Point-to-Point Strategic Communication

(1)

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Point-to-Point Strategic Communication

Mael Le Treust, Tristan Tomala

To cite this version:

Mael Le Treust, Tristan Tomala. Point-to-Point Strategic Communication. IEEE ITW, Oct 2020,

Riva del Garda (virtual), Italy. �hal-03219028�

(2)

Point-to-Point Strategic Communication

Maël Le Treust

ETIS UMR 8051, CY Cergy Paris Université, ENSEA, CNRS, 6, avenue du Ponceau,

95014 Cergy-Pontoise CEDEX, France Email: mael.le-treust@ensea.fr

Tristan Tomala

HEC Paris, GREGHEC UMR 2959 1 rue de la Libération, 78351 Jouy-en-Josas CEDEX, France

Email: tomala@hec.fr

Abstract

—We investigate a strategic formulation of the joint source-channel coding problem in which the encoder and the decoder are endowed with distinct distortion functions. We provide the solutions in four different scenarios. First, we assume that the encoder and the decoder cooperate in order to achieve a certain pair of distortion values. Second, we suppose that the encoder commits to a strategy whereas the decoder implements a best response, as in the persuasion game where the encoder is the Stackelberg leader. Third, we consider that the decoder commits to a strategy, as in the mismatched rate-distortion problem or as in the mechanism design framework. Fourth, we study the cheap talk game in which the encoding and the decoding strategies form a Nash equilibrium.

I. I

NTRODUCTION

Strategic communication takes place when an informed sender communicates with a receiver that takes an action, given that the sender and the receiver optimize different metrics. This question was originally formulated in the Game Theory literature were the messages are costless and the communication is unrestricted. Crawford and Sobel [1] investigate the Nash equilibrium of the cheap talk game, whereas Kamenica and Gentzkow [2] introduce the Bayesian persuasion game in which the sender commits to an information disclosure policy, as the leader of the Stackelberg game. In a previous work [3], we characterize the solution of the Bayesian persuasion game when the communication channel is noisy.

U

ⁿ

X

ⁿ

Y

ⁿ

V

ⁿ

P

U

σ T

Y|X

τ

d

^e

(u, v) d

d

(u, v)

Fig. 1. The source is i.i.d. and the channel is memoryless. The encoder and the decoder have mismatched distortion functionsde(u, v)6=dd(u, v).

The strategic communication problem has attracted attention in Computer Science [4], in Control Theory [5], in Information Theory [6], [7], [8], [9] and is related to the lossy source coding with mismatch distortion functions [10], [11]. Recently, Vora and Kulkarni investigate a strategic communication problem in which the receiver is the Stackelberg leader that must recover the source sequence [12]. The authors introduce the notion of the “information extraction capacity” and formulate an elegant solution in terms of the zero error capacity of “the sender graph” [13].

In this paper, we compare four different solutions for the point-to-point strategic communication problem and we characterize the limit set of ε-Nash equilibrium distortions.

II. S

YSTEM

M

ODEL

We denote by U , X , Y , V, the finite sets of information source, channel inputs, channel outputs and decoder’s outputs.

Uppercase letters U

ⁿ

= (U

1

, . . . , U

n

) ∈ U

ⁿ

and X

ⁿ

, Y

ⁿ

, V

ⁿ

stand for n-length sequences of random variables with n ∈ N

^⋆

= N\ {0}, whereas lowercase letters u

ⁿ

= (u

1

, . . . , u

n

) ∈ U

ⁿ

and x

ⁿ

, y

ⁿ

, v

ⁿ

, stand for sequences of realizations. We denote by ∆(X) the set of probability distributions Q

X

over X , i.e. the probability simplex. The support of a distribution Q

X

is denoted by supp Q

X

= {x ∈ X, Q(x) > 0}.

We consider an i.i.d. information source and a memoryless channel distributed according to P

U

∈ ∆(U ) and T

_Y|X

: X →

∆(Y), as depicted in Fig. 1.

