# Changing Simplistic Worldviews††thanks: We would like to thank Ole Jann,
Artyom Jelnov, Pavel Kocourek, Fedor Sandomirskiy, and Jan Zapal for their
valuable comments and suggestions.

Maxim Senkov
European Research University, U Haldy 200/18, 700 30 Ostrava, Czech Republic.
email<EMAIL_ADDRESS>Toygar T. Kerman Institute of Economics,
Corvinus University of Budapest, Fővám tér 8, 1093 Budapest, Hungary. email:
<EMAIL_ADDRESS>ORCID: 0000-0003-3038-3666. The author
acknowledges funding by the Hungarian National Research, Development and
Innovation Office, Project Number K-143276.

###### Abstract

We study a Bayesian persuasion model with two-dimensional states of the world,
in which the sender (she) and receiver (he) have heterogeneous prior beliefs
and care about different dimensions. The receiver is a naive agent who has a
_simplistic worldview_ : he ignores the dependency between the two dimensions
of the state. We provide a characterization for the sender’s gain from
persuasion both when the receiver is naive and when he is rational. We show
that the receiver benefits from having a simplistic worldview if and only if
it makes him perceive the states in which his interest is aligned with the
sender as less likely.

Keywords: Bayesian persuasion; misspecified prior; correlation neglect

JEL Classification: D82, D83, D91

##  1 Introduction

Standard models of Bayesian persuasion assume that the sender (she) and
receiver(s) (he) care about the same dimension of the true state of the world.
For example, in their seminal paper Kamenica and Gentzkow (2011) consider a
prosecutor (sender) and a judge (receiver), both of whom are only interested
in one dimension of the true state, that is whether the defendant is guilty or
innocent. What if the sender and receiver are concerned about different
dimensions of the true state?

Consider a company that wishes to build a new factory and only cares about the
project’s profitability. In order to be perceived as environmentally friendly,
the company asks a sustainability consultant to evaluate the project and
provide a recommendation on whether to continue or terminate it. However, the
consultant only cares about a different aspect of the project: its
sustainability. How should the consultant conduct her research to convince the
company to continue the project only when it is sustainable?

When considering multidimensional states of the world, however, a behavioral
aspect emerges: thinking about how multiple issues relate to each other is
cognitively demanding. Experimental evidence suggests that people find it
challenging to work with joint distributions of random variables and tend to
underestimate or ignore the correlation between state variables or observed
signals (Kallir and Sonsino, 2009; Eyster and Weizsäcker, 2011; Enke and
Zimmermann, 2019).

In this paper, we study how the receiver’s _simplistic worldview_ (i.e.
treating dependent state variables as independent) affects the sender’s
ability to influence his decision. We extend Kamenica and Gentzkow (2011) by
$(i)$ allowing for two-dimensional states of the world and $(ii)$ considering
a receiver who ignores the dependency of state variables (which leads to
heterogeneous priors). We model the receiver’s disregard of state dependency
as a misspecified prior belief; while the receiver knows the correct marginal
distribution for each dimension, he assumes that they are independent. We
characterize optimal disclosure both when the receiver is _rational_ (i.e.
does not ignore dependency) and _naive_ (i.e. assumes that dimensions are
independent), and analyze the welfare effects of the receiver’s simplistic
worldview.

###  1.1 Illustrative example

Let us return to our example of a company and sustainability consultant. Both
the company and the consultant are initially uncertain whether the project is
_profitable_ $(P)$ or _loss-making_ $(L)$ and _sustainable_ $(S)$ or
_unsustainable_ $(U)$. The company chooses to either _continue_ or _terminate_
the project and only cares about its profitability. Continuing the project
when it is profitable and terminating it when it is loss-making provide the
company a utility of 1, otherwise, the company’s utility is 0. On the other
hand, the consultant only cares about the sustainability of the project.
Continuing the project when it is sustainable and terminating it when it is
unsustainable provide the consultant a utility of 1, otherwise the
consultant’s utility is 0.

Hence, the true state of the world has two dimensions, and the conflict of
interest depends on the joint distribution of the dimensions. First, suppose
that the company is rational. That is, it shares a common prior with the
consultant, where the prior distribution of the states is given as follows
(with marginal distributions on the sides).

| $L$ | $P$ |
---|---|---|---
$U$ | 0.25 | 0.1 | 0.35
$S$ | 0.35 | 0.3 | 0.65
| 0.6 | 0.4 |
Table 1: Prior belief distribution.

The company asks the consultant to investigate the profitability and
sustainability of the project and provide a recommendation to either
_continue_ $(c)$ or _terminate_ $(t)$ it. Suppose that if the company is
indifferent between continuing and terminating the project, it continues.
Since the project is initially more likely to be loss-making, the company
would (ex ante) prefer the project to be terminated (which yields expected
payoff $(0.65)\cdot 0+(0.35)\cdot 1=0.35$ to the consultant and $(0.6)\cdot
1+(0.4)\cdot 0=0.6$ to the company). However, the consultant can improve her
expected payoff by strategically disclosing information about the true state
of the world. Mathematically, the information disclosure can be represented by
a _signal_ $\pi$ that sends recommendations contingent on the state of the
world. Consider an optimal signal given as follows.

$\pi$ | $(U,L)$ | $(U,P)$ | $(S,L)$ | $(S,P)$
---|---|---|---|---
$c$ | 0 | 0.5 | 1 | 1
$t$ | 1 | 0.5 | 0 | 0

First, observe that $\pi$ recommends continuing with probability 1 when the
project is both sustainable and profitable (i.e. in state $(S,P)$) and
recommends terminating with probability 1 when the project is both
unsustainable and loss-making (i.e. in state $(U,L)$). This is intuitive since
in these states the preferred actions of the sender and receiver are aligned.
However, observe that $\pi$ recommends to continue the project with
probability $1$ also when it is sustainable and loss-making.111Note that the
optimal signal is not unique. The consultant chooses the probability of $c$ in
state $(U,P)$ such that the company’s posterior belief that the project is
profitable is exactly $1/2$ and thus it chooses to continue. The consultant’s
expected payoff of employing $\pi$ is $(0.3)\cdot 1+(0.35)\cdot
1+0.1\cdot(0.5)+(0.25)\cdot 1=0.95$, improving upon the case of no disclosure.
Similarly, the company’s expected payoff from $\pi$ is $(0.3)\cdot
1+(0.35)\cdot 0+(0.1)\cdot(0.5)+(0.25)\cdot 1=0.6$, which is equal to the case
of no disclosure.

Now suppose that the company is naive and ignores the dependency between the
states of the world due to insufficient knowledge of sustainability. In other
words, the company assumes that the dimensions of the true state are
independent, and thus has a misspecified prior. While the company knows the
marginal distributions $\Pr(S)=0.65$ and $\Pr(P)=0.4$, it assumes that
$\Pr(S,P)=\Pr(S)\Pr(P)=(0.65)\cdot(0.4)=0.26$.222That is, the company assumes
that $\Pr(P)=\Pr(P|S)$. The company’s _simplistic worldview_ regarding the
prior distribution is given as follows.

| $L$ | $P$ |
---|---|---|---
$U$ | 0.21 | 0.14 | 0.35
$S$ | 0.39 | 0.26 | 0.65
| 0.6 | 0.4 |
Table 2: Misspecified prior belief distribution.

Notice that the company now perceives states $(S,L)$ and $(U,P)$ as relatively
more likely than before. Therefore, the sender has two options to keep the
recommendation persuasive: $(i)$ recommend to terminate more often than before
when the project is loss-making (i.e. decrease probability of $c$ in $(S,L)$)
or $(ii)$ recommend to continue more often than before when the project is
profitable (i.e. increase probability of $c$ in $(U,P)$). In other words, the
consultant needs to make the recommendation to continue _more informative_
regarding the project’s profitability. It turns out that given the
misspecified prior of the receiver, it is optimal for the sender to follow the
second route.

$\hat{\pi}$ | $(U,L)$ | $(U,P)$ | $(S,L)$ | $(S,P)$
---|---|---|---|---
$c$ | 0 | 0.93 | 1 | 1
$t$ | 1 | 0.07 | 0 | 0

Indeed, when the project is unsustainable and profitable, the recommendation
to continue is sent with probability $0.93$, while it was sent with
probability $0.5$ in this state before. The consultant’s expected payoff of
employing $\hat{\pi}$ is $(0.3)\cdot 1+(0.35)\cdot
1+(0.1)\cdot(0.07)+(0.25)\cdot 1=0.907$, still improving upon the case of no
disclosure, albeit less than when the receiver is rational. On the other hand,
the company’s expected payoff from $\hat{\pi}$ is $(0.3)\cdot 1+(0.35)\cdot
0+(0.1)\cdot(0.93)+(0.25)\cdot 1=0.643$, which is higher relative to the case
of having a correct prior belief (and also higher relative to the case of no
disclosure).333We use misspecified model framework, which implies existence of
correct prior belief (corresponding here to the sender’s belief). Thus, the
naive receiver’s expected payoff is evaluated using the correct (sender’s)
prior belief. Hence, while the company’s simplistic worldview _benefits_ the
company, it is _detrimental_ to the consultant.

The example illustrates how the receiver _benefits_ from ignoring dependency
between the state variables. However, both $(i)$ the sender’s approach
regarding disclosure in states with misaligned preferences and $(ii)$ whether
the receiver benefits from ignoring dependency crucially depend on the prior
belief distribution. In fact, the receiver might also be _worse off_ due to
ignoring the dependency of states. In our analysis, we consider all possible
prior belief distributions, provide a full characterization of the sender’s
gain from persuasion, and pin down the conditions under which the receiver
benefits from having a simplistic worldview.

