ ∈BR d (σ) u n ,z n ,v n n i=1 P u i , z i × τ v n z n • 1 n n i=1 φ e (u i , z i , v i ) (74) = 1 n n i=1 ui,zi,vi P u i , z i × 1(v ⋆ i (z i , Q(u|z))) • φ e (u i , z i , v i ) (75)
= E P(u,z) φ e (U, Z, V ⋆ (z, P(u|z))) .

Equation ( 73) comes from the zero capacity that imposes that the channel outputs Y n are independent of the channel inputs X n .

Equation (74) comes from removing the random variables (X n , Y n ) and noting that the decoder's bestreply τ v n z n does not depend on y n anymore, since y n is independent of (u n , z n ).

Equation ( 75) is a reformulation based on the best-reply action v ⋆ z, P(u|z) of Definition III.4, for the symbol z ∈ Z and the belief P(u|z).

Equation (76) comes from the i.i.d. property of (U, Z) and concludes the proof of Lemma 3.

B. Positive Capacity

We now assume that the channel capacity is strictly positive max P(x) I(X; Y ) > 0. We consider an auxiliary concavification in which the information constraint is satisfied with strict inequality and the sets of decoder's best-reply actions are always singletons. For the proof of Lemma 4, we refers directly to the similar proof of [START_REF] Treust | Persuasion with limited communication capacity[END_REF]Lemma A.7,pp. 46]. We denote by Q n (u, z, w) the empirical distribution of the sequence (u n , z n , w n ) and we denote by A δ the set of typical sequences with tolerance δ > 0, defined by:

A δ = (u n , z n , w n , x n , y n ), s.t. ||Q n (u, z, w) -P(u, z) × Q(w|u)|| 1 ≤ δ,
and

||Q n (x, y) -P ⋆ (x) × T (y|x)|| 1 ≤ δ . (78)
We define T α (w n , y n , z n ) and B α,γ,δ depending on parameters α > 0 and γ > 0:

T α (w n , y n , z n ) = i ∈ {1, . . . , n}, s.t. D P σ (U i |y n , z n ) Q(U i |w i , z i ) ≤ α 2 2 ln 2 , ( 79
) B α,γ,δ = (w n , y n , z n ), s.t. |T α (w n , y n , z n )| n ≥ 1 -γ and (w n , y n , z n ) ∈ A δ . ( 80
)
The notation B c α,γ,δ stands for the complementary set of B α,γ,δ ⊂ W n × Y n × Z n . The sequences (w n , y n , z n ) belong to the set B α,γ,δ if: 1) they are typical and 2) if the corresponding posterior belief

P σ (U i |y n , z n ) is close in K-L divergence to the target belief Q(U i |w i , z i ),
V ⋆ (z, Q(u|z, w)) is a singleton ∀(z, w) ∈ Z × W, (81)
then ∀ε > 0, ∀α > 0, ∀γ > 0, ∃ δ > 0, ∀δ < δ, ∃n ∈ N ⋆ , ∀n ≥ n, ∃σ, s.t. P σ (B c α,γ,δ ) ≤ ε. ( 82
)
The proof of proposition B.1 is stated in App. B-C.

Proposition B.2 For any encoding strategy σ, we have:

min τ ∈BR d (σ)
Φ n e (σ, τ ) -Φ e ≤ (α + 2γ + δ)

• φe + (1 -P σ (B α,γ,δ )) • φe , (83)
where φe = max u,z,v φ e (u, z, v) is the largest absolute value of encoder's utility.

For the proof of Proposition B.2, we refers directly to the similar proof of [49, Corollary A.18, pp. 53].

Corollary B.3

For any ε > 0, there exists n ∈ N ⋆ such that for all n ≥ n there exists an encoding strategy σ such that:

min τ ∈BR d (σ) Φ n e (σ, τ ) -Φ e ≤ ε. (84)
The proof of Corollary B.3 comes from combining Proposition B.1 with Proposition B.2 and choosing parameters α, γ, δ small and n ∈ N ⋆ large. This concludes the achievability proof of Theorem III.3.