Definition 1 We define the encoding strategy σ : U

ⁿ

−→

∆(X

ⁿ

) and the decoding strategy τ : Y

ⁿ

−→ ∆(V

ⁿ

), and we denote by P

^σ,τ

the distribution defined by

P

^σ,τ

= Y

ⁿ

t=1

P

Ut

σ

_Xⁿ|Uⁿ

Y

ⁿ

t=1

T

_Y_t|Xt

τ

_Vⁿ|Yⁿ

, (1) where σ

_Xⁿ|Uⁿ

, τ

_Vⁿ|Yⁿ

denote the distributions of σ, τ.

Definition 2 The encoder and decoder distortion functions d

e

: U × V −→ R and d

d

: U × V −→ R induce long-run distortion functions d

eⁿ

(σ, τ ) and d

_dⁿ

(σ, τ) defined by

d

dⁿ

(σ, τ ) = X

uⁿ,vⁿ

P

^σ,τ

u

ⁿ

, v

ⁿ

· "

1 n

X

n

t=1

d

(u

t

, v

t

)

# . (2) III. C

OOPERATIVE

S

CENARIO

Definition 3 The pair (D

e

, D

d

) is achievable if

∀ε > 0, ∃¯ n ∈ N

^⋆

, ∀n ≥ n, ¯ ∃(σ, τ ), (3)

|D

e

− d

eⁿ

(σ, τ )| + |D

d

− d

dⁿ

(σ, τ )| ≤ ε (4) We denote by C the set of achievable pairs (D

e

, D

d

).

We define the set of distributions e

Q = n

P

U

Q

_V|U

s.t. max

PX

I(X; Y ) − I(U ; V ) ≥ 0 o . (5) Theorem 1 (Cooperative scenario)

C = n

E

_Q

[d

^e

(U, V )], E

_Q

[d

d

(U, V )]

Q ∈ Q e o

. (6)

The proof of Theorem 1 follows from Shannon’s separation

result [15, Theorem 3.7] with two distortion functions.

(3)

IV. P

ERSUASION

G

AME

: E

NCODER

C

OMMITMENT

In this section, the encoder chooses first a strategy σ and the decoder selects a best response strategy τ accordingly. This corresponds to the Bayesian persuasion game [2], where the encoder is the Stackelberg leader.

Definition 4 Given n ∈ N

^⋆

, we define

1. the set of decoder best responses to strategy σ by

BR

_d

(σ) =argmin

τ

d

dⁿ

(σ, τ ), (7) 2. the long-run encoder distortion value by

D

eⁿ

= inf

σ

max

τ∈BR_d(σ)

d

eⁿ

(σ, τ ). (8) In case BR

_d

(σ) is not a singleton, we assume that the decoder selects the worst strategy for the encoder distortion max

τ∈BR_d(σ)

d

eⁿ

(σ, τ ), so that the solution is robust to the exact specification of the decoding strategy.

We aim at characterizing the asymptotic behavior of D

eⁿ

. Definition 5 We consider an auxiliary random variable W ∈ W with |W| = min |U| + 1, |V|

and we define

Q = n

P

U

Q

_W|U

, max

PX

I(X ; Y ) − I(U ; W ) ≥ 0 o . (9) Given Q

U W

, we define the single-letter decoder best responses

A

d

Q

U W

=argmin

Q_V|W

E

QU W QV|W

h d

d

(U, V ) i

. (10)

The encoder optimal distortion D

e^⋆

is given by D

e^⋆

= inf

QU W∈Q

max

QV|W∈ Ad(QU W)

E

QU W QV|W

h

d

^e

(U, V ) i

. (11)

Theorem 2 (Encoder commitment, Theorem 3.1 in [3])

∀n ∈ N

^⋆

, D

ⁿe

≥ D

e^⋆

, (12)

∀ε > 0, ∃¯ n ∈ N

^⋆

, ∀n ≥ n, ¯ D

ⁿe

≤ D

e^⋆

+ ε. (13) Theorem 2 is a particular case of [9, Theorem III.3] when no side information is available at the decoder. Note that the sequence (D