###  1.2 Related literature

Our paper extends Kamenica and Gentzkow (2011) in two aspects and contributes
to three strands of literature. First, it relates to the literature on
misspecified models. The receiver in our model knows the correct marginal
distribution of each dimension, yet, assumes that the dimensions are
independent. Hence, the receiver possesses a misspecified prior belief about
the states of the world. This naturally relates to studies that consider
correlation neglect, as they motivate the behavioral assumption we make about
the receiver. One main difference from such models (e.g. Levy, Barreda, and
Razin (2022)) is that our receiver ignores dependency between _states_ ,
whereas most papers in the literature focus on the receiver neglecting
correlation between _messages_.444Eyster and Rabin (2005) introduce _cursed
equilibrium_ which assumes that players underestimate the correlation between
other players’ actions and information.

Our findings share similarities with those from studies focusing on the
political applications of correlation neglect. Ortoleva and Snowberg (2015)
show that correlation neglect might lead to overconfidence in voters and that
overconfidence is an important predictor for ideological extremeness. In our
model, the misspecified prior belief of the receiver leads to the receiver
having over-optimistic or over-pessimistic beliefs about the state of the
world. Levy and Razin (2015) show that voters neglecting the correlation
between their information sources might contribute to higher political
polarization in society, howbeit, voters might benefit from it. Similarly, we
illustrate that if the receiver ignores dependency between states then the
optimal signal might be more informative.555This also relates to studies that
investigate how a receiver can strategize to increase the informativeness of
the signal (Austen-Smith and Banks, 2000; Kartik, 2007; Ambrus and Egorov,
2017; Tsakas, Tsakas, and Xefteris, 2021) and how different kinds of biases
might affect persuasion (Hagmann and Loewenstein, 2017; Augias and Barreto,
2020; de Clippel and Zhang, 2022).

Our behavioral assumption regarding the receiver alters his perception of the
joint belief distribution. Yet, the receiver has no ambiguity about the
message he observes. Eliaz, Spiegler, and Thysen (2021a) consider a sender who
can redact his message once it is chosen, but the receiver only knows the
message strategy and not the redaction strategy. In a related study, Eliaz,
Spiegler, and Thysen (2021b) consider a sender who can attach interpretations
to messages. Unlike our model where the receiver’s perception of the signal
structure remains correct, both studies assume a distortion in the perception
of the signal structure.

Second, our paper contributes to the literature on Bayesian persuasion with
heterogeneous priors. Alonso and Câmara (2016) assume that the sender and
receiver have subjective prior beliefs and analyze its effect on the set of
inducible posterior beliefs. Galperti (2019) focuses on a similar setup and
considers receivers who might be reluctant to change their worldviews. In a
related paper, Kosterina (2022) considers a sender who is ignorant about the
receiver’s prior and characterizes optimal information structures. In contrast
to these studies, our model posits that the sender is aware of the accurate
prior distribution while the receiver is not, and this discrepancy is common
knowledge. Moreover, we contrast the receiver’s payoff to the case in which he
has the correct prior belief and and study the implications of incorrect prior
for welfare.

Third, our study relates to the literature on persuasion with multidimensional
state variables. Rayo and Segal (2010) consider a model of information
disclosure and assume that the agent is rational, in which the sender’s
expected payoff is the product of posterior mean of the sender’s and
receiver’s dimension of state, and characterize optimal disclosure. Tamura
(2018) studies optimal disclosure in a more general setup similar to the model
in Rayo and Segal (2010). In a related study, Dworczak and Kolotilin (2019)
characterize the optimal persuasion mechanism when the state is two-
dimensional and the objective is quadratic. Malamud and Schrimpf (2021) study
a model where the sender observes the multidimensional state and show that it
is optimal for the sender to “reduce” the dimensions of the state. All studies
above consider a rational sender and receiver. In contrast, our paper
considers implications of the receiver’s ignorance regarding the connection
between the dimensions of state for the optimal signal and welfare.
Babichenko, Talgam-Cohen, Xu, and Zabarnyi (2022) consider a sender who
provides information to a receiver about a product, where each dimension
represents a different attribute of the product. The authors extend the
standard model also by assuming the sender does not know the utility of the
receiver and wishes to minimize regret and show that ignorance of the utility
of the receiver is extremely harmful to the sender. We, on the other hand,
assume that agents have complete information about the utility functions, yet,
the receiver incorrectly assumes independence of the dimensions.

##  2 Preliminaries

###  2.1 The model

A sender (she) wishes to persuade a receiver (he) to take an action depending
on the true state of the world. Let $\Omega=\Sigma\times R$ be the set of
_states_ of the world, such that the sender’s payoff depends only on
$\Sigma=\\{\sigma_{0},\sigma_{1}\\}$ and the receiver’s payoff depends only on
$R=\\{\rho_{0},\rho_{1}\\}$. For each $k,\ell\in\\{0,1\\}$, let
$\omega_{k\ell}\equiv(\sigma_{k},\rho_{\ell})\in\Omega$.

A _signal_ consists of a finite set of _messages_ $M$ and a family of
distributions $\\{\pi(\cdot|\omega)\\}_{\omega\in\Omega}$ over $M$. That is,
$\pi:\Omega\rightarrow\Delta(M)$ maps each state of the world to a probability
distribution over $M$. Denote the set of all signals by $\Pi$.

We start by considering a rational receiver, hence, the sender and receiver
share a common prior belief $\mu\in\Delta(\Omega)$ about the true state of the
world. For each $k,\ell\in\\{0,1\\}$, let
$\mu_{k\ell}\equiv\Pr(\sigma_{k},\rho_{\ell})$ and let the marginal
distributions be denoted as $\mu(\sigma_{k})\equiv\Pr(\sigma_{k})$ and
$\mu(\sigma_{\ell})\equiv\Pr(\rho_{\ell})$. The distribution $\mu$ is
presented in Table 3 with marginal distributions given on the sides.

$(\Sigma,\mathrm{P})$ | $\rho_{0}$ | $\rho_{1}$ |
---|---|---|---
$\sigma_{0}$ | $\mu_{00}$ | $\mu_{01}$ | $\mu(\sigma_{0})$
$\sigma_{1}$ | $\mu_{10}$ | $\mu_{11}$ |
| $\mu(\rho_{0})$ |  |
Table 3: Common prior $\mu$.

Given $\mu\in\Delta(\Omega)$ and $\pi\in\Pi$, a message $m\in M^{\pi}$
generates the posterior belief $\mu^{m}\in\Delta(\Omega)$. For each
$k,\ell\in\\{0,1\\}$, the marginal posterior beliefs upon observing message
$m$ are given by $\mu^{m}(\sigma_{k})$ and $\mu^{m}(\rho_{\ell})$, where

$\mu^{m}(\rho_{\ell})=\frac{\mu_{k\ell}\pi(m|\omega_{k\ell})+\mu_{\ell\ell}\pi(m|\omega_{\ell\ell})}{\mu_{k\ell}\pi(m|\omega_{k\ell})+\mu_{\ell\ell}\pi(m|\omega_{\ell\ell})+\mu_{\ell
k}\pi(m|\omega_{\ell k})+\mu_{kk}\pi(m|\omega_{kk})}$

and $\mu^{m}(\sigma_{k})$ is given analogously.666We drop $\pi$ when denoting
$\mu^{m}(\sigma_{k})$ and $\mu^{m}(\rho_{\ell})$ to ease notation. We denote
the posterior belief for the joint distribution upon observing message $m$ by
$\mu^{m}(\sigma_{k},\rho_{\ell})$.

The set of actions that are available to the receiver is given by
$A=\\{0,1\\}$. The receiver wishes his action to match the dimension of the
state he cares about, i.e., the receiver wants to choose 0 when the state is
$(\sigma_{k},\rho_{0})$ and wants to choose 1 when the state is
$(\sigma_{k},\rho_{1})$, for any $k\in\\{0,1\\}$. The receiver’s utility
function $u:A\times R\rightarrow\\{0,1\\}$ is given by

$u\left(a,\rho\right)=\begin{cases}1&\text{if
}\left(a,\rho\right)=\left(0,\rho_{0}\right)\text{ or
}\left(a,\rho\right)=\left(1,\rho_{1}\right),\\\
0&\text{otherwise.}\end{cases}$

Given the receiver’s preferences, it follows that the receiver is indifferent
between $0$ and $1$ if $\mu^{m}(\rho_{0})=\mu^{m}(\rho_{1})=1/2$.

The sender wishes the chosen action to match the dimension of the state she
cares about. The sender’s utility function
$v:A\times\Sigma\rightarrow\\{0,1\\}$ is given by

$v\left(a,\sigma\right)=\begin{cases}1&\text{if
}\left(a,\sigma\right)=\left(0,\sigma_{0}\right)\text{ or
}\left(a,\sigma\right)=\left(1,\sigma_{1}\right),\\\
0&\text{otherwise.}\end{cases}$

We assume that the the default action of the receiver is $0$, i.e.
$\mu(\rho_{1})<\mu(\rho_{0})$. Given $\pi\in\Pi$, the _choice rule_
$\alpha^{\pi}:M\rightarrow A$ of the receiver is given by

$\alpha^{\pi}\left(m\right)=\begin{cases}1&\text{if
}\mu^{m}(\rho_{1})\geq\frac{1}{2},\\\ 0&\text{otherwise.}\end{cases}$ (1)

That is, the receiver switches his action when he is indifferent upon
observing a message.

###  2.2 Benchmark

We start by characterizing optimal disclosure when the receiver does not
ignore the dependency of states, i.e., when he is rational. This will serve as
a benchmark for the next sections.

First, note that since both the receiver’s action set and the payoff-relevant
state are binary, we can restrict attention to _direct_ signals without loss
of generality.777A signal is _direct_ if it holds that ($i$) $M\subseteq A$
and ($ii$) for all $a\in M$, $\alpha^{\pi}(a)=a$. Therefore, we hereon assume
that $M=\\{0,1\\}$ and that $\Pi$ is the set of all direct signals. We denote
a signal by $\pi=(\pi_{00},\pi_{01},\pi_{10},\pi_{11})$ where
$\pi_{k\ell}\equiv\pi(1|\omega_{k\ell})$, for $k,\ell\in\\{0,1\\}$.

