Thesis
Name: Kristian Bakker
Student number: 10631135
Specialisation: Economics
Field: Organizational Economics
Number of credits: 12 EC
Title: Fake News and the Market for Lemons
2 Statement of originality
This document is written by Student Johannes Kristian Bakker (10631135) who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision and completion of the work, not for the contents.
3
Fake News and the Market for Lemons
Kristian Bakker
Abstract
A reliable flow of news reporting is vital to any society, and is especially important for political processes and policymaking. The current prevalence of 'fake news' puts pressure on the availability of reliable news reporting. In this thesis a mathematical model will be constructed which aims to capture the information asymmetry consumers face when consuming news items. This model is constructed in the style of Akerlof's famous article The Market for Lemons, which assumes that consumers act according to their inferences about the distribution of a desired quality. The model set forth in this thesis contains two types of firms, which either produce genuine news, or fake news. The truthfulness of the two types of news (its primary quality) are distributed differently. Consumers thus need to infer about a mixture of distributions. Each period consumer choices influence the proportion of each type of firm active on the market in the next period. From this model follows that the information asymmetry that leads to the disappearance of markets in Akerlof's model, need not occur when consumers infer about a mixture of distributions. Fake news, if modelled correctly, does not seem to be a problem for the flow of reliable news.
4 1. Introduction
It is now nearly fifty years ago that George Akerlof published the famous article ‘The Market for ‘Lemons’: Qualitative Uncertainty and Market Mechanisms’. In which he models certain markets, most famously the market for used cars, where information asymmetry degrades the market to a state where only inferior goods are supplied, or where the market disappears entirely. Other markets, such as the market for news are usually not studied in this way. Although researchers have studied the influence of bias, credibility, and reputation building within the market (Gentzkov & Shapiro, 2006; Mullainathan & Shleifer, 2005), this possible disruption remains unaddressed. However, fake news, or more generally, disinformation, might introduce asymmetries into the market for news so that the market for news may start to look like the market for “lemons” with similar outcomes.
As fake news, or disinformation, is by its very nature difficult to detect, an empirical investigation is difficult to conduct. Also, an experimental setup would most likely involve induced values expressed in terms of monetary valuations, which may not correspond to the inherent utility consumers derive from the consumption of news. A theoretical approach, like most of the literature on this topic, and related topics, is the most suited.
This thesis investigates whether the market for news, under the influence of the presence of fake news, can be suitably modelled like Akerlof’s famous Market for “Lemons" and whether or not it leads to the same result.
This question might be of scientific interest as the current existing body of scientific theory concerning economic models of news markets, have not, for as far as I am aware, taken the disruptive role of fake news or rather disinformation, into account. Although there is some literature on the (political) bias of news reporting, and numerous literature on the spread of (mis)information in (social) networks (Acemoglu, Ozdaglar & Parandehgheibi, 2009; Azzimonti & Fernandes, 2018). Literature taking both information as a product, as well as the spread of fake information on into account, is still missing.
Given that reliable news is relevant for issues of political economics, policy, legislation and policymaking, and finance, a proper model of the market and possible failures is justified. It must be noted that reliable news is relevant via at least two channels. Firstly, reliable news is relevant for those active in the fields of policymaking, legislating, and finance. Secondly, it is also relevant for all other actors in society as far as they decide upon the actions of those active in the aforementioned field. For example, voters might need reliable news about the state of their country so as to make an informed decision about which representative to support, and which policies to endorse.
In order to investigate the influence of fake news in a market for news, the available literature and related models, such as the news model by Mullainathan and Shleifer (2005), will be reviewed. Then, a mathematical model will be constructed in which the likelihood of a news story to be true, the proportion of firms creating fake news, and the valuation of consumers, are taken into account as the most important variables. After this follows a critical examination of Akerlof’s model for the market for "lemons" and his assumptions
5 concerning uncertainty and quality. This is followed by a discussion on the literature, the basis of these models, and the best way to model the link between uncertainty and truthfulness. It will appear that the market for (fake) news can be successfully modelled in the style of Akerlof’s market for “lemons”, but will result in the opposite outcome. This is because consumers infer about the distribution of real news correctly. I will conclude by discussing the assumptions that have lead to this result as well as suggesting some possible additions to the model.
