# Strategic Successive Refinement Coding for Bayesian Persuasion with Two
Decoders

Rony Bou Rouphael and Maël Le Treust

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).  ETIS UMR 8051, CY Cergy-Paris Université, ENSEA,
CNRS,
6, avenue du Ponceau, 95014 Cergy-Pontoise CEDEX, FRANCE
Email: {rony.bou-rouphael<EMAIL_ADDRESS>
###### Abstract

We study the multi-user Bayesian persuasion game between one encoder and two
decoders, where the first decoder is better informed than the second decoder.
We consider two perfect links, one to the first decoder only, and the other to
both decoders. We consider that the encoder and both decoders are endowed with
distinct and arbitrary distortion functions. We investigate the strategic
source coding problem in which the encoder commits to an encoding while the
decoders select the sequences of symbols that minimize their long-run
respective distortion functions. We characterize the optimal encoder
distortion value by considering successive refinement coding with respect to a
specific probability distribution which involves two auxiliary random
variables, and captures the incentive constraints of both decoders.

##  I Introduction

The optimization of distinct and arbitrary distortion functions resulting from
the communication between several autonomous devices with non-aligned
objectives is under study. This problem was originally formulated in the game
theory literature and referred to as the sender-receiver game, where the
amount of information transmitted is generally unrestricted. In the seminal
paper [crawford1982], Crawford and Sobel investigate the Nash equilibrium
solution of the cheap talk game in which the encoder and the decoder have
distinct objectives and choose their coding strategies simultaneously. In
[KamenicaGentzkow11], Kamenica and Gentzkow formulate the Bayesian persuasion
game in which the encoder is the Stackelberg leader and the decoder is the
Stackelberg follower. More recently, Koessler et al. in
[KoesslerLaclauTomala21] investigate games of information design where
multiple encoders influence the behavior of several decoders. As a motivating
example, one could think of a company trying to convince investors into
putting money on a certain number of projects, or a job seeker trying to
persuade recruiters to be hired.

This problem is an attractive multi-disciplinary subject of study. The Nash
equilibrium solution is investigated for multi-dimensional sources and
quadratic distortion functions in [sar1],
[SaritasFurrerGeziciLinderYukselISIT2019], whereas the Stackelberg solution is
studied in [sar2]. The computational aspects of the persuasion game are
considered in [dughmi]. The strategic communication problem with a noisy
channel is investigated in [AkyolLangbortBasar15], [akyol2017information],
[LeTreustTomala(Allerton)16], [jet], and four different scenarios of strategic
communication are studied in [pointtopoint]. The case where the decoder
privately observes a signal correlated to the state, also referred to as the
Wyner-Ziv setting [wyner-it-1976], is studied in [akyol2016role],
[corsica2020] and [LeTreustTomalaISIT21]. Vora and Kulkarni investigate the
achievable rates for the strategic communication problem in [voraachievable],
[voraextraction] where the decoder is the Stackelberg leader.

In this paper, we investigate a Bayesian persuasion game with two decoders and
restricted communication. We consider an i.i.d. source of information and we
suppose that the observation of the first decoder contains the observation of
the second decoder, as in Fig. 1. More specifically, we assume that the
encoder $\mathcal{E}$ selects and announces beforehand the compression scheme
to be implemented. Upon receipt of the indices, the decoders $\mathcal{D}_{1}$
and $\mathcal{D}_{2}$ update their Bayesian beliefs over the source sequence
and select the action sequences that minimizes their respective distortion
functions. We characterize the optimal encoder distortion value obtained via
the successive refinement coding with respect to the distribution that
involves two auxiliary random variables, and that satisfies both decoders
incentive constraints.

$\mathcal{E}$$d_{e}^{n}(\sigma,\tau_{1},\tau_{2})$$d_{1}^{n}(\sigma,\tau_{1})$$d_{2}^{n}(\sigma,\tau_{2})$$\mathcal{D}_{1}$$\mathcal{D}_{2}$$V^{n}_{2}$$V^{n}_{1}$$U^{n}$$M_{1}\in\\{1,..2^{\lfloor
nR_{1}\rfloor}\\}$$M_{2}\in\\{1,..2^{\lfloor nR_{2}\rfloor}\\}$ Figure 1:
Strategic successive refinement source coding.

###  I-A Notations

Let $\mathcal{E}$ denote the encoder and $\mathcal{D}_{i}$ denote the decoder
$i\in\\{1,2\\}$. Notations $U^{n}$ and $V^{n}_{i}$ denote the $n$-sequences of
random variables of source information
$u^{n}=(u_{1},...,u_{n})\in\mathcal{U}^{n}$, and decoder $\mathcal{D}_{i}$
actions $v^{n}_{i}\in\mathcal{V}^{n}_{i}$ respectively for $i\in\\{1,2\\}$.
Calligraphic fonts $\mathcal{U}$ and $\mathcal{V}_{i}$ denote the alphabets
and lowercase letters $u$ and $v_{i}$ denote the realizations. For a discrete
random variable $X,$ we denote by $\Delta(\mathcal{X})$ the probability
simplex, i.e. the set of probability distributions over $\mathcal{X},$ and by
$\mathcal{P}_{X}(x)$ the probability mass function $\mathbb{P}\\{X=x\\}$.
Notation
$X-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-Y-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-Z$ stands
for the Markov chain property $\mathcal{P}_{Z|XY}=\mathcal{P}_{Z|Y}$.

##  II System Model

In this section, we aim at formulating the coding problem. We assume that the
information source $U$ follows the independent and identically distributed
(i.i.d) probability distribution $\mathcal{P}_{U}\in\Delta(\mathcal{U})$.

######  Definition 1.

Let $R_{1},R_{2}\in\mathbb{R}_{+}^{2}=[0,+\infty[^{2}$, where $[0,+\infty[$
denotes the set of non-negative real numbers, and
$n\in\mathbb{N}^{\star}=\mathbb{N}\backslash\\{0\\}$. The encoding $\sigma$
and decoding $\tau_{i}$ strategies of the encoder $\mathcal{E}$ and decoders
$\mathcal{D}_{i}$, $i\in\\{1,2\\}$ are defined by

$\displaystyle\sigma:$ $\displaystyle
U^{n}\longrightarrow\Delta(\\{1,2,..2^{\lfloor
nR_{1}\rfloor}\\}\times\\{1,2,..2^{\lfloor nR_{2}\rfloor}\\}),$ (1)
$\displaystyle\tau_{1}:$ $\displaystyle\\{1,2,..2^{\lfloor
nR_{1}\rfloor}\\}\times\\{1,2,..2^{\lfloor
nR_{2}\rfloor}\\}\longrightarrow\Delta(\mathcal{V}_{1}^{n}),$ (2)
$\displaystyle\tau_{2}:$ $\displaystyle\\{1,2,..2^{\lfloor
nR_{2}\rfloor}\\}\longrightarrow\Delta(\mathcal{V}_{2}^{n}),$ (3)

where $\lfloor x\rfloor=\max\\{m\in\mathbb{Z},\ m\leq x\\}$ for
$x\in\mathbb{R}$. We denote by $\mathcal{S}(n,R_{1},R_{2})$ the set of coding
triplets ($\sigma,\tau_{1},\tau_{2})$.