ⁿe

)

n∈N^⋆

is sub-additive. Indeed, when σ is the concatenation of several encoding strategies, the concatenation of the corresponding optimal decoding strategies still belongs to BR

_d

(σ). Theorem 2 and Fekete’s lemma show that

D

^⋆e

= lim

n→+∞

D

ⁿe

= inf

n∈N^⋆

D

eⁿ

. (14) Remark 1 The decoder long-run distortion d

dⁿ

(σ, τ ) obtained with σ asymptotically optimal for (8) and τ ∈ BR

_d

(σ), converges to E

QU W

QV|W

d

(U, V )

, where Q

_V|W

∈ A

d

Q

U W

and Q

U W

is a limit of a minimizing sequence of (11).

V. M

ECHANISM

D

ESIGN

: D

ECODER

C

OMMITMENT

In this section, it is the decoder that chooses first a strategy τ and then the encoder selects a strategy σ accordingly.

This corresponds to the mismatched rate-distortion problem in Information Theory [10], [11] and to the mechanism design problem [14] in Game Theory, where the decoder is the Stackelberg leader.

Definition 6 Given n ∈ N

^⋆

, we define

1. the set of encoder best responses to strategy τ by BR

_e

(τ ) =argmin

σ

d

eⁿ

(σ, τ ), (15) 2. the long-run decoder distortion value by

D

ⁿd

= inf

τ

max

σ∈BR_e(τ)

d

dⁿ

(σ, τ ). (16) The value D

ⁿd

corresponds to the best distortion the decoder can obtain for fixed n ∈ N

^⋆

. In case there are several best responses, we assume the encoder selects the worst strategy σ for the decoder distortion.

We aim at characterizing the asymptotic behaviour of D

dⁿ

Definition 7 Given an auxiliary random variable W ∈ W with |W| = min |U|+1, |V|

with distribution P

_W

, we define P(P

W

) = n

Q

U W

∈ ∆(U × W), Q

U

= P

U

, Q

_W

= P

_W

, max

PX

I(X; Y ) − I(U; W ) ≥ 0 o

. (17) Given P

_{W V}

, we define the single-letter encoder best responses

A

e

(P

W V

) = argmin

QU W∈P(PW)

E

QU W PV|W

h d

e

(U, V ) i

. (18)

The decoder optimal distortion D

^⋆d

is given by D

d^⋆

= inf

PW V Q

max

U W∈ Ae(PW V)

E

QU W PV|W

h d

d

(U, V ) i

. (19)

In both (11) and (19), it is the Stackelberg leader that selects the marginal distribution P

W

, whereas the incentive constraints affect the Stackelberg follower. In both settings, the encoder selects the distribution Q

U W

∈ P(P

W

) that satisfies the information constraint and the decoder selects P

_V|W

. Theorem 3 (Decoder commitment)

∀n ∈ N

^⋆

, D

ⁿd

≥ D

^⋆d

, (20)

∀ε > 0, ∃¯ n ∈ N

^⋆

, ∀n ≥ ¯ n, D

ⁿd

≤ D

^⋆d

+ ε. (21) The achievability proof of Theorem 3 is provided in App.

B, and relies on similar arguments as in [10, Step 1] and [11, Lemma 4.3]. The converse proof is based on the identification of the auxiliary random variables U = U

T

, W = (Y

ⁿ

, T ), V = V

T

where T ∈ {1, . . . , n} is uniformly distributed.

The sequence (D

dⁿ

)

n∈N^⋆

is sub-additive, thus Theorem 3 and Fekete’s lemma show that

D

d^⋆

= lim

n→+∞

D

dⁿ

= inf

n∈N^⋆

D

ⁿd

. (22)

(4)

Remark 2 The encoder long-run distortion d

eⁿ

(σ, τ ) obtained with τ asymptotically optimal for (16) and σ ∈ BR

_e

(τ), converges to E

QU W

PV|W

d

e

(U, V )

, where Q

U W

∈ A

e

P

W V

and P

W V

is a limit of a minimizing sequence of (19).