C. Proof of Proposition B.1

We assume that the probability distribution P(u, z) × Q(w|u) satisfies the two following conditions:

     max P(x) I(X; Y ) -I(U ; W |Z) > 0, V ⋆ (z, Q(u|z, w)) is a singleton ∀(z, w) ∈ Z × W, (85)
The strict information constraint ensures there exists a small parameter η > 0 and rates R ≥ 0, R L ≥ 0, such that:

R + R L = I(U ; W ) + η, (86)
R L ≤ I(Z; W ) -η, (87)
R ≤ max

P(x) I(X; Y ) -η. (88)
We now recall the random coding construction of Wyner-Ziv [START_REF] Wyner | The rate-distortion function for source coding with side information at the decoder[END_REF] and we investigate the corresponding posterior beliefs. We note by Σ the random encoding/decoding, defined as follows:

• • Error Event. We introduce the event of error E δ ∈ {0, 1} defined as follows:

E δ = 0 if (M, L) = ( M , L) and U n , Z n , W n , X n , Y n ∈ A δ , 1 otherwise.
(89)

Expected error probability of the random encoding/decoding Σ. For all ε 2 > 0, for all η > 0, there exists a δ > 0, for all δ ≤ δ there exists n such that for all n ≥ n, the expected probability of the following error events are bounded by ε 2 :

E Σ P ∀(m, l), U n , W n (m, l) / ∈ A δ ≤ ε 2 , (90)
E Σ P ∃l ′ = l, s.t. Z n , W n (m, l ′ ) ∈ A δ ≤ ε 2 , (91)
E Σ P ∃m ′ = m, s.t. Y n , X n (m ′ ) ∈ A δ ≤ ε 2 , (92)
Eq. (90) comes from (86) and the covering lemma [58, pp. 208].

Eq. (91) comes from (87) and the packing lemma [58, pp. 46].

Eq. (92) comes from (88) and the packing lemma [58, pp. 46].

There exists a coding strategy σ with small error probability:

∀ε 2 > 0, ∀η > 0, ∃ δ > 0, ∀δ ≤ δ, ∃n > 0, ∀n ≥ n, ∃σ, P σ E δ = 1 ≤ ε 2 . ( 93
)
July 13, 2018 DRAFT H(Z n |U n ), the i.i.d. property of the source (U, Z) that implies H(Z|U ) and the Markov chain Z--U --W that implies H(Z|U, W ).

Equations ( 98)-(102) shows that on average, the posterior beliefs P σ (u i |y n , z n , E δ = 0) induced by strategy σ is close to the target probability distribution Q(u|w, z).

E σ 1 n n i=1 D P σ (U i |Y n , Z n , E δ = 0) Q(U i |W i , Z i ) ≤2δ + η + 2 n + 2 log 2 |U | • P σ E δ = 1 := ǫ. (103)
Then we have:

P σ (B c α,γ,δ ) =1 -P σ (B α,γ,δ ) =P σ (E δ = 1)P σ (B c α,γ,δ |E δ = 1) + P σ (E δ = 0)P σ (B c α,γ,δ |E δ = 0) ≤P σ (E δ = 1) + P σ (B c α,γ,δ |E δ = 0) ≤ε 2 + P σ (B c α,γ,δ |E δ = 0). (104)
Moreover:

P σ (B c α,γ,δ |E δ = 0) = w n ,y n ,z n P σ (w n , y n , z n ) ∈ B c α,γ,δ E δ = 0 (105) = w n ,y n ,z n P σ (w n , y n , z n ) s.t. |T α (w n , y n , z n )| n < 1 -γ E δ = 0 (106) =P σ 1 n • i, s.t. D P σ (U i |y n , z n ) Q(U i |w i , z i ) ≤ α 2 2 ln 2 < 1 -γ E δ = 0 (107) =P σ 1 n • i, s.t. D P σ (U i |y n , z n ) Q(U i |w i , z i ) > α 2 2 ln 2 ≥ γ E δ = 0 (108) ≤ 2 ln 2 α 2 γ • E σ 1 n n i=1 D P σ (U i |y n , z n ) Q(U i |w i , z i ) (109) ≤ 2 ln 2 α 2 γ • η + δ + 2 n + 2 log 2 |U | • P σ E δ = 1 . (110)
Eq. ( 105) to (108) are simple reformulations.

