IIMB-WP N0. 580

WORKING PAPER NO: 580

Social Media Advertising and Fake Followers

Abhinav Anand Assistant Professor

Finance and Accounting Indian Institute of Management Bangalore Bannerghatta Road, Bangalore – 5600 76

abhinav.anand@iimb.ac.in

Souvik Dutta Assistant Professor

Economics and Social Sciences Indian Institute of Management Bangalore Bannerghatta Road, Bangalore – 5600 76

souvik@iimb.ac.in

Prithwiraj Mukherjee

Assistant Professor Marketing

Indian Institute of Management Bangalore Bannerghatta Road, Bangalore – 5600 76

pmukherjee@iimb.ac.in

Year of Publication – December 2018

IIMB-WP N0. 580

Social Media Advertising and Fake Followers Abstract Marketing promotions via paid social media influencers are now commonplace across the world. These influencers are minor celebrities with self-selected audiences based on their domains of interest, and often make thousands of dollars per sponsored post. However, this practice has led to the emergence of click farms - businesses that sell fake followers to such influencers to artificially inflate their follower counts, thus signaling higher reach and thus more fees from advertisers using their services. In this paper, we develop an optimal contract between an influencer and advertiser that maximizes advertiser profit given that the influencer can buy fake followers from click farms. We model two cases: (1) an ex ante rational influencer, i.e. a celebrity who does not yet have a social media account but may be incentivised to open one given a potential follower count and (2) an interim rational influencer who privately knows her true follower count but may inflate the number to increase her fee for posting promotional content. In the former case, we show that the optimal contract involves the advertiser paying out a lump sum amount based on expected gains, and that there is no incentive for the influencer to buy fake followers. However, in the latter scenario, the optimal contract actually involves the influencer inflating her follower count, which the advertiser can take into account while deciding the payment. In both cases, the contract incorporates the influencer's outside option which we suggest must be determined by market research that is independent of the influencer's social media follower count. Keywords: Social media, influencer marketing, analytical modeling, contract theory

Social Media Advertising and Fake Followers

Abhinav Anand∗

Souvik Dutta†

Prithwiraj Mukherjee‡

December 20, 2018

WORKING PAPER

Abstract

Marketing promotions via paid social media in�uencers are now

commonplace across the world. These in�uencers are minor celebrities

with self-selected audiences based on their domains of interest, and

often make thousands of dollars per sponsored post. However, this

practice has led to the emergence of click farms - businesses that sell

fake followers to such in�uencers to arti�cially in�ate their follower

∗Assistant Professor of Finance, Indian Institute of Management Bangalore; abhi-nav.anand@iimb.ac.in†Assistant Professor of Economics, Indian Institute of Management Bangalore; sou-

vik@iimb.ac.in‡Assistant Professor of Marketing, Indian Institute of Management Bangalore;

pmukherjee@iimb.ac.in

1

counts, thus signaling higher reach and thus more fees from advertisers

using their services. In this paper, we develop an optimal contract

between an in�uencer and advertiser that maximizes advertiser pro�t

given that the in�uencer can buy fake followers from click farms. We

model two cases: (1) an ex ante rational in�uencer, i.e. a celebrity who

does not yet have a social media account but may be incentivised to

open one given a potential follower count and (2) an interim rational

in�uencer who privately knows her true follower count but may in�ate

the number to increase her fee for posting promotional content. In the

former case, we show that the optimal contract involves the advertiser

paying out a lump sum amount based on expected gains, and that there

is no incentive for the in�uencer to buy fake followers. However, in the

latter scenario, the optimal contract actually involves the in�uencer

in�ating her follower count, which the advertiser can take into account

while deciding the payment. In both cases, the contract incorporates

the in�uencer's outside option which we suggest must be determined

by market research that is independent of the in�uencer's social media

follower count.