The stochastic coding strategies
($\sigma,\tau_{1},\tau_{2})\in\mathcal{S}(n,R_{1},R_{2})$ induce a joint
probability distribution
$\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}\in\Delta(U^{n}\times\\{1,2,..2^{\lfloor
nR_{1}\rfloor}\\}\times\\{1,2,..2^{\lfloor nR_{2}\rfloor}\\}\times
V^{n}_{1}\times V^{n}_{2})$ defined by

$\displaystyle\forall(u^{n},m_{1},m_{2},v_{1}^{n},v_{2}^{n}),\quad\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}(u^{n},m_{1},m_{2},v_{1}^{n},v_{2}^{n})=$
$\displaystyle\bigg{(}\prod_{t=1}^{n}\mathcal{P}_{U}(u_{t})\bigg{)}\sigma(m_{1},m_{2}|u^{n})\tau_{1}(v_{1}^{n}|m_{1},m_{2})\tau_{2}(v_{2}^{n}|m_{2}).$
(4)

######  Definition 2.

We consider arbitrary single-letter distortion functions
$d_{e}:\mathcal{U}\times\mathcal{V}_{1}\times\mathcal{V}_{2}\longrightarrow\mathbb{R}$
for the encoder $\mathcal{E}$,
$d_{1}:\mathcal{U}\times\mathcal{V}_{1}\longrightarrow\mathbb{R}$ for the
decoder $\mathcal{D}_{1}$ and
$d_{2}:\mathcal{U}\times\mathcal{V}_{2}\longrightarrow\mathbb{R}$ for the
decoder $\mathcal{D}_{2}$. The long-run distortion functions are defined by

$\displaystyle
d_{e}^{n}(\sigma,\tau_{1},\tau_{2})=\mathbb{E}_{\sigma,\tau_{1},\tau_{2}}\Bigg{[}\frac{1}{n}\sum_{t=1}^{n}d_{e}(U_{t},V_{1,t},V_{2,t})\Bigg{]}$
$\displaystyle=\sum_{u^{n},v_{1}^{n},v_{2}^{n}}\mathcal{P}_{U^{n}V_{1}^{n}V_{2}^{n}}^{\sigma,\tau_{1},\tau_{2}}(u^{n},v_{1}^{n},v_{2}^{n})\cdot\Bigg{[}\frac{1}{n}\sum_{t=1}^{n}d_{e}(u_{t},v_{1,t},v_{2,t})\Bigg{]},$
$\displaystyle
d^{n}_{1}(\sigma,\tau_{1})=\sum_{u^{n},v_{1}^{n}}\mathcal{P}_{U^{n}V_{1}^{n}}^{\sigma,\tau_{1}}(u^{n},v_{1}^{n})\cdot\Bigg{[}\frac{1}{n}\sum_{t=1}^{n}d_{1}(u_{t},v_{1,t})\Bigg{]},$
$\displaystyle
d^{n}_{2}(\sigma,\tau_{2})=\sum_{u^{n},v_{2}^{n}}\mathcal{P}_{U^{n}V_{2}^{n}}^{\sigma,\tau_{2}}(u^{n},v_{2}^{n})\cdot\Bigg{[}\frac{1}{n}\sum_{t=1}^{n}d_{2}(u_{t},v_{2,t})\Bigg{]}.$

In the above equations,
$\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}_{U^{n}V_{1}^{n}V_{2}^{n}}$,
$\mathcal{P}^{\sigma,\tau_{1}}_{U^{n}V_{1}^{n}}$ and
$\mathcal{P}^{\sigma,\tau_{2}}_{U^{n}V_{2}^{n}}$ denote the marginal
distributions of $\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}$ defined in (4) over
$(U^{n},V_{1}^{n},V_{2}^{n})$, $(U^{n},V_{1}^{n})$, and $(U^{n},V_{2}^{n})$
respectively.

######  Definition 3.

For any encoding strategy $\sigma,$ the set of best-response strategies of
decoder $i\in\\{1,2\\}$ is defined by

$BR_{i}(\sigma)=\Big{\\{}\tau_{i},{d_{i}}^{n}(\sigma,\tau_{i})\leq{d_{i}}^{n}(\sigma,\tilde{\tau_{i}}),\forall\
\tilde{\tau_{i}}\Big{\\}}.$ (5)

If several pairs of best-response strategies $(\tau_{1},\tau_{2})\in
BR_{1}(\sigma)\times BR_{2}(\sigma)$ are available, we assume that the worst
pair $(\tau_{1},\tau_{2})$, from the encoder perspective, is selected.
Therefore, the solution is robust to the exact specification of the decoding
strategies. For $(R_{1},R_{2})\in\mathbb{R}_{+}^{2}$ and
$n\in\mathbb{N}^{\star}$, the coding problem under study is

$D_{e}^{n}(R_{1},R_{2})=\underset{\sigma}{\inf}\underset{\tau_{1}\in
BR_{1}(\sigma),\atop\tau_{2}\in
BR_{2}(\sigma)}{\max}d_{e}^{n}(\sigma,\tau_{1},\tau_{2}).$ (6)

######  Remark 1.

Suppose that the decoders choose, among their best-response strategies, the
pair that also minimizes the encoder distortion. This “optimistic” coding
problem writes

$D_{o}^{n}(R_{1},R_{2})=\underset{\sigma}{\min}\underset{\tau_{1}\in
BR_{1}(\sigma),\atop\tau_{2}\in
BR_{2}(\sigma)}{\min}d_{e}^{n}(\sigma,\tau_{1},\tau_{2}).$ (7)

For generic problems $D_{o}^{n}(R_{1},R_{2})=D_{e}^{n}(R_{1},R_{2})$ [jet, pp.
8].

The operational significance of (6) corresponds to the persuasion game that is
played in the following steps:

  * •

Encoder $\mathcal{E}$ chooses, announces the encoding $\sigma$.

  * •

Sequence $U^{n}$ is drawn i.i.d with distribution $\mathcal{P}_{U}$.

  * •

Messages $(M_{1},M_{2})$ are encoded according to
$\mathcal{P}^{\sigma}_{M_{1}M_{2}|U^{n}}$.

  * •

Knowing $\sigma$, decoder $\mathcal{D}_{1}$ observes $(M_{1},M_{2})$ and draws
$V_{1}^{n}$ according to $\tau_{1}\in BR_{1}(\sigma)$, and decoder
$\mathcal{D}_{2}$ observes $M_{2}$ and draws $V_{2}^{n}$ according to
$\tau_{2}\in BR_{2}(\sigma)$.

  * •

Distortion values are $d_{e}^{n}(\sigma,\tau_{1},\tau_{2})$,
$d_{1}^{n}(\sigma,\tau_{1})$, $d_{2}^{n}(\sigma,\tau_{2})$.

######  Lemma 1.

The sequence $\big{(}nD_{e}^{n}(R_{1},R_{2})\big{)}_{n\in\mathbb{N}^{\star}}$
is sub-additive.

The proof is stated in Appendix A.

##  III Main Result

In this section, we characterize the asymptotic behaviour of
$D_{e}^{n}(R_{1},R_{2})$. Our solution combines the decoders incentive
constraints with the information constraints of the successive refinement
source coding.

