Strategic Communication via Cascade Multiple-Description Network Rony Bou Rouphael and Maël Le Treust

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Strategic Communication via Cascade

Multiple-Description Network

Rony Bou Rouphael and Maël Le Treust

ETIS UMR 8051, CY Cergy-Paris Université, ENSEA, CNRS,

6, avenue du Ponceau, 95014 Cergy-Pontoise CEDEX, FRANCE

Email: {rony.bou-rouphael ; mael.le-treust}@ensea.fr

Abstract—In decentralized decision-making problems, agents

choose their actions based on locally available information and

knowledge about decision rules or strategies of other agents.

We consider a three-node cascade network with an encoder,

a relay and a decoder, having distinct objectives captured by

cost functions. In such a cascade network, agents choose their

respective strategies sequentially, as a response to the former

agent’s strategy and in a way to inﬂuence the decision of the

latter agent in the network. We assume the encoder commits to a

strategy before the communication takes place. Upon revelation of

the encoding strategy, the relay commits to a strategy and reveals

it. The communication starts, the source sequence is drawn and

processed by the encoder and relay. Then, the decoder observes

a sequences of symbols, updates its Bayesian posterior beliefs

accordingly, and takes the optimal action. This is an extension of

the Bayesian persuasion problem in the Game Theory literature.

In this work, we provide an information-theoretic approach to

study the fundamental limit of the strategic communication via

three-node cascade network. Our goal is to characterize the

optimal strategies of the encoder, the relay and the decoder, and

study the asymptotic behavior of the encoder’s minimal long-run

cost function.

I. INTRODUCTION

We study a decentralized decision-making problem with re-

stricted communication between three agents with non-aligned

objectives. As depicted in Fig. 1, we consider a Cascade

channel where information travels from a strategic encoder

to a decoder through a strategic relay. We are interested in

designing an achievable multiple description coding scheme

that minimizes the encoder’s long run cost function subject to

the challenges imposed by the Cascade channel.

The problem of strategic communication originally emerged

in the game theory literature to address situations in economics

(lobbying, advertising, sales, negotiations, etc.). The game was

referred to as the sender-receiver game, and communication

was assumed to be perfect and unconstrained by any limits on

the amount of information transmitted. The Nash equilibrium

solution of the cheap talk game was investigated by Crawford

and Sobel in their seminal paper [1], in which the encoder

and the decoder are endowed with distinct objectives and

choose their coding strategies simultaneously. The Stackelberg

Maël Le Treust gratefully acknowledges ﬁnancial support from INS2I

CNRS, DIM-RFSI, SRV ENSEA, UFR-ST UCP, INEX Paris Seine Initiative

and IEA Cergy-Pontoise. This research has been conducted as part of the

project Labex MME-DII (ANR11-LBX-0023-01).

PUσ µ τ

c2(U, V )c3(U, V )c1(U, V )

UnM1M2

R1R2

Fig. 1: Strategic Source Coding for Cascade Channel with

Successive Commitment.

version of the strategic communication game, referred to as

the Bayesian persuasion game, was formulated by Kamenica

and Gentzkow in [2], where the encoder is the Stackelberg

leader and the decoder is the Stackelberg follower choosing

its strategies as a response to the encoder’s strategy . In this

paper, we assume that the encoder commits to an encoding and

announces its commitment before observing the source. Then,

the relay commits to and announces a strategy accordingly.

If the relay was assumed to commit to a strategy before

the encoder, then the problem boils down to a strategic joint

source-channel coding of Shannon, like the one investigated in

[3]. Cascade source coding consists of compressing a source

sequence through an intermediate or relay node which then

reconstructs the source and transmits it to the next node. In [4],

Yamamoto considered the source coding problem for cascade

and branching communication systems, and established the

region of achievable rates for cascade systems and bounds

for the branching systems. Lossy source coding for cascade

communication systems was also considered in [5] where both

the relay and the terminal node have access to side information

and wish to reconstruct the source with certain ﬁdelities.

In Bayesian persuasion, the encoder is considered to be an

information designer. In [6], information design with multiple

designers interacting with a set of agents is studied. In [7],

[8], the Nash equilibrium solution is investigated for multi-

dimensional sources and quadratic cost functions, whereas

the Stackelberg solution is studied in [9]. The computational

aspects of the persuasion game are considered in [10]. In

several recent contributions [11], [12], [13], Vora and Kulkarni

addressed the problem of extracting truthful information from

a strategic sender with an incentive to misreport information.

Modeled as a Stackelberg game in which the decoder is the

Stackelberg leader, authors investigate the region of achievable

rates of strategic communication between agents with distinct

arXiv:2210.00533v1 [cs.IT] 2 Oct 2022

utility functions. The strategic communication problem with

a noisy channel is investigated in [14], [15], [16], [3], and

[17]. The case where the decoder privately observes a signal

correlated to the state, also referred to as the Wyner-Ziv setting

[18], is studied in [19], [20] and [21].

