# On The Role of Social Identity in the Market for (Mis)information

Vijeth Hebbar and Cédric Langbort This work was supported by the ARO MURI
grant W911NF-20-0252 (76582 NSMUR).All authors are with the Coordinated
Science Lab, University of Illinois at Urbana–Champaign, Urbana, IL 61801,
USA<EMAIL_ADDRESS>
###### Abstract

Motivated by recent works in the communication and psychology literature, we
model and study the role social identity – a person’s sense of belonging to a
group – plays in human information consumption. A hallmark of Social Identity
Theory (SIT) is the notion of ‘status’, i.e., an individual’s desire to
enhance their and their ‘in-group’s’ utility relative to that of an ‘out-
group’. In the context of belief formation, this comes off as a desire to
believe positive news about the in-group and negative news about the out-
group, which has been empirically shown to support belief in misinformation
and false news.

We model this phenomenon as a Stackelberg game being played over an
information channel between a news-source (sender) and news-consumer
(receiver), with the receiver incorporating the ‘status’ associated with
social identity in their utility, in addition to accuracy. We characterize the
strategy that must be employed by the sender to ensure that its message is
trusted by receivers of all identities while maximizing their overall quality
of information. We show that, as a rule, this optimal quality of information
at equilibrium decreases when a receiver’s sense of identity increases. We
further demonstrate how extensions of our model can be used to quantitatively
estimate the level of importance given to identity in a population.

##  I INTRODUCTION

Social Identity Theory (SIT), proposed by Tajfel and Turner [1], seeks to
formalize the notion that humans see the world through an ‘us’ vs ‘them’ lens.
In doing so, it proposes the existence of three cognitive processes -
categorization (breaking down society into categories e.g. races, genders),
identification (assigning a set of categories for oneself and adopting
pertinent norms) and comparison (comparing the categories assigned to oneself
with others to determine one’s status in the society).

This last element of comparison hallmarks the theory of social identity and
differentiates it from the idea of homophily (the tendency to form connections
with similar individuals) in that, the status of an individual and their in-
group is measured relative to an out-group. Under the lens of social identity
theory, a decrease in the welfare of the out-group and an increase in welfare
of the in-group both lead to an increased social status for the individual.
Accordingly, any information that reflects negatively on the receiver’s in-
group affects their posterior beliefs differently than information that speaks
positively of it. This variation in posteriors differs, however, from
confirmation bias, as belief updates under the latter do not depend on the
‘nature’ of the received information with respect to receiver’s type, but only
on its closeness to the their prior.

Many models have been presented in a bid to capture the effect of identity in
economic behaviour, in applications ranging from income redistribution and
immigration [2] to poverty and social exclusion [3]. Shayo [2] proposes a game
theoretic-model where the ‘identification’ and ‘comparison’ steps are taken
simultaneously in equilibrium. In this framework, agents trade-off the
relative ‘distance’ they feel to an identity group with the social status they
accrue from identifying with it. Akerlof and Kranton [3] consider a similar
model but with the additional feature that payoffs from the actions depend
explicitly on the identity group chosen. In our work, we will assume that
identities have already been assigned to agents and will seek to model the
effect these identities have on the agent’s beliefs.

Identity is known to play a cornerstone role in belief of information, in
particular, false news and misinformation. Bavel and Pereira [4] provide
empirical evidence suggesting that belief in misinformation is better
explained by an identity driven processes than through confirmation bias or
ideological value (belief in news that agree with one’s ideologies) processes.
Kahan [5] theorises that belief in misinformation is driven by the
psychological desire to signal and re-affirm ones identity, and thus status,
in society. A prominent modern day example of socio-political identities
driving belief in false news is seen among the deniers of climate change [6].
Thus, modeling the effect of social identity on misinformation can better
inform our search for counter mechanisms to this ubiquitous issue.

In our work, we propose a model with three elements – source, sender and
receiver – acting over an information channel, with the source and the
receiver having ‘types’ (or in our context ‘identities’). Both the sender and
receiver have their respective utilities and are thus incentivized to send and
interpret information in strategic ways. This approach is closely related to
the idea of strategic information transmission [7, 8]. More precisely, we view
the interaction between sender and receiver as a Stackelberg game with the
sender as the leader and the receiver as the follower. The role of social
identity in this interaction is captured through the utilities of the agents.

Our work parallels the growing literature on (mis)information dynamics over
networks, with many works using notions of antagonism and homophily among
interacting agents to explain partisanship. The role of homophily in
information propagation over networks, as well as extensions that incorporate
misinformation explicitly have been studied recently [9, 10]. Much existing
work seeking to explain the creation and growth of polarization consider
opinion dynamical models with antagonistic agents [11]—[13]. Leonard et al.
[14] consider opinion dynamical model incorporating political self-
reinforcement (party polarization driven by internal mechanisms, e.g.
eliminating moderating voices) and reflexive partisanship (party elites
shunning a policy because the other side supports it) to explain the growth of
polarization among political elite and validate their findings using
historical data. Both the mechanisms considered in their work can be closely
tied to the notion of political identity. Deviating from the standard DeGroot
model for opinion dynamics that takes weighted averaging opinion updates, Mei
et al. [15] propose a weighted median opinion update model to better explain
polarization and echo chamber formation.

Such studies over networks give us useful insights into the ‘virality’ and
persistence of inaccurate opinions and fake news, and can be used to arrive at
counter-mechanisms to curb this spread. While all such models seek to
understand how (mis)information (or the beliefs induced by them) spreads, we
seek to look more closely at how an individual’s beliefs are shaped by the
presence of misinformation. In particular, our contribution lies in modeling
how identity influences this process. Our work is closest in spirit to the
work by Mullainathan and Shleifer [16] (which also inspired the reference to a
’market’ in our paper’s title) wherein, they try to model media slant stemming
from readers’ biases and study the effect of reader heterogeneity on the same.

In Section II, we will outline the different elements of our model and
introduce the utilities of the agents in our model. We also introduce an
illustrative example to help us better interpret the results we obtain
throughout this paper. In Section III, we will detail the structure of the
game induced between the receiver and the sender and characterize the
corresponding equilibrium. In Section IV we will highlight and visualize some
deeper implications of our model. We will see that these results act both to
validate our model and to offer insight into a property counter-measures to
curb belief in misinformation may possess. Finally, in Section V, we will
augment our model with an observation mechanism allowing us to estimate
parameters that define the receiver behaviour when they are a priori unknown.
In doing so, we will illustrate the versatility of our model in being able to
estimate an imperfectly characterized receiver’s behaviour.

This process of estimating a receiver’s behavioural parameters from
observations is tied to the broader literature of learning human preferences
by observing their actions [17]. These approaches usually assume humans are
Bayesian utility maximizing agents to learn their reward functions given a
sequence of optimal actions. Such methods have found wide applicability in
domains ranging from robotics [18] to natural language processing (NLP) [19].