VI. C

HEAP

T

ALK

G

AME

: N

O

C

OMMITMENT

Definition 8 Given ε ≥ 0 and n ∈ N

^⋆

, an ε-Nash equilibrium is a pair of strategies (σ, τ) such that

σ ∈ BR

_e^ε

(τ) and τ ∈ BR

_d^ε

(σ) where, (23) BR

^ε

e

(τ) = n

σ, d

eⁿ

(σ, τ ) ≤ min

˜

σ

d

eⁿ

(˜ σ, τ ) + ε o

, (24) BR

^ε_d

(σ) = n

τ, d

dⁿ

(σ, τ ) ≤ min

˜

τ

d

dⁿ

(σ, τ) + ˜ ε o

. (25) We denote by NE

ⁿ

ε

the set of distortion pairs (D

^εe

, D

^εd

) for which there exists a ε-Nash equilibrium (σ, τ ) such that

D

^εe

= d

eⁿ

(σ, τ ) and D

d^ε

= d

dⁿ

(σ, τ ). (26) We denote by NE

ⁿ

the set NE

ⁿ

ε

with ε = 0.

Definition 9 For ε ≥ 0, we define the set of distributions that are ε-best responses for both encoder and decoder.

D

^ε

= n

Q

U W V

= P

U

Q

_W|U

Q

_V|W

s.t.

Q

U W

∈ A

^ε_e

(Q

W V

), Q

_V_|W

∈ A

^ε_d

(Q

U W

) o , (27) A

^ε_e

(Q

_{W V}

) = n

Q

_{U W}

∈ P (Q

_W

) s.t. E

QU W QV|W

h d

e

(U, V ) i

≤ min

QeU W

∈P(QW)

E

QeU W QV|W

h

d

^e

(U, V ) i + ε o

, (28)

A

^ε_d

(Q

_{U W}

) = n

Q

_V_|W

s.t. E

QU W QV|W

h

d

(U, V ) i

≤ min

Pe_V_|W

E

QU W e PV|W

h d

d

(U, V ) i + ε o

. (29)

Then, we define N

^ε

= n

E

Q

[d

e

(U, V )], E

Q

[d

d

(U, V )]

, Q ∈ D

^ε

o . (30) We denote by N (resp. D) the set N

^ε

(resp. D

^ε

) with ε = 0.

Theorem 4 (Nash equilibrium distortions)

∀ε ≥ 0, ∀n ∈ N, NE

ⁿ

ε

⊂ N

^ε

, (31)

ε→0

lim lim

n→+∞

NE

ⁿ

ε

= N . (32)

Theorem 4 is a consequence of the arguments in the achievability proofs of Theorems 2 and 3, which show that Shannon’s encoding and decoding schemes based a distribution P

U

Q

W|U

Q

V|W

∈ D

^ε

, form an ε-Nash equilibrium.

Conjecture 1

n→+∞

lim lim

ε→0

NE

ⁿ

ε

= N . (33)

Proposition 1 We compare the solutions (6), (11), (19), (30).

min

Q_V|U∈eQ

E

d

e

(U, V )

≤ D

e^⋆

≤ inf

QU W V∈D

E

d

e

(U, V ) , (34) min

Q_V_|U∈eQ

E

d

(U, V )

≤ D

d^⋆

≤ inf

QU W V∈D

E

d

(U, V ) . (35)

A

PPENDIX

A P

RELIMINARY

R

ESULTS

Definition 10 Given P

U W

∈ ∆(U × W), tolerance δ > 0, let B

δ

(P

U W

) = n

Q

U W

s.t. ||Q

U W

− P

U W

||

1

≤ δ o

. (36) We define the set of typical sequences by

T

δ

(P

U W

) = n

(u

ⁿ

, w

ⁿ

) s.t. Q

ⁿ_{U W}

∈ B

δ

(P

U W

) o

, (37) where Q

ⁿ_{U W}

denotes the empirical distribution of (u

ⁿ

, w

ⁿ

).

Definition 11 We consider two distributions P

U

∈ ∆(U ), P

W

∈ ∆(W), a rate parameter R ≥ 0 and a tolerance δ ≥ 0.