Depending on the prior, the sender may choose $\pi\in\Pi$ to induce a switch
of the receiver’s default action $0$ to $1$. For this to be successful,
message $1$ should induce a posterior belief $\mu^{1}(\rho_{1})\geq 1/2$. We
formalize the obedience constraint in the following lemma.

######  Lemma 1.

Given $\pi\in\Pi$, the receiver chooses action $1$ if and only if

$\mu_{00}\pi_{00}+\mu_{10}\pi_{10}\leq\mu_{01}\pi_{01}+\mu_{11}\pi_{11}.$

Note that Lemma 1 implies that the obedience constraint for action $0$ given
by
$\mu_{01}(1-\pi_{01})+\mu_{11}(1-\pi_{11})<\mu_{00}(1-\pi_{00})+\mu_{10}(1-\pi_{10})$
is also satisfied. See Lemma A1 in the appendix for the formal statement and
proof.

We can now state the sender’s optimization problem, which is to maximize the
ex-ante expected utility by choosing a signal:

$\displaystyle\max_{\pi\in\Pi}\>\mu_{11}\pi_{11}+\mu_{10}\pi_{10}$
$\displaystyle-\mu_{01}\pi_{01}-\mu_{00}\pi_{00}$ s.t.
$\displaystyle\mu_{00}\pi_{00}+\mu_{10}\pi_{10}\leq\mu_{01}\pi_{01}+\mu_{11}\pi_{11}.$
(2)

Let $\pi^{*}$ denote an optimal signal when the receiver is rational.

######  Proposition 1.

An optimal signal when the receiver is rational is given by

$(\pi^{*}_{00},\pi^{*}_{01},\pi^{*}_{10},\pi^{*}_{11})=\begin{cases}(0,0,1,1)&\text{if
}\mu_{11}\geq\mu_{10},\\\ (0,\beta,\gamma,1)&\text{otherwise},\end{cases}$

where $\mu_{10}\gamma-\mu_{01}\beta=\mu_{11}$.

There are a few aspects to note about Proposition 1. First, it is intuitive
that $\pi^{*}_{00}=0$ and $\pi^{*}_{11}=1$, since in states
$(\sigma_{0},\rho_{0})$ and $(\sigma_{1},\rho_{1})$ the sender’s and
receiver’s preferences are aligned. In other words, it is optimal for the
sender to _disclose the truth_ in these states, i.e., she sends message $1$
with probability 0 in state $(\sigma_{0},\rho_{0})$ and with probability 1 in
state $(\sigma_{1},\rho_{1})$. More precisely, when the state is
$(\sigma_{1},\rho_{1})$ the sender has an incentive to _increase_ the
frequency of message $1$, which switches the receiver’s action. Similarly,
when the state is $(\sigma_{0},\rho_{0})$ the sender has an incentive to
_decrease_ the frequency of message $1$. Note that both increasing $\pi_{11}$
and decreasing $\pi_{00}$ _relaxes_ the obedience constraint in (2).

Second, observe that while the sender has an incentive to _decrease_
$\pi_{01}$ in state $(\sigma_{0},\rho_{1})$ and _increase_ $\pi_{10}$ in state
$(\sigma_{1},\rho_{0})$, this _tightens_ the obedience constraint since the
sender’s and receiver’s preferences are misaligned. Therefore, the sender
chooses $\pi_{01}$ and $\pi_{10}$ such that the obedience constraint binds.
However, if the prior for $(\sigma_{1},\rho_{1})$ is (weakly) higher than the
prior for $(\sigma_{1},\rho_{0})$ (i.e. $\mu_{11}\geq\mu_{10}$), then the
effect of relaxing the constraint is stronger than the effect of tightening
the constraint. In this case, since $\pi^{*}_{01}=0$ and $\pi^{*}_{10}=1$, the
sender can implement her desired outcome with certainty and obtain a payoff of
1.

##  3 Optimal signal when the receiver is naive

From hereon we assume that the receiver is naive. More precisely, while the
receiver knows $(\mu(\sigma_{0}),\mu(\sigma_{1}))$ and
$(\mu(\rho_{0}),\mu(\rho_{1}))$, he perceives $\sigma$ and $\rho$ as
independent random variables. This gives rise to the misspecified prior
$\hat{\mu}$ about $\Omega$, where the receiver assumes that each element of
the support of $\hat{\mu}$ is the product of corresponding marginals, i.e.
$\hat{\mu}_{k\ell}=\mu^{\sigma}_{k}\mu^{\rho}_{\ell}$. We present the
misspecified prior in the following table with marginal distributions written
on the sides.

$(\Sigma,\mathrm{P})$ | $\rho_{0}$ | $\rho_{1}$ |
---|---|---|---
$\sigma_{0}$ | $\mu^{\sigma}_{0}\mu^{\rho}_{0}$ | $\mu^{\sigma}_{0}\mu^{\rho}_{1}$ | $\mu(\sigma_{0})$
$\sigma_{1}$ | $\mu^{\sigma}_{1}\mu^{\rho}_{0}$ | $\mu^{\sigma}_{1}\mu^{\rho}_{1}$ |
| $\mu(\rho_{0})$ |  |
Table 4: Misspecified prior $\hat{\mu}$.

We call such a misspecified prior $\hat{\mu}$ a _simplistic worldview_. Lemma
2 below demonstrates that the simplistic worldview relates to the correct
prior in a systematic way.

######  Lemma 2.

Let $\mu\in\Delta(\Omega)$ and $k,\ell\in\\{0,1\\}$. Let

$\hat{\mu}_{kk}-\mu_{kk}\equiv c.$

In this case,

$\hat{\mu}_{k\ell}-\mu_{k\ell}=-c,\text{ for }k\neq\ell.$

Lemma 2 shows that the amount of change in the priors about the states (due to
the simplistic worldview) in which preferences are aligned and misaligned is
the same but in opposite directions. That is, if the naive receiver perceives
the states in which preferences are aligned as more likely than the rational
receiver (i.e. if $c>0$), then he perceives the states in which preferences
are misaligned as less likely than the rational receiver, by the same amount
(and vice versa). Table 5 demonstrates the misspecified prior $\hat{\mu}$ in
terms of the correct prior $\mu$.

$(\Sigma,\mathrm{P})$ | $\rho_{0}$ | $\rho_{1}$ |
---|---|---|---
$\sigma_{0}$ | $\mu_{00}+c$ | $\mu_{01}-c$ | $\mu(\sigma_{0})$
$\sigma_{1}$ | $\mu_{10}-c$ | $\mu_{11}+c$ |
| $\mu(\rho_{0})$ |  |
Table 5: $\hat{\mu}$ expressed in terms of $\mu$.

The naive receiver observes the signal chosen by the sender, who knows that
the receiver is naive. Moreover, he updates his belief according to the signal
chosen by the sender and the misspecified prior $\hat{\mu}$. Let
$\hat{\mu}^{m}(\rho_{\ell})$ denote the posterior belief about state
$\rho_{\ell}$ when the receiver updates according to $\hat{\mu}$. Thus, the
receiver’s posterior that the state is $\rho_{1}$ upon observing message $1$
is

$\hat{\mu}^{1}(\rho_{1})=\frac{\hat{\mu}_{01}\pi_{01}+\hat{\mu}_{11}\pi_{11}}{\hat{\mu}_{00}\pi_{00}+\hat{\mu}_{10}\pi_{10}+\hat{\mu}_{01}\pi_{01}+\hat{\mu}_{11}\pi_{11}}.$
(3)

While the sender calculates her expected utility according to $\mu$, she knows
that the naive receiver updates according to $\hat{\mu}$. Therefore, the
sender’s optimization problem in (2) becomes

$\displaystyle\max_{\pi\in\Pi}\>\mu_{11}\pi_{11}+\mu_{10}\pi_{10}$
$\displaystyle-\mu_{01}\pi_{01}-\mu_{00}\pi_{00}$ s.t.
$\displaystyle\hat{\mu}_{00}\pi_{00}+\hat{\mu}_{10}\pi_{10}\leq\hat{\mu}_{01}\pi_{01}+\hat{\mu}_{11}\pi_{11}.$
(4)

Let $\hat{\pi}^{*}$ denote the optimal signal when the receiver is naive. Note
that in (4), similar to (2), the sender’s optimal choice in states
$(\sigma_{0},\rho_{0})$ and $(\sigma_{1},\rho_{1})$ relaxes the obedience
constraint and thus $\hat{\pi}^{*}_{00}=0$ and $\hat{\pi}^{*}_{11}=1$, which
allows us to simplify (4).

######  Lemma 3.

The sender’s simplified problem is given by

$\displaystyle\max_{\pi\in\Pi}\mu_{10}\pi_{10}$
$\displaystyle-\mu_{01}\pi_{01}$ (5) $\displaystyle s.t.$
$\displaystyle\quad\hat{\mu}_{10}\pi_{10}-\hat{\mu}_{01}\pi_{01}\leq\hat{\mu}_{11}.$

We now provide the full characterization of the optimal signal for a naive
receiver.