##  II Our Model

We consider a communication theoretic model with three elements - source,
sender and receiver. Associated with the source are two boolean random
variables $(X,\theta)$, (we will call this the ‘state’ and the ‘type’
respectively) with some joint distribution $\pi$. For the purpose of this
work, we will assume both $X\in\\{0,1\\}$ and $\theta\in\\{A,B\\}$ are
Bernoulli random variables with a uniform distribution. We also assume that
this distributional information is known to both the sender and the receiver.

The sender encodes the information about the source (that is, $(X,\theta)$)
into a Boolean ‘message’ $Y\in\\{a,b\\}$ and sends it to the receiver. The
receiver then decodes the message to make an ‘estimate’ $\hat{X}$ on the true
state of the source. The receiver also has a type $\bar{\theta}\in\\{A,B\\}$
which will influence how receiver chooses to decode the message. Figure 1
depicts the standard information theoretic representation for our
communication channel.

The illustration also highlights the encoding and decoding strategies
available to sender and receiver respectively. For instance, $m_{A}$ denotes
the probability that the sender sends out the message $Y=a$ when the source
has state $X=1$ and $p_{\bar{\theta}}$ signifies the probability that a
receiver of type $\bar{\theta}$ infers $\hat{X}=1$ from the message $Y=a$.

Source $\displaystyle\mathbf{(X,\theta)}$
$\displaystyle(1,A)$$\displaystyle(1,B)$$\displaystyle(0,A)$$\displaystyle(0,B)$
Sender $m_{A}$$m_{B}$$n_{B}$$n_{A}$ Receiver $\displaystyle
a$$\displaystyle\mathbf{Y}$$\displaystyle
b$$p_{\bar{\theta}}$$q_{\bar{\theta}}$$\displaystyle
1$$\displaystyle\mathbf{\hat{X}}$$\displaystyle 0$ Figure 1: (A classical
communication channel representation of our model.) The sender plays the role
of the encoder while the receiver plays the role of a decoder.

The types (both for the source and the receiver) will signify the identity of
agents in our model. The way in which they impact the receiver’s utility is
best motivated in the context described below.

Let the source be a politician and let the state $X$ denote whether they are
corrupt ($X=1$) or not $(X=0)$. The type $\theta$ then naturally indicates
their political affiliation to one of two political groups $A$ or $B$. The
sender can be viewed as a news agency who seeks to inform the receiver about
the true nature of a politician through a news article (the ‘message’). Let us
assume that the message $Y=a$ loosely indicates that the politician in
question is corrupt while $Y=b$ signifies that the politician is not corrupt.
Finally, the receiver may simply be an individual seeking to form an opinion
on the politician. The receiver’s type is an indication of their partisan
nature and reflects their political identity. $\hat{X}=1$ ($\hat{X}=0$)
indicates the receiver’s opinion that the politician in question is corrupt
(not corrupt). We will revisit this illustrative example repeatedly to
highlight key aspects of our model as well as to elucidate the implications of
our results.

###  II-A Receiver Utility

We model the receiver utility to include three components - accuracy utility,
social identity utility and economic utility. Mathematically, for a receiver
with type $\bar{\theta}$ we have

$\displaystyle
U^{R}(x,\hat{x},\theta,\bar{\theta})=\lambda_{a}^{\bar{\theta}}u^{a}(x,\hat{x})+\lambda_{s}^{\bar{\theta}}u^{s}(\hat{x},\theta,\bar{\theta})+u^{e}(x),$

where $\lambda_{a}^{\theta}\in\mathbb{R}_{\geq 0}$ is the receivers accuracy
weight and $\lambda_{s}\in\mathbb{R}_{\geq 0}$ is the receivers identity
weight. The accuracy utility $u^{a}$, as the name suggests, offers a utility
when the receiver’s estimate of the state of the source is accurate.
Mathematically,

$\displaystyle u^{a}(x,\hat{x})=\delta_{x=\hat{x}}$

where $\delta$ is the usual indicator function. The social identity utility is
mathematically given by

$\displaystyle
u^{s}(\hat{x},\theta,\hat{\theta})=\begin{cases}-\Delta_{I}^{\bar{\theta}}&\hat{x}=1,\theta=\bar{\theta}\\\
-\Delta_{O}^{\bar{\theta}}&\hat{x}=0,\theta\neq\bar{\theta}\\\
0&\text{otherwise.}\end{cases}$

where $\Delta_{O}^{\bar{\theta}},\Delta_{I}^{\bar{\theta}}\in\mathbb{R}_{\geq
0}$. The motivation for this utility is best seen by revisiting our
illustrative example from Section II. $\Delta_{I}^{A}$, for instance, is the
dis-utility to receiver of type $A$ when they believe that a politician of
type $A$ is corrupt. Note that this utility does not depend on the true state
of the source but only on the receiver’s estimate of it and is thus, a
psychological dis-utility. Likewise, $\Delta_{O}^{A}$ is the dis-utility to
receiver of type $A$ when they believe that a politician of type $B$ is not
corrupt. It is easy to see that this simple utility function captures the
essence of social identity. Since the status of a group under Social Identity
Theory is always seen as a relative metric, belief in both – positive news
about out-group politician and negative news about in-group politician – can
cause a decrease in in-group’s social status.

Finally, we introduce the economic utility as

$\displaystyle u^{e}(x)=\delta_{x=0}$

which, in our politician story, simply indicates that every individual
receives a utility from the politician in power being non-corrupt. Note that
this utility depends only on the true state of the source and not on the
receivers estimate. As such, this utility will be exogenous and independent of
the receivers decoding strategy. Likewise, we could have included an additive
component in the social identity utility that depends on the true state of the
source, say an additional dis-utility if the true state of the in-group
politician is corrupt. But once again, such a dis-utility will be exogenous to
receiver’s strategy and can be neglected for the purpose of our model. Going
forward we will work with the truncated utility for the receiver as

$\displaystyle
U^{R}(x,\hat{x},\theta,\bar{\theta})=\lambda_{a}^{\bar{\theta}}u^{a}(x,\hat{x})+\lambda_{s}^{\bar{\theta}}u^{s}(\hat{x},\theta,\bar{\theta}).$
(1)

###  II-B Sender Utility

We consider the sender to require two things – receiver’s belief in their
message and the receiver estimating the state of the true source accurately.
We consider two receivers (one of either types) and we model the utility of
the sender as

$\displaystyle
U^{S}(x,\hat{x},\mathbf{p},\mathbf{q})=\begin{cases}\delta_{x=\hat{x}}&\mathbf{p}>\mathbf{\frac{1}{2}},\mathbf{q}>\mathbf{\frac{1}{2}}\\\
0&\text{otherwise},\end{cases}$ (2)

where $\mathbf{p}=[p_{A}\;p_{B}]^{T},\mathbf{q}=[q_{A}\;q_{B}]^{T}$ and
$\mathbf{\frac{1}{2}}=[\frac{1}{2}\;\frac{1}{2}]^{T}$. We assume that if a
receiver chooses not to believe in the sender
($p_{\bar{\theta}}\leq\frac{1}{2}$ or $q_{\bar{\theta}}\leq\frac{1}{2}$), they
will unsubscribe from the sender leading to loss in sender utility. In our
illustrative example, an individual may choose to unsubscribe from a news
agency if they stop believing in the veracity of the message $Y$. On the other
hand, the sender also receives utility from the receiver being well-informed
when they are ‘subscribed’ to the sender and this is captured using the
indicator function $\delta$ over the receiver estimate matching the true state
of the source.