Eq. (109) comes from the double use of Markov's inequality as in [START_REF] Treust | Persuasion with limited communication capacity[END_REF]Lemma A.22,pp.60].

Eq. (110) comes from (103).

Combining equations (93), (104), (110) and choosing η > 0 small, we obtain the following statement:

∀ε > 0, ∀α > 0, ∀γ > 0, ∃ δ > 0, ∀δ < δ, ∃n ∈ N ⋆ , ∀n ≥ n, ∃σ, s.

Eq. (120) comes from the Markov chain Y n --X n --(U n , Z n ).

Eq. (121) comes from the memoryless property of the channel and from removing the positive term

I(U n ; Z n ) ≥ 0.
Eq. (122) comes from taking the maximum P(x) and chain rule.

Eq. (123) comes from the i.i.d. property of the source (U, Z) that implies I(U i , Z i ; Z -i , U i-1 ) = I(U i ; Z -i , U i-1 |Z i ) = 0.

Eq. (124) comes from removing I(U i ; U i-1 |Y n , Z -i , Z i ) ≥ 0.

Eq. ( 125) comes from the uniform random variable T ∈ {1, . . . , n}.

Eq. ( 126) comes from the independence between T and the source (U, Z), that implies I(U T , Z T ; T ) = I(U T ; T |Z T ) = 0.

Eq. (127) comes from the identification W = (Y n , Z -T , T ).

Eq. (128) comes from the Markov chain W --U T --Z T . This proves that the distribution P σ (u, z, w) belongs to the set Q 0 .

July 13, 2018 DRAFT Therefore, for any encoding strategy σ and all n, we have:

min τ ∈BR d (σ)
Φ n e (σ, τ ) (133)

Assume that q 1 ≤ q 2 , then we have:

0 ≤ α ⇐⇒ 0 ≤ (1 -q 2 ) • (q 1 -p 0 ) (1 -p 0 ) • (q 1 -q 2 ) ⇐⇒ q 1 -p 0 ≤ 0, (134)
α ≤ 1 ⇐⇒ (1q 2 ) • (q 1p 0 ) (1p 0 ) • (q 1q 2 ) ≤ 1 ⇐⇒ q 2p 0 ≥ 0, (135)

0 ≤ β ⇐⇒ 0 ≤ q 1 • (p 0 -q 2 ) p 0 • (q 1 -q 2 ) ⇐⇒ p 0 -q 2 ≤ 0, (136)
β ≤ 1 ⇐⇒ q 1 • (p 0 -q 2 ) p 0 • (q 1 -q 2 ) ≤ 1 ⇐⇒ p 0 -q 1 ≥ 0. (137)
(138)

This proves the equivalence:

             q 1 ≤ q 2 α ∈ [0, 1] β ∈ [0, 1] ⇐⇒ q 1 ≤ p 0 ≤ q 2 . ( 139
)
Assume that q 1 ≥ q 2 , then we have:

0 ≤ α ⇐⇒ 0 ≤ (1 -q 2 ) • (q 1 -p 0 ) (1 -p 0 ) • (q 1 -q 2 ) ⇐⇒ q 1 -p 0 ≥ 0, (140)
α ≤ 1 ⇐⇒ (1q 2 ) • (q 1p 0 ) (1p 0 ) • (q 1q 2 ) ≤ 1 ⇐⇒ q 2p 0 ≤ 0, (141) 0 ≤ β ⇐⇒ 0 ≤ q 1 • (p 0q 2 ) p 0 • (q 1q 2 ) ⇐⇒ p 0q 2 ≥ 0, (142)

β ≤ 1 ⇐⇒ q 1 • (p 0 -q 2 ) p 0 • (q 1 -q 2 ) ≤ 1 ⇐⇒ p 0 -q 1 ≤ 0. (143)
(144)

This proves the equivalence:

             q 1 ≥ q 2 α ∈ [0, 1] β ∈ [0, 1] ⇐⇒ q 2 ≤ p 0 ≤ q 1 . (145)
Hence there exists probability parameters α ∈ [0, 1] and β ∈ [0, 1] if and only if q 1 ≤ p 0 ≤ q 2 or q 2 ≤ p 0 ≤ q 1 .