######  Proposition 2.

An optimal signal when the receiver is naive is given by

$(\hat{\pi}^{*}_{00},\hat{\pi}^{*}_{01},\hat{\pi}^{*}_{10},\hat{\pi}^{*}_{11})=\begin{cases}(0,0,1,1)&\text{if
}\>\hat{\mu}_{11}\geq\hat{\mu}_{10},\\\
(0,\beta,\gamma,1)&\text{otherwise},\end{cases}$

where

$\left(\beta,\gamma\right)=\begin{cases}\left(0,\frac{\hat{\mu}_{11}}{\hat{\mu}_{10}}\right)&\text{if
}\>\frac{\hat{\mu}_{01}}{\hat{\mu}_{10}}<\frac{\mu_{01}}{\mu_{10}},\\\
\left(1,\frac{\hat{\mu}_{01}+\hat{\mu}_{11}}{\hat{\mu}_{10}}\right)&\text{if
}\>\frac{\hat{\mu}_{01}}{\hat{\mu}_{10}}>\frac{\mu_{01}}{\mu_{10}}\text{ and
}\frac{\hat{\mu}_{01}+\hat{\mu}_{11}}{\hat{\mu}_{10}}\leq 1,\\\
\left(\frac{\hat{\mu}_{10}-\hat{\mu}_{11}}{\hat{\mu}_{01}},1\right)&\text{if
}\>\frac{\hat{\mu}_{01}}{\hat{\mu}_{10}}>\frac{\mu_{01}}{\mu_{10}}\text{ and
}\frac{\hat{\mu}_{01}+\hat{\mu}_{11}}{\hat{\mu}_{10}}\geq
1,\end{cases}\vspace{0.25cm}$

and $\hat{\mu}_{10}\gamma-\hat{\mu}_{01}\beta=\hat{\mu}_{11}$ if
$(\hat{\mu}_{01}/\hat{\mu}_{10})=(\mu_{01}/\mu_{10})$.

The optimal signal in Proposition 2 has several features similar to the one in
Proposition 1. First, the sender always recommends switching the default
action in state $(\sigma_{1},\rho_{1})$ (i.e. $\hat{\pi}^{*}_{11}=1$) and
never recommends switching in state $(\sigma_{0},\rho_{0})$ (i.e.
$\hat{\pi}_{00}^{*}=0$). Second, when $\hat{\mu}_{11}\geq\hat{\mu}_{10}$, the
sender can ensure that the receiver takes her desired action with probability
1 (i.e. $\hat{\pi}_{01}^{*}=0$ and $\hat{\pi}_{10}^{*}=1$). On the other hand,
if $\hat{\mu}_{11}<\hat{\mu}_{10}$ then $\hat{\pi}^{*}_{01}$ and
$\hat{\pi}^{*}_{10}$ are pinned down by the relation between the slope of the
obedience constraint $(\hat{\mu}_{01}/\hat{\mu}_{10})$ and the slope of the
objective function $(\mu_{01}/\mu_{10})$.

One very important difference from Proposition 1, however, is that when
$\mu_{10}\neq\mu_{01}$ (which implies
$(\hat{\mu}_{01}/\hat{\mu}_{10})\neq(\mu_{01}/\mu_{10})$) the optimal signal
in Proposition 2 is unique. This is a direct consequence of the receiver’s
naivete, as it leads to the sender and receiver having different priors. More
precisely, if they have the same prior then the sender is indifferent between
marginally increasing $\pi_{10}$ or marginally decreasing $\pi_{01}$. On the
other hand, if they have different priors then this indifference no longer
holds as the two options tighten the obedience constraint to a different
extent. Hence, the sender chooses the option that tightens the obedience
constraint less as to optimally persuade the receiver.

##  4 Welfare implications of a simplistic worldview

In this section, we proceed beyond characterizing the optimal signal and study
a number of general properties of the sender’s problem (4) in comparison to
the benchmark (2). This allows us to characterize the welfare implications of
the receiver’s naivete and explain the economic intuition behind them. Lemma 4
demonstrates how the receiver’s naivete affects the sender’s ability to
manipulate the receiver’s actions.

######  Lemma 4.

Assume that $c>0$.

  1. (i)

For any $\pi\in\Pi$ with $\pi_{00}=0$ and $\pi_{11}=1$ it holds that
$\hat{\mu}^{1}(\rho_{1})>\mu^{1}(\rho_{1})$. That is, recommendation 1 induces
a higher belief about state $\rho_{1}$ for a receiver with a simplistic
worldview than for a rational receiver.

  2. (ii)

For any $\pi\in\Pi$ it holds that
$(\hat{\mu}_{11}/\hat{\mu}_{10})+(\hat{\mu}_{01}/\hat{\mu}_{10})\pi_{01}>(\mu_{11}/\mu_{10})+(\mu_{01}/\mu_{10})\pi_{01}$.
That is, the obedience constraint in (5) is more relaxed relative to the
benchmark (2).

If $c<0$, the converse of the inequalities in $(i)$ and $(ii)$ hold.

The intuition of Lemma 4 is as follows. First, if $c>0$ then by Lemma 2 the
naive receiver perceives the aligned states $\left(\sigma_{0},\rho_{0}\right)$
and $\left(\sigma_{1},\rho_{1}\right)$ as more likely (and the two misaligned
states as less likely) than the rational receiver. Hence, the naive receiver
associates recommendation 1 with $\rho_{1}$ more than the rational receiver
and thus has a higher posterior belief.

Part $(ii)$ of the lemma follows easily from $(i)$. Since the sender can
induce a posterior belief of at least 1/2 that the state is $\rho_{1}$ when
the receiver is naive by recommending 1 more frequently than before, the
obedience constraint in (5) is more relaxed relative to the benchmark (2). It
is easy to see that the converse of the above arguments hold if $c<0$.

Importantly, Lemma 4 has direct implications on the sender’s and receiver’s
welfare as it pins down the conditions under which it is “easier” for the
sender to persuade a naive receiver. In particular, if $c>0$ then by $(ii)$
the obedience constraint is more slack than in the benchmark. Hence, the
sender can decrease $\pi_{01}$ and increase $\pi_{10}$ so that the receiver
chooses the sender-preferred actions in the misaligned states
$(\sigma_{0},\rho_{1})$ and $(\sigma_{1},\rho_{0})$ with a higher probability.
Clearly, this benefits the sender. Moreover, since total welfare is the same
at the optimum in both cases, it follows that naivete harms the receiver.
Intuitively, converse arguments hold if $c<0$.

The welfare implications of the receiver’s simplistic worldview are summarized
in Proposition 3.

######  Proposition 3.

Let $c>0$. A comparison of expected utilities yields the following:

  1. (i)

The receiver’s simplistic worldview harms him and benefits the sender.

  2. (ii)

The differences in their expected utilities are strict if and only if
$\mu_{11}<\mu_{10}$.

If $c<0$, then the converse of $(i)$ holds and the differences in their
expected utilities are strict if and only if $\hat{\mu}_{11}<\hat{\mu}_{10}$.

Part $(ii)$ of Proposition 3 follows from the structure of the optimal
signals. If the sender is facing a rational receiver then she can get her
preferred outcome with certainty whenever $\mu_{11}\geq\mu_{10}$. Hence, in
this case if $c>0$ the sender can only weakly benefit from the receiver being
naive. Otherwise, the sender can always guarantee a strict improvement. On the
other hand, if $c<0$ and the sender is facing a naive receiver then she can
get her preferred outcome with certainty whenever
$\hat{\mu}_{11}\geq\hat{\mu}_{10}$. But since the receiver’s naivete is
detrimental for the sender if $c<0$, it follows that the sender obtains her
preferred outcome with certainty also when the receiver is rational. In this
case, the sender is only (weakly) harmed by the receiver’s naivete.

Proposition 3 can also be interpreted in terms of the receiver’s misperception
of ex ante preference alignment. Recall that by Lemma 2 if $c>0$ then the
naive receiver perceives the aligned states as more likely than the rational
receiver. Therefore, Lemma 4 implies that it is easier for the sender to
persuade the receiver. Hence, _naivete makes the receiver more manipulable and
is detrimental_. Intuitively, if $c<0$ the naivete of the receiver constrains
the sender more and thus the “correct” recommendation is sent with a higher
probability. In this case, _naivete makes the receiver less manipulable and is
beneficial_.

##  5 Conclusion

Real-world examples of Bayesian persuasion problems, such as corporate
consulting, often involve the sender and receiver caring about different, but
connected issues of a problem. This paper studies the impact of the receiver’s
naivete regarding the interconnected nature of the two dimensions of the state
of the world on the optimal outcomes for both parties. First, as a benchmark,
we study the case of a rational receiver, i.e., the sender and receiver both
know the correct prior distribution of the states. We show that regardless of
the receiver being rational or naive, it is optimal for the sender to
truthfully disclose the states whenever her preference is aligned with the
receiver. On the other hand, optimal disclosure in states where preferences
are misaligned crucially depends on the receiver’s perception of the prior.

Our findings reveal that the receiver’s naivete can disadvantage him while
favoring the sender, particularly when it leads the receiver to overestimate
the likelihood of states where both parties prefer the same action and
underestimate states where they have conflicting preferences. Intuitively,
this makes the receiver “easier” to persuade; the sender can commit to sending
the “wrong” message with a higher probability. Conversely, if the naivete
leads the receiver to perceive aligned states as less likely and misaligned
states as more probable, the receiver benefits at the expense of the sender.

##  Appendix A Proofs

_Proof of Lemma 1_. For the obedience constraint to be satisfied, it must hold
that $\mu^{1}(\rho_{1})\geq 1/2$, which is equivalent to

$\frac{\mu_{01}\pi_{01}+\mu_{11}\pi_{11}}{\mu_{01}\pi_{01}+\mu_{11}\pi_{11}+\mu_{10}\pi_{10}+\mu_{00}\pi_{00}}\geq\frac{1}{2}.$

Rearranging the inequality yields the obedience constraint.∎

######  Lemma A1.

Let $\pi\in\Pi$. If
$\mu_{00}\pi_{00}+\mu_{10}\pi_{10}\leq\mu_{01}\pi_{01}+\mu_{11}\pi_{11}$, then
$\mu_{01}(1-\pi_{01})+\mu_{11}(1-\pi_{11})<\mu_{00}(1-\pi_{00})+\mu_{10}(1-\pi_{10})$.