##  III Game Over Information Channel

Having laid out the receiver’s and the sender’s utilities and strategy sets,
we now dwell into the information structure of our game. For now, we assume
that the utility function of the receiver is common knowledge. The
distribution on the true state and the type of the source is also common
knowledge, but the type of the receiver is unknown to the sender. Before the
game plays out, the sender commits to an encoding strategy and announces it to
the receiver. This assumption of commitment is best understood in the context
of our politician story, in which the sender (news agency) has to uphold a
stance (a certain bias or lack thereof in reporting style). For instance, an
openly partisan news agency would be expected to continue being so or risk
losing credibility among its readership.

###  III-A Receiver Optimal Strategy

For every message $Y$ generated by the sender strategy, the receiver picks
their best response to maximize expected receiver utility

$\displaystyle\max_{p_{\bar{\theta}},q_{\bar{\theta}}}\underbrace{{\mathbf{E}}[u^{R}(X,\hat{X},\theta,\bar{\theta})|Y]}_{\triangleq
f(\bar{\theta},Y)},$ (3)

where the expectation is taken over the distribution of messages $Y$ generated
by the sender, distribution of the source state and type, and the distribution
generated by the receiver strategy. Noting that the expectation above is
linear in receiver strategy, it is easy to see that the optimal strategy for
the receiver is given by111Strictly speaking, $p^{*}_{\bar{\theta}}$ is not
unique when $\frac{\partial f(\bar{\theta},a)}{\partial p}=0$ but we pick
$p^{*}_{\bar{\theta}}=1$ as a convention. Likewise for $q^{*}_{\bar{\theta}}$.

$\displaystyle p^{*}_{\bar{\theta}}$
$\displaystyle=\begin{cases}1&\frac{\partial f(\bar{\theta},a)}{\partial
p}\geq 0\\\ 0&\frac{\partial f(\bar{\theta},a)}{\partial p}<0,\end{cases}$
(4a) $\displaystyle q^{*}_{\bar{\theta}}$
$\displaystyle=\begin{cases}1&\frac{\partial f(\bar{\theta},b)}{\partial
q}\geq 0\\\ 0&\frac{\partial f(\bar{\theta},b)}{\partial q}<0.\end{cases}$
(4b)

Thus, we see that the receiver employs pure strategies.

###  III-B Sender Optimal Strategy

The sender on the other hand maximises its own expected utility assuming that
the receiver is playing their best response. Mathematically, the sender picks
its strategy to optimize

$\displaystyle\max_{m_{A},m_{B},n_{A},n_{B}\in[0,1]}\;$
$\displaystyle{\mathbf{E}}[\delta_{X=\hat{X}^{*}(Y)}]={\mathbf{P}}(X=\hat{X}^{*}(Y))$
(5a) $\displaystyle S.T\quad$ $\displaystyle\frac{\partial f(A,a)}{\partial
p}\geq 0,\quad\frac{\partial f(A,b)}{\partial q}\geq 0$ (5b)
$\displaystyle\frac{\partial f(B,a)}{\partial p}\geq 0,\quad\frac{\partial
f(B,b)}{\partial q}\geq 0$ (5c)

where the expectation is taken over the distribution of the sources state and
type, and over the distribution over messages generated by the sender’s
strategy.

In (5a), $\hat{X}^{*}(Y)$ denotes the state estimate generated by the
receiver’s best response upon receiving message $Y$. Indeed, based on (4),
such a best response is nothing but a deterministic map from $\\{a,b\\}$ to
$\\{0,1\\}$. (Simply, $a\mapsto\hat{X}^{*}(a)=1$ and
$b\mapsto\hat{X}^{*}(b)=0$.) Also note that the constraints in (5b) and (5c)
are a result of incorporating the receiver’s best response into the sender
utility, and act to ensure that the optimal receiver response is to always
‘believe’ the sender. This results in a Stackelberg game between the sender
(leader) and the receiver (follower), in accordance with the assumption of
commitment on part of the sender.

With these elements in place, we can prove the following

###### Lemma III.1

The optimization problem in (5) is an linear program in
$m_{A},m_{B},n_{A},n_{B}$. Moreover, under our assumption of uniform
distribution over the source states and types we can show that maximizing the
expectation in (5a) is equivalent to maximizing the sum

$\displaystyle Q(m_{A},n_{A},m_{B},n_{B})\triangleq m_{A}+m_{B}+n_{A}+n_{B},$
(6)

where $Q$ denotes the ‘quality of information‘ available to the receiver.

Clearly if $Q=4$, then the sender is always truthful and the receiver can
perfectly recreate the state of the source. On the other hand, if $Q<4$ the
receiver loses information about the source due to the sender’s encoding
strategy. While this linear program can be solved using any one of the widely
available numerical solvers, it can also be readily solved in closed form
under an additional assumption on the receiver’s identity parameters
($\Delta_{O}^{\bar{\theta}},\Delta_{I}^{\bar{\theta}}$) which we introduce in
the next section. We see that our assumption offers us a large gain in
analytical mileage while being only mildly restrictive.

###  III-C Equilibrium in a Restricted Game

Going forward we will make the following

###### Assumption III.1

$\Delta_{O}^{\bar{\theta}}\geq\Delta_{I}^{\bar{\theta}}\quad\forall\bar{\theta}\in\\{A,B\\}.$

Before making use of this assumption222We could have done a similar analysis
if we instead had
$\Delta_{O}^{\bar{\theta}}\leq\Delta_{I}^{\bar{\theta}}\quad\forall\bar{\theta}\in\\{A,B\\}$.
The crucial element of our assumption is the existence of the same ordering
between the two identity parameters holding true regardless of $\bar{\theta}$.
to simplify the linear program in (5), we note that it is qualitatively
supported by a number of works in the Communication literature. Indeed, the
study [20] finds that negative language about out-group played a strong role
in social media engagement through an empirical study on Facebook and Twitter.
Moreover, works such as [21] have shown that negative news content spreads
faster and wider on social media. While the intent to share a negative post
about an out-group may not directly indicate the utilities gleaned from the
process, we hypothesize that the relative gain in social status from doing so
is higher than from sharing news about the in-group.