Fig. 1 .

 1 Fig. 1. The information source is i.i.d. P(u, z) and the channel T (y|x) is memoryless. The encoder E and the decoder D are endowed with distinct utility functions φe(u, v) ∈ R and φ d (u, v) ∈ R.

  ) is |U |. Caratheodory's Lemma (see [60, Corollary 17.1.5, pp. 157] and [49, Corollary A.4, pp. 39]) provides the cardinality bound: |W| = |U | + 1.

Fig. 3 .

 3 Fig. 3. Utility function of the encoder φe(u, v).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Utility function of the decoder φ d (u, v).

4 Fig. 6 .

 46 Fig.6. Posterior beliefs (p1, p2) depending on q1 ∈ [0, p0] and posterior beliefs (p3, p4) depending on q2 ∈ [p0, 1], for p0 = 0.5, δ1 = 0.7, δ2 = 0.9 and γ = 0.6.

Fig. 9 .

 9 Fig. 9. Regions of posterior beliefs (q1, q2) satisfying the information constraints: I(U ; W |Z) ≤ C in dark blue, and I(U ; W ) ≤

Fig. 11 .

 11 Fig. 11. Optimal encoder's utility depending on the belief p(u2) ∈ [0, 1] after observing (W, Z), for parameters p0 = 0.5, δ1 = 0.7, δ2 = 0.9 and γ = 0.6. The curve represents the binary entropy H b (•), as a function of p(u2) ∈ [0, 1].

ΦLemma 4

 4 e = sup w λ w • Ψ e (p w ) s.t. w λ w • p w = P(u) ∈ ∆(U ), and w λ w • h(p w ) > H(U |Z)max P(x) I(X; Y ), and ∀(z, w) ∈ Z × W, V ⋆ (z, Q(u|z, w)) is a singleton , (77) If max P(x) I(X; Y ) > 0, then Φ e = Φ ⋆ e .

  for a large fraction of stages i ∈ {1, . . . , n}. July 13, 2018 DRAFT The cornerstone of this achievability proof is Proposition B.1, which refines the analysis of Wyner-Ziv's source coding, by characterizing its posterior beliefs. Proposition B.1 (Wyner-Ziv's Posterior Beliefs) If the probability distribution P(u, z) × Q(w|u) satx) I(X; Y ) -I(U ; W |Z) > 0,

•

  Random codebook. We defines the indices m ∈ M with |M| = 2 nR and l ∈ M L with |M L | = 2 nR L . We draw |M×M L | = 2 n(R+R L ) sequences W n (m, l) with the i.i.d. probability distribution Q ⊗n (w), and |M| = 2 nR sequences X n (m), with the i.i.d. probability distribution P ⊗n (x) that maximizes the channel capacity in (88). Encoding function. The encoder observes the sequence of symbols of source U n ∈ U n and finds a pair of indices (m, l) ∈ M × M L such that the sequences U n , W n (m, l) ∈ A δ are jointly typical. It sends the sequence X n (m) corresponding to the index m ∈ M. • Decoding function. The decoder observes the sequence of channel output Y n ∈ Y n . It returns an index m ∈ M such that the sequences Y n , X n ( m) ∈ A δ are jointly typical. Then it observes the sequence of state information Z n ∈ Z n and returns an index l ∈ M L such that the sequences Z n , W n ( m, l) ∈ A δ are jointly typical.

  t. P σ (B c α,γ,δ ) ≤ ε.(111)This concludes the proof of Proposition B.1.July 13, 2018 DRAFTWe now prove that the distribution P(u, z, w) defined in (112), satisfies the information constraint of theset Q 0 . 0 ≤I(X n ; Y n ) -I(U n , Z n ; Y n ) i |X i ) -I(U n ; Y n |Z n ) i ; Y n |Z n , U i-1 ) i ; Y n , Z -i , U i-1 |Z i ) i ; Y n , Z -i |Z i ) (124) =n • max P(x) I(X; Y )n • I(U T ; Y n , Z -T |Z T , T ) (125) =n • max P(x) I(X; Y )n • I(U T ; Y n , Z -T , T |Z T ) (126) =n • max P(x) I(X; Y )n • I(U ; W |Z)(127)=n • max P(x) I(X; Y ) -I(U ; W ) + I(Z; W ) .