###### Proof.

Let $\mu_{00}\pi_{00}+\mu_{10}\pi_{10}\leq\mu_{01}\pi_{01}+\mu_{11}\pi_{11}$,
which implies
$-\mu_{01}\pi_{01}-\mu_{11}\pi_{11}\leq-\mu_{00}\pi_{00}-\mu_{10}\pi_{10}$.
Recall that $\mu(\rho_{1})<\mu(\rho_{0})$, i.e.
$\mu_{01}+\mu_{11}<\mu_{00}+\mu_{10}$. Hence, it follows that

$\mu_{01}+\mu_{11}-\mu_{01}\pi_{01}-\mu_{11}\pi_{11}<\mu_{00}+\mu_{10}-\mu_{00}\pi_{00}-\mu_{10}\pi_{10}.$

Rewriting the above inequality provides the desired result. Hence, the
obedience constraint for $0$ is satisfied. That is,

$\mu^{0}(\rho_{0})>\frac{\mu_{10}(1-\pi_{10})+\mu_{00}(1-\pi_{00})}{\mu_{10}(1-\pi_{10})+\mu_{00}(1-\pi_{00})+\mu_{01}(1-\pi_{01})+\mu_{11}(1-\pi_{11})}>\frac{1}{2}.$

∎

_Proof of Proposition 1_. First suppose that $\mu_{10}\leq\mu_{11}$. It is
clear that setting
$\pi^{*}=(\pi^{*}_{00},\pi^{*}_{01},\pi^{*}_{10},\pi^{*}_{11})=(0,0,1,1)$
satisfies the obedience constraint in (2) and maximizes the sender’s expected
utility since it provides the upper bound of 1. Hence, $\pi^{*}$ is the unique
solution to the sender’s optimization problem.

Now suppose that $\mu_{10}>\mu_{11}$. First observe that it is optimal in this
case to set $\pi^{*}_{11}=1$ and $\pi^{*}_{00}=0$ as well, since it increases
the sender’s expected utility and relaxes the obedience constraint. Hence,
substituting $\pi^{*}_{11}=1$ and $\pi^{*}_{00}=0$ into the sender’s
optimization problem and simplifying we obtain

$\displaystyle\max_{\pi\in\Pi}\>\mu_{10}\pi_{10}$
$\displaystyle-\mu_{01}\pi_{01}$ (6) s.t.
$\displaystyle\mu_{10}\pi_{10}\leq\mu_{01}\pi_{01}+\mu_{11}.$

It is easy to see from (6) that the objective function is increasing in
$\pi_{10}$ and decreasing in $\pi_{01}$. Hence, if (6) is not binding then the
sender can marginally increase $\pi_{10}$ or decrease $\pi_{01}$ to achieve a
higher expected payoff. Thus, the constraint is binding at the optimum.

Therefore, since both the objective function and the constraint are linear (in
$\pi_{10}$ and $\pi_{01}$), there exists a continuum of solutions to (6).∎

_Proof of Lemma 2_. Let $\hat{\mu}_{kk}-\mu_{kk}=c$ and assume that
$k\neq\ell$. Then,

$\displaystyle\hat{\mu}_{k\ell}-\mu_{k\ell}=\mu(\sigma_{k})(1-\mu(\rho_{k}))-\mu_{k\ell}$
$\displaystyle=\mu_{k\ell}+\mu_{kk}-\mu(\sigma_{k})\mu(\rho_{k})-\mu_{k\ell}$
$\displaystyle=\mu(\sigma_{k})\mu(\rho_{k})-c-\mu(\sigma_{k})\mu(\rho_{k})=-c.$

∎

_Proof of Lemma 3_. As one can observe from (4), the sender’s payoff is
increasing in $\pi_{11}$, which relaxes the obedience constraint. Thus,
$\hat{\pi}_{11}^{*}=1$. Similar reasoning leads to $\hat{\pi}_{00}^{*}=0$.
Substituting $\hat{\pi}_{00}^{*}=0,\hat{\pi}_{11}^{*}=1$ into (4) yields the
result. ∎

_Proof of Proposition 2_. The sender’s problem is given by

$\displaystyle\max_{\pi\in\Pi}\mu_{10}\pi_{10}$
$\displaystyle-\mu_{01}\pi_{01}$ $\displaystyle s.t.$
$\displaystyle\quad\hat{\mu}_{10}\pi_{10}-\hat{\mu}_{01}\pi_{01}\leq\hat{\mu}_{11}.$

As the problem is a linear program, the solution lies either on the obedience
constraint or on one of the boundaries of the set $\Pi$. First, suppose that
$\hat{\mu}_{11}\geq\hat{\mu}_{10}$. Then the solution lies on the boundary and
the obedience constraint is inactive for $\hat{\mu}_{11}>\hat{\mu}_{10}$. It
follows easily that $\pi_{01}^{*}=0,\pi_{10}^{*}=1$.

Next, suppose that $\hat{\mu}_{11}<\hat{\mu}_{10}$. All three possible types
of solutions for this case are presented in Figure 1. We proceed with
characterizing each of the three cases.

Figure 1: Three possible forms of solutions to the problem (5). Indifference
curves are in blue and dashed. Active constraints are in orange.

Case (a) The active constraints are $\pi_{01}\geq 0$ and
$\hat{\mu}_{10}\pi_{10}-\hat{\mu}_{01}\pi_{01}\leq\hat{\mu}_{11}$. Hence,
$\hat{\pi}_{01}^{*}=0$ and $\hat{\pi}_{10}^{*}$ solves
$\hat{\mu}_{10}\pi_{10}-\hat{\mu}_{01}\pi_{01}=\hat{\mu}_{11}$, i.e.
$\pi^{*}_{10}=\hat{\mu}_{11}/\hat{\mu}_{10}$. The solution to (5) has this
particular form if and only if the slope of the objective function is steeper
than the slope of the obedience constraint, i.e.
$(\hat{\mu}_{01}/\hat{\mu}_{10})<(\mu_{01}/\mu_{10})$.

Case (b) The active constraints are $\pi_{01}\leq 1$ and
$\hat{\mu}_{10}\pi_{10}-\hat{\mu}_{01}\pi_{01}\leq\hat{\mu}_{11}$. Hence,
$\hat{\pi}_{01}^{*}=1$ and $\hat{\pi}_{10}^{*}$ solves
$\hat{\mu}_{10}\pi_{10}-\hat{\mu}_{01}\pi_{01}=\hat{\mu}_{11}$, i.e.
$\hat{\pi}_{10}^{*}=(\hat{\mu}_{01}+\hat{\mu}_{11})/\hat{\mu}_{10}$. The
solution to (5) has this form if and only if (i)
$(\hat{\mu}_{01}/(\hat{\mu}_{10})>(\mu_{01}/\mu_{10})$ (i.e. the slope of the
objective function is flatter than the slope of the obedience constraint) and
(ii) $(\hat{\mu}_{01}+\hat{\mu}_{11})/\hat{\mu}_{10}\leq 1$ (which implies
that the constraint $\pi_{10}\leq 1$ is inactive).

Case (c) The active constraints are $\pi_{10}\leq 1$ and
$\hat{\mu}_{10}\pi_{10}-\hat{\mu}_{01}\pi_{01}\leq\hat{\mu}_{11}$. Hence,
$\hat{\pi}_{10}^{*}=1$ and $\hat{\pi}_{01}^{*}$ solves
$\hat{\mu}_{10}\pi_{10}-\hat{\mu}_{01}\pi_{01}=\hat{\mu}_{11}$, i.e.
$\hat{\pi}_{01}^{*}=(\hat{\mu}_{10}-\hat{\mu}_{11})/\hat{\mu}_{01}$. The
solution to (5) has this form if and only if (i)
$\hat{\mu}_{01}/\hat{\mu}_{10}>\mu_{01}/\mu_{10}$, and (ii)
$(\hat{\mu}_{01}+\hat{\mu}_{11})/\hat{\mu}_{10}\geq 1$ (which implies that the
constraint $\pi_{10}\leq 1$ is active).

Finally, if $(\hat{\mu}_{01}/\hat{\mu}_{10})=(\mu_{01}/\mu_{10})$, then any
$\pi_{10}$ and $\pi_{01}$ such that
$\hat{\mu}_{10}\pi_{10}-\hat{\mu}_{01}\pi_{01}=\hat{\mu}_{11}$ solves (5). ∎

_Proof of Lemma 4_. We begin by proving part $(i)$. Since it is optimal to set
$\hat{\pi}^{*}_{00}=0$ and $\hat{\pi}^{*}_{11}=1$, (3) can be rewritten as

$\hat{\mu}^{1}\left(\rho_{1}\right)=\frac{\hat{\mu}_{01}\pi_{01}+\hat{\mu}_{11}}{\hat{\mu}_{01}\pi_{01}+\hat{\mu}_{11}+\hat{\mu}_{10}\pi_{10}}.$

By Lemma 2, it follows that

$\displaystyle\hat{\mu}^{1}\left(\rho_{1}\right)$
$\displaystyle=\frac{\left(\mu_{01}-c\right)\pi_{01}+\left(\mu_{11}+c\right)}{\left(\mu_{01}-c\right)\pi_{01}+\left(\mu_{11}+c\right)+\left(\mu_{10}-c\right)\pi_{10}}$
$\displaystyle=\frac{\mu_{01}\pi_{01}+\mu_{11}+c\left(1-\pi_{01}\right)}{\mu_{01}\pi_{01}+\mu_{11}+\mu_{10}\pi_{10}+c\left(1-\pi_{01}-\pi_{10}\right)}.$

Suppose that $c>0$. Then it holds that
$c\left(1-\pi_{01}\right)>c\left(1-\pi_{01}-\pi_{10}\right)$, and thus

$\mu^{1}\left(\rho_{1}\right)=\frac{\mu_{01}\pi_{01}+\mu_{11}}{\mu_{01}\pi_{01}+\mu_{11}+\mu_{10}\pi_{10}}<\frac{\mu_{01}\pi_{01}+\mu_{11}+c\left(1-\pi_{01}\right)}{\mu_{01}\pi_{01}+\mu_{11}+\mu_{10}\pi_{10}+c\left(1-\pi_{01}-\pi_{10}\right)}=\hat{\mu}^{1}\left(\rho_{1}\right).$

Next suppose that $c<0$. Then it holds that
$c\left(1-\pi_{01}\right)<c\left(1-\pi_{01}-\pi_{10}\right)$, and thus