2

1 Introduction

Social media provides advertisers with the unique opportunity to promote

their brands via targeted advertising, either through a given platform (e.g.

promoted tweets on Twitter, sponsored posts on Facebook etc), or via celebri-

ties who have large numbers of followers. Such celebrities, known as in�u-

encers in digital marketing practitioner jargon, usually have niche appeals

- many of them are experts in cooking, travel, fashion, electronic gadgets

etc, and usually have large numbers of self-selected followers who share these

interests, often looking up to them for advice in these domains. This is a ris-

ing trend among advertisers today. According to one source (Quoc, 2017;

The Economist, 2016), Youtube in�uencers with over 7 million followers

command as much as $300,000 per sponsored post, while the correspond-

ing �gures for their counterparts on Instagram, Facebook and Twitter are

$150,000, $187,500 and $60,000 respectively, allowing social media followings

to be monetized quite pro�tably. Even in�uencers with less than 250,000 fol-

lowers make hundreds of dollars per sponsored post. A survey by in�uencer

marketing agency Linqia (2018) across various industry sectors including con-

sumer packaged goods, food and beverage and retail in the US �nds that 86%

of marketers surveyed used some form of in�uencer marketing in 2017, and

of them, 92% reported �nding it e�ective. 39% of those surveyed by Lin-

qia planned to increase their in�uencer marketing budgets. Similar trends

reported by Whosay (2018), eMarketer (2017) and IRI (2018) suggest that

3

this practice is on an upswing, and possibly here to stay.

The rise of in�uencer marketing has its own dark side. Sponsored posts

by social media in�uencers often do not reveal that they are promotional in

nature; rather, they masquerade as word of mouth, where in�uencers pretend

to genuinely appreciate the brand that sponsors them. This has fueled a lot of

debate, and the US Federal Trade Commission (FTC) has also taken notice.

FTC (2017) guidelines now require in�uencers to clearly tag sponsored posts

as such, and di�erentiate them from genuine word of mouth. Some businesses

have reported that self-proclaimed in�uencers have demanded freebies and

even money in exchange for either publicity, as well as occasionally threaten-

ing bad reviews if they were not entertained. In an open letter that went viral,

an Irish luxury hotel owner lambasted a young Instagram celebrity for de-

manding a free holiday in lieu of posts promoting his business, further fueling

the debate surrounding unattributed in�uencer marketing and consumer wel-

fare (Ritschel, 2018). Some platforms like Instagram and Facebook now even

insist that all sponsored in�uencer posts be tagged as such, even introducing

algorithms to do this automatically, notwithstanding the fact that revealing

a commercial tie between a brand and in�uencer decreases a consumer's trust

and intentions to engage in electronic word of mouth (Boerman et al., 2017).

Many in�uencers do this voluntarily, and some in�uencer marketing agencies

like Bzzagent request all their brand promoters to be transparent about the

remuneration they have received to promote a product, even if it may be as

4

little as a free sample of a fast moving consumer good1.

Another interesting consequence of in�uencer marketing is the rise of click

farms, which o�er fake followers in the form of bots, hacked pro�les of genuine

social media users and thousands of fake pro�les generated by employees

paid to do so. For a price, these �y-by-night operators o�er in�uencers a

range of services, including being followed by these fake pro�les, in�ating

the number of " likes" on their fan or brand pages, and even generating

spurious comments on content posted by them. While reliable data on these

are hard to come by, there have been allegations that these click farms have

been used by politicians to retweet their messages on a large scale and sway

public opinion, by tech startups to in�ate their app download statistics before

seeking venture capitalist or angel investor funding, and by social media

in�uencers themselves to in�ate their follower counts, and thus command

higher fees from advertisers for promoting their brands via sponsored posts.

A recent New York Times expose (Confessore et al., 2018) veri�ed some of

these anecdotal claims. In a story covering an agency called Devumi, they

found that several people including businessmen, politicians, academics, PR

professionals and social media in�uencers had bought hundreds of thousands

1Bzzagent has a fairly unique in�uencer marketing business model that involves givingfree samples of clients' products to empaneled in�uencers (anybody can sign up), who maynot be obligated to post only positive word of mouth. Their model is extensively describedin Berger and Schwartz (2011) and Berger (2016, chapter 2). However, typical in�uencermarketing campaigns across the world involve brands or digital marketing agencies solic-iting social media users with large follower counts, and paying them to post about theirbrands. Though such in�uencers increasingly disclose �nancial ties with the sponsoringbrand, a large number of social media in�uencers still do not explicitly disclose that theirposts may be sponsored

5

of fake followers and in�ated their follower counts. Devumi is now the subject

of a legal inquiry initiated by the New York Attorney General (Confessore,

2018).