######  Definition 4.

We consider two auxiliary random variables $W_{1}\in\mathcal{W}_{1}$ and
$W_{2}\in\mathcal{W}_{2}$ with $|\mathcal{W}_{i}|=|\mathcal{V}_{i}|$, for
$i\in\\{1,2\\}$. For $(R_{1},R_{2})\in\mathbb{R}^{2}_{+}$, we define

$\displaystyle\mathbb{Q}_{0}(R_{1},R_{2})=\big{\\{}$
$\displaystyle\mathcal{Q}_{W_{1}W_{2}|U},\quad R_{2}\geq I(U;W_{2}),$
$\displaystyle\quad R_{1}+R_{2}\geq I(U;W_{1},W_{2})\big{\\}}.$ (8)

For every distribution
$\mathcal{Q}_{W_{1}W_{2}|U}\in\Delta(\mathcal{W}_{1}\times\mathcal{W}_{2})^{|\mathcal{U}|}$,
we define

$\displaystyle\mathbb{Q}_{1}(\mathcal{Q}_{W_{1}W_{2}|U})=$
$\displaystyle\operatorname*{arg\,min}_{\mathcal{Q}_{V_{1}|W_{1}W_{2}}}\mathbb{E}_{\mathcal{Q}_{W_{1}W_{2}|U}\atop\mathcal{Q}_{V_{1}|W_{1}W_{2}}}\Big{[}d_{1}(U,V_{1})\Big{]},$
(9) $\displaystyle\mathbb{Q}_{2}(\mathcal{Q}_{W_{2}|U})=$
$\displaystyle\operatorname*{arg\,min}_{\mathcal{Q}_{V_{2}|W_{2}}}\mathbb{E}_{\mathcal{Q}_{W_{2}|U}\atop\mathcal{Q}_{V_{2}|W_{2}}}\Big{[}d_{2}(U,V_{2})\Big{]}.$
(10)

Note that
$\mathbb{Q}_{1}(\mathcal{Q}_{W_{1}W_{2}|U})\in\Delta(\mathcal{V}_{1})^{|\mathcal{W}_{1}\times\mathcal{W}_{2}|}$
and
$\mathbb{Q}_{2}(\mathcal{Q}_{W_{2}|U})\in\Delta(\mathcal{V}_{2})^{|\mathcal{W}_{2}|}$.
The encoder’s optimal distortion is defined by

$\displaystyle D^{\star}_{e}(R_{1},R_{2})$ $\displaystyle=$
$\displaystyle\underset{\mathcal{Q}_{W_{1}W_{2}|U}\atop\in\mathbb{Q}_{0}(R_{1},R_{2})}{\inf}\underset{\mathcal{Q}_{V_{1}|W_{1}W_{2}}\in\mathbb{Q}_{1}(\mathcal{Q}_{W_{1}W_{2}|U})\atop\mathcal{Q}_{V_{2}|W_{2}}\in\mathbb{Q}_{2}(\mathcal{Q}_{W_{2}|U})}{\max}\mathbb{E}\Big{[}d_{e}(U,V_{1},V_{2})\Big{]},$
(11)

where the expectation in (11) is evaluated with respect to
$\mathcal{P}_{U}\mathcal{Q}_{W_{1}W_{2}|U}\mathcal{Q}_{V_{1}|W_{1}W_{2}}\mathcal{Q}_{V_{2}|W_{2}}$.

######  Remark 2.

The random variables $U,W_{1},W_{2},V_{1},V_{2}$ satisfy

$\displaystyle(U,V_{2})-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-(W_{1},W_{2})-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-V_{1},\quad(U,W_{1},V_{1})-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-W_{2}-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-V_{2}.$

Given $\mathcal{Q}_{W_{1}W_{2}|U}$, we denote by
$\mathcal{Q}_{U|W_{1}W_{2}}\in\Delta(\mathcal{U})^{|\mathcal{W}_{1}\times\mathcal{W}_{2}|}$
and $\mathcal{Q}_{U|W_{2}}\in\Delta(\mathcal{U})^{|\mathcal{W}_{2}|}$ the
posterior beliefs of decoders $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$.
Moreover, for $(w_{1},w_{2})\in\mathcal{W}_{1}\times\mathcal{W}_{2}$, we
introduce the notations
$\mathcal{Q}^{w_{1}w_{2}}_{U}=\mathcal{Q}_{U|W_{1}W_{2}}(.|w_{1},w_{2})\in\Delta(\mathcal{U})$
and
$\mathcal{Q}^{w_{2}}_{U}=\mathcal{Q}_{U|W_{2}}(.|w_{2})\in\Delta(\mathcal{U})$.

######  Theorem 1.

Let $(R_{1},R_{2})\in\mathbb{R}_{+}^{2}$, we have

$\displaystyle\forall\varepsilon>0,\exists\hat{n}\in\mathbb{N},\forall
n\geq\hat{n},D^{n}_{e}(R_{1},R_{2})$ $\displaystyle\leq
D_{e}^{\star}(R_{1},R_{2})+\varepsilon,$ $\displaystyle\forall
n\in\mathbb{N},D_{e}^{n}(R_{1},R_{2})$ $\displaystyle\geq
D_{e}^{\star}(R_{1},R_{2}).$

The proof of Theorem 1 is stated in Sec. IV and V. Together with Fekete’s
Lemma for the sub-additive sequence
$\big{(}nD_{e}^{n}(R_{1},R_{2})\big{)}_{n\in\mathbb{N}^{\star}}$ (see Lemma
1), we obtain

$\displaystyle\lim_{n\rightarrow\infty}D^{n}_{e}(R_{1},R_{2})=\underset{n\in\mathbb{N}^{\star}}{\inf}D^{n}_{e}(R_{1},R_{2})=D^{\star}_{e}(R_{1},R_{2}).$
(12)

##  IV Converse Proof of Theorem 1

Let $(R_{1},R_{2})\in\mathbb{R}^{2}_{+}$ and $n\in\mathbb{N}^{\star}$. We
consider $(\sigma,\tau_{1},\tau_{2})\in\mathcal{S}(n,R_{1},R_{2})$ and a
random variable $T$ uniformly distributed over $\\{1,2,...,n\\}$ and
independent of $(U^{n},M_{1},M_{2},V_{1}^{n},V_{2}^{n})$. We introduce the
auxiliary random variables $W_{1}=(M_{1},T)$, $W_{2}=(M_{2},T)$,
$(U,V_{1},V_{2})=(U_{T},V_{1,T},V_{2,T})$, distributed according to
$\mathcal{P}_{UW_{1}W_{2}V_{1}V_{2}}^{\sigma\tau_{1}\tau_{2}}$ defined for all
$(u,w_{1},w_{2},v_{1},v_{2})=(u_{t},m_{1},m_{2},t,v_{1,t},v_{2,t})$ by