2. Literature Review
The presence of similar information asymmetry is discussed in the work of Akerlof (1970) whom concludes that when consumers have no way of observing the real value of a product, consumers would pay less to account for the uncertainty of buying a low value product. Sellers with high value products will then leave the market. Subsequently, the market may vanish entirely due to this uncertainty and lack of information (Akerlof, 1970). The combination of uncertainty and quality is the key insight for the purpose of the thesis. I will discuss Akerlof's paper in more detail, when outlining the proposed model.
However, when considered more generally, what does the market for news require in order to function properly? Gentzkow and Shapiro (2008) discuss various factors influencing the prevalence of truth in the market for news, within the context of possible government intervention on the free press and the subsequent ‘marketplace of ideas’. They conclude that the number of news firms is not a relevant factor, as information may be distributed among diverse sources, which should be included so as to produce truthful news articles. However, this could also be done by a monopolist (Gentzkow and Shapiro, 2008, p. 135). The number of firms decreases the likelihood of the suppression of news by bribe, or by collusion (2008, pp. 135-139).
Mullainathan and Shleifer (2005) also find that increased competition does not result in more accuracy. Mullainathan and Shleifer (2005) assume that people are biased, but also buy news because they have some preference for the truth. They model the market for news in a typical adaptation of the Hotelling model. They find that news firms produce news according to the (political) bias of their customers. They also conclude that increased competition increases the importance of producing ‘biased’ news, as this is the prime quality by which news firms compete. On the other hand, they found that, assuming people dislike obvious distortions of the facts, greater competition and consumer diversity can increase truthfulness of news production (Mullainathan & Shleifer, 2005). The distortion of the facts is captured by the variable 'slanting'. They have effectively explained how any news item can be framed into opposite political contexts (pp. 1032-1033), however they introduce the variable 'slanting' into their model, which consumers dislike. A 'slant' is a deliberate attempt to omit or fabricate some details so as to frame an event according to any one particular political bias (Mullainathan & Shleifer, 2005, p. 1032). This variable is one of the weak spots of Mullainathan and Shleifer's model, as it raises the question whether a news story might
6 be slanted because it has been deliberately made inaccurate, or because the news item has been made inaccurate so as to frame it according to a political bias.
Gentzkow and Shapiro (2006) try to model a Bayesian approach to credibility of certain news firms. They distinguish between two types of firms, high-quality and normal firms. They assume that for each news item the possibilities for reporting are binary. One of the binary options is the 'true' state of the world, and consumers only derive utility from news that reports the actual state of the world. High quality firms always report the true state of the world, normal firms are free to report whatever they wish. Consumers have some probability of learning the true state of the world independently of news reporting, which feeds back into their subsequent consumption choices. Consumers base their choices on prior beliefs about the true state of the world. They balance the likelihood that a firm is 'high-quality' and thus reports the truth. They find that in the case of a monopoly, news firms will report the prior belief of the consumer, even it the world is not in this state, depending on the likelihood of feedback. In a competitive market, the bias of news reporting decreases with the number of firms, and the likelihood of feedback. (Gentzkow and Shapiro, 2006). The model of Gentzkow and Shapiro corresponds to the results obtained by Mullainathan and Shleifer (2006), and presents an effective way of modelling the prevalence of bias in news markets. However, while a model for news market which looks at the quality of reporting should have some feedback mechanism, it cannot be the case that consumers learn the true state of the independently of the news reporting, as consumer rely on news reporting to learn about the state of the world. Also, the reliance on prior beliefs about the state of the world is a wrong assumption as news is always new, any prior belief about the true state of the world is necessarily false or inaccurate.
In a short paper by Kshetri and Voas (2018), the business dimensions of fake news are examined. Kshetri and Voas (2018) identify that the monetary, and non-monetary benefits, as well as the the opportunity costs of creating fake news and the probability of being arrested and fined are all relevant factors. They provide useful examples for each variable except the probability of arrest. Although they establish an identity that describes that fake news will be produced as long as it is profitable enough, i.e. the benefits exceed the (opportunity) costs, they include the probability of arrest. This is a strange addition, given that (deliberately) inaccurate news reporting is not a criminal act. Therefore, this key insight is somewhat superficial. The main contribution of the paper by Kshetri and Voas (2018) are their examples of intermediaries that can aid in the reduction of the prevalence of fake news. They mention social arbiters, such as a network of third party fact checkers, commercial, and legal arbiters (Kshetri & Voas (2018).