Research by Paquet-Clouston et al. (2017) on fake followers gives further

information on the fake follower economy. They report that customers of

click farms pay an average of $49 for every thousand YouTube followers.

The corresponding �gures are $34 for Facebook, $16 for Instagram and $15

for Twitter. They also report that the corresponding average prices for a

thousand likes on these platforms are $50, $20, $14 and $15 respectively. A

simple Google search can easily corroborate these claims - hundreds of click

farms and potentially millions of fake followers are accessible to anyone with

a social media account, a credit card and money to spare. This practice has

even found its way to pop culture, with Silicon Valley, a hit show on HBO

also featuring fake app users as a major plot element in an episode. Recent

revelations suggest that click farms are getting more and more sophisticated;

shell companies that calibrate their bots based on the behavior of genuine

users on their own apps may have siphoned o� hundreds of millions of dollars

from online advertisers. A study by Juniper research estimates online ad

fraud at about $19 billion for 2018, and projects it to grow to $44 billion by

2022.

In light of the above discussion, it is clear that ethical issues of in�u-

encer marketing notwithstanding, fake followers pose a signi�cant problem

to advertisers trying to reach genuine audiences. Assuming that commercial

6

ties between the advertiser and in�uencer are clearly conveyed in in�uencer

posts, the latter still has an incentive to in�ate her follower count to attract

higher fees from the advertiser. This paper tackles this particular problem -

we build an optimal contract between an advertiser and a social media in�u-

encer (both of whom are risk neutral), such that the advertiser's pro�ts are

maximized subject to the in�uencer's own participation constraints, keeping

in mind that the latter could in�ate her follower count. In the following

section, we develop this contract for two scenarios:

1. An ex ante rational in�uencer, i.e. an individual who does not yet have

a social media account but may be incentivised to open one given the

expectation of a certain follower count and,

2. An interim rational in�uencer who has private information about her

true follower count, but may report an in�ated number of followers to

the advertiser

In the �rst scenario, we demonstrate that the optimal contract does not allow

the in�uencer to in�ate her follower count by buying fake followers because

the advertiser should o�er only a �xed lum sum payment equaling the in�u-

encer's outside option. In the second, there is still incentive for the in�uencer

to buy fake followers even under optimal settings. The contract here toler-

ates small amounts of fraud but disproportionately penalizes in�uencers who

in�ate their follower count too much.

7

2 Literature survey

In this section, we outline three brief streams of literature germane to the

issue at hand: the practice of in�uencer marketing, models of economic fraud

(both online and o�ine) and a brief overview of applications of contract

theory to problems in the marketing and allied domains. While some of

these domains are admittedly large, we outline only a few studies in each,

that we deem to be most useful to readers of this particular paper.

2.1 In�uencer marketing

While the literature on social in�uence and opinion leadership is huge, we

focus here only those papers talking explicitly about social media and spon-

sored posts, i.e. in�uencer marketing. Jin and Phua (2014) present experi-

mental results about consumers' reactions to celebrity posts on Twitter; they

�nd that celebrities with more followers tend to be perceived as more credible

and have a higher in�uence on consumers' product involvement and buying

intentions. Similar results are found by Djafarova and Rushworth (2017),

in a qualitative study of Instagram users. Interestingly, they �nd that non-

traditional celebrities like popular bloggers and Youtube video creators seem

to have a more powerful in�uence than traditional celebrities like movie stars

on their target demographic of women aged 18-30 years. De Veirman et al.

(2017) �nd that Instagram users with high follower counts are likely to be

perceived as more likeable, though this may be lessened if they themselves do

8

not follow too many people. Furthermore, the �nd that in�uencer marketing

marketing may not be as e�ective if in�uencers with high follower counts for

promoting divergent products, as this practice may reduce the perception

of brand uniqueness. Casaló et al. (2018) observe that in�uencer posts on

Instagram are most e�ective when the perceived �ty between the in�uencer

and brand is high.