$\displaystyle\mathcal{P}_{UW_{1}W_{2}V_{1}V_{2}}^{\sigma\tau_{1}\tau_{2}}(u,w_{1},w_{2},v_{1},v_{2})$
$\displaystyle=$
$\displaystyle\mathcal{P}_{U_{T}M_{1}M_{2}TV_{1T}V_{2T}}^{\sigma\tau_{1}\tau_{2}}(u_{t},m_{1},m_{2},t,v_{1,t},v_{2,t})$
$\displaystyle=$ $\displaystyle\frac{1}{n}\sum_{u^{t-1}\atop
u_{t+1}^{n}}\sum_{v_{1}^{t-1},v_{1,t+1}^{n}\atop
v_{2}^{t-1},v_{2,t+1}^{n}}\\!\\!\\!\\!\\!\\!\bigg{(}\prod_{t=1}^{n}\mathcal{P}_{U}(u_{t})\bigg{)}\sigma(m_{1},m_{2}|u^{n})\times\tau_{1}(v_{1}^{n}|m_{1},m_{2})\tau_{2}(v_{2}^{n}|m_{2}).$
(13)

######  Lemma 2.

The distribution
$\mathcal{P}_{UW_{1}W_{2}V_{1}V_{2}}^{\sigma\tau_{1}\tau_{2}}$ has marginal on
$\Delta(\mathcal{U})$ given by $\mathcal{P}_{U}$ and satisfies the Markov
chain properties

$\displaystyle(U,V_{2})-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-(W_{1},W_{2})-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-V_{1},\quad(U,W_{1},V_{1})-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-W_{2}-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-V_{2}.$

Proof. [Lemma 2] The i.i.d. property of the source ensures that the marginal
distribution is $\mathcal{P}_{U}$. By the definition of the decoding functions
$\tau_{1}$ and $\tau_{2}$ we have

$\displaystyle(U_{T},V_{2,T})-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-(M_{1},M_{2},T)-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-V_{1,T},$
$\displaystyle(U_{T},M_{1},V_{1,T})-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-(M_{2},T)-\\!\\!\\!\\!\minuso\\!\\!\\!\\!-V_{2,T}.$

Therefore
$\mathcal{P}_{UW_{1}W_{2}V_{1}V_{2}}^{\sigma\tau_{1}\tau_{2}}=\mathcal{P}_{U}\mathcal{P}_{W_{1}W_{2}|U}^{\sigma}\mathcal{P}_{V_{1}|W_{1}W_{2}}^{\tau_{1}}\mathcal{P}_{V_{2}|W_{2}}^{\tau_{2}}$.

######  Lemma 3.

For all $\sigma$, the distribution
$\mathcal{P}_{W_{1}W_{2}|U}^{\sigma}\in\mathbb{Q}_{0}$.

Proof. [Lemma 3] We consider an encoding strategy $\sigma$, then

$\displaystyle nR_{2}\geq H(M_{2})\geq I(M_{2};U^{n})$ (14)
$\displaystyle=\sum_{t=1}^{n}I(U_{t};M_{2}|U^{t-1})$ (15)
$\displaystyle=nI(U_{T};M_{2}|U^{T-1},T)$ (16)
$\displaystyle=nI(U_{T};M_{2},U^{T-1},T)$ (17) $\displaystyle\geq
nI(U_{T};M_{2},T)$ (18) $\displaystyle=nI(U;W_{2}).$ (19)

In fact, (16) follows from the introduction of the uniform random variable
$T\in\\{1,\ldots,n\\}$, (17) comes from the i.i.d. property of the source, and
(19) follows from the identification of the auxiliary random variables
$(U,W_{2})$ and the independence between $T$ and $U_{T}$. Similarly,

$\displaystyle n(R_{1}+R_{2})\geq H(M_{1},M_{2})\geq I(U^{n};M_{1},M_{2})$
$\displaystyle=$ $\displaystyle\sum_{t=1}^{n}I(U_{t};M_{1},M_{2}|U^{t-1})$
(20) $\displaystyle=$ $\displaystyle nI(U_{T};M_{1},M_{2}|U^{T-1},T)$
$\displaystyle\geq$ $\displaystyle nI(U_{T};M_{1},M_{2},T)$ (21)
$\displaystyle=$ $\displaystyle nI(U;W_{1},W_{2}).$ (22)

######  Lemma 4.

For all $(\sigma,\tau_{1},\tau_{2})$ and $i\in\\{1,2\\}$, we have
$d_{e}^{n}(\sigma,\tau_{1},\tau_{2})=\mathbb{E}\big{[}d_{e}(U,V_{1},V_{2})\big{]}$
and $d_{i}^{n}(\sigma,\tau_{i})=\mathbb{E}\big{[}d_{i}(U,V_{i})\big{]}$
evaluated with respect to
$\mathcal{P}_{U}\mathcal{P}_{W_{1}W_{2}|U}^{\sigma}\mathcal{P}_{V_{1}|W_{1}W_{2}}^{\tau_{1}}\mathcal{P}_{V_{2}|W_{2}}^{\tau_{2}}$.
Moreover, for all $\sigma$, we have

$\displaystyle\mathbb{Q}_{1}(\mathcal{P}^{\sigma}_{W_{1}W_{2}|U})=$
$\displaystyle\Big{\\{}\mathcal{Q}_{V_{1}|W_{1}W_{2}},\;\qquad\exists\tau_{1}\in
BR_{1}(\sigma),\;\mathcal{Q}_{V_{1}|W_{1}W_{2}}=\mathcal{P}^{\tau_{1}}_{V_{1}|W_{1}W_{2}}\Big{\\}},$
(23) $\displaystyle\mathbb{Q}_{2}(\mathcal{P}^{\sigma}_{W_{2}|U})=$
$\displaystyle\Big{\\{}\mathcal{Q}_{V_{2}|W_{2}},\;\qquad\exists\tau_{2}\in
BR_{2}(\sigma),\;\mathcal{Q}_{V_{2}|W_{2}}=\mathcal{P}^{\tau_{2}}_{V_{2}|W_{2}}\Big{\\}}.$
(24)

Proof. [Lemma 4] By Definition 2 and (4), (13), we have

$\displaystyle d_{e}^{n}(\sigma,\tau_{1},\tau_{2})=$
$\displaystyle\sum_{u^{n},m_{1},m_{2},\atop
v_{1}^{n},v_{2}^{n}}\bigg{(}\prod_{t=1}^{n}\mathcal{P}_{U}(u_{t})\bigg{)}\sigma(m_{1},m_{2}|u^{n})\times\tau_{1}(v_{1}^{n}|m_{1},m_{2})\tau_{2}(v_{2}^{n}|m_{2})\Bigg{[}\frac{1}{n}\sum_{t=1}^{n}d_{e}(u_{t},v_{1,t},v_{2,t})\Bigg{]}$
$\displaystyle=$ $\displaystyle\sum_{t=1}^{n}\sum_{u_{t},m_{1},m_{2},\atop
v_{1,t},v_{2,t}}\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}(u_{t},m_{1},m_{2},t,v_{1,t},v_{2,t})\times
d_{e}(u_{t},v_{1,t},v_{2,t})$ $\displaystyle=$
$\displaystyle\mathbb{E}\big{[}d_{e}(U_{T},V_{1,T},V_{2,T})\big{]}=\mathbb{E}\big{[}d_{e}(U,V_{1},V_{2})\big{]}.$
(25)