Azzimonti and Fernandes (2018) present a model of bots or users spreading fake news on social media. They investigate the measure of the spread of polarizing information, some of which might be false. The relevance of this paper are the insights about the behaviour of consumers whom infer about the reliability of their sources of information. Azzimonti and Fernandes (2018) distinguish between consumers being polarized and consumers being misinformed, which are not mutually exclusive. Polarized consumers have
7 strong differences of preference. Misinformed consumers are subject to missing information, or are confronted with biased information. They build a dynamic graph model of a social network and find that the main factor influencing the prevalence of polarization and misinformation, which is propagated through the network for a long time, is the amount of influence they can exert over consumers. Vice versa, the success rate of polarizing and misinforming bots is affected by the consumer's ability to detect and dismiss polarizing bots as a source of information. When the influence of bots is large enough, it can even "… prevent information aggregation and consensus in the population" (Azzimonti & Fernandes, 2018, p. 42). For the task of modelling a market for news under influence of fake news, a dynamic graph model is unavailable for a number of reasons, however the intuitions behind Azzimonti and Fernandes' model are sound.
Related to this model is the model proposed by Acemoglu, Ozdaglar, and ParandehGheibi (2009). They assume all consumers exchange information pairwise, letting each exchange influence their beliefs. Acemoglu et al. (2009) introduce misinformation by pairwise exchanges whereby there is an asymmetric exchange of information. In this asymmetric exchange, one of the consumers does not change his beliefs according to what the other communicates. They find three sets of conclusions, the first of which may be dismissed as they conclude that ultimately, society has some control over the beliefs of consumers who spread misinformation (while such a consumer was defined as to have beliefs that were not subject to influence by others), and believes will thus converge towards a consensus. The other two concern the composition of the graph of the (social) network, which are not relevant to discuss here.
The finding of Azzimonti and Fernandes (2018), also concur with the findings of Aymanns, Foerster and Georg (2017), whose computational model with reinforced learning leads them to conclude that the key factor influencing the prevalence is the ability for consumer to learn the truth independently and thus to filter out fake news.
Generally, the literature has two approaches to the market for news. It either examines the news as politically biased on a two-dimensional axis, or looks at the spread of misinformation on social networks. No paper as of yet has examined truthfulness as a quality of a news product, although they touch on many important aspects necessary to construct such model.
3. The Model
This section outlines a simple model of news firms and consumers with information asymmetry. Two types of news firms populate the market, one type producing normal news and another producing fake news. Both types of firms will produce news items with some degree of truthfulness. In the case of regular firms, possible deviations from the truth are accidental, whereas in fake news firms these are deliberate. Consumers will select a sample of news items from multiple sources and they prefer their sample to be coherent. Consumers cannot observe the truth independently of the news items they consume.
8 While Akerlof (1970) tries to relate "quality and uncertainty" (p. 488), it is of importance what the 'quality' of a news article is. In this paper, it will be assumed that the 'quality' of a news item lies in the accuracy of the description of the event, or the state of the world, to which the news item pertains. The main insight that has been borrowed from Akerlof to construct the model below is the fact that when the consumer (or buyer) is faced with an information asymmetry, the consumer infers based on the distribution of the variable that determines his behaviour.
This move potentially allows for very delicate consumer behaviour, which is now bounded by the distribution. The 'simplest' consumer will act randomly, but is still constrained to a particular distribution, while it allows for modelling of consumers that are more complex. For these reasons the market for news, under the influence of fake news, will be modelled like Akerlof's market for “lemons”. This will also test, whether the supply of news is influenced by the uncertainty of the accuracy of whatever is being reported.
Unlike Mullainathan and Shleifer (2005) and Akerlof (1970), there are no prices in this model. This model best describes news items that are freely available, such as internet news, and thus does not include subscription newspapers or television broadcasts. This allows for simplification of the model although it foregoes a possible price strategy dimension of market interaction. The external validity however does not suffer greatly as an increasing number of people receive their news from (free) online sources (Newman, Fletcher, Kalogeropoulos, Levy, & Kleis Nielsen, 2018). Also unlike Mullainathan and Shleifer (2005), there is no role for politically biased news reporting as the spread of disinformation can be a political motive in itself. Furthermore, this avoids a difficult discussion about the relation between the accuracy of the description of an event, and the political bias of that description, as mentioned above. Political bias will thus be omitted from this model
What exactly is fake news? Don Fallis (2015) argues for the following definition: "disinformation is misleading information that has the function of misleading" (p. 422). This includes fake news articles and doctored photographs. This definition will be assumed so as to differentiate fake news firms from regular news firms. This indicates that, where regular firms are motivated by profit, and face a trade-off between the quality of their products, and the costs of production. Although profit maximization and market competition are omitted from this model, expanded or adapted models should take this distinction into consideration.