2.2 Economic fraud

We now present a brief survey of models of economic fraud that may be rel-

evant in our general scenario. We include models where an agent misreports

a certain state variable, tied to an economic incentive, to a principal. On-

line advertisers often encounter the problem of click fraud, where they must

pay a platform for every click on their ads. Wilbur and Zhu (2009) demon-

strates that for a constant rate of spurious clicks, advertisers can combat it

by lowering their bids, but for uncertain rates, search engine revenues can

�uctuate. They suggest the use of a third party to audit clicks to keep the

market sustainable. Another common kind of online fraud is the practice of

leaving fake reviews - either positive reviews for oneself or negative reviews

for competitors. A signi�cant body of research exists in both detecting fake

reviews as well as estimating their e�ect on consumer and �rm outcomes.

For example, Lappas et al. (2016) demonstrate that even a small number

of fake reviews can signi�cantly help a hotel surpass its competitors on on-

line platforms. In our scenario too, many social media platforms' algorithms

9

tend to boost the visibility of already-popular posts; using fake followers to

like or retweet one's posts could actually induce algorithms to further boost

their visibility to genuine viewers. However, Zhuang et al. (2018) �nd some

deletrious e�ects of excessive review faking for weaker brands. While a rig-

orous empirical study like the ones above may be an ideal scenario, lack of

reliable data sets on social media handles with fake followers is a signi�cant

impediment. Thus, we focus on a theoretical approach of contract design.

Outside the domain of online businesses, insurance is another domain

where fraud is a major concern. While making claims, insurance clients have

an intensity to in�ate their losses and claim higher payouts. A model of in-

surance fraud by Crocker and Morgan (1998) models this particular scenario.

Their approach is that of developing an optimal contract between a principal

and agent, where the agent may have incentive to lie about a certain state

variable (their particular paper explores two scenarios - insurance claimants

and sharecroppers who must declare their yields to a principal). Their gen-

eralized model yields interesting results that have been extended to other

fraud scenarios, like misreporting of earnings by CEOs (Crocker and Slem-

rod, 2007; Sun, 2014), multiple contexts of insurance fraud (eg. Crocker and

Tennyson, 2002; Dionne et al., 2009; Doherty and Smetters, 2005) and even

in designing optimal product returns policies (Crocker and Letizia, 2014).

Our own model is based on this particular approach suggested by Crocker

and Morgan. Employee theft in retail is another kind of fraud that may be

tackled using elements of contract theory and optimal control. Mishra and

10

Prasad (2006) demonstrate that a complete elimination of theft may be eco-

nomically infeasible; they derive an optimal frequency of random inspections

that may be a better approach to minimizing losses due to theft by retail

employees.

2.3 Contract theory

The principle of contract theory (see Bolton and Dewatripont 2005 for a de-

tailed exposition) is the following: a principal must design a contract between

herself and an agent such that her utility (or pro�t in case of risk-neutrality)

is maximized. However, the agent herself imposes some constraints by her

own utility maximizing considerations, that may a�ect both her decision to

participate as well as further actions if she does participate. Thus, the incen-

tives laid out in the contract must be aligned according to these constraints.

These settings usually incorporate some information asymmetry, where the

agent may not reveal some state variable truthfully (like the true number

of followers in our case) to the principal. As the domain of contract theory

itself is too vast to meaningfully cover in this essay, we will mention a few

applications of this approach only in the marketing literature, going beyond

the domain of economic fraud. Contract theoretic approaches have been

used successfully in marketing scenarios such as designing warranties and

extended service contracts (Padmanabhan and Rao, 1993) and delegation of

pricing decisions to salespersons (eg. Lal and Staelin, 1984; Bhardwaj, 2001;

Joseph, 2001; Mishra and Prasad, 2004, 2005), to name a few.

11

To the best of our knowledge however, most research on fake followers

in social media has been directed at identifying fake followers via machine

learning, rather than designing contracts that may minimize this problem

at the outset, at least among those making sponsored posts. Given the

magnitude of this problem, as discussed in the previous section, we posit a

contract design approach that seeks to minimize the fake follower problem

via incentives, rather than curtail it after it has occurred.