Now we prove the second part of lemma 4. For any $\sigma$ and any
$\mathcal{Q}_{V_{1}|W_{1}W_{2}}\in\mathbb{Q}_{1}(\mathcal{P}^{\sigma}_{W_{1}W_{2}|U})$,
we define $\tilde{\tau}_{1}$ by

$\displaystyle\tilde{\tau}_{1}(v_{1}^{n}|m_{1},m_{2})=\prod_{s=1}^{n}\mathcal{Q}_{V_{1}|W_{1}W_{2}}(v_{1,s}|m_{1},m_{2},s),$
(26)

$\forall(m_{1},m_{2},v_{1}^{n})$. Then
$\forall(w_{1},w_{2},v_{1})=(m_{1},m_{2},t,v_{1,t})$

$\displaystyle\mathcal{P}^{\tilde{\tau}_{1}}_{V_{1}|W_{1}W_{2}}(v_{1}|w_{1},w_{2})=\mathcal{P}^{\tilde{\tau}_{1}}_{V_{1}|W_{1}W_{2}}(v_{1,t}|m_{1},m_{2},t)$
$\displaystyle=$
$\displaystyle\sum_{v_{1}^{t-1},v_{1,t+1}^{n}}\tilde{\tau}_{1}(v_{1}^{n}|m_{1},m_{2})$
$\displaystyle=$
$\displaystyle\sum_{v_{1}^{t-1},v_{1,t+1}^{n}}\prod_{s=1}^{n}\mathcal{Q}_{V_{1}|W_{1}W_{2}}(v_{1,s}|m_{1},m_{2},s)$
$\displaystyle=$
$\displaystyle\mathcal{Q}_{V_{1}|W_{1}W_{2}}(v_{1,t}|m_{1},m_{2},t)\times\sum_{v_{1}^{t-1},v_{1,t+1}^{n}}\prod_{s\neq
t}\mathcal{Q}_{V_{1}|W_{1}W_{2}}(v_{1,s}|m_{1},m_{2},s)$ $\displaystyle=$
$\displaystyle\mathcal{Q}_{V_{1}|W_{1}W_{2}}(v_{1,t}|m_{1},m_{2},t)=\mathcal{Q}_{V_{1}|W_{1}W_{2}}(v_{1}|w_{1},w_{2}).$
(27)

Moreover assume that $\tilde{\tau}_{1}\notin BR_{1}(\sigma)$, then there
exists $\bar{\tau}_{1}\neq\tilde{\tau}_{1}$ such that

$\displaystyle\mathbb{E}_{\mathcal{P}^{\sigma}_{W_{1}W_{2}|U}\atop\mathcal{P}^{\bar{\tau}_{1}}_{V_{1}|W_{1}W_{2}}}\big{[}d_{1}(U,V_{1})\big{]}=d_{1}^{n}(\sigma,\bar{\tau}_{1})<d_{1}^{n}(\sigma,\tilde{\tau}_{1})$
$\displaystyle=\mathbb{E}_{\mathcal{P}^{\sigma}_{W_{1}W_{2}|U}\atop\mathcal{P}^{\tilde{\tau}_{1}}_{V_{1}|W_{1}W_{2}}}\big{[}d_{1}(U,V_{1})\big{]}=\mathbb{E}_{\mathcal{P}^{\sigma}_{W_{1}W_{2}|U}\atop\mathcal{Q}_{V_{1}|W_{1}W_{2}}}\big{[}d_{1}(U,V_{1})\big{]},$
(28)

which contradicts
$\mathcal{Q}_{V_{1}|W_{1}W_{2}}\in\mathbb{Q}_{1}(\mathcal{P}^{\sigma}_{W_{1}W_{2}|U})$.
Therefore, $\tilde{\tau}_{1}\in BR_{1}(\sigma)$ and thus
$\mathcal{Q}_{V_{1}|W_{1}W_{2}}$ belongs to the right-hand side of (23). For
the other inclusion, we assume that $\mathcal{Q}_{V_{1}|W_{1}W_{2}}$ belongs
to the right-hand side of (23) and does not belong to
$\mathbb{Q}_{1}(\mathcal{P}^{\sigma}_{W_{1}W_{2}|U})$, then we show that it
leads to a contradiction. Similar arguments imply (24).

For any strategy $\sigma$, we have

$\displaystyle\underset{\tau_{1}\in BR_{1}(\sigma),\atop\tau_{2}\in
BR_{2}(\sigma)}{\max}\;d_{e}^{n}(\sigma,\tau_{1},\tau_{2})$ $\displaystyle=$
$\displaystyle\underset{\tau_{1}\in BR_{1}(\sigma),\atop\tau_{2}\in
BR_{2}(\sigma)}{\max}\;\mathbb{E}_{\mathcal{P}^{\sigma}_{W_{1}W_{2}|U}\atop\mathcal{P}^{\tau_{1}}_{V_{1}|W_{1}W_{2}}\mathcal{P}^{\tau_{2}}_{V_{2}|W_{2}}}\Big{[}d_{e}(U,V_{1},V_{2})\Big{]}$
(29) $\displaystyle=$
$\displaystyle\underset{\mathcal{Q}_{V_{1}|W_{1}W_{2}}\in\mathbb{Q}_{1}(\mathcal{P}^{\sigma}_{W_{1}W_{2}|U})\atop\mathcal{Q}_{V_{2}|W_{2}}\in\mathbb{Q}_{2}(\mathcal{P}^{\sigma}_{W_{2}|U})}{\max}\mathbb{E}_{\mathcal{P}^{\sigma}_{W_{1}W_{2}|U}\atop\mathcal{Q}_{V_{1}|W_{1}W_{2}}\
\mathcal{Q}_{V_{2}|W_{2}}}\Big{[}d_{e}(U,V_{1},V_{2})\Big{]}$ (30)
$\displaystyle\geq$
$\displaystyle\underset{\mathcal{Q}_{W_{1}W_{2}|U}\atop\in\mathbb{Q}_{0}(R_{1},R_{2})}{\inf}\underset{\mathcal{Q}_{V_{1}|W_{1}W_{2}}\in\mathbb{Q}_{1}(\mathcal{Q}_{W_{1}W_{2}|U})\atop\mathcal{Q}_{V_{2}|W_{2}}\in\mathbb{Q}_{2}(\mathcal{Q}_{W_{2}|U})}{\max}\mathbb{E}\Big{[}d_{e}(U,V_{1},V_{2})\Big{]}$
(31) $\displaystyle=$ $\displaystyle D^{\star}_{e}(R_{1},R_{2}).$ (32)

Equations (29) and (30) come from Lemma 4, whereas (31) comes from Lemma 3 and
(32) follows from (11). Therefore,
$D^{\star}_{e}(R_{1},R_{2})\leq\underset{\sigma}{\inf}\underset{\tau_{1}\in
BR_{1}(\sigma),\atop\tau_{2}\in
BR_{2}(\sigma)}{\max}d_{e}^{n}(\sigma,\tau_{1},\tau_{2})=D_{e}^{n}(R_{1},R_{2})$,
$\forall n$.

##  V Sketch of Achievability Proof of Theorem 1

###  V-A Alternative Formulation

######  Definition 5.

For any distributions $q_{1}\in\Delta(\mathcal{U})$ and
$q_{2}\in\Delta(\mathcal{U})$, we denote by $V_{1}^{\star}(q_{1})$ and
$V_{2}^{\star}(q_{2})$, the sets of optimal actions of decoders
$\mathcal{D}_{1}$ and $\mathcal{D}_{2}$.