Q 2 (Q 2 (

 22 P(u,z,w)) u,z,w P(u, z, w)v τ (v|w, z) • φ e (u, z, v) Q(u,z,w)) E Q(u,z,w) ×Q(v|z,w) φ e (U, Z, V ) = Φ ⋆ e .(132)The last inequality comes from the probability distribution P(u, z, w) that satisfies the information constraint of the set Q 0 . The first cardinality bound |W| = |U | + 1 comes from [49, Lemma 6.1]. The second cardinality bound for |W| = |V| |Z| comes from [49, Lemma 6.3], by considering the encoder tells to the decoder which action v ∈ V to play in each state z ∈ Z. This conclude the proof of (20) in Theorem III.3. APPENDIX D PROOF OF LEMMA 2 By inverting the system of equations, we have the following equivalence: p0)•(q1-q2) β = q1•(p0-q2) p0•(q1-q2) .

  stands for the Markov chain property

	corresponding to P(y|x, u) = P(y|x), for all (u, x, y). The encoder and the decoder and denoted by E
	and D.
	Definition II.1 (Encoding and Decoding Strategies)
	July 13, 2018	DRAFT

  EXAMPLE WITH BINARY SOURCE AND STATEWe consider a binary source U ∈ {u 1 , u 2 } with probability P(u 2 ) = p 0 ∈ [0, 1]. The binary state information Z ∈ {z 1 , z 2 } is drawn according to the conditional probability distribution P(z|u) with parameter δ 1 ∈ [0, 1] and δ 2 ∈ [0, 1], as depicted in Fig.2. For the clarity of the presentation, we consider a binary auxiliary random variable W ∈ {w 1 , w 2 }, even if this choice might be sub-optimal since: |W| = min |U | + 1, |V| |Z| = 3. The solutions provided below implicitly refer to this special case of |W| = 2. The random variable W is drawn according to the conditional probability distribution Q(w|u) with parameters α ∈ [0, 1] and β ∈ [0, 1]. The joint probability distribution P(u, z) × Q(w|u) is represented by Fig. 2. The utility functions of the encoder φ e (u, v) and decoder φ d (u, v) are given by

The proof follows directly from [49, Proposition A.2, pp. 37]. Equation (34) corresponds to a bi-variate concavification where the information constraint requires an additional dimension. The proof follows directly from [49, Lemma A.1, pp. 36]. July 13, 2018 DRAFT IV.

July 13, 2018 DRAFT

* Maël Le Treust gratefully acknowledges the support of the Labex MME-DII (ANR11-LBX-0023-01) and the DIM-RFSI of the Région Île-de-France, under grant EX032965. † Tristan Tomala gratefully acknowledges the support of the HEC foundation and ANR/Investissements d'Avenir under grant ANR-11-IDEX-0003/Labex Ecodec/ANR-11-LABX-0047. Conference, Monticello, Illinois, Sept. 2016 [1]Institute Henri Poincaré (IHP) in Paris,

) Fig. 7. Optimal encoder's utility depending on the belief parameter q ∈ [0, 1] after observing W , for parameters p0 = 0.5, δ1 = 0.7, δ2 = 0.9 and γ = 0.6.

B. Concavification with Information constraint

In this section, we assume that the channel capacity is equal to: C = max P(x) I(X; Y ) = 0.1, so the information constraint of Q 0 is active. The average entropy stated in [START_REF] Cuff | Coordination using implicit communication[END_REF] reformulates as a function of q ∈ [0, 1]:

where H b (•) denotes the binary entropy. The dark blue region in Fig. 9 represents the set of posterior distributions (q 1 , q 2 ) with q 1 ≤ p 0 ≤ q 2 or q 2 ≤ p 0 ≤ q 1 , that satisfy the information constraint:

The green region in Fig. 9 corresponds to the information constraint I(U ; W ) ≤ max P(x) I(X; Y ), i.e.

when the decoder does not observe the state information Z. The case without decoder's state information July 13, 2018 DRAFT APPENDIX A PROOF OF THEOREM III.7

We identify the weight parameters λ w = Q(w) and p w = Q(u|w) ∈ ∆(U ) and ( 26) becomes:

and

= sup

Equation ( 63) comes from the identification of the weight parameters λ w = Q(w) and

Equation (64) comes from the definitions of Ψ e (p w ) and h(p w ) in ( 24) and ( 25) and from:

, due to the Markov chain property Z --U --W and the fixed distribution of the source P(u, z).