We define the sets

Q

⁻_δ

( R ) = n

Q

_{U W}

∈ ∆(U × W ) s.t. ||Q

_U

− P

_U

||

1

≤ δ,

||Q

W

− P

W

||

1

≤ δ and I(U ; W ) ≤ R o

, (38) Q

⁺_δ

( R ) = n

Q

_{U W}

∈ ∆(U × W ) s.t. ||Q

_U

− P

_U

||

1

≤ δ,

||Q

W

− P

W

||

1

≤ δ and I(U ; W ) ≥ R o

. (39) We use the notation Q

⁻₀

( R ) and Q

⁺₀

( R ) when δ = 0.

Lemma 1 (see Step 1 in [10] and Lemma 4.3 in [11]) We consider two distributions P

U

∈ ∆(U ) and P

W

∈ ∆(W), a rate R ≥ 0, a small parameter η > 0 and n ∈ N

^⋆

.

• We generate a sequence U

ⁿ

according to P

_U^⊗n

.

• Independently, we generate a family of sequences W

ⁿ

(m)

m∈{1,...,2^nR}

according to P

_W^⊗n

.

There exists δ, for all ¯ δ < δ ¯ and for all ε > 0, there exists n, ¯ for all n ≥ ¯ n,

Pr

∃m ∈ {1, . . . , 2

ⁿ^R

}, Q

ⁿ_m

∈ Q

⁺_δ

( R + η)

≤ ε,

where Q

ⁿ_m

denotes the empirical distribution of (U

ⁿ

, W

ⁿ

(m)).

The provide the proof of Lemma 1 in App. C.

Lemma 2 (Covering lemma, see Lemma 3.3 in [15]) We consider a distribution P

U W

∈ ∆(U × W), a rate parameter R = I(U ; W ) + η with η > 0, n ∈ N.

• We generate a sequence U

ⁿ

according to P

_U^⊗n

.

• Independently, we generate a family of sequences W

ⁿ

(m)

m∈{1,...,2ⁿ^R}

according to P

_W^⊗n

.

There exists ¯ δ > 0, for all δ < δ ¯ and for all ε > 0, there exists

¯

n, such that for all n ≥ n, ¯ Pr

∃m ∈ {1, . . . , 2

ⁿ^R

}, ||Q

ⁿ_m

− P

U W

||

1

≤ δ

≥ 1 − ε.

Definition 12 For P

_U

∈ ∆(U ), P

_{W V}

∈ ∆(W × V ), δ > 0, R ≥ 0, and D ≥ 0 we define

Q

_δ

( R , D ) = n

Q

U W

∈ ∆(U × W ) s.t. ||Q

U

− P

U

||

1

≤ δ,

||Q

W

− P

W

||

1

≤ δ, I(U ; W ) ≤ R , E

QU W PV|W

h d

e

(U, V ) i

≤ D o

.

(40)

(5)

We have Q

_δ

( R , D ) = Q

⁻_δ

( R ) ∩ Q

^◦_δ

( D ) with Q

^◦_δ

( D ) = n

Q

U W

∈ ∆(U × W) s.t. ||Q

U

− P

U

||

1

≤ δ,

||Q

W

− P

W

||

1

≤ δ and E

QU W PV|W

h d

e

(U, V ) i

≤ D o . (41) A

PPENDIX

B

A

CHIEVABILITY

P

ROOF OF

T

HEOREM

3 If the channel capacity is equal to zero, then a trivial coding scheme satisfies (21). From now on, we assume that the channel capacity is strictly positive. Therefore, for all ε

0

> 0 there exists η

0

> 0 and a distribution P

W V

such that

D

d^⋆

− max

QU W∈A^ηe⁰(PW V)

E

QU W PV|W

h d

d

(U, V ) i ≤ ε

0

, (42)

where

A

^ηe⁰

(P

W V

) = argmin

QU W∈P^η⁰(PW)

E

QU W PV|W

h d

e

(U, V ) i

, (43)