Under Assumption III.1 we can make the following

###### Proposition III.2

The optimal solution to (5) is given by $m_{A}^{*}=m_{B}^{*}=1$, and
$(n_{A}^{*},n_{B}^{*})$ the optimal point of the linear program,

$\displaystyle\max_{n_{A},n_{B}\in[0,1]}n_{A}+n_{B}$ (7a) $\displaystyle
S.T\;$ $\displaystyle
n_{A}(\lambda_{s}^{A}\Delta_{I}^{A}+\lambda_{a}^{A})+n_{B}(\lambda_{a}^{A}-\lambda_{s}^{A}\Delta_{O}^{A})\geq
0$ (7b) $\displaystyle
n_{B}(\lambda_{s}^{B}\Delta_{I}^{B}+\lambda_{a}^{B})+n_{A}(\lambda_{a}^{B}-\lambda_{s}^{B}\Delta_{O}^{B})\geq
0$ (7c)

First, note that the linear program in (7) always has a solution as
$n_{A}=n_{B}=0$ is always a feasible point to the problem. The LP has two
decision variables and two linear inequality constraints and can easily be
solved analytically giving us

###### Corollary III.3

The optimizer to (7) is given by,

$\displaystyle(n_{A}^{*},n_{B}^{*})=\begin{cases}(1,1)&k_{A}<0,k_{B}<0\\\
(0,0)&k_{B}>k_{A}>0\\\ (1,k_{A})&1>k_{A}>k_{B}\\\ (1,1)&k_{A}>1>k_{B}\\\
(\frac{1}{k_{B}},1)&k_{A}>k_{B}>1\\\
\bigg{(}\min\\{1,\frac{1}{k_{B}}\\},1\bigg{)}&k_{B}>0>k_{A}\end{cases}$

where

$\displaystyle
k_{A}\triangleq\frac{\lambda_{s}^{A}\Delta_{I}^{A}+\lambda_{a}^{A}}{\lambda_{s}^{A}\Delta_{O}^{A}-\lambda_{a}^{A}},\;k_{B}\triangleq\frac{\lambda_{s}^{B}\Delta_{O}^{B}-\lambda_{a}^{B}}{\lambda_{s}^{B}\Delta_{I}^{B}+\lambda_{a}^{B}}.$
(8)

The most important consequence of Corollary III.3 for our purposes is that the
closed-form solution makes it possible to discuss the sender’s optimal
strategy qualitatively and, in particular, to study the influence of the
various parameters accounting for the receiver’s identity.

##  IV Analytical Results

In this section, we will interpret some of the direct consequences of the
equilibrium obtained in Corollary 3.3. In doing so, we will routinely revisit
the illustrative example introduced in Section II to give more context and
illustrate the implications of our results. First we have

###### Corollary IV.1

When $\Delta_{O}^{\bar{\theta}}>\Delta_{I}^{\bar{\theta}}$, the senders are
_always_ completely truthful about ‘negative’ states.

Here, by ‘negative’, we mean any state that causes loss in utility to the in-
group, if realized. In our corrupt politician story, the ‘negative’ state was
‘corrupt’ ($X=1$) and so Corollary IV.1 implies that the news agency always
truthfully reports when the source is corrupt.

While Corollary IV.1 gives us a robust result on the nature of sender’s
equilibrium strategy, we would also like to know how the equilibrium varies
with change in receiver characteristics (i.e change in
$\lambda^{\bar{\theta}}$’s and $\Delta^{\bar{\theta}}$’s). This is the content
of the following

###### Claim IV.2

The quality of information $Q(m_{A}^{*},m_{B}^{*},n_{A}^{*},n_{B}^{*})$
available to receiver at equilibrium

  * •

decreases with increase in social identity levels
($\lambda_{s}^{\bar{\theta}}$) among the receivers,

  * •

increases with increase in accuracy seeking levels
($\lambda_{a}^{\bar{\theta}}$) among the receivers,

  * •

increases with separation in relative importance of group identities
($\Delta_{O}^{\bar{\theta}}-\Delta_{I}^{\bar{\theta}}$) among the receivers.

This claim very much agrees with intuition in the sense that a sender who
seeks to maximise the quality of information available to receivers will do
worse when the receivers themselves care more about their identity goals (and
less about accuracy). While Claim IV.2 is a result of our modelling, it can
also be viewed as a validation of our model as it agrees with empirical
evidence on the role of social identity in misinformation [4, 22]. Figure 2
illustrates the monotonic variation of quality of information with changes in
social identity levels and accuracy seeking behaviour among receivers.

(a) $\lambda_{a}^{B}=0.6,\lambda^{B}_{s}=0.4$

(b) $\lambda_{a}^{B}=0.55,\lambda^{B}_{s}=0.45$

(c) $\lambda_{a}^{B}=0.4,\lambda^{B}_{s}=0.6$

Figure 2: (Variation of quality of information with levels of identity.) In
the plots above, we have picked $\Delta_{I}^{A}=\Delta_{I}^{B}=1$,
$\Delta_{O}^{A}=2$ and $\Delta_{O}^{B}=3.5$.

For fixed accuracy seeking and social identity levels in out-group (consider
Figure 2(b)) we see that $Q$ drops sharply for high levels of in-group
identity ($\lambda_{s}^{A}$) and low accuracy seeking
levels($\lambda_{a}^{A}$). Further, with an increase in identity levels in an
out-group receiver (going from Figure 2(b) to ) we see a sharp drop in $Q$ for
some ranges of in-group identity levels. At the same time we see that in the
general range where $\lambda_{a}^{A}>\lambda_{s}^{A}$ (yellow region in all
plots in Figure 2), increase in identity levels of an out-group receiver makes
no difference to the quality of information available to the receivers. This
shows that the sender can communicate a higher quality of information to a
receiver of any type, even if only one of the identity groups cares enough
about accuracy. In turn, interventions aimed at increasing the focus on
accuracy (such as those proposed in [22, 23]) can be targeted at only one
identity group.

##  V Exploiting Our Model

We sought to model the effect of social identity with the specific aim of
analyzing its role in misinformation belief. In Section IV, we obtained some
results that aim to do so under the assumption that the sender has knowledge
of the receiver payoffs. While we have made other assumptions, the knowledge
of parameters defining the receiver utility
($\lambda_{a}^{\bar{\theta}},\lambda_{s}^{\bar{\theta}},\Delta_{I}^{\bar{\theta}}$
and $\Delta_{O}^{\bar{\theta}}$) may be particularly difficult for the sender
to obtain.