Equations ( 65) -(67) are reformulations.

APPENDIX B ACHIEVABILITY PROOF OF THEOREM III.3

This proof is built on Wyner-Ziv's source coding [START_REF] Wyner | The rate-distortion function for source coding with side information at the decoder[END_REF] and the achievability proof stated in [49, App. Control of the posterior beliefs. We assume that the event E δ = 0 is realized and we investigate the posterior beliefs P σ (u i |y n , z n , E δ = 0) induced by Wyner-Ziv's encoding strategy σ.

Eq. ( 94)-( 95) come from the hypothesis E δ = 0 of typical sequences (u n , z n , w n , y n ) ∈ A δ and the definition of the conditional K-L divergence [54, pp. 24].

Eq. (96) comes from property of typical sequences [58, pp. 26] and the conditioning that reduces entropy.

Eq. ( 97) comes from the Markov chain Z n --U n --W n --Y n induced by the strategy σ, that implies

Eq. ( 98) is a reformulation of (97).

Eq. (99) comes from the i.i.d. source and Fano's inequality.

Eq. (100) comes from the cardinality of codebook given by (86). This argument is also used in [61, Eq. ( 23)].

Eq. ( 101) comes from the cardinality of A δ (z n |w n ), see also [58, pp. 27].

Eq. ( 102) comes from Fano's inequality H(Z n |U n , W n ), and the Markov chain

We consider an encoding strategy σ of length n ∈ N. We denote by T the uniform random variable {1, . . . , n} and the notation Z -T stands for (Z 1 , . . . , Z t-1 , Z t+1 , . . . Z n ), where Z T has been removed.

We introduce the auxiliary random variable W = (Y n , Z -T , T ) whose joint probability distribution P(u, z, w) with (U, Z) is defined by:

This identification ensures that the Markov chain W --U T --Z T is satisfied. Let us fix a decoding strategy τ (v n |y n , z n ) and define τ (v|w, z) = τ (v|y n , z -i , i, z) = τ i (v i |y n , z n ) where τ i denotes the i-th coordinate of τ (v n |y n , z n ). The encoder's long-run utility writes:

Eq. ( 113) -( 115) are reformulations and re-orderings.

Eq. ( 116) comes from replacing the random variables (Y n , Z -T , T ) by W whose distribution is defined in (112).

Equations ( 113


A circular economy classification of mine waste

A circular economy classification of mine waste

Circular economy is a framework for an array of critical changes in the global economy trends, aimed at reducing its pressure on the Earth's resources, on the environment and on the climate. A continuous decrease is observed in minable grades for most commodities, implying that waste rates increase in ore beneficiation. The move from underground to open pit extraction increases also massively the amount of waste rock. Waste is one of the biggest costs and burdens of mining. A systematic increase of waste recycling, reducing together the need in primary resources and the waste generation, is critical for mining future. Waste reprocessing aims at better recovery of primary commodities, recovery of overlooked or critical metals, and extracting reusable mineral fractions. Recovery and reuse do not always pay for themselves but reduce the cost of site remediation. Mine waste has a higher sustainability potential when compared with mineral extraction, especially in energy and land use. It however often contains residual metals and undesirable or potentially toxic elements. Reuse options for residual mineral fractions include civil engineering, cement and building materials. Waste classification by hazardousness or by ore grade is not suitable for their development. A new classification is proposed for reusable mineral fractions, based on physical and chemical properties and on suitability for the main potential applications.


est destinée au dépôt

Application of pXRF to environmental investigations and geochemical baseline of abandoned mines

Bruno Lemiere

To cite this version:

Bruno Lemiere. Application of pXRF to environmental investigations and geochemical baseline of abandoned mines. Resources for Future Generations -RFG2018, Jun 2018, Vancouver, Canada. hal-

Application of pXRF to environmental investigations and geochemical baseline of abandoned mines

We developed a method based on field measurements by portable XRF to distinguish anthropogenic anomalies resulting from mining activities, and other anthropogenic anomalies, from geogenic anomalies. It is derived from the traditional exploration strategy, based on soil and stream sediment. Instead of starting from a large and weak anomaly and focusing towards a target orebody, we start from a known mine and we observe its outwards decreasing anomaly, which includes the natural geochemical anomaly and the anthropogenic impacts. This method was applied to determine the environmental legacy of abandoned metal mines in France.

Most of the differences in geochemical baselines are due to the dissimilarities in basic geology. Metals and metalloids in soils are derived from the soil parent material (lithogenic source) and from various anthropogenic sources. Transport mechanisms comprise waste erosion, airborne particles, soil creep, solifluction and surface water transport. Flood plain deposition of contaminated stream sediment and mine waste was observed and identified as one of the most extensive anomalies of anthropogenic origin. Natural and anthropogenic anomalies can be distinguished on the basis of elemental signatures, speciation and grain size patterns. Soil anomalies need to be related with background element ranges determined for each lithofacies. Large extension anomalies are usually geogenic or related with large mines. Determination of geochemical signatures of mineralisation and of mining waste allows recognition of these signatures in soil. Anthropogenic elemental anomalies are observed in stream sediments much further away than in soils, except in flood plains.

Environmental geochemical baselines taking into account lithofacies are needed to assess the present state of the surface environment and distinguish the "natural part" from the "anthropogenic part" (mine, smelter…), to follow the extent and impact of a contamination, and to define remediation objectives.


Jérémie Melleton -Brgm

Bruno Lemière -Brgm

Virginie Derycke -Brgm

Anne-Sophie Serrand -Brgm

Eric Fournier -Brgm

Eric Gloaguen -Brgm

Frédéric Lacquement -Brgm

Pascal Auger -Brgm

Maarit Middleton -Gtk

Vesa Nykänen -Gtk

395 -Exploration geochemistry: comparison between classic trace elements geochemistry, soil partial leaches, portable XRF, on soils and biogeochemistry in Western Europe Environment. Example from Li-Ta-Sn and W deposits

In Europe, exploration geochemistry is mainly restricted to classical soils geochemical analyses based on total or near total digestion of samples, despite the good results obtained in other areas with approaches related to selective and partial leaches, in particular to reveal anomalies linked to deeply buried deposits. Similarly, examples of biogeochemistry usage in continental Europe remain anecdotic. However, with the actual detection levels in analytical methods and the knowledge about metals accumulation reached in a large spectrum of plants, biogeochemistry has already been successfully tested in various environments and over distinct deposit types.

Within the UpDeep project (European Institute of Innovation and Technology -Raw Materials) framework which aims to develop the geochemical expertise on deep buried exploration in Europe, a comparison of some of these surface geochemical methods were performed in the vicinity of the Beauvoir rare-metals (Li-Ta-Sn-Be) granite and La Bosse stockwork (W), located in the Variscan French Massif Central.

One of the challenging questions of exploration geochemistry in Western Europe is the long-term anthropogenic contamination related to past mining activities but also agricultural and forestry activities, which can strongly affect the primary geochemical anomalies especially in soil horizons and plants. In the studied area, mining activities extended from Antiquity to the middle of the last century, and forestry has been conducted intensively.

Within the scope of testing different well documented approaches, around 160 samples of soils were collected, on which has been performed separately the following treatment: deionized water leach, sodium pyrophosphate leach, and Aqua Regia digestion on Ah horizon; and hot hydroxylamine leach, Ionic LeachTM (ALS) and Aqua Regia digestion on B horizon. Meanwhile, soil samples have also been analyzed by pXRF.

Moreover 84 samples of fern, 80 samples of bramble and 64 samples of Douglas fir were sampled in order to compare their respective geochemical signal. In this Western European context, selection of a homogenously distributed vegetation media represents a complex issue.