P

^η⁰

(P

W

) = n

Q

U W

∈ ∆(U × W) s.t. Q

U

= P

U

, Q

W

= P

W

and max

PX

I(X; Y ) − I(U; W ) ≥ 2η

0

o . (44) We use the notation Q

U W

to refer to the distribution that achieves the maximum in (42), and without loss of generality, we assume that I(U ; W ) = max

PX

I(X ; Y ) − 2η

0

. We introduce the rate parameter R = I(U ; W ) + η

0

and the tolerance of the typical sequences δ > 0. We consider that the decoder implements Shannon’s channel decoding and lossy source decoding, see [15, Sec. 3.1 and 3.6], that we denote by τ

^⋆

. We denote by M and m the indexes selected by the encoder, whereas M ˆ and m ˆ refer to the indexes selected by the decoder.

• The random codebooks (W

ⁿ

(m), X

ⁿ

(m))

_m∈{1,...,2^nR_}

are drawn independently according to P

_W^⊗n

and P

_X^⊗n

, where P

_X

maximizes the channel capacity.

• The decoder observes the sequence of channel output Y

ⁿ

∈ Y

ⁿ

and returns the unique index m ˆ such that the sequences Y

ⁿ

, X

ⁿ

( ˆ m)

∈ T

δ

(P

X

T

_Y_|X

) are jointly typical. Otherwise it returns the index 1.

• Then the decoder returns the sequence W

ⁿ

( ˆ m) corresponding to m ˆ and draws V

ⁿ

i.i.d. according to P

_V|W

. Standard channel coding arguments ensures that

∃ δ ¯

1

, ∀δ < δ ¯

1

, ∀ε

1

, ∃¯ n

1

, ∀n ≥ n ¯

1

, Pr( ˆ M 6= M ) ≤ ε

1

. (45) Since the encoder is strategic, it selects a best response σ ∈ BR

_e

(τ

^⋆

) that, for a given u

ⁿ

, returns x

ⁿ

in order to minimize

X

yn ,vn ˆ m

T

^⊗n

(y

ⁿ

|x

ⁿ

)Pr( ˆ m|y

ⁿ

)P

^⊗n

(v

ⁿ

|w

ⁿ

( ˆ m)) 1 n

X

n

t=1

d

^e

(u

t

, v

t

)

= X

mˆ

Pr( ˆ m|x

ⁿ

) · X

u,w

Q

ⁿ_m_ˆ

(u, w) X

v

P(v|w)d

e

(u, v), (46) where Q

ⁿ_m_ˆ

∈ ∆(U × W) denotes the empirical distribution of (u

ⁿ

, w

ⁿ

( ˆ m)). We denote by x

^n⋆

the sequence that minimizes (46) and we denote by

Q

^xⁿ

= X

mˆ

Pr( ˆ m|x

ⁿ

) · Q

ⁿ_m_ˆ

∈ ∆(U × W), (47)

the average empirical distribution induced by the input sequence x

ⁿ

. By Lemma 1, for all η

2

> 0, there exists δ ¯

2

, for all δ < δ ¯

2

and for all ε

2

> 0, there exists ¯ n

2

, for all n ≥ n ¯

2

,

Pr

Q

^X^n⋆

∈ / Q

⁻_δ

( R + η

2

)

≤ Pr

Q

^X^n⋆

∈ Q

⁺_δ

( R + η

2

)

(48) + Pr

||Q

^X_U^n⋆

− P

U

||

1

+ ||Q

^X_W^n⋆

− P

W

||

1

> δ

(49)

≤Pr

∃x

ⁿ

∈ X

ⁿ

, Q

^xⁿ

∈ Q

⁺_δ

( R + η

2

)

+ ε

2

(50)

≤Pr

∃m ∈ {1, . . . , 2

ⁿ^R

}, Q

ⁿ_m

∈ Q

⁺_δ

( R + η

2

)

+ ε

2

(51)

≤2ε

2

. (52)