$\displaystyle V_{1}^{\star}(q_{1})=\operatorname*{arg\,min}_{v_{1}\in
V_{1}}\sum_{u}q_{1}(u)d_{1}(u,v_{1}),$ (33) $\displaystyle
V_{2}^{\star}(q_{2})=\operatorname*{arg\,min}_{v_{2}\in
V_{2}}\sum_{u}q_{2}(u)d_{2}(u,v_{2}).$ (34)

######  Definition 6.

Given a strategy $\mathcal{Q}_{W_{1}W_{2}|U}$ and symbols $(w_{1},w_{2})$, we
denote by $\mathcal{Q}_{U}^{w_{1}w_{2}}\in\Delta(\mathcal{U})$ and
$\mathcal{Q}_{U}^{w_{2}}\in\Delta(\mathcal{U})$ the Bayesian posterior beliefs
defined by

$\displaystyle\mathcal{Q}_{U}^{w_{1}w_{2}}(u)=$
$\displaystyle\frac{\mathcal{P}_{U}(u)\mathcal{Q}_{W_{1}W_{2}|U}(w_{1},w_{2}|u)}{\sum_{u^{\prime}}\mathcal{P}_{U}(u^{\prime})\mathcal{Q}_{W_{1}W_{2}|U}(w_{1},w_{2}|u^{\prime})}.$
(35)

Among the set of optimal actions of $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$,
we denote the worst pairs for the encoder distortion by

$\displaystyle\tilde{A}(\mathcal{Q}_{W_{1}W_{2}|U},w_{1},w_{2})=\underset{(v_{1},v_{2})\in
V_{1}^{\star}(\mathcal{Q}_{U}^{w_{1}w_{2}})\times\atop
V_{2}^{\star}(\mathcal{Q}_{U}^{w_{2}})}{\operatorname*{arg\,max}}\Big{\\{}\sum_{u}\mathcal{Q}_{U}^{w_{1},w_{2}}(u)d_{e}(u,v_{1},v_{2})\Big{\\}}.$
(36)

######  Definition 7.

Given $(R_{1},R_{2})\in\mathbb{R}^{2}_{+}$, we define

$\displaystyle\tilde{\mathbb{Q}}_{0}(R_{1},R_{2})=\Big{\\{}\mathcal{Q}_{W_{1}W_{2}|U}\
s.t.\ R_{2}>I(U;W_{2})\ ,$ $\displaystyle
R_{1}+R_{2}>I(U;W_{1},W_{2}),\max_{w_{1},w_{2}}|\tilde{A}(\mathcal{Q}_{W_{1}W_{2}|U},w_{1},w_{2})|=1\Big{\\}}.$

######  Definition 8.

Consider the following problem

$\displaystyle\tilde{D}_{e}(R_{1},R_{2})=\underset{\mathcal{Q}_{W_{1}W_{2}|U}\atop\in\tilde{\mathbb{Q}}_{0}(R_{1},R_{2})}{\inf}\underset{\mathcal{Q}_{V_{1}|W_{1}W_{2}}\in\mathbb{Q}_{1}(\mathcal{Q}_{W_{1}W_{2}|U})\atop\mathcal{Q}_{V_{2}|W_{2}}\in\mathbb{Q}_{2}(\mathcal{Q}_{W_{2}|U})}{\max}\mathbb{E}\Big{[}d_{e}(U,V_{1},V_{2})\Big{]},$

where the expectation is evaluated with respect to
$\mathcal{P}_{U}\mathcal{Q}_{W_{1}W_{2}|U}\mathcal{Q}_{V_{1}|W_{1}W_{2}}\mathcal{Q}_{V_{2}|W_{2}}$.

######  Lemma 5.

For $(R_{1},R_{2})\in\mathbb{R}_{+}^{2}$,
$D_{e}^{\star}(R_{1},R_{2})=\tilde{D}_{e}(R_{1},R_{2}).$

Similarly to the proof of [jet, Lemma A.5], this proof relies on showing that
$\tilde{\mathbb{Q}}_{0}(R_{1},R_{2})$ is dense in
$\mathbb{Q}_{0}(R_{1},R_{2})$. It is provided in the full version of the paper
[rouphael2021strategic, Lemma 6].

###  V-B Achievability Scheme

$\mathcal{E}$$\mathcal{D}_{1}$$\mathcal{D}_{2}$$U^{n}$$M_{1}$$M_{2}$$V_{1}^{n}$$V_{2}^{n}$$\mathcal{P}_{V_{1}|W_{1}W_{2}}$$\mathcal{P}_{V_{2}|W_{2}}$$d_{e}(U,V_{1},V_{2})$$W_{1}^{n},W_{2}^{n}$$W_{2}^{n}$$d_{2}(U,V_{2})$$d_{1}(U,V_{1})$
Figure 2: Achievability of successive refinement source coding.

####  V-B1 $R_{1}>0,R_{2}>0$

Fix a conditional probability distribution $\mathcal{Q}_{W_{1},W_{2}|U}$.
There exists $\eta>0$ such that

$\displaystyle R_{2}=I(U;W_{2})+\eta,\quad R_{1}=I(U;W_{1}|W_{2})+\eta.$ (37)

Codebook generation: Randomly and independently generate $2^{\lfloor
nR_{2}\rfloor}$ sequences $w_{2}^{n}(m_{2})$ for
$m_{2}\in[1:2^{\lfloor{nR_{2}}\rfloor}]$, according to the i.i.d distribution
$\mathcal{P}_{W_{2}^{n}}=\Pi_{t=1}^{n}\mathcal{P}_{W_{2}}(w_{2t})$. For each
$(m_{1},m_{2})\in[1:2^{\lfloor nR_{1}\rfloor}]\times[1:2^{\lfloor
nR_{2}\rfloor}]$ generate a sequence $w_{1}^{n}(m_{1},m_{2})$ randomly and
conditionally independently according to the i.i.d conditional distribution
$\mathcal{P}_{W_{1}^{n}|M_{1}W_{2}^{n}}=\Pi_{t=1}^{n}\mathcal{P}_{W_{1}|M_{1}W_{2}}(w_{1t}|m_{1},w_{2t}(m_{2}))$.

Encoding strategy $\sigma$: Encoder $\mathcal{E}$ observes $u^{n}$ and looks
in the codebook for a pair $(m_{1},m_{2})$ such that
$(u^{n},w_{1}^{n}(m_{1},m_{2}),w^{n}_{2}(m_{2}))\in\mathcal{T}_{\delta}^{n}(\mathcal{P}_{U}\mathcal{P}_{W_{1}W_{2}|U})$,
i.e. the sequences are jointly typical with tolerance parameter $\delta>0$. If
such a jointly typical tuple doesn’t exist, the source encoder sets
$(m_{1},m_{2})$ to $(1,1)$. Then, it sends $m_{2}$ to decoder
$\mathcal{D}_{2}$, and $(m_{1},m_{2})$ to decoder $\mathcal{D}_{1}$.