In our politician story for instance, this simply means that news agencies do
not have information about the identity and accuracy seeking levels among
their audience. This would naturally be ‘learnt’ over time through a sequence
of ‘trial-and-error’ attempts to know what their audience believes. In the
same spirit, we will propose a thought experiment in Section V-A that,
together with the structure of our problem, can be used to estimate these
unknown quantities. These estimates can then be used by the sender to obtain
its encoding strategy. Going forward we will assume that the parameters
defining receiver utility are unknown constants, but we will continue to work
under Assumption III.1.

###  V-A A Thought Experiment

Not knowing the identity parameters of the receiver makes it impossible for
the sender to solve (7) and obtain the optimal $n_{A}^{*}$ and $n_{B}^{*}$ for
equilibrium. And so, in an attempt to estimate these unknowns, we postulate
the existence of a mechanism for the sender to announce their strategy (i.e.
$n_{A}$ and $n_{B}$, we already know $m_{A}=m_{B}=1$ for equilibrium) to the
receiver before the game is played out. The sender then responds whether or
not they ‘believe’ in the messages induced by this encoding strategy. A
receiver of type $A$ (type $B$) believing in the message amounts to constraint
(7b) (constraint (7c) respectively) being satisfied. Here we make the implicit
assumptions that receiver’s response is not noisy and can be used to
accurately ascertain that the corresponding constraints are satisfied.

Now returning to the problem of estimation, we would like to design a
procedure that takes as input a sequence of Boolean (‘believe’/‘don’t
believe’) replies from the receiver and outputs the unknown parameters.
Unfortunately, such a procedure would be ill-defined as the mapping from the
set of identity parameters ($\lambda^{\bar{\theta}}$’s and
$\Delta^{\bar{\theta}}$’s) to the sequence of replies is not one-to-one.
However, the ‘augmented’ identity parameters $k_{A}$ and $k_{B}$ as defined in
(8) can be identified uniquely. Fortunately, we can show that the knowledge of
these augmented parameters is sufficient to ascertain receiver behaviour (i.e.
to ascertain feasibility of (7b) and (7c)) and, thus, for the sender to pick
its optimal encoding strategy according to (7).

In Section V-B, we propose an algorithmic approach to estimating the augmented
identity parameters ($k_{A}$ and $k_{B}$). In doing so, we also justify their
sufficiency in evaluating receiver behaviour. Then in Section V-C, we use
these estimates to arrive at sender’s optimal encoding strategy.

###  V-B Estimating augmented identity parameters $k_{A}$ and $k_{B}$

Note that we simply want feasible solutions to the set of linear inequalities
given by (7b) and (7c). It is easy to see that if either
$\lambda_{s}^{A}\Delta_{O}^{A}-\lambda_{a}^{A}\leq 0$ or
$\lambda_{s}^{B}\Delta_{O}^{B}-\lambda_{a}^{B}\leq 0$ the corresponding
receiver will believe the sender for any encoding strategy. In this case, we
see that the end goal of obtaining the optimal sender strategy can be achieved
even without explicitly computing the augmented identity parameters.

So it suffices to consider the case when
$\lambda_{s}^{A}\Delta_{O}^{A}-\lambda_{a}^{A}>0$ and
$\lambda_{s}^{B}\Delta_{O}^{B}-\lambda_{a}^{B}>0$ and treat the former
conditions as special cases. We can rework (7b) and (7c) to get,

$\displaystyle 0<k_{B}\leq\frac{n_{B}}{n_{A}}\leq k_{A}$ (9)

when $n_{A}\neq 0$. Note that if $k_{A}<k_{B}$ then $n_{A}=n_{B}=0$ is the
only feasible encoding strategy available to the sender. So we see that
obtaining $k_{A}$ and $k_{B}$ is indeed sufficient to evaluate feasibility of
an encoding strategy according to (9). We now propose an algorithm that
implements bisection search to obtain estimates for the augmented identity
parameters. The algorithm takes two inputs - resolution $\delta>0$ to which we
want our estimate to be accurate and upper bound $M>1$ on our search space for
the parameters.

###### Theorem V.1

The output $\hat{k}_{A}$ (and $\hat{k}_{B}$) of Algorithm $1$ satisfies
$|k_{A}-\hat{k}_{A}|<\delta$ ($|k_{B}-\hat{k}_{B}|<\delta$) and is obtained in
under $\big{\lceil}\log_{2}{\big{(}\frac{M}{\delta})}\big{\rceil}+1$ steps.

Algorithm 1 Algorithm for estimating identity parameter $k_{A}$ ($k_{B}$) when
it is non-negative.

0: $\delta>0,M>1$

0: $\hat{k}_{A}$ Initialization:
$\underline{k}_{A}^{1}=0,\bar{k}_{A}^{1}=M,\eta^{1}=1,i=1$

1: while $|\bar{k}_{A}^{i}-\underline{k}_{A}^{i}|<\delta$ do

2: $n_{B}^{i}=\min\\{1,\eta^{i}\\},n_{A}^{i}=\min\\{1,\frac{1}{\eta^{i}}\\}$

3: if (7b) ((7c) for $k_{B}$) satisfied for this $n_{A}^{i},n_{B}^{i}$ then

4: $\underline{k}_{A}=\eta^{i}$ $\quad(\bar{k}_{B}=\eta^{i})$

5: else

6: $\bar{k}_{A}=\eta^{i}$ $\quad(\underline{k}_{B}=\eta^{i})$

7: end if

8: $\eta^{i+1}=\frac{\underline{k}_{A}+\bar{k}_{A}}{2}$

9: $i=i+1$

10: end while

11: return $\hat{k}_{A}=\frac{\underline{k}_{A}+\bar{k}_{A}}{2}$

To obtain $k_{B}$ we will replace every instance of $k_{A}$ by $k_{B}$ in
Algorithm 1 and change lines 3,4 and 6 as indicated.

###### Example V.1

Let the true identity parameters for the receiver of either type be as given
in Figure 2(b) (take $\lambda_{s}^{A}=0.45$, $\lambda_{a}^{A}=0.55$). Choosing
$M=10^{4}$ and $\delta=0.01$, Algorithm 1 converges to estimates
$\hat{k}_{A}=2.855$ and $\hat{k}_{B}=1.024$ in $21$ steps. True augmented
parameters are $k_{A}=2.857$ and $k_{B}=1.025$.