3 Model

We consider a game with two players - an advertiser (principal) and a social

media in�uencer (agent). The number of followers n of the in�uencer is

only privately known to herself. The publicly observed number of followers

however, is the reported number u(n) which could in general be di�erent

from n. The advertiser only knows that the number of original followers

varies between [nL, nH ] with probability density f .2

The advertiser is risk-neutral and wishes to reach as large an audience as

possible. We specify the contract between the advertiser and the in�uencer

as a 3-tuple: C ≡ {v1, v2, u} where v1(n) is the variable payment made by

the advertiser to the in�uencer depending on her followers; v2 is a constant,

�xed payment made to the in�uencer; and u(n) is the reporting function

2While n is a natural number in real life, we assume hereon for the sake of mathematicalconvenience that it is a continuous parameter

12

used by the in�uencer.3

The advertiser wishes to maximize its pro�t, which we denote as Π(v1, v2, u).

This is the di�erence between revenue from reaching the in�uencer's n fol-

lowers and the payment made to the in�uencer,

Π(v1(n), v2, u(n)) = A(n)− v1(n)− v2 (1)

The utility of the in�uencer however, depends on the realization of n ∈

[nL, nH ] and is denoted as Y (v1(n), v2, u(n)|n). The in�uencer's utility de-

pends on payments received from the advertiser. However, she can in�ate

follower count, which entails a cost. The in�uencer's cost of in�ation depends

upon her own degree of misrepresentation. We thus specify the in�uencer's

utility as,

Y (v1(n), v2, u(n)) = v1(n) + v2 − g(u(n)− n) (2)

where g(·) represents the costs of in�ating the follower count. We assume

that g ≥ 0 (costs are non-negative), g(0) = 0 (no in�ation entails no cost),

g′(0) = 0 and g′′ > 0 (more misrepresentation gets progressively more costly

to hide).

We consider direct, incentive-compatible mechanisms to characterize the

solution. In order for the contract to be incentive compatible, the in�uencer

should not be able to increase her utility by mimicking someone with n 6= n∗

3The reporting function u(n) = n, the in�uencer reports her number of followers truth-fully. Any other reporting function indicates some degree of falsi�cation

13

when n∗ is her true number of followers. This necessitates that for all distinct

n, n∗ ∈ [nL, nH ]:

Y (v1(n∗), v2, u(n∗)|n∗) ≥ Y (v1(n), v2, u(n)|n∗) (3)

The function Y (·) attains its maximum value at n∗ which leads to the

�rst order necessary condition,

dY

dn≡ Yv1v

′

1 + Yuu′= 0

at n = n∗. Totally di�erentiating Y with respect to n∗ and substituting the

�rst order condition from above, we obtain

dY

dn∗=∂Y

∂n∗(4)

The risk-neutral advertiser wishes to maximize its expected pro�t:

maxv1(n),v2,u(n)

∫ nH

nL

Π(v1(n), v2, u(n))f(n)dn

However, the solution must obey the in�uencer's incentive-compatibility

constraint: equation 3 or alternatively equation 4. In addition, the solution to

the advertiser's maximum pro�t must also obey the participation constraint

of the in�uencer, i.e. her individual rationality constraint.

In this paper, we consider two alternatives of individual rationality. We

14

�rst consider the ex-ante individual rationality and then followed by interim

individual rationality constraint.

3.1 Ex-ante individual rationality

The ex-ante individual rationality constraint of the in�uencer is given by

∫ nH

nL

Y (v1(n), v2, u(n)|n)f(n)dn ≥ Y (5)

This constraint ensures that it is worthwhile for the in�uencer to partic-

ipate in the hiring process. This suggests that the ex-ante expected utility

from participation is more than that of a reservation level of utility given by

Y .

Hence the optimization problem now is,

maxv1,v2,u

∫ nH

nL

Π(v1, v2, u)f(n)dn (6)

subject to the in�uencer's incentive compatibility constraint (equation 4) and

her ex-ante individual rationality constraint(equation 5).