$\mu^{1}\left(\rho_{1}\right)=\frac{\mu_{01}\pi_{01}+\mu_{11}}{\mu_{01}\pi_{01}+\mu_{11}+\mu_{10}\pi_{10}}>\frac{\mu_{01}\pi_{01}+\mu_{11}+c\left(1-\pi_{01}\right)}{\mu_{01}\pi_{01}+\mu_{11}+\mu_{10}\pi_{10}+c\left(1-\pi_{01}-\pi_{10}\right)}=\hat{\mu}^{1}\left(\rho_{1}\right).$

Now we prove part $(ii)$. Recall the sender’s simplified problems (2) for the
rational receiver and (5) for the naive receiver. Rewriting the constraints in
(2) and (5), respectively, yields

$\pi_{10}\leq\frac{\mu_{01}}{\mu_{10}}\pi_{01}+\frac{\mu_{11}}{\mu_{10}},\hskip
10.0pt\pi_{10}\leq\frac{\hat{\mu}_{01}}{\hat{\mu}_{10}}\pi_{01}+\frac{\hat{\mu}_{11}}{\hat{\mu}_{10}}.$

Define
$g_{1}\left(\pi_{01}\right)\equiv(\mu_{01}/\mu_{10})\pi_{01}+(\mu_{11}/\mu_{10})$
and
$g_{2}\left(\pi_{01}\right)\equiv(\hat{\mu}_{01}/\hat{\mu}_{10})\pi_{01}+(\hat{\mu}_{11}/\hat{\mu}_{10})$.
By Lemma 2 it follows that

$g_{2}\left(\pi_{01}\right)=\frac{\mu_{01}-c}{\mu_{10}-c}\pi_{01}+\frac{\mu_{11}+c}{\mu_{10}-c}.$

Consider $g_{1}\left(\cdot\right)$ and $g_{2}\left(\cdot\right)$ at the
boundaries of the set of feasible signals:
$g_{1}\left(0\right)=(\mu_{11}/\mu_{10}),$
$g_{2}\left(0\right)=(\mu_{11}+c)/(\mu_{10}-c)$ and
$g_{1}\left(1\right)=(\mu_{01}+\mu_{11})/\mu_{10},$
$g_{2}\left(1\right)=(\mu_{01}+\mu_{11})/(\mu_{10}-c)$.

First suppose that $c>0$. It follows that
$g_{1}\left(0\right)<g_{2}\left(0\right)$ and
$g_{1}\left(1\right)<g_{2}\left(1\right)$. Moreover, by linearity of
$g_{1}\left(\cdot\right)$ and $g_{2}\left(\cdot\right)$, for all
$\pi_{01}\in[0,1]$ it holds that

$g_{1}\left(\pi_{01}\right)<g_{2}\left(\pi_{01}\right).$

Thus, the constraint in (2) is relatively more slack than in (5).

Next suppose that $c<0$. It follows that
$g_{1}\left(0\right)>g_{2}\left(0\right)$ and
$g_{1}\left(1\right)>g_{2}\left(1\right)$. Again, by linearity of
$g_{1}\left(\cdot\right)$ and $g_{2}\left(\cdot\right)$, for all
$\pi_{01}\in[0,1]$ it holds that

$g_{1}\left(\pi_{01}\right)>g_{2}\left(\pi_{01}\right).$

Thus, the constraint in (2) is relatively more tight than in (5). ∎

_Proof of Proposition 3_. Let $V^{*}$ and $\hat{V}^{*}$ denote the sender’s
optimal (ex-ante) expected payoff when the receiver is rational and when he is
naive, respectively. Then,

$\displaystyle
V^{*}\equiv\mu_{11}\pi_{11}^{*}+\mu_{10}\pi_{10}^{*}+\mu_{01}(1-\pi_{01}^{*})+\mu_{00}(1-\pi_{00}^{*}),$
$\displaystyle\hat{V}^{*}\equiv\mu_{11}\hat{\pi}_{11}^{*}+\mu_{10}\hat{\pi}_{10}^{*}+\mu_{01}(1-\hat{\pi}_{01}^{*})+\mu_{00}(1-\hat{\pi}_{00}^{*}).$

As the form of optimal signal in Propositions 1 and 2 depends on
$sign(\mu_{11}-\mu_{10})$, we consider four cases based on
$sign(\mu_{11}-\mu_{10})$ and $sign(c)$. First, assume $\mu_{11}\geq\mu_{10}$.
If $c\geq 0$, then from Proposition 1 and 2 and Lemma 2,
$(\pi_{00}^{*},\pi_{01}^{*},\pi_{10}^{*},\pi_{11}^{*})=$
$(\hat{\pi}_{00}^{*},\hat{\pi}_{01}^{*},\hat{\pi}_{10}^{*},\hat{\pi}_{11}^{*})=(0,0,1,1)$.
Thus, $V^{*}=\hat{V}^{*}=1$. If $c<0$, then either
$\hat{\mu}_{11}\geq\hat{\mu}_{10}$ and, as above, $V^{*}=\hat{V}^{*}=1$, or
$\hat{\mu}_{11}<\hat{\mu}_{10}$ and (as, according to Proposition 2, the
sender does not get her first best payoff) $\hat{V}^{*}<V^{*}=1$. Thus, it
follows that

$\hat{V}^{*}\geq V^{*}\text{ if and only if }c\geq 0.$ (7)

Second, assume $\mu_{11}<\mu_{10}$. By Proposition 1, the difference in the
sender’s optimal expected payoffs between the benchmark and the naive receiver
case is given by

$\displaystyle\nu\equiv\hat{V}^{*}-V^{*}$
$\displaystyle=\mu_{10}\left(\hat{\pi}_{10}^{*}-\pi_{10}^{*}\right)+\mu_{01}\left(\pi_{01}^{*}-\hat{\pi}_{01}^{*}\right)$
(8)
$\displaystyle=\mu_{10}\left(\hat{\pi}_{10}^{*}-\gamma\right)+\mu_{01}\left(\frac{\mu_{10}}{\mu_{01}}\gamma-\frac{\mu_{11}}{\mu_{01}}-\hat{\pi}_{01}^{*}\right)$
$\displaystyle=\mu_{10}\hat{\pi}_{10}^{*}-\mu_{01}\hat{\pi}_{01}^{*}-\mu_{11}.$

Plugging the optimal signal for case (a) from Proposition 2 in (8) yields

$\displaystyle\nu$
$\displaystyle=\mu_{10}\frac{\hat{\mu}_{11}}{\hat{\mu}_{10}}-\mu_{11}=\mu_{10}\frac{\mu_{11}+c}{\hat{\mu}_{10}}-\mu_{11}$
(9)
$\displaystyle=\frac{\mu_{10}\mu_{11}+\mu_{11}c-\mu_{10}\mu_{11}+\mu_{10}c}{\hat{\mu}_{10}}=\frac{c\left(\mu_{11}+\mu_{10}\right)}{\hat{\mu}_{10}}.$

Similarly, plugging the optimal signal for case (b) from Proposition 2 in (8)
yields

$\displaystyle\nu=$
$\displaystyle\mu_{10}\frac{\hat{\mu}_{01}+\hat{\mu}_{11}}{\hat{\mu}_{10}}-\mu_{01}-\mu_{11}$
(10)
$\displaystyle=\mu_{10}\frac{\mu_{01}-c+\mu_{11}+c}{\hat{\mu}_{10}}-\mu_{01}-\mu_{11}=\frac{c\left(\mu_{11}+\mu_{01}\right)}{\hat{\mu}_{10}}.$

Finally, plugging the optimal signal for case (c) from Proposition 2 in (8)
yields

$\displaystyle\nu$
$\displaystyle=\mu_{10}-\mu_{01}\frac{\hat{\mu}_{10}-\hat{\mu}_{11}}{\hat{\mu}_{01}}-\mu_{11}$
(11)
$\displaystyle=\frac{2c\mu_{01}+\left(\mu_{10}-\mu_{11}\right)\left(\hat{\mu}_{01}\right)-\mu_{01}\left(\mu_{10}-\mu_{11}\right)}{\hat{\mu}_{01}}=\frac{c\left(2\mu_{01}+\mu_{11}-\mu_{10}\right)}{\hat{\mu}_{01}}$
$\displaystyle=\frac{c\left(\mu(\rho_{1})-\mu_{10}+\mu_{01}\right)}{\hat{\mu}_{01}}.$

From the parametric condition for the case (c),
$(\hat{\mu}_{01}+\hat{\mu}_{11})/\hat{\mu}_{10}\geq 1$ and thus it follows
that $\mu(\rho_{1})\geq\hat{\mu}_{10}$. By Lemma 2, this can be rewritten as
$\mu(\rho_{1})-\mu_{10}\geq-c$. Since we also have $\mu_{01}\geq c$ by Lemma
2, it holds that

$\mu(\rho_{1})-\mu_{10}+\mu_{01}\geq 0.$ (12)

By (9), (10), (11), and (12) it follows that,

$\hat{V}^{*}\geq V^{*}\text{ if and only if }c\geq 0.$ (13)

Note that the sum of optimal expected payoffs of the sender and the receiver
in the benchmark and the case of naive receiver is given by the same constant:

$\displaystyle\underbrace{\mu_{11}\pi_{11}^{*}+\mu_{00}(1-\pi_{00}^{*})+\mu_{10}\pi_{10}^{*}+\mu_{01}(1-\pi_{01}^{*})}_{\text{the
sender's expected payoff}}$ (14)
$\displaystyle\quad\quad+\underbrace{\mu_{11}\pi_{11}^{*}+\mu_{00}(1-\pi_{00}^{*})+\mu_{10}\left(1-\pi_{10}^{*}\right)+\mu_{01}\pi_{01}^{*}}_{\text{the
receiver's expected payoff}}$
$\displaystyle=2\left(\mu_{11}\pi_{11}^{*}+\mu_{00}(1-\pi_{00}^{*})\right)+\mu_{10}+\mu_{01}$
$\displaystyle=2\left(\mu_{11}+\mu_{00}\right)+\mu_{10}+\mu_{01},$

where both use the correct (sophisticated) prior belief for evaluating the
receiver’s expected utility when he is naive.