###  V-C Obtaining optimal encoding strategy for the sender

For the purpose of this section we will treat the estimates $\hat{k}_{A}$ and
$\hat{k}_{B}$ as the true augmented identity parameters. Note that the
constraint in (9) acts on the ratio of $n_{A}$ and $n_{B}$. Noting that both
$n_{A}$ and $n_{B}$ are probabilities we can make the following

###### Claim V.2

$n^{*}_{A}=n^{*}_{B}=0$ if $k_{A}<k_{B}$. Otherwise,

$\displaystyle\max\\{n^{*}_{A},n^{*}_{B}\\}=1$ (10a) and
$\displaystyle\frac{n^{*}_{B}}{n^{*}_{A}}=\operatorname*{arg\,min}_{\gamma\in[\hat{k}_{B},\hat{k}_{A}]}|\gamma-1|.$
(10b)

Recall that the sender is trying to maximize the quality of information
available to the receivers and thus, wishes to pick $n_{A}$ and $n_{B}$ as
close to one as possible. Conditions (10a) and (10b) together ensure that this
indeed happens. Moreover, from (10b) it is easy to see that the maximum value
of Q (equal to 4) is attained when $1\in[k_{B},k_{A}]$.

Note that in Algorithm 1, we do not explicitly check if $k_{A}>0$ and
$k_{B}>0$. When $k_{A}$ ($k_{B}$) is negative our algorithm will return
$\hat{k}_{A}=M$ ($\hat{k}_{B}=0$) and it can be shown that the optimal
encoding strategy is still given by (10).

##  VI CONCLUSIONS AND FUTURE WORKS

We set out to understand how social identity plays a role in formation and
consumption of misinformation and proposed a communication model to achieve
our goal. Despite its seeming simplicity, the predictions from our model are
well aligned with empirical evidence and offer some useful insights in the
search for counter-mechanisms to misinformation. We also showed how this model
lends itself to estimation of some behavioural parameters of the receiver.

The work presented in Section V can be viewed as a parameter estimation
problem. In doing so, we essentially considered a noiseless measurement
setting in our thought experiment of Section V-A. A more realistic mechanism
to infer whether receivers ‘believe’ a certain sender strategy would likely
involve ‘bandit feedback’ instead, whereby the various probabilities defining
the sender’s strategy, and the receiver’s ’believe/not believe’ choices, must
be inferred from realized time series. Additionally, uncertainty may be
introduced in this inference process ‘in two ways - the receiver may infer a
different encoding strategy as generating the sampled messages and thus take
actions responding to this strategy, or the receiver may simply respond sub-
optimally to the ‘correct’ strategy because they are boundedly rational. So
further work needs to be done to address such ‘noisy’ feedback.

In the same spirit, the natural next step in any control theoretic framework
is to look at possible avenues of control. There is a vast literature that
studies ‘nudging’ - a form of ‘soft’ control - to influence human choice
architecture. We are currently looking to answer how ‘nudges’ can be
incorporated naturally into our framework.

Finally, our model looks at a single receiver (having one of two types) with
the idea that the receiver can be a prototype for a large population (of
either type). This viewpoint can be enriched by considering a population of
receivers with identity parameters picked from a distribution. We intend to
look in this direction in subsequent work as well.

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##  Appendix A Proofs For Selected Results

###  A-A Proof for Lemma III.1

From (5), we see that the sender wishes to maximize $P(X=\hat{X}^{*}(Y))$
given $p^{*}_{\bar{\theta}}=q^{*}_{\bar{\theta}}=1$. Let us look more closely
at the objective function,

$\displaystyle P(X=\hat{X})=$
$\displaystyle\sum_{\begin{subarray}{c}x\in\\{0,1\\}\\\ \Theta\in\\{A,B\\}\\\
y\in\\{a,b\\}\end{subarray}}{\mathbf{P}}(X=x,\hat{X}=x,\theta=\Theta,Y=y)$
$\displaystyle=$ $\displaystyle\sum_{\begin{subarray}{c}x\in\\{0,1\\}\\\
\Theta\in\\{A,B\\}\\\
y\in\\{a,b\\}\end{subarray}}{\mathbf{P}}(\hat{X}=x|Y=y,X=x,\theta=\Theta)$
$\displaystyle{\mathbf{P}}(Y=y|X=x,\theta=\Theta){\mathbf{P}}(X=x,\theta=\Theta).$

Note that the receiver can only observe the message $Y$ in estimating
$\hat{X}$ and so $\hat{X}$ in independent of $X$ and $\theta$ given $Y$. The
mapping $Y$ to $\hat{X}$ is captured through the receivers decoding strategy
i.e. through $p_{\bar{\theta}}$ and $q_{\bar{\theta}}$. The generation of
message $Y$ from the source state $X$ and type $\theta$ is determined by the
encoder strategy i.e. through $m_{A},n_{A},m_{B}$ and $n_{B}$. Finally, noting
from our assumption that source state and types are uniformly distributed we
get

$\displaystyle P(X=\hat{X})=$
$\displaystyle\frac{1}{4}\big{(}p_{\bar{\theta}}(m_{A}+m_{B})+(1-p_{\bar{\theta}})(2-m_{A}-m_{B})$
$\displaystyle+q_{\bar{\theta}}(n_{A}+n_{B})+(1-q_{\bar{\theta}})(2-n_{A}-n_{B})\big{)}.$

Incorporating receiver best response as $p^{*}_{\bar{\theta}}=1$ and
$q^{*}_{\bar{\theta}}=1$ justifies the definition of $Q$ in Lemma III.1. A
similar expansion of (3) allows us to express the constraints in (7b) and (7c)
as

$\displaystyle\lambda_{s}^{A}(\Delta_{O}^{A}$
$\displaystyle(m_{B}+1-n_{B})-\Delta_{I}^{A}(m_{A}+1-n_{A}))$
$\displaystyle+\lambda_{a}^{A}(n_{A}+n_{B}+m_{A}+m_{B}-2)\geq 0$ (11)
$\displaystyle\lambda_{s}^{A}(\Delta_{I}^{A}$
$\displaystyle(n_{A}+1-m_{A})-\Delta_{O}^{A}(n_{B}+1-m_{B}))$
$\displaystyle+\lambda_{a}^{A}(n_{A}+n_{B}+m_{A}+m_{B}-2)\geq 0$ (12)
$\displaystyle\lambda_{s}^{B}(\Delta_{O}^{B}$
$\displaystyle(m_{A}+1-n_{A})-\Delta_{I}^{B}(m_{B}+1-n_{B}))$
$\displaystyle+\lambda_{a}^{B}(n_{A}+n_{B}+m_{A}+m_{B}-2)\geq 0$ (13)
$\displaystyle\lambda_{s}^{B}(\Delta_{I}^{B}$
$\displaystyle(n_{B}+1-m_{B})-\Delta_{O}^{B}(n_{A}+1-m_{A}))$
$\displaystyle+\lambda_{a}^{B}(n_{A}+n_{B}+m_{A}+m_{B}-2)\geq 0.$ (14)

So we see that optimization (5) is indeed a linear program in
$m_{A},n_{A},m_{B}$ and $n_{B}$.