On the other hand, we assume that the encoder implements Shannon’s coding scheme σ

c

, by selecting the unique m such that (U

ⁿ

, W

ⁿ

(m)) ∈ T

δ

(Q

U W

), and 1 otherwise. By Lemma 2, there exists δ ¯

3

> 0, for all δ < δ ¯

3

and for all ε

3

> 0, there exists n ¯

3

, such that for all n ≥ ¯ n

3

,

Pr

∀m ∈ {1, . . . , 2

ⁿ^R

}, ||Q

ⁿ_m

− Q

U W

||

1

> δ

≤ ε

3

. (53) The bounds given in (45), (53) imply

1 − ε

1

− ε

3

≤ Pr

Q

^Xⁿ^(m)

∈ Q

^◦_δ

( D + µ)

(54)

≤ Pr

Q

^X^n⋆

∈ Q

^◦_δ

( D + µ)

, (55)

with D = min

QU W∈P^η⁰(PW)

E

d

e

(U, V )

and µ = δd

e

where d

e

= max

u,v

d

e

(u, v). Thus for all δ ≤ min(¯ δ

1

, δ ¯

2

, δ ¯

3

) and n ≥ max(¯ n

1

, n ¯

2

, n ¯

3

) we have

Pr

Q

^X^n⋆

∈ Q

_δ

( R + η

2

, D + µ)

(56)

≥1 − Pr

Q

^X^n⋆

∈ / Q

⁻_δ

( R + η

2

)

− Pr

Q

^X^n⋆

∈ / Q

^◦_δ

( D + µ)

≥ 1 − ε

1

− 2ε

2

− ε

3

. (57) This shows the existence of a strategy τ

^⋆

with codebook (w

ⁿ

(m), x

ⁿ

(m))

_m∈{1,...,2^nR_}

such that (57) is satisfied. We consider σ ∈ BR

_e

(τ

^⋆

) that achieves the maximum in (16) and we denote d

d

= max

u,v

d

(u, v). The correspondance (δ, R , D ) 7→ Q

_δ

( R , D ) is continuous, from Berge’s Maximum Theorem we have

d

ⁿd

(σ, τ

^⋆

) = E

_QX^n⋆

PV|W

h

d

^d

(U, V ) i

≤ sup

PU W∈ Qδ(R+η2,D+µ)

E

PU W PV|W

h

d

^d

(U, V ) i

+ (ε

1

+ 2ε

2

+ ε

3

)d

^d

≤ sup

PU W∈ Q(R−η0,D)

E

PU W PV|W

h d

d

(U, V ) i

+ (ε

1

+ 2ε

2

+ ε

3

+ ε

4

)d

d

= max

PU W∈ Aη0

e (PW V)

E

PU W PV|W

h d

d

(U, V ) i

+ (ε

1

+ 2ε

2

+ ε

3

+ ε

4

)d

d

≤D

d^⋆

+ ε

0

+ (ε

1

+ 2ε

2

+ ε

3

+ ε

4

)d

d

.

We take ε

0

, ε

1

, ε

2

, ε

3

, ε

4

, δ, η

2

, η

0

small and n ∈ N

^⋆

large

and the achievability result of Theorem 3 follows.

(6)

A

PPENDIX

C P

ROOF OF

L

EMMA

1 Lemma 3 below ensures for all δ > 0, there exists a family of distributions (Q

^k_{U W}

)

k∈K

⊂ int ∆(U × W) with |K| < +∞

such that

∆(U × W) ⊂ [

k∈K

T

δ

(Q

^k_{U W}

), (58) min

k∈K

min

(u,w)∈U ×W

Q

^k

(u, w) ≥ δ

4(|U × W| − 1) . (59) Thus for all δ > 0, there exists a family of distributions (Q

^k_{U W}^˜

)

˜k∈Ke

⊂ Q

⁺_δ

( R + η) ∩ int ∆(U × W) with | K| e < +∞

such that (59) is satisfied and Q

⁺_δ

( R + η) ⊂ [

˜k∈Ke

T

δ

(Q

^k_{U W}^˜

). (60) We choose δ < δ ¯ such that 3¯ δ log

4(|U ×W|−1)

¯δ

< η.