Denote the receiver’s sender-optimal expected payoff in the benchmark and the
case of naive receiver by $U^{*}$ and $\hat{U}^{*}$, respectively. Since by
(14) the sum of sender’s and receiver’s expected payoffs does not change
between the benchmark and the case of naive receiver and the sender’s expected
payoff changes according to (7) and (13), it follows that

$\hat{U}^{*}\leq U^{*}\text{ if and only if }c\geq 0,$ (15)

i.e. when the receiver’s naivete yields a higher expected payoff to the
sender, it yields a lower expected payoff to the receiver.

Finally, given $c>0$, from Lemma 4, the receiver’s naivete relaxes the
obedience constraint. Thus, the difference between the expected payoffs in
(7), (13), and (15) is strict if and only if in the benchmark case the sender
does not obtain her first best payoff of $1$, which is the case if and only if
$\mu_{11}<\mu_{10}$. Similarly, given $c<0$, by Lemma 4, the receiver’s
naivete tightens the obedience constraint. Thus, the difference between the
expected payoffs in (7), (13), and (15) is strict if and only if the sender
does not obtain her first best payoff of $1$ in the naive receiver case, which
is true if and only if $\hat{\mu}_{11}<\hat{\mu}_{10}$.∎

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# Blockchain-Oriented Software Variant Forks:
A Preliminary Study

Henrique Rocha  Department of Computer Science
Loyola University Maryland
Baltimore, USA
<EMAIL_ADDRESS>John Businge  Department of Computer Science /
ANSYMO
University of Antwerp
Antwerp, Belgium
<EMAIL_ADDRESS>
###### Abstract

In collaborative social development platforms such as GitHub, forking a
repository is a common activity. A variant fork wants to split the development
from the original repository and grow towards a different direction. In this
preliminary exploratory research, we analyze the possible reasons for creating
a variant fork in blockchain-oriented software. By collecting repositories in
GitHub, we created a dataset with repositories and their variants, from which
we manually analyzed 86 variants. Based on the variants we studied, the main
reason to create a variant in blockchain-oriented software is to support a
different blockchain platform (65%).

###### Index Terms:

Blockchain-oriented software, BOS, variant forks, hard forks, software family.

##  I Introduction

Due to the popularity of cryptocurrencies [1], such as BitCoin [2] and Ether
[3], blockchain platforms have also become more popular [4, 5]. Initially
designed as a distributed ledger, nowadays blockchain has grown to address
many different scenarios. For example, blockchain can be used in supply-chain
[6, 7], voting [8, 9], vehicular ad-hoc networks [10], and others [11, 12].

Since there are many applications for blockchain [6, 7, 8, 9, 10, 11, 12, 13],
with the rising popularity of blockchain platforms there is also an increase
in the development of blockchain-oriented software (BOS) [4, 14].

Many of these BOS are developed using coding platforms such as GitHub. The
advent of social coding platforms like GitHub, BitBucket, and GitLab have
substantially improved software reuse through forking. Developers may fork a
mainline repository into a new forked repository and take governance over the
latter while preserving the full revision history of the former. The social
coding platforms comprise a number of software ecosystems i.e., large
collections of interdependent software components that are maintained by large
and geographically distributed communities of collaborating contributors [15].
A software family is subset of the a software ecosystem. A software family
comprises two or more software projects sharing a common code base as well as
variable code. An example of a family is a variant fork and the original
project (e.g., MariaDB is a variant fork of the upstream project MySQL). As
opposed to the very common bug-fixing/feature social forks that are usually
integrated back in the upstream, variant forks are very rare and usually
created to diverge away from the upstream with no intention to contribute back
[16, 17].

Many cryptocurrency applications inherit code from the mainline project. For
example, the variant fork repository PIVX-Project/PIVX is a cryptocurrency
project hosted on Github that was forked from dashpay/dash. Moreover,
dashpay/dash was also forked from the mainline bitcoin/bitcoin. The downstream
applications continuously monitor their immediate upstream and others in the
hierarchy for important updates like bug and security fixes as well as other
specific updates [18]. These three cryptocurrency BOS can be considered as a
software family comprising three variants (a mainline and two variant forks)
and all other related cryptocurrency applications form software ecosystem [16,
19].

In this short paper, we conduct a preliminary study on BOS repositories with
variant forks. We analyze the characteristics of the repositories and find the
reason why the variant fork was created. For example, someone may fork a
Wallet application to support a different blockchain platform than the
original. For this study, we collected 124 mainline BOS repositories in
GitHub, and 237 variant forks, for a total of 361 repositories. From those, we
manually analyzed 86 variant forks which led to discovering the most common
reason for creating a variant in BOS is to support another blockchain platform
(65% of the investigated cases).

The remainder of this paper is organized as follows. Section II describes our
collection process for the GitHub repositories which we use in this study.
Section III presents our preliminary results. Section IV discuss some related
work research. Finally, in Section V, we present our final remarks and outline
future research goals and challenges.

##  II Data Collection

To collect the repositories, we employed a similar method adopted by another
research that studied variants and software families [16, 18]. We started by
looking for mainline projects (i.e., original repositories) using the Github’s
Rest API v3 search endpoint
111https://docs.github.com/en/rest/reference/search. We identify the
repository as being a BOS if in their GitHub name / description / readme
contains the following keywords: blockchain, ethereum, bitcoin, or
cryptocurrency. We then filtered out _popular_ ($>5$ stars and forks), _long-
lived_ (created before 2018) and _active_ (still updated in 2020)
repositories. Then, for all the mainline projects we found, we tried to
identify and collect variant forks. This process is subject to a known threat
to validity since previous studies revealed that the majority of forks on
GitHub are inactive [20, 21] or are social forks [22]. To reduce this threat,
we filtered forks based on the following heuristics: $\geq 3$ stars, $\geq 3$
commits ahead of the mainline. A fork with stars implies that it is liked by
other persons other than developers. Commits ahead of the mainline implies
that the fork has added extra functionality that is not in the original
project. The is no criteria for the thresholds of the stars and commits,
however, the lower numbers results in a lot of noise and higher numbers
results missing possibly interesting cases. To reduce on the noise, we further
filtered the mainline–variant fork pairs by ensuring they had diverging README
files presenting BOS applications (manually verified). Based on the criteria
detailed above, we gathered 124 mainline repositories, and 237 variant forks.
A spreadsheet with the dataset is publicly available.222https://bit.ly/3GV2PeO

##  III Preliminary Results

We start our analysis by looking at the number of variant forks that each
mainline repository has. From the 124 mainline repositories, in 91 of them
(72%) we only identified one variant. Therefore, the median value for number
of variants is one, in our collected data. The average number of variants is
1.91. Table I shows the top-ten mainline BOS repositories with the most number
of variants as identified by us.

TABLE I: Top-10 BOS Mainlines according to the number of Variants. Repository |  Domain | # of Variants
---|---|---
ethereum/go-ethereum |  Protocol / Node | 19
Uniswap/uniswap-interface |  Protocol / Token Exchange | 15
bitcoin/bitcoin |  Protocol / Node / Crytpcurrency | 12
monero-project/monero |  Cryptocurrency | 9
ethereum/solidity |  Compiler | 6
MetaMask/metamask-extension |  Browser Add-on | 6
EOSIO/eosio.contracts |  Smart Contracts | 5
tendermint/tendermint |  BFT Consensus | 5
turtlecoin/turtlecoin |  Cryptocurrency | 5
bitcoinj/bitcoinj |  Library/Protocol | 4

We like to note that the mainline repositories we gathered (exemplified in
Table I) have more forks than the ones used in this study. For instance,
ethereum/go-ethereum repository, has more than 12 thousand forks. The 19 forks
we showed for this repository are the ones we were able to identify as
variants.

We manually analyze the mainline–variant fork pairs, by comparing the
repository descriptions and README files, to find the possible reason for
variant creation. We discarded pairs where the reason variant creation is not
clear. We will analyze the forks for the top-ten mainline repositories we
presented in Table I.

Table II shows the variant forks we identified for the ethereum/go-ethereum
repository. The ahead by column shows how many commits the variant is ahead of
the original repository. This table also shows the possible reason for the
variant creation, based on our manual analysis.

TABLE II: Variants for ethereum/go-ethereum. Variant | Ahead by |  Reason
---|---|---
tomochain/tomochain-v1 | 1595 |  Support a different blockchain.
ShyftNetwork/go-empyrean | 748 |  Support a different blockchain.
expanse-org/go-expanse | 532 |  Support a different blockchain.
XinFinOrg/XDPoSChain | 508 |  Support a different blockchain.
ubiq/go-ubiq | 320 |  Support a different blockchain.
NoteGio/go-ethereum | 268 |  Unspecified.
EthereumCommonwealth/go-callisto | 150 |  Support a different blockchain.
second-state/lityvm | 122 |  Extending the protocol.
Ethersocial/go-ethersocial | 103 |  Support a different blockchain.
BrightID/IDChain | 47 |  Support a different blockchain.
blocknative/go-ethereum | 36 |  Unspecified.
nebulaai/nbai-node | 28 |  Support a different blockchain.
AlexeyAkhunov/turbo-geth-archive | 26 |  Efficiency improvement.
Onther-Tech/go-ethereum | 22 |  Unspecified.
ewasm/go-ethereum | 19 |  Support another technology.
flashbots/mev-geth | 11 |  Support another technology.
Giulio2002/go-ethereum | 11 |  Unspecified.
loomnetwork/go-ethereum | 9 |  Unspecified.
eth4nos/go-ethereum | 4 |  Efficiency improvement.