The news product
There are two types of news producers. There are regular producers who produce normal news, which may contain inaccuracies. Regular producers aim to produce news items that describe the world and current events in an accurate manner. The other type produces fake
9 news, which aims to disrupt other (true) descriptions of the world. The proportion of regular news firms is given by 𝑛𝑦.
The true state of the world is represented by θ. For any given event there can be only limited information, especially in news reporting. Therefore, every news article, item or item is only an approximation of θ. Every producer produces its own approximation, denoted by 𝑦𝑖. This allows the computation for a mean, given by ∑𝑦𝑁𝑖= 𝑦̅, where N is the number of firms. This mean signifies the approximation of all regular news item of a given θ. The mean 𝑦̅ converges to θ when the number of firms is increased.
Any news item describes the world approximately. Given the available pieces of information that constitute each item 𝑦𝑖 is normally distributed, so that 𝑌 ~ N(θ, 1) . This results in
possible mistakes, or deliberate omissions of information being further away from θ than genuine attempts at reporting the truth, though this will never be equal to θ. This also takes into account that given a limited amount of time, there can be only limited data gathering which would lead to a less than perfect description of the world and events.
Each (fake) news story is thus always some distance away from θ, meaning that there are some elements in the story that are missing or incorrect. The description of an event can be false in an infinite number of ways, which would imply that the description can be differentiated along an infinite number of axes.
To simplify, I am using just one dimension, with two signs (+ and -). This means that stories that are distanced on one side of θ are not in the same way incorrect as stories on the other side. For example, stories with a negative value may lack some information, while stories with a positive value may contain information that is false.
Fake news items are denoted by 𝑤𝑖 and are not drawn from a normal distribution,
but instead from a uniform distribution Unif[-3, 3], where the value of 0 is equated by θ. Furthermore, it will be assumed that the normal distribution is flat beyond 3 standard deviations on either side, so that N(θ, 1) is only defined for [-3, 3]. Although the normal distribution is never 0 at any one point, a discrete interval is needed in order to construct a mixed interval with a uniform and a standard distribution as its components, as uniform distributions are only defined for a given interval. The area between 3 standard deviations on either side should contain more than 99% of the mass, so it will be assumed that this interval contains all of the mass.
Although fake news stories with a value close to θ seem paradoxical, this is due to the simplification of the ways a story can be incorrect and thus distanced from θ. The difference is in the malicious aspect of the fake news. While the news story with a value close to θ, might have only very few flaws, this model allows for fake news stories that are very close to the truth, but have a few minor details deliberately altered. To model this, a distribution where stories that are further removed from theta are becoming more likely, relative to real news stories, is sufficient. The uniform distribution meets this criterion, and makes it easier to solve the model.
10 Consumer behaviour
I will assume that consumers get their news from a multiple sources. They will always consume, at least 2 news items. News items, whether fake or real, are denoted by 𝑥𝑖. The
number of items selected is denoted by: 𝑛𝑥.
Because both the normal distribution and the uniform distribution are symmetrical around θ, the value of any sample of news items would converge towards θ. This need not necessarily be the case, because for any event there is limited information available. Genuine news reporting thus need not closely approximate θ. Also, consumers, like firms, cannot observe θ. Therefore, they will let their measure for truthfulness depend on the standard deviation of the sample sources of news they consume. Consumers thus do not try to consolidate unfounded or contradictory information, but rather choose a more or less consistent set of information that is supported by multiple sources.