In this paper, we study the Bayesian persuasion game via

a Cascade multiple description network. The objectives of the

players are captures by distinct cost functions that depend on

the source and the action taken by the decoder. For some

particular cases, we are able to characterize the encoder’s op-

timal cost obtained with strategic cascade multiple description

coding that satisﬁes the decoder’s incentives constraints.

A. Notations

Let n∈N?=N\{0}denote a sequence block length. We

denote by Unthe n-sequences of random variables of source

information un= (u1, ..., un)∈ Un, and by Vnthe sequences

of decoders’ actions vn∈ Vn. Sequences M1and M2denote

the channel inputs of encoders 1and 2respectively. Calli-

graphic fonts U, and Vdenote the alphabets and lowercase

letters uand vdenote the realizations. For a discrete random

variable X, we denote by ∆(X)the probability simplex, i.e.

the set of probability distributions over X,and by PX(x)the

probability mass function P{X=x}. Notation X−−Y−−Z

stands for the Markov chain property PZ|XY =PZ|Y.

II. SYSTEM MODEL

In this section, we formulate the coding problem. Let

n∈N?, and (R1, R2)∈R2

+denote the rate pair. We

assume that the information source Ufollows the indepen-

dent and identically distributed (i.i.d) probability distribution

PU∈∆(U).

Deﬁnition 1. The coding strategies σ,µand τof the encoder,

relay, and decoder respectively are deﬁned by

σ:Un−→ ∆{1, ..2bnR1c},(1)

µ:{1, ..2bnR1c} −→ ∆{1, ..2bnR2c},(2)

τ:{1, ..2bnR2c} −→ ∆Vn.(3)

The stochastic coding strategies (σ, µ, τ)induce a joint

probability distribution Pσµτ ∈∆Un× {1,2, ..2bnR1c} ×

{1,2, ..2bnR2c}×Vndeﬁned for all (un, m1, m2, vn)by

Pσµτ (un, m1, m2, vn) =

n

t=1

PU(ut)σ(m1|un)µ(m2|m1)τ(vn|m2).(4)

Deﬁnition 2. We consider arbitrary single-letter cost functions

c1:U × V −→ Rfor the encoder E,c2:U × V −→ Rfor

the relay, and c3:U × V −→ Rfor the decoder. The long-run

cost functions are deﬁned by

1(σ, µ, τ) =Eσ,µ,τ "1

t=1

c1(Ut, Vt)#

un,vn

Pσ,µ,τ

UnVn(un, vn)·"1

t=1

c1(ut, vt)#,

2(σ, µ, τ) =Eσ,µ,τ "1

t=1

c2(Ut, Vt)#

un,vn

Pσ,µ,τ

UnVn(un, vn)·"1

t=1

c2(ut, vt)#,

3(σ, µ, τ) =Eσ,µ,τ "1

t=1

c3(Ut, Vt)#

un,vn

Pσ,µ,τ

UnVn(un, vn)·"1

t=1

c3(ut, vt)#,

In the above equations, Pσµτ

UnVndenote the marginal distri-

butions over the sequences (Un, V n)of Pσµτ deﬁned in (4)

over the n-sequences (Un, M1, M2, V n).

Deﬁnition 3. For any strategy pair (σ, µ), the set of best-

response strategies τof the decoder is deﬁned by

A3(σ, µ) = arg min

3(σ, µ, τ).(5)

For any strategy σ, the set of best-response strategies µof the

relay is deﬁned by

A2(σ) = arg min

(µ,τ )s.t.

τ∈A3(σ,µ)

2(σ, µ, τ).(6)

Therefore, the encoder has to solve the following coding

problem,

Γn

e(R1, R2) = inf

σmax

(µ,τ )∈

A2(σ)

1(σ, µ, τ).(7)

Remark 1. In order to get a robust solution concept, we

assume that the encoder solves the problem for the worst case

scenario, i.e. if more than one pair of strategies are available

in A2(σ), we consider the one that maximizes the encoder’s

cost.

The operational signiﬁcance of (7) corresponds to the per-

suasion game that is played in the following steps:

•The encoder chooses, announces the encoding σ.

•knowing σ, the relay chooses, announces the encoding µ.

•Knowing (σ, µ), the decoder compute its best-response

strategy τ.

•Sequences Unare drawn i.i.d with distribution PU.

•Message sequence M1are encoded according to σM1|Un.

•Message sequence M2are encoded according to µM2|M1.

•The decoder observes M2and draws Vnaccording to

τVn|M2.

•Cost functions cn

1(σ, µ, τ),cn

2(σ, µ, τ),cn

3(σ, µ, τ)are

computed.

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StrategicCommunicationviaCascadeMultiple-DescriptionNetworkRonyBouRouphaelandMaëlLeTreustETISUMR8051,CYCergy-ParisUniversité,ENSEA,CNRS,6,avenueduPonceau,95014Cergy-PontoiseCEDEX,FRANCEEmail:{rony.bou-rouphael;mael.le-treust}@ensea.frAbstractIndecentralizeddecision-makingproblems,agentschoosetheira...

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