The augmented Hamiltonian expression associated with this optimization

can be written as

H = Π(v1, v2, u)f + λ(n)∂Y

∂n+ µ · (Y (v1, v2, u|n)f (7)

In the above augmented Hamiltonian formulation, Y (·), the in�uencer's

15

utility function, is the state variable with its equation of motion represented

by condition 4. The control variable is u(·); λ(n) is the co-state variable cor-

responding to the incentive compatibility contraint 4; and µ is the Lagrangian

multiplier associated with the individual rationality constraint 5. Proposition

1 below gives the necessary conditions for the optimization problem.

Proposition 1. The necessary conditions to characterize the solution to

above problem can be obtained from the Pontryagin's Maximum Principle

and is given by

1. f ·(Πu − Πv1

YuYv1

)+ λ ·

(Yu,n − Yv1,n Yu

Yv1

)= 0

2. λ = dλdn

= −f · Πv1

Yv1− λ · Yv1,n

Yv1− µf

3.∫ nHnL

(Πv2 + µ · Yv2) fdn = 0

Proof in the Appendix

Together the three equations characterize the necessary conditions for the

optimal control problem. Using the results from proposition 1, we now char-

acterize the optimal contract in the next proposition given the advertiser's

pro�t function as in equation 1 and in�uencer's payo� given by 2.

Proposition 2. An optimal contract in the case of an ex-ante individual

rationality constraint is characterized by the following conditions:

1. v1(n) = 0

16

2. v2 = Y

3. u(n) = n

Proof in the Appendix

Proposition 2 states that the in�uencer will always report her follower

count truthfully and will not engage in any falsi�cation. The advertiser will

pay the in�uencer her outside option and the variable payment will be zero

and independent of the actual number of followers.

3.2 Interim Individual Rationality

In this section, we impose that the advertiser maximizes pro�ts subject to

her incentive compatibility and interim individual rationality constraint. The

interim constraint means that the in�uencer knows her actual number of

followers before getting into the contract. The interim individual rationality

constraint is given by:

v1(n) + v2 − g(u(n)− n) ≥ Y (8)

for every n ∈ [nL, nH ].

The following proposition states the necessary conditions for an optimal

contract in this regime,

Proposition 3. When the in�uencer is constrained by interim individual

17

rationality as in equation 8, an optimal contract must satisfy the following

conditions:

1.g′(u(n)−n)g′′(u(n)−n)

=(

1−Ff

)(u′ − 1)

2. v1(n) + v2 = Y + g(u(n)− n)−∫ nnLg′(u(t)− t)dt

3. u(n) > n for each n ∈ [nL, nH) and u(nH) = nH

Proof in the Appendix

We can only determine v1(n)+v2 uniquely from the optimization problem.

Alternatively we can substitute v2 = 0 and then v1(n) is uniquely determined.

The optimal contract allows for some level of falsi�cation or misrepresentation

by the in�uencer, which we explain in the following section.

4 Discussion

The above section outlines optimal contracts between an advertiser and an

in�uencer, both of whom are risk neutral. We show that in the ex ante case,

where an individual has some promise of a social media following, but has

not yet signed up on a platform, there is no incentive to in�ate her follower

count because the optimal contract is just a �xed payment. On the other

hand, for an in�uencer already on a platform, some amount of in�ation is

inevitable, a fact that the advertiser knows, and designs the contract keeping

this in mind.

18

In both the above scenarios, we see that the optimal contract involves

payments incorporating the in�uencer's outside option Y . This is an ex-

tremely important feature: it is tempting to value an individual's potential

outreach based on their social media following. However, given the possibility

of them buying fake followers, this is not necessarily the wisest thing to do.

A genuine celebrity could have fan followings independent of social media,

and their outside options are usually much higher than ordinary individuals.

There are often individuals one may not have heard much of outside of social

media circles, but they are seen to have hundreds of thousands of followers.

While it may be tempting to harness such individuals for in�uencer cam-

paigns, it may be a wiser strategy for the advertiser to conduct independent

research into their true popularity outside of social media, to determine if

they are worthwhile candidates for an in�uencer marketing campaign.