The utility function of the consumer is given by: 𝑈 = 𝑀𝑖𝑛(𝑠)1 , where s is the standard deviation of the sample of news items selected. Given the utility function, consumers maximize their utility by minimizing the size of their sample, and the difference of the descriptions of each news item. Consumers will select one news item and find one more news item that results in the lowest possible standard deviation for a sample of two. Thus, s is given by: 𝑠 = √(𝑥1− ∑ 𝑥𝑖 𝑛𝑥) 2 + (𝑥2− ∑𝑥𝑖 𝑛𝑥) 2 𝑛𝑥− 1
Given that 𝑛𝑥= 2 this simplifies to:
𝑠 = √(𝑥1−𝑥1+ 𝑥2 2 ) 2 + (𝑥2−𝑥1+ 𝑥2 2 ) 2
In order to minimize s, the consumer seeks out values of x that are close together. This is given by: 𝑀𝑖𝑛(𝑠) = 𝑀𝑖𝑛 (√(𝑥1−𝑥1+ 𝑥2 2 ) 2 + (𝑥2−𝑥1+ 𝑥2 2 ) 2 ) = 𝑀𝑖𝑛 ( 𝑥𝑖−𝑥1+ 𝑥2 2 ) = 𝑀𝑖𝑛 (12( 𝑥𝑖− 𝑥1) + 1 2( 𝑥𝑖− 𝑥2)) = 𝑀𝑖𝑛 (12( 𝑥1− 𝑥1) +1 2( 𝑥2− 𝑥1) + 1 2( 𝑥2− 𝑥2) + 1 2( 𝑥1− 𝑥2))
11 In the formula for s, any non-negative result will be squared and thus positive. This means that ( 𝑥1− 𝑥2) = ( 𝑥2− 𝑥1). Therefore: 𝑀𝑖𝑛(𝑠) = ( 𝑥1− 𝑥2) . Thus, the consumer will try to
minimize the difference between the two items selected.
The market
After firms have produced their (fake) news items, each consumer will randomly select the first from all items produced with equal likelihood, so with 𝑁1. Given this first item, they will select another that produces a sample with the lowest possible variance. This will result in a cluster of two news items that are next to one another.
The proportion changes according to the proportion of (fake) news item selected for each. So for each event or state of the world θ, there is an expected number of news items selected, denoted by: 𝐸[𝑌]. The proportion of firms on the market adapt to consumer selection of items. The proportion of firms will match the proportion of items selected. The proportion of firms for the next event or state of the world θ, is given by:
(1) 𝑛𝑦 𝑡+1=
𝐸[𝑌] 𝑛𝑥
In order to determine whether fake news firms will thrive, or fade out of the market, it is necessary to determine the factors determining 𝐸[𝑌] and subsequently the probability for each of the selected items to be a (fake) news item.
The first item (𝑥1) is selected at random with probability 𝑁1. This probability depends
on the proportion of firms active on the market. Thus, the value of 𝑥1 also depends on this
proportion.
News items are drawn from the normal distribution, weighted by the proportion of firms. This is given by:
(2) 𝑛𝑦(𝜎√2𝜋1 𝑒−12(𝑥1−𝜇𝜎 ) 2
) = 𝑛𝑦( 1
√2𝜋𝑒
−12𝑥12); Given that σ = 1 and μ = θ = 0.
Fake news items are selected from the weighted uniform distribution: Unif[a, b]. This is given by:
(1 − 𝑛𝑦) (𝑥1− 𝑎 𝑏 − 𝑎)
where, a and b are the bounds of the interval. As mentioned before, the interval used here is [-3, 3]. Thus, this part becomes:
(3) (1 − 𝑛𝑦) (
𝑥1+ 3
12 The news items are thus drawn from the mixed distribution, which depends on the proportion of news firms, 𝑛𝑦. Combining (2) and (3) gives the expression for this
distribution. This is given by:
(4) 𝑓(𝑥) = 𝑛𝑦( 1 √2𝜋𝑒−12𝑥 12) + (1 − 𝑛 𝑦) ( 𝑥1+ 3 6 )
Plotting this mixed distribution for 𝑛𝑦=12, gives:
The consumer will now select the second item. The consumer will select the value closest to 𝑥1. On average, the closest possible 𝑥𝑖 will be in the remaining part of the distribution that is
largest, given the distance of 𝑥1 from θ. The mass of the remaining part of the distribution is
given by the integral of the mixed distribution, so:
(5) { ∫ 𝑓(𝑥) 𝑥1 −3 𝑑𝑥, 𝑥1> θ (1 − ∫ 𝑓(𝑥) 𝑥1 −3 𝑑𝑥) , 𝑥1< θ
13 If we assume that 𝑥1 > θ, we can calculate the probability that 𝑥2 is also a news item. In
order to do this, we need find the probability that a news item is drawn from the remaining part of the distribution. The relative probability is dependent on the surface area of the normal distribution, over the entire distribution. The proportion of real news on a given interval is given by:
(6) ∫ 𝑓(𝑥−3𝑥1 1)𝑑𝑥 − (1 − 𝑛𝑦)(𝑥16+3)