###  A-B Proof for Proposition III.2

We consider the linear program form for (5) with (6) as the objective function
and constraints (11)—(14), and show that it can be simplified to (7). First,
we consider (11) and see that,

$\displaystyle\lambda_{s}^{A}(\Delta_{O}^{A}(m_{B}+1-n_{B})-\Delta_{I}^{A}(m_{A}+1-n_{A}))$
$\displaystyle=$
$\displaystyle\lambda_{s}^{A}(\Delta_{I}^{A}(n_{A}+1-m_{A})-\Delta_{O}^{A}(n_{B}+1-m_{B}))$
$\displaystyle+\underbrace{2\lambda_{s}^{A}(\Delta_{O}^{A}-\Delta_{I}^{A})}_{\geq
0},$

which shows that (12) implies (11) under Assumption III.1. Similarly we can
show that (14) implies (13). So it suffices to solve the LP with objective
function (6) and constraints (12) and (14). For convenience we redefine the
left-hand side in (12) and (14) as $g_{A}(m_{A},n_{A},m_{B},n_{B})$ and
$g_{A}(m_{A},n_{A},m_{B},n_{B})$ respectively.

Also, note that $m_{A}=m_{B}=1$ and $n_{A}=n_{B}=0$ is a feasible solution to
(12) and (14) and so, the value of our LP is lower bounded by $2$ giving us,

###### Corollary A.1

$m_{A}^{*}+n_{A}^{*}+m_{B}^{*}+n_{B}^{*}\geq 2$

where $(m_{A}^{*},n_{A}^{*},m_{B}^{*},n_{B}^{*})$ denotes the optimal encoding
strategy and will be used as such throughout our proof. In continuing our
proof, we first state

###### Lemma A.2

$\displaystyle n_{B}^{*}>0\implies m_{A}^{*}=1$ (15) $\displaystyle
n_{A}^{*}>0\implies m_{B}^{*}=1$ (16)

###### Proof:

We prove here, (15) and (16) can be shown similarly. Let us assume instead
that $m_{A}^{*}<1$ and $n_{B}^{*}>0$ and define

$\displaystyle\zeta\triangleq\min\bigg{\\{}n_{B}^{*},\frac{(\lambda_{s}^{A}\Delta_{I}^{A}-\lambda_{a}^{A})}{(\lambda_{s}^{A}\Delta_{O}^{A}-\lambda_{a}^{A})}(1-m_{A}^{*})\bigg{\\}}>0$

Now consider an alternate set of decisions variables obtained as,

$\displaystyle(m_{A}^{\prime},n_{A}^{\prime},m_{B}^{\prime},n_{B}^{\prime})$
$\displaystyle=$
$\displaystyle\bigg{(}m_{A}^{*}+\frac{(\lambda_{s}^{A}\Delta^{A}_{O}-\lambda_{a}^{A})}{(\lambda_{s}^{A}\Delta^{A}_{I}-\lambda_{a}^{A})}\zeta,n_{A}^{*},m_{B}^{*},n_{B}^{*}-\zeta\bigg{)}$

The definition of $\zeta$ ensures that
$m_{A}^{\prime},n_{B}^{\prime}\in[0,1]$. We will now evaluate the feasibility
of this new set of points.

$\displaystyle
g_{A}(m_{A}^{\prime},n_{A}^{\prime},m_{B}^{\prime},n_{B}^{\prime})=g_{A}(m_{A}^{*},n_{A}^{*},m_{B}^{*},n_{B}^{*})$
$\displaystyle+(\lambda_{s}^{A}\Delta^{A}_{I}-\lambda_{a}^{A})\frac{(\lambda_{s}^{A}\Delta^{A}_{O}-\lambda_{a}^{A})}{(\lambda_{s}^{A}\Delta^{A}_{I}-\lambda_{a}^{A})}\zeta-(\lambda_{s}^{A}\Delta^{A}_{O}-\lambda_{a}^{A})\zeta$
$\displaystyle=g_{A}(m_{A}^{*},n_{A}^{*},m_{B}^{*},n_{B}^{*})\geq 0,$
$\displaystyle
g_{B}(m_{A}^{\prime},n_{A}^{\prime},m_{B}^{\prime},n_{B}^{\prime})=g_{B}(m_{A}^{*},n_{A}^{*},m_{B}^{*},n_{B}^{*})$
$\displaystyle+(\lambda_{s}^{B}\Delta_{O}^{B}+\lambda_{a}^{B})\frac{(\lambda_{s}^{A}\Delta_{O}^{A}-\lambda_{a}^{A})}{(\lambda_{s}^{A}\Delta^{A}_{I}-\lambda_{a}^{A})}\zeta-(\lambda_{s}^{B}\Delta_{I}^{B}+\lambda_{a}^{B})\zeta$
$\displaystyle=g_{B}(m_{A}^{*},n_{A}^{*},m_{B}^{*},n_{B}^{*})+(\lambda_{s}^{B}\Delta_{I}^{B}+\lambda_{a}^{B})$
$\displaystyle\times\bigg{(}\underbrace{\frac{(\lambda_{s}^{B}\Delta_{O}^{B}+\lambda_{a}^{B})}{(\lambda_{s}^{B}\Delta_{I}^{B}+\lambda_{a}^{B})}}_{>1\;\because\Delta^{B}_{O}>\Delta^{B}_{I}}\times\underbrace{\frac{(\lambda_{s}^{A}\Delta_{O}^{A}-\lambda_{a}^{A})}{(\lambda_{s}^{A}\Delta_{I}^{A}-\lambda_{a}^{A})}}_{>1\;\because\Delta^{A}_{O}>\Delta^{A}_{I}}-1\bigg{)}\zeta$
$\displaystyle>g_{B}(m_{A}^{*},n_{A}^{*},m_{B}^{*},n_{B}^{*})\geq 0.$

Note the explicit usage of Assumption III.1 above. Finally we compute the
value of objective as,

$\displaystyle
Q(m_{A}^{\prime},n_{A}^{\prime},m_{B}^{\prime},n_{B}^{\prime})=Q(m_{A}^{*},n_{A}^{*},m_{B}^{*},n_{B}^{*})+$
$\displaystyle\underbrace{\frac{(\lambda_{s}^{A}\Delta_{O}-\lambda_{a}^{A})}{(\lambda_{s}^{A}\Delta_{I}-\lambda_{a}^{A})}}_{>1\;\because\Delta^{A}_{O}>\Delta^{A}_{I}}\zeta-\zeta>Q(m_{A}^{*},n_{A}^{*},m_{B}^{*},n_{B}^{*}).$

The last inequality above again relies on Assumption III.1 and contradicts the
fact that we picked $(m^{*}_{A},m_{B}^{*},n_{A}^{*},n_{B}^{*})$ to be an
optimizer. And so, $\zeta$ must be zero. A direct implication from the
definition of $\zeta$ is then that if $n_{A}^{*}>0$ we must have
$m_{B}^{*}=1$. ∎

We are now in position to prove Proposition III.2. First, we show that both
$m_{A}^{*}$ and $m_{B}^{*}$ are equal to $1$. We will split this in two
different cases.