While the ex ante indivdual rationality does not involve falsi�cation, the

interim rationality condition does. The amount paid to the in�uencer is

Y + g(u(n)−n)−∫ nnLg′(u(t)− t)dt. The term g(u(n)−n)−

∫ nnLg′(u(t)− t)dt

indicates that it is optimal for the advertiser to compensate the in�uencer

for some degree of follower count in�ation, but not all of it, which would

incentivize the latter to in�ate her following all the way up to nH . Thus, the

advertiser tolerates some degree of fraud, but progressively grows stricter as

the magnitude increases.

Finally, our model guides the digital marketer on the type of in�uencer

to choose. Though we have speci�ed a revenue function A(n), we see that

19

the optimal contract does not incorporate it. However it is obvious that

the advertiser enlists those in�uencers such that its own pro�t is above zero.

Thus, niche brands may not �nd it pro�table to enlist celebrities with very

high outside options Y . However, this necessitates even more independent

validation of what Y is, to avoid wasting money on in�uencer campaigns

with shoestring budgets. An interesting avenue for future research itself is

modifying this setup with an extra budget constraint.

Our model adds to a nascent but upcoming literature on in�uencer mar-

keting, modeling the ever-increasing phenomenon of in�uencers using fake

followers. While we have limited ourselves to this particular scenario, this

can extended to generic scenarios of online advertising fraud, an increas-

ingly sophisticated genre where shell companies make it increasingly harder

for machine learning algorithms by calibrating their bots to mimic usage

patterns of genuine human users. Another possible avenue for further in-

vestigation is how social media platforms themselves could intervene - given

that their knowledge of a pro�le's authenticity could be much higher than

the advertiser's. In the presence of reliable data, this domain could also be

an encouraging venue for empirical research opportunities.

20

Appendix

Proof of Proposition 1

Proof. See Crocker and Morgan (1998) Theorem 1

Proof of Proposition 2

Proof. We use the three necessary conditions derived from Proposition 1 to

derive the optimal contract. For the functional speci�cations we can compute

the partial derivatives as follows:

Πv2 = −1

Yv2 = 1

Using the above and inserting in the third condition we get:

∫ nH

nL

(µ− 1)f(n)dn = 0

(µ− 1)∫ nH

nL

f(n)dn = 0

Hence µ = 1.

Additionally, we know that Yv1 = 1. Using this we can compute Yn,v1 =

Yv1,n = 0.

Using this information along with µ = 1 in the second equation, we get:

21

λ =dλ

dn= −f−1

1− λ · 0− 1 · f

λ =dλ

dn= f − f = 0

This implies that λ(n) = λ a �xed constant.

From the transversality condition λ(nH) = 0, we get the value of the

constant λ = 0.

Reinserting λ = 0 in the �rst equation gives us:

f ·(

0− (−1)Yu1

)+ 0 = 0

fYu = 0

Evaluating Yu

Yu =∂

∂u(v1(n) + v2 − g(u(n)− n)) = 0

−g′ = 0

This necessitates that u(n) = n given our assumptions on g(·).

Moreover, since µ = 1 this implies that the inequality 5 binds. From

there in order to maximize the pro�t of the advertiser we get v1(n) = 0 and

v2 = Y .

22

Proof of Proposition 3

Proof. Since equation 8 implies equation 5, the �rst two conditions of propo-

sition 1 applies with µ = 0. From the �rst condition, we obtain that

fg′ = λg′′(u′ − 1)

From the second condition of proposition 1, we obtain that

λ = f

which along with the transversality condition λ(nH) = 0 gives us that λ(n) =

F (n) − 1. Substituting this result in fg′ = λg′′(u′ − 1) gives us the �rst

condition.

Since F (nH) = 1 and from the assumption on g(·) we obtain that u(nH) =

nH . Given the assumptions on g(·) and(

1−Ff

)≥ 0 with the assumption that

u′ > 1 ensures that u(n) < n for each n ∈ [nL, nH)

The second condition can be derived easily so that the surplus from in-

�ation is divided between the advertiser's share and that of the in�uencer.

23

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