In other words, the entire surface area bounded by the value of 𝑥1, minus the probability of a
fake news item being selected. This can be rewritten as:
(7) 𝐹(𝑥1) = ∫ (√2𝜋1 𝑒−
1 2𝑥12) 𝑑𝑥
𝑥1
−3 = 𝜙(𝑥1)
Equation (7) is also known as the Cumulative Distribution Function (CDF) of a normal distribution. The relative chance is then given by the weighted proportion of the normal distribution over the whole distribution. This is given by:
(8) 𝑛𝑦 𝜙(𝑥1) ∫ 𝑓(𝑥−3𝑥1 1)𝑑𝑥
This allows for a derivation of the expected value of Y given a certain sample of items. Because the first item is selected at random, the probability that 𝑥1, is a news item is simply
given by:
(9) 𝑃(𝑥1∈ 𝑌) = 𝑛𝑦
The probability that 𝑥2, is a news item is given by (8). Combining (8) and (9) gives:
(10 ) 𝐸[𝑌] = 𝑃(𝑥1∈ 𝑌) + 𝑃(𝑥2∈ 𝑌) = 𝑛𝑦+ 𝑛𝑦 𝜙(𝑥1) ∫ 𝑓(𝑥−3𝑥1 1)𝑑𝑥
; 𝑔𝑖𝑣𝑒𝑛 𝑥1> θ
This is a function of 𝑥1 and 𝑛𝑦. From (10) results the proportion of news item selected out of
all items. This proportion is then taken to be representative for all consumers’ choices. Combining (1) and (10) gives the proportion of firms for the next state of the world 𝑛𝑦 𝑡+1,
which is dependent on the current proportion of firms (𝑛𝑦 𝑡) and the value of 𝑥1. This gives:
(11) 𝑛𝑦 𝑡+1= 𝑛𝑦 𝑡+ 𝑛𝑦 𝑡 𝜙(𝑥1) ∫ 𝑓(𝑥−3𝑥1 1)𝑑𝑥 𝑛𝑥 = 1 2(𝑛𝑦 𝑡+ 𝑛𝑦 𝑡 𝜙(𝑥1) ∫ 𝑓(𝑥−3𝑥1 1)𝑑𝑥 ) ; 𝑥1> 𝜃
14 It must be noted here that the results obtained via this method will differ from results obtained from a simulation of this model, since it is not possible to calculate the nearest value to 𝑥1 from the distribution, as there are at least N-1 possibilities. Subsequently, it will
be the case that the nearest value to 𝑥1is in the smaller remainder of the distribution.
However, the results given here should be unbiased towards the real value of 𝐸[𝑌].
A numerical example
If we assume that the market is saturated with both types of firm in equal proportion (𝑛𝑦= 12 ), we can calculate the expected number of news stories to be selected, 𝐸[𝑌], and the
proportion of firms of each type (𝑛𝑦𝑡+1) for the next period.
The main difficulty comes from finding the value of 𝑥1. If we assume 𝑥1 ≠ 𝜃 and is
drawn from either [-3, θ], or [θ, 3], one can find an average value of 𝑥1. This is given by
calculating the expected value given the interval. For [θ, 3] this is given by the average of the interval, thus, where there are equal amounts of mass in 𝑓(𝑥) on either side between the bounds of the interval. For the normally distributed component the value of 𝑥1 where there
is equal mass on both sides in the interval [θ, 3] represents 75% of all mass in the distribution, hence this gives:
(12) 𝑥1= 1 2( 𝜙−1( 3 4)) + 1 2( 𝑏 − 𝜃 2 ) = 1 2(0.67449) + 1 2( 3 2) = 1.087245
The 𝜙−1(𝑝) used here is the quantile function, which produces the value of x for a given percentage of probability. Plugging this value into (7) gives us the expected proportion of real news selected. This results in:
(13) 𝐸[𝑌] = 𝑛𝑦+ 𝑛𝑦 𝜙(𝑥1) ∫ 𝑓(𝑥−3𝑥1 1)𝑑𝑥 =1 2+ 1 2 𝜙(𝑥1) 0.75 = 1 2+ 0.430768 … 0.75 ≈ 1.148713
Substituting the result obtained from (13) into (11) gives the new proportion of firms. This is given by: 𝑛𝑦𝑡+1= 𝐸[𝑌]2 = 1.148713 ∙12= 0.5743565. Meaning that for a starting proportion of 50%, the result is that the proportion of real news firms changes to 57.44%
Equilibrium
To arrive at an equilibrium for which the proportion of both types of firms remains stable, there must be a value of 𝑛𝑦, for which holds that 𝐸[𝑌]𝑛
𝑥 = 𝑛𝑦𝑡+1. Thus, we need to solve:
(11) 𝑛𝑦 𝑡+ 𝑛𝑦 𝑡 𝜙(𝑥1 ) ∫ 𝑓(𝑥−3𝑥1 1)𝑑𝑥 𝑛𝑥 = 1 2(𝑛𝑦 𝑡+ 𝑛𝑦 𝑡 𝜙(𝑥1) ∫ 𝑓(𝑥−3𝑥1 1)𝑑𝑥 ) = 𝑛𝑦𝑡+1 ; 𝑥1> 𝜃
15 This formula is dependent on both 𝑛𝑦 𝑡 and 𝑥1. Given that 𝑥1 is randomly selected and its
value is thus randomly determined as well. Recall that 𝑥1 is drawn with equal probability
out of all items. This draw is itself dependent on 𝑛𝑦 𝑡. In order to find a value for 𝑛𝑦 for
which holds that 𝑛𝑦 𝑡 = 𝑛𝑦 𝑡+1, we need to reduce the left hand side of (11) to a function that
only depends on 𝑛𝑦.