Pr

∃m ∈ {1, . . . , 2

ⁿ^R

} s.t. Q

ⁿ_m

∈ Q

⁺_δ

( R + η)

(61)

≤Pr

∃m s.t. Q

ⁿ_m

∈ [

˜k∈Ke

T

δ

(Q

^˜^k_{U W}

)

(62)

=Pr

∃ k ˜ ∈ K, e ∃m s.t. Q

ⁿ_m

∈ T

δ

(Q

^k_{U W}^˜

)

(63)

≤ X

k∈˜ Ke

X

m∈{1,...,2^nR}

X

(un ,wn)∈

Tδ(Q˜k U W)

P

_U^⊗n

(u

ⁿ

)P

_W^⊗n

(w

ⁿ

) (64)

≤| K| · e 2

^n(R−I(U;W^)+3δ^log4(|U×W |−1)

δ )

(65)

≤| K| · e 2

^{−n(η−3δ}^log4(|U×W |−1)

δ )

. (66)

Equation (62) comes from (60). Equation (65) comes from (59) with min

u,w

Q

^k^˜

(u, w) ≥

4(|U ×W|−1)^δ

, and Proposition 2 and 3 below. Equation (66) comes from Q

^˜^k_{U W}

∈ Q

⁺_δ

( R + η), that induce R ≤ I(U ; W ) − η.

Since | K| e < +∞ and η − 3δ log

4(|U ×W|−1)

δ

> 0, we choose n large such that | K| · e 2

^{−n(η−3δ}^log4(|U×W |−1)

δ )

≤ ε.

This concludes the proof of Lemma 1.

Proposition 2 (see 1. pp. 27 in [15]) We consider P

_U

∈

∆(U), n ∈ N, δ > 0. For all u

ⁿ

∈ T

δ

(P

_U

) we have

2

^−n(H(U)+δ¹⁾

≤ P

_U^⊗n

(u

ⁿ

) ≤ 2

^{−n(H(U)−δ}¹⁾

, (67) with δ

1

= log

_min¹

u∈suppPU

P(u)

· δ.

Proposition 3 (see 2. pp. 27 in [15]) We consider P

U W

∈

∆(U × W), n ∈ N, δ > 0. Then T

_δ

(P

U W

) ≤ 2

^n(H(U,W)+δ²⁾

with δ

2

= log

_min ¹

(u,w)∈suppPU W

P(u,w)

· δ.

Lemma 3 We consider a set U such that 2 ≤ |U| < +∞. For all δ > 0, there exists a family of distributions (Q

^k_U

)

k∈K

⊂ int ∆(U ) with |K| < +∞ such that

∆(U ) ⊂ [

k∈K

T

δ

(Q

^k_U

), min

k∈K

min

u∈U

Q

^k

(u) ≥ δ 4(|U| − 1) .

Proof. [Lemma 3] We consider a symbols u ˜ ∈ U and we define the distributions

P

U

=

( 1 if U = ˜ u,

0 otherwise, Q

^u_U^˜

=

( 1 −

^δ₄

if U = ˜ u,

δ

4(|U |−1)

otherwise.

Then,

||Q

^u_U^˜

− P

U

||

1

= X

u

|Q

^u^˜

(u) − P(u)|

= δ

4 + δ

4(|U| − 1) (|U| − 1) = δ

2 < δ. (68) This shows that P

U

∈ T

δ

(Q

^u_U^˜

). The same construction applies to any other symbol u ˆ ∈ U , and this generates a collection of distributions (Q

^u_U^ˆ

)

u∈Uˆ

. We construct a family of distributions (Q

^k_U

)

k∈K

⊂ int ∆(U ) based on the lattice with steps

_{4(|U |−1)}^δ

that connects the elements of (Q

^u_U^ˆ

)

u∈Uˆ

. Since

∆(U ) ⊂ [0, 1]

^{|U |−1}

, we have

|K| ≤ 4(|U| − 1) δ

|U |−1

< +∞. (69)

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