Here comes the main difference with the successive refinement coding, which is
due to the strategic nature of the problem. Instead of declaring
$w_{1}^{n}(m_{1},m_{2})$ and $w_{2}^{n}(m_{2})$ and selecting $V_{1}^{n}$ and
$V_{2}^{n}$ i.i.d. with respect to
$\mathcal{Q}_{V_{1}|W_{1}W_{2}}\in\mathbb{Q}_{1}(\mathcal{Q}_{W_{1}W_{2}|U})$
and $\mathcal{Q}_{V_{2}|W_{2}}\in\mathbb{Q}_{2}(\mathcal{Q}_{W_{2}|U})$, at
each stage $t\in\\{1,\ldots,n\\}$ the decoders $\mathcal{D}_{1}$ and
$\mathcal{D}_{2}$ compute their Bayesian posterior beliefs
$\mathcal{P}^{\sigma}_{U_{t}|M_{1}M_{2}}(\cdot|m_{1},m_{2})$ and
$\mathcal{P}^{\sigma}_{U_{t}|M_{2}}(\cdot|m_{2})$ and select the actions
$v_{1,t}\in V_{1}^{\star}(\mathcal{P}^{\sigma}_{U_{t}|M_{1}M_{2}})$ and
$v_{2,t}\in V_{2}^{\star}(\mathcal{P}^{\sigma}_{U_{t}|M_{2}})$ that minimize
their own distortion function. If several pairs are available, they select the
worst one for the encoder distortion.

Error Event: Given a tolerance $\delta>0$, the error event is given by
$\mathcal{F}=\\{(U^{n},W^{n}_{2}(m_{2}),W_{1}^{n}(m_{2},m_{1})\notin\mathcal{T}_{\delta}^{n}\\}$.
We have by the union of events bound
$\mathcal{P}(\mathcal{F})\leq\mathcal{P}(\mathcal{F}_{1})+\mathcal{P}(\mathcal{F}_{2}(M_{2})\cap\mathcal{F}_{1}^{c})$,
where
$\mathcal{F}_{1}=\\{(U^{n},W_{2}^{n}(m_{2}))\notin\mathcal{T}_{\delta}^{n}\
\forall m_{2}\\}$,
$\mathcal{F}_{2}(m_{2})=\\{(U^{n},W^{n}_{2}(m_{2}),W_{1}^{n}(m_{2},m_{1}))\notin\mathcal{T}_{\delta}^{n}\
\forall m_{1}\\}$. By [elgamal, Lemma 3.3, pp. 62],
$\mathcal{P}(\mathcal{F}_{1})$ tends to zero as $n\to\infty$ if $R_{2}>\
I(U;W_{2})+\eta.$ By [elgamal, Lemma 3.3, pp. 62],
$\mathcal{P}(\mathcal{F}_{1}^{c}\cap\mathcal{F}_{2}(M_{2}))$ goes to zero if
$R_{1}+R_{2}>\ I(U;W_{1},W_{2})+\eta.$

Since the expected error probability evaluated with respect to the random
codebook is small, we have that for all $\varepsilon_{2}>0$, for all $\eta>0$,
there exists $\bar{\delta}>0$, for all $\delta\leq\bar{\delta}$, there exists
$\bar{n}\in\mathbb{N}$ such that for all $n\geq\bar{n}$, we have

$\displaystyle\mathbb{E}\big{[}\mathcal{P}(\mathcal{F}_{1})\big{]}\leq\varepsilon_{2},\quad$
$\displaystyle\mathbb{E}\big{[}\mathcal{P}(\mathcal{F}_{2}(m_{2}))\big{]}\leq\varepsilon_{2}.$
(38)

####  V-B2 Control of beliefs

We introduce the indicator of error events $E^{1}_{\delta}\in\\{0,1\\}$ for
decoder $\mathcal{D}_{1}$ defined as follows

$\displaystyle E^{1}_{\delta}=$ $\displaystyle\begin{cases}1,&\text{if
$(u^{n},w_{1}^{n},w_{2}^{n})\notin\mathcal{T}_{\delta}^{n}(\mathcal{P}_{U}\mathcal{Q}_{W_{1}W_{2}|U})$}.\\\
0,&\text{otherwise}.\end{cases}$ (39)

We denote the Bayesian posterior beliefs
$\mathcal{P}^{\sigma}_{U_{t}|M_{1}M_{2}}(\cdot|m_{1},m_{2})\in\Delta(\mathcal{U})$
and $\mathcal{P}^{\sigma}_{U_{t}|M_{2}}(\cdot|m_{2})\in\Delta(\mathcal{U})$ by
$\mathcal{P}^{m_{1},m_{2}}_{t}$ and $\mathcal{P}^{m_{2}}_{t}$. We show that on
average, the Bayesian beliefs are close in KL distance to the target beliefs
$\mathcal{Q}_{U|W_{1}W_{2}}$ and $\mathcal{Q}_{U|W_{2}}$ induced by the
single-letter distribution $\mathcal{Q}_{W_{1}W_{2}|U}$. Assuming the
distribution $\mathcal{Q}_{U|W_{1}W_{2}}$ is fully supported, the beliefs of
decoder $\mathcal{D}_{1}$ are controlled as follows

$\displaystyle\mathbb{E}\Big{[}\frac{1}{n}\sum_{t=1}^{n}D(\mathcal{P}^{m_{1},m_{2}}_{t}||\mathcal{Q}_{U|W_{1}W_{2}}(\cdot|W_{1t},W_{2t}))\Big{|}E^{1}_{\delta}=0\Big{]}$
$\displaystyle=$ $\displaystyle\sum_{m_{1},m_{2},\atop
w_{1}^{n},w_{2}^{n}}\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}(m_{1},m_{2},w_{1}^{n},w_{2}^{n}\Big{|}E^{1}_{\delta}=0)$
$\displaystyle\times\frac{1}{n}\sum_{t=1}^{n}\sum_{u}\mathcal{P}_{t}^{m_{1}m_{2}}(u)\log_{2}\frac{\mathcal{P}_{t}^{m_{1}m_{2}}(u)}{\mathcal{Q}_{U|W_{1}W_{2}}(u|w_{1t},w_{2t})}$
$\displaystyle=$ $\displaystyle\sum_{m_{1},m_{2},\atop
w_{1}^{n},w_{2}^{n}}\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}(m_{1},m_{2},w_{1}^{n},w_{2}^{n}\Big{|}E^{1}_{\delta}=0)$
$\displaystyle\times\frac{1}{n}\sum_{t=1}^{n}\sum_{u}\mathcal{P}_{t}^{m_{1}m_{2}}(u)\log_{2}\frac{1}{\mathcal{Q}_{U|W_{1}W_{2}}(u|w_{1t},w_{2t})}$
$\displaystyle-\sum_{m_{1},m_{2},\atop
w_{1}^{n},w_{2}^{n}}\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}(m_{1},m_{2},w_{1}^{n},w_{2}^{n}\Big{|}E^{1}_{\delta}=0)\times$
$\displaystyle\frac{1}{n}\sum_{t=1}^{n}\sum_{u}\mathcal{P}_{t}^{m_{1}m_{2}}(u)\log_{2}\frac{1}{\mathcal{P}_{t}^{m_{1}m_{2}}(u)}$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}I(U^{n};M_{1},M_{2}\Big{|}E^{1}_{\delta}=0)-I(U;W_{1},W_{2})+\delta$
$\displaystyle+\frac{1}{n}+\log_{2}|\mathcal{U}|\cdot\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}(E^{1}_{\delta}=1)$
$\displaystyle\leq$
$\displaystyle\eta+\delta+\frac{1}{n}+\log_{2}|\mathcal{U}|\cdot\mathcal{P}^{\sigma,\tau_{1},\tau_{2}}(E^{1}_{\delta}=1).$
(40)

####  V-B3 Conclusion

By combining the equations (38), (40) with [jet, Lemma A.21, equations
(40)-(46), Lemma A.8 ], we obtain $\forall\varepsilon>0$, $\exists\hat{n}$,
$\forall n\geq\hat{n}$, $D_{e}^{n}(R_{1},R_{2})\leq
D^{\star}_{e}(R_{1},R_{2})+\varepsilon$. More details are provided in Appendix
LABEL:Appendix_B.