  * •

Case I:($(\lambda_{a}^{A}-\lambda_{s}^{A}\Delta_{O}^{A})<0$ and
$(\lambda_{a}^{B}-\lambda_{s}^{B}\Delta_{O}^{B})<0$) Let us assume WLOG that
$m_{B}^{*}<1$. By Lemma A.2 we must have $n_{A}^{*}=0$ as well. From Corollary
A.1 we also know that $m_{A}^{*}+n_{B}^{*}>1$. For this to hold, once again
invoking Lemma A.2 we must have $m_{A}*=1,n_{B}>0$. For our point to be
feasible it must satisfy the constraint,

$\displaystyle 0\leq g_{A}(1,0,m_{B},n_{B})=$
$\displaystyle-(\lambda_{a}^{A}+\lambda_{s}^{A}\Delta_{O}^{A})(1-m_{B})$
$\displaystyle+(\lambda_{a}^{A}-\lambda_{s}^{A}\Delta_{O}^{A})n_{B}.$ (17)

It is then easy to see that we must have $m_{B}^{*}=1,n_{B}^{*}=0$ when
$\lambda_{a}^{A}-\lambda_{s}^{A}\Delta_{O}^{A})<0$ to satisfy Equation (17).
This is in contradiction with our assumption of $m_{B}^{*}<1$, so $m_{B}^{*}$
must be $1$.

  * •

Case II:($(\lambda_{a}^{A}-\lambda_{s}^{A}\Delta_{O}^{A})\geq 0$ or
$(\lambda_{a}^{B}-\lambda_{s}^{B}\Delta_{O}^{B})\geq 0$). WLOG assume
$(\lambda_{a}^{A}-\lambda_{s}^{A}\Delta_{O}^{A})\geq 0$ then since
$\Delta_{O}^{A}>\Delta_{I}^{A}$ we also have
$(\lambda_{a}^{A}-\lambda_{s}^{A}\Delta_{I}^{A})>0$. Then we observe that
$g_{A},g_{B}$ and $f$ are increasing in $m_{A}$ and $n_{B}$. Then it is easy
to see that $m_{A}^{*}=1$ and $n_{B}*=1$. Now suppose $m_{B}^{*}<1$. Recall
that this then means $n_{A}^{*}=0$ by Lemma A.2. Once again we obtain the
constraints to be satisfied as,

$\displaystyle 0\leq
g_{A}(1,0,m_{B}^{*},1)=(\lambda_{a}^{A}+\lambda_{s}^{A}\Delta_{O}^{A})m_{B}^{*}-2\lambda_{s}^{A}\Delta_{O}^{A}$
$\displaystyle 0\leq
g_{B}(1,0,m_{B}^{*},1)=(\lambda_{a}^{B}-\lambda_{s}^{B}\Delta_{I}^{B})m_{B}^{*}+2\lambda_{s}^{B}\Delta_{I}^{B}$

We see that $g_{B}\geq 0$ holds for all values of $m_{B}^{*}$ but to ensure
that $g_{A}\geq 0$ for all values of $m_{B}^{*}$ we must have

$\displaystyle
m_{B}^{*}\geq\frac{2\lambda_{s}^{A}\Delta_{O}^{A}}{(\lambda_{a}^{A}+\lambda_{s}^{A}\Delta_{O}^{A})}$

for all values of $\Delta_{O}^{A},\lambda_{a}^{A}$ and $\lambda_{s}^{A}$ such
that $\lambda_{a}^{A}-\lambda_{s}^{A}\Delta_{O}^{A}\geq 0$. Taking
$\lambda_{a}^{A}-\lambda_{s}^{A}\Delta_{O}^{A}=\epsilon\geq 0$ we then have

$\displaystyle
m_{B}^{*}\geq\frac{2(\lambda_{a}^{A}-\epsilon)}{(2\lambda_{a}^{A}-\epsilon)}\quad\forall\epsilon\geq
0,\quad\forall\lambda_{a}^{A}\implies m_{B}^{*}\geq 1$

We arrive at a contradiction. So we must have $m_{B}^{*}=1$ and Proposition
III.2 is proven.

##  Appendix B Illustrating Algorithm 1

Let us suppose WLOG that the true $k_{A}>1$. We sequentially improve the lower
bound $\underline{k}_{A}$ and the upper bound $\bar{k}_{A}$ for our estimate
$\hat{k}_{A}$ using our ‘observations’. We initialize
$\underline{k}_{A}^{1}=0$ and $\bar{k}_{A}^{1}=\infty$. We will call
$[\underline{k}^{i}_{A},\bar{k}^{i}_{A}]$ our region of interest and
sequentially shrink this region until it has Lebesgue measure less than
$\delta$.

We will aim to find $n_{A}$ and $n_{B}$ that satisfies (9) and we note that
only the ratio $\frac{n_{B}}{n_{A}}$ needs to satisfy certain properties to do
so. Algorithm 1 in fact iterates over values of this ratio (which we indicate
by $\eta$). Initializing $\eta^{1}=1$, we ask receiver A if they will believe
the sender for the corresponding encoding strategy. We see in Figure 3 that
receiver $A$ indeed believes the sender and so our estimate
$\hat{k}_{A}>\eta^{1}$. Accordingly we update the lower threshold as
$\underline{k}_{A}^{2}=\eta^{1}$.

$\displaystyle\dots$$\infty$$\displaystyle
0$$\displaystyle\eta^{1}$$\displaystyle k_{A}$ Receiver A believes. Receiver A
does not believe. Figure 3: (First iteration of Algorithm 1.) The first guess
for encoding strategy lies in the region where the receiver A believes.

We now generate our next guess for $\eta_{2}$ by bisecting the new region of
interest according to Line 8. Note that the upper threshold is currently
$\bar{k}_{A}^{2}=\infty$. In reality, we will use some large and finite value
for this upper threshold so this bisection step is well-defined. Having
obtained the next guess $\eta^{2}$, we will again evaluate whether the
receiver believes the sender. From Figure 4 we see that the receiver does not
and so we re-set the upper threshold as $\bar{k}_{A}^{3}=\eta^{2}$.

$\displaystyle\dots$$\infty$$\displaystyle
0$$\displaystyle\eta^{1}$$\displaystyle\eta^{2}$$\displaystyle k_{A}$ Receiver
A believes. Receiver A does not believe. Figure 4: (Second iteration of
Algorithm 1.) The second guess for encoding strategy lies in the region where
the receiver A does not believes.

We carry on bisecting our regions of interest to obtain subsequent guesses
$\eta^{i}$ and the algorithm terminates when the region of interest
$[\underline{k}_{A}^{i},\bar{k}_{A}^{i}]$ is small enough.