The ethereum/go-ethereum is the official Go implementation of the Ethereum [3]
protocol and its client node. The most popular reason for the variant forks is
to support a different blockchain platform (eight out of 19 forks). Since the
original repository contains the core of the Ethereum implementation, it is
possible that some of those variants to be sidechains333A sidechain is a
separate blockchain that runs in parallel to another blockchain platform
(referred as mainnet). The sidechain is connected to the mainnet and provides
a bridge for transactions between both blockchains.
$<$https://ethereum.org/en/developers/docs/scaling/sidechains/$>$ as well. We
also found two forks that specified some sort of efficiency or performance
improvement (e.g., one is a lightweight client node for Ethereum). Moreover,
we found two forks to support another technology but unrelated to the
blockchain. For instance, ewasm/go-ethereum adds support to Ewasm, which is
the primary candidate to replace EVM (Ethereum Virtual Machine) as part of the
Ethereum 2.0 ”Serenity” roadmap. Just one variant we found was to extend the
Ethereum protocol. From the 19 forks, three did not specify in their
description or readme file any difference from the original repository. Based
on their commits, we flagged them as variants, but we are unable to find the
reason for their creation.

The repository Uniswap/uniswap-interface is an open-source interface for
Uniswap which is a protocol for decentralized exchange of Ethereum tokens. It
shares a similar domain of ethereum/go-ethereum, but it serves a different
purpose. Tokens (or Non-fungible Tokens) are very popular in Ethereum and, by
consequence, so are token exchange contracts and websites. From the 15
variants, 13 are just to support a different exchange protocol or contract.
The fork NavidGoalpure/persian-uniswap-interface had the same basic
functionality as the original but it was internationalized and and support for
another language was added. The other remaining fork was not accessible
anymore.

The third repository, in Table I, is bitcoin/bitcoin which is an open source
BitCoin [2] implementation of its protocol, client node, and cryptocurrency.
Therefore it is very similar to ethereum/go-ethereum. Ten out of the 12 forks
are to support a different blockchain platform and create a new
cryptocurrency. This is a much higher amount of forks to support another
platform than the Ethereum variants. Probably, because the Ethereum repository
has other functionalities, while the BitCoin one is more limited and tied to
its cryptocurrency. The remaining two forks are extensions to the regular
BitCoin. For instance, OmniLayer/omnicore adds a communication layer to enable
more sophisticated smart contracts in Bitcoin; and jlopp/statoshi claims to
bring more transparency to the nodes.

In Table III, we show the variant forks for Monero (monero-project/monero),
which is a cryptocurrency. We were expecting similar results from
bitcoin/bitcoin, as most variant forks for a cryptocurrency are probably to
create another type of cryptocurrency, which is similar in principle to
support a new blockchain platform.

TABLE III: Variants for monero-project/monero. Variant | Ahead By |  Reason
---|---|---
oxen-io/oxen-core | 3894 |  Support a different Cryptocurrency
Beldex-Coin/beldex | 1034 |  Support a different Cryptocurrency
aeonix/aeon | 980 |  Support a different Cryptocurrency
electroneum/electroneum | 808 |  Support a different Cryptocurrency
EquilibriaCC/Equilibria | 558 |  Support a different Cryptocurrency
X-CASH-official/xcash-core | 420 |  Support a different Cryptocurrency
swap-dev/swap | 129 |  Support a different Cryptocurrency
dweab/haven-do-not-use | 103 |  Support a different Cryptocurrency
toints/moneroclassic | 45 |  Maintain the classic version of the original cryptocurrency.

As we expected, the majority of the variant forks for monero-project/monero
are to support or create another type of cryptocurrency (eight out of nine
forks). There is one fork that is used to maintain and support the classic
version of Monero.

The fifth repository (according to Table I) is ethereum/solidity which
contains the language definition, documentation, and compiler for Solidity
[23]. Solidity is one of the major languages for coding smart contracts in
Ethereum. From its variants (Table IV), two forks are extensions for the
compiler (model checking, and static verifier). One variant is to support a
different platform, by compiling Solidity into Tron Virtual Machine. Another
fork is a translation for the documentation of Solidity. Finally, there were
two forks we could not identify the reason for their creation.

TABLE IV: Variants for ethereum/solidity. Variant | Ahead By |  Reason
---|---|---
SRI-CSL/solidity | 1348 |  Compiler Extension
ScottWe/solidity-to-cmodel | 364 |  Compiler extension
tronprotocol/solidity | 174 |  Support a different platform
akira-19/solidity | 68 |  Unspecified
Karocyt/solidity-fr | 62 |  Translation
PlatONnetwork/solidity | 4 |  Unspecified

In Table I, the sixth repository with most variants is MetaMask/metamask-
extension where we identified six forks (Table V). MetaMask is a browser
extension (Chrome and Firefox) that facilitates browsing Ethereum blockchain
on certain websites. This repository is from a different domain than the
previous ones, by being a browser add-on.

TABLE V: Variants for MetaMask/metamask-extension. Variant | Ahead By |  Reason
---|---|---
poanetwork/nifty-wallet | 1480 |  Enhancement
Conflux-Chain/conflux-portal | 869 |  Support a different blockchain
smilofoundation/SmiloWallet-extension | 142 |  Support a different blockchain
ubiq/sparrow-extension-OLD | 90 |  Support a different blockchain
CoboVault/metamask-extension | 64 |  Unspecified
dsrvlabs/celo-extension-wallet | 29 |  Support a different blockchain

Four out of six MetaMask variants were to support a different blockchain
platform. One fork was a usability enhancement for the add-on. And one fork,
we could not discover the reason for its creation.

The seventh repository (Table I) we looked was EOSIO/eosio.contracts with five
variants. This repository contains smart contracts to provide basic functions
of the EOSIO blockchain. Four out of five variants were created to support
another blockchain platform that uses EOSIO as a basis. One fork did not
specify its purpose on the description or readme file.

The eighth repository (Table I) is a middleware for the application-based
blockchain platform that uses a Byzantine Fault Tolerant (BFT) consensus
algorithm, Tendermint [24] (tendermint/tendermint). Since application-based
blockchains (such as Tendermint, and Hyperledger Fabric) have major
differences from public blockchain platforms (such as Ethereum, and BitCoin),
we were expecting to observe different reasons for the creation of variants.
However, four of the five variants did not specify anything to justify their
creation, and the description and readme files were identical to the original
repository. One fork, QuarkChain/tendermintx, was created as an extension to
provide greater flexibility for Tendermint.

The crytocurrency Turtlecoin is the ninth repository (turtlecoin/turtlecoin)
in number of variant forks. All five variants we found for this repository are
to support or create a new cryptocurrency based on the Turtlecoin original
code.

Finally, the final repository on our top-ten list with most variants (Table I)
is the bitcoinj/bitcoinj. BitCoinJ is a Java library that implements the
BitCoin protocol, has a built-in wallet, and can communicate with the BitCoin
blockchain without the need for any other external code. Three out of the four
forks were created to support a different blockchain platform and
cryptocurrency than BitCoin. One fork does not specify the reason for its
creation.

Summarizing. We aimed to identify the possible reason for the creation of BOS
variants by analyzing the top-ten mainline repositories with most variant
forks. In total, we manually investigated 86 variant forks from those top-ten
repositories. We can see the most popular reason for creating a variant in BOS
is to support a different blockchain platform (65% of the variants or 56 out
of 86). The second most popular reason is the extend the original with
additional features or technologies (10% or nine out of 86). Approximately 14%
of the variants (12 out of 86) did not specify any difference with the main
repository in their description or readme file.

##  IV Related Work

Most studies analysed forking on Sourceforge, pre-dating the advent of social
coding platforms like GitHub [25, 26]. Several of those early studies report
perceived controversy around variant forks [27]. Jiang et al. [28] state that,
although forking may have been controversial in the open source software (OSS)
community, it is now encouraged and a built-in feature on GitHub.

We have encountered few studies analysing variant forks in the social coding
era [22, 17, 29, 16]. Businge et al. [22] study focused on the Android
ecosystem and found that re-branding, simple customizations, feature
extension, and implementation of different but related features are the main
motivations to create a fork for Android apps. Zhou et al. [17] interviewed 18
developers of hard forks on GitHub to understand reasons for forking in modern
social coding environments that explicitly support forking. Businge et al.
[16] investigated the interaction between mainline and variants. The authors
quantitatively investigated code propagation among variants and their mainline
in three software ecosystems. They found that only about 11% of the 10,979
mainline–variant pairs had integrated code among themselves.

To the best of our knowledge, there are no studies investigating variant forks
for blockchain-oriented software.

##  V Final Remarks

In this short paper, we presented a preliminary exploratory research on the
possible reasons for creating a variant fork in blockchain-oriented software.
We collected repositories from GitHub and manually analyzed 86 variant forks
from the top-ten mainlines in our dataset.

We discovered in our dataset that for BOS variants, 65% were created to
support a different blockchain platform. This seems like a reasonable and fast
way to develop software for a specific blockchain platform. Instead of coding
everything from scratch, a developer finds a similar BOS repository that
accomplishes its main goal, forks it, and then just changes the protocol to
interact with a different blockchain. However, before our study, we did not
know how common this practice is.

The second most common reason for the creation of a variant is to extend the
mainline with additional features or technologies, occurring in 10% of the
analyzed forks. Why these extensions were not integrated into the original
repository remains an open question.

For future research, we plan to investigate all the variants in our current
dataset. Additionally, we will analyze other aspects of the variants compared
to the mainline such as number of commits, commit size, community size, number
of stars, technical debt, etc. We also plan to contrast our findings in BOS
repositories with non-blockchain software to properly compare the differences
between them. Moreover, we would like to contact the main developers or owners
from the variants to know the reasons why the variant was created. However,
since GitHub policy rules prohibit sending unsolicited emails, we are not sure
how to contact those developers.

## Acknowledgment

John Businge’s work is supported by the FWO-Vlaanderen and F.R.S.-FNRS via the
EOS project 30446992 SECO-ASSIST.

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