###  V-C Special Cases

####  V-C1 $R_{1}=R_{2}=0$

The auxiliary random variables $(W_{1},W_{2})$ are independent of $U$. The
message sets are singletons, and the only possible encoding strategy
$\sigma_{0}$ is given by
$\sigma_{0}:\mathcal{U}^{n}\longrightarrow\\{1\\}\times\\{1\\}$. The codebook
consists of two sequences $W_{2}^{n}(1)$ and $W_{1}^{n}(1,1)$ only. Therefore,
$\forall n\in\mathbb{N}^{\star}$, $D_{e}^{\star}(0,0)=D_{e}^{n}(0,0).$

####  V-C2 $R_{1}>0\And R_{2}=0$

Random variables $W_{2}$ and $U$ are independent for $R_{1}>0$ and $R_{2}=0$,
i.e.
$\mathcal{Q}_{W_{1}W_{2}|U}=\mathcal{Q}_{W_{2}}\mathcal{Q}_{W_{1}|W_{2}U}$.
This means that decoder $\mathcal{D}_{2}$ will repeatedly chose the action
$v_{2,0}\in V^{\star}(\mathcal{P}_{U})$ that corresponds to its prior belief
$\mathcal{P}_{U}$ and maximizes the encoder’s distortion. The persuasion game
is thus reduced to the point-to-point problem with one decoder
$\mathcal{D}_{1}$, as in [jet].

####  V-C3 $R_{1}=0\And R_{2}>0$

The auxiliary random variable $W_{1}$ is independent of $U$. Hence, the
encoder transmits the same index to both decoders. Therefore, both decoders
will have the same posterior belief
$\mathcal{Q}^{w_{2}}_{U}\in\Delta(\mathcal{U})$, $\forall
w_{2}\in\mathcal{W}_{2}$.

In that case, the optimal distortion can be reformulated in terms of a
convexification of its expected distortion as in [jet],
$D^{\star}_{e}(0,R_{2})=\underset{(\lambda_{w_{2}},\mathcal{Q}_{U}^{w_{2}})_{w_{2}\in\mathcal{W}_{2}}}{\inf}\sum_{w_{2}\in\mathcal{W}_{2}}\lambda_{w_{2}}\Psi_{e}(\mathcal{Q}_{U}^{w_{2}})$
where $\Psi_{e}(q)=\underset{(v_{1},v_{2})\in\atop V_{1}^{\star}(q)\times
V_{2}^{\star}(q)}{\max}\mathbb{E}_{q}\big{[}d_{e}(U,v_{1},v_{2})\big{]}.$

##  Appendix A Proof of Lemma 1

Proof. [Lemma 1] Let $n,m\in\mathbb{Z}$. We denote by ${\sigma}_{c}^{n+m}$,
the concatenation of the strategies ${\sigma}^{n}$, $\sigma^{m}$ where
${\sigma}^{n}$ is implemented over the first $n$ stages and ${\sigma}^{m}$ is
implemented over the last $m$ stages. For decoder $i\in\\{1,2\\}$, consider
the best responses ${\tau_{i}}^{n}\in BR_{i}({\sigma}^{n})$ and
${\tau_{i}}^{m}\in BR_{i}({\sigma}^{m})$. Then, the concatenation
$\tau_{i,c}^{n+m}$ of ${\tau_{i}}^{n}$ and ${\tau_{i}}^{m}$ is also a best
response $\tau_{i,c}^{n+m}\in BR_{i}({\sigma}_{c}^{n+m})$. Therefore, we have
the inequality

$\displaystyle nD_{e}^{n}(R_{1},R_{2})+mD_{e}^{m}(R_{1},R_{2})$
$\displaystyle=$
$\displaystyle\underset{\sigma^{n}}{\inf}\underset{{\tau_{1}}^{n}\in
BR_{1}(\sigma^{n}),\atop{\tau_{2}}^{n}\in
BR_{2}(\sigma^{n})}{\max}\mathbb{E}\Big{[}\sum_{t=1}^{n}d_{e}(U_{t},V_{1,t},V_{2,t})\Big{]}$
$\displaystyle+\underset{\sigma^{m}}{\inf}\underset{{\tau_{1}}^{m}\in
BR_{1}(\sigma^{m}),\atop{\tau_{2}}_{m}\in
BR_{2}(\sigma^{m})}{\max}\mathbb{E}\Big{[}\sum_{t=1}^{m}d_{e}(U_{t},V_{1,t},V_{2,t})\Big{]}$
(41) $\displaystyle=$
$\displaystyle\underset{\sigma^{n+m}_{c}}{\inf}\underset{\tau_{1}^{n+m}\in
BR_{1}(\sigma_{c}^{n+m}),\atop\tau_{2}^{n+m}\in
BR_{2}(\sigma_{c}^{n+m})}{\max}\mathbb{E}\Big{[}\sum_{t=1}^{n+m}d_{e}(U_{t},V_{1,t},V_{2,t})\Big{]}$
(42) $\displaystyle\geq$
$\displaystyle\underset{\sigma^{n+m}}{\inf}\underset{{\tau_{1}}^{n+m}\in
BR_{1}(\sigma^{n+m}),\atop{\tau_{2}}^{n+m}\in
BR_{2}(\sigma^{n+m})}{\max}\mathbb{E}\Big{[}\sum_{t=1}^{n+m}d_{e}(U_{t},V_{1,t},V_{2,t})\Big{]}$
(43) $\displaystyle=$ $\displaystyle(n+m)D_{e}^{n+m}(R_{1},R_{2}),$ (44)

where the notation $\sigma^{n+m}_{c}$stands for the encoding strategies
obtained by concatenation.

######  Definition 9 (KL Divergence).

The Kullback-Leiber (KL) Divergence for distributions $P$ and $Q$ on
$\Delta(U)$ with respective supports $\mathrm{supp}P$ and $\mathrm{supp}Q$ is
given by

$\displaystyle
D(P||Q)=\begin{cases}\sum_{u\in\mathrm{supp}P}P(u)\log_{2}\frac{P(u)}{Q(u)}&,\text{
if $\mathrm{supp}Q\ \subset\ \mathrm{supp}P$}.\\\ +\infty&,\text{
otherwise.}\end{cases}$ (45)