Both parts of the distribution are symmetrical around θ and thus the value of 𝑥1 is on
average equal to θ, the true state of the world. This is what we would expect for the real news part of the distribution, but not for the uniformly distributed part that represents the fake news items.
A possible method for finding a general solution is to use the same approach used in the numerical example set out above. Although values of x would on average be equal to θ, the variance of the mixed distribution is not 0. Thus, any concrete sample would be at least some distance removed from θ. As both distributions are symmetrical around θ, the likelihood for each value of 𝑥1 can be calculated. This allows us to determine the value for 𝑥1 for which it is
equally likely than not, that x indeed assumes the value of 𝑥1. In other words, if we divide
the distribution in the two halves [-3, θ], and [θ, 3], we can for each half calculate the value for which 𝑥1 indicates the point where there is an equal amount of mass in both sides of the
interval. This value indicates the 1st and 3rd quartiles of the mixed distribution. This value is
dependent on 𝑛𝑦.
This means that, for 𝑥1 > 𝜃, we need to set (4) equal to 34 and solve for 𝑥1. This is given by:
(14) 𝑛𝑦( 1 √2𝜋𝑒−12𝑥 12) + (1 − 𝑛 𝑦) ( 𝑥1+ 3 6 ) = 3 4
16 Unfortunately, this equation has no solution. However, we can calculate the value for 𝑥1 for
both the normal distribution and the uniform distribution which give 75% of the mass of the distribution, given 𝑥1> 𝜃. These values are the same as the ones used in the numerical
example. This gives 𝜙−1(34) = 0.67449 for the normal distribution, and (3−𝜃2 ) =32 for the uniform distribution. We can the construct a linear approximation of 𝑥1 by constructing a
weighted average of these values. Using a linear approximation for the normal distribution is justified because on the interval, the normal distribution decreases approximately linearly. The value for 𝑥1 is thus given by:
(15) 𝑥1= 𝑛𝑦(0.67449) + (1 − 𝑛𝑦) (3 2)
This is a function that is only dependent on 𝑛𝑦. Substituting (15) into (11) gives:
(16) 1 2 ( 𝑛𝑦 𝑡+ 𝑛𝑦 𝑡 𝜙 (𝑛𝑦 𝑡(0.67449) + (1 − 𝑛𝑦 𝑡) (32)) ∫ 𝑓 (𝑛−3𝑥1 𝑦 𝑡(0.67449) + (1 − 𝑛𝑦) (32))𝑑𝑥 ) = 𝑛𝑦𝑡+1 ; 𝑥1> 𝜃
Now that (11) has been reduced to a function dependent only on 𝑛𝑦, we can find the
equilibrium proportion of firms. If we plot (16) we find that:
The only points where the line corresponding to 𝐸[𝑌]2 intersects the line that describes the values where 𝑛𝑦 𝑡 = 𝑛𝑦 𝑡+1, are 0 and 1. Thus, if we assume that there is at least one regular
news firm active on the market, the proportion of regular news firms should converge to 1. Thus, fake news firms will slowly be driven out of the market as it is always more likely consumers will select real news items. Thus: 𝑛𝑦 𝑡 = 𝑛𝑦 𝑡+1 = 1.
17 4. Discussion
Given the outcome of the model found in (16) it would appear that fake news, when modelled correctly, is no inherent problem for the market for news. The information asymmetry consumers face does not seem to lead to the same results as Akerlof’s market for lemons. However, the model does constitute a successful attempt to model this asymmetric situation as it derives it solutions from the way consumers infer about the distribution of the relevant quality, which in this case is truthfulness. It must be noted that, where in Akerlof’s model, only a uniform distribution is employed. In contrast, the model set forth in this thesis leads the consumer to infer about a mixture of two distributions. As such, the disappearance of fake news from the market might be attributed to consumers inferring about the right distribution. This model was constructed by assuming that minor fabrications of news (fake news close to θ) are equally likely as major fabrications. However, this may not be how firms producing fake news operate in reality, and by this assumption, consumers are thus always ‘right’; consumers always infer about the correct distribution.
Yet, assuming real news is distributed normally is a strong assumption. Another reason for assuming that fake news was distributed uniformly was for simplicity of calculations. In the case that fake news stories with large fabrications (fake news far removed from θ) are more frequent, such as would be the case with a quadratic or exponential function describing fake news, the result obtained in (16) might not hold. Indeed, the opposite result might also be found, depending on the distribution of fake news used.
Another important caveat is the difference from the result obtained in (16) and the result that would be obtained by running a simulation. Calculating the value for 𝑥1 might be
reasonable, according to the solution set forth above, the consumer always seeks the second news item in the largest part of the distribution remaining. While on average, the item closest to 𝑥1 should be in this part of the distribution, it ignores the part of the distribution
where fake news is relatively more prevalent. A simulation would often select an item from the shorter part of the distribution, which is more likely to be a fake news item. Thus, a simulation would find a weaker convergence towards the equilibrium, or no convergence at all. Unfortunately, there is no way to compensate for this problem in an analytic solution as averaging the long and short parts of the distribution should result in an average result of 𝑛𝑦 , which would not capture possible change in this variable which will be present in a
simulation of this model.
In its current form the model is still simple, which leaves very little room for firm strategy. Apart from the factors already omitted, such as prices and bias, another few dimensions of depth are impossible in the current model.
An interesting addition to the model would be the case where real news firms would observe some signal 𝑦̃, which is an approximate value about the state of the world. News items would then be symmetrical around 𝑦̃ , instead of θ. If we assume fake news is distributed in the same fashion, the results obtained might not hold. This allows for societies
18 at large to be 'misinformed' about a certain topic or event, just as in the model developed by Azzimonti & Fernandes (2018). Depending on the value of 𝑥1, fake news might have an
advantage compared to real news as it is not symmetrical around θ.
Another interesting addition would be to model the relation between the state of the world in one instance 𝜃𝑡, and study the influence of the outcome of the market for news on a
next state of the world 𝜃𝑡+1. Given the pre-existing relation between these two states of the
world, the items the consumer selects have some average value, which corresponds to the state of the world the consumers infers to world to be in. Consumers would also act on this perceived state of the world. Although the model found that on average, this perceived state should be equal to the real state of the world θ, in the case where fake news is prevalent or regular news is skewed towards some signal 𝑦̃, consumers would act on false information, thus the change between 𝜃𝑡 and 𝜃𝑡+1 would be influenced by the proportion of firms on the
market.
Although fake news does not seem to be a problem for the market for news in the model developed in this thesis. In the case a different distribution for fake news results in a convergence to a market with only fake news firms, thought must be put into a possible solution.
Firstly, firms, could modify the parameters of distribution of the news stories. Firms could influence the variance of the population of news stories by reporting more thoroughly, or laxly. In the model it was assumed that news items were distributed N(θ, 1). Given that the standard deviation is also variable, and possibly different from 1, the equilibrium is then given by: 𝑛𝑦𝑡+1=1 2 ( 𝑛𝑦 𝑡+ 𝑛𝑦 𝑡∫ ( 1 𝜎√2𝜋𝑒−12( 𝑥1−𝜃 𝜎 ) 2 ) 𝑑𝑥 𝑥1 −3 ∫ 𝑓(𝑥−3𝑥1 1)𝑑𝑥 )
Should firms decide to report less thoroughly, 𝜎 should rise above 1, and the probability that 𝑥2 is a news item is lowered. Should firms decide more thoroughly, the opposite happens.
However, this may only influence the speed of convergence towards the equilibrium, but does not change the equilibrium proportion of firms.
Akerlof (1970) briefly discusses four real world cases where information asymmetry is a relevant factor and discusses that for most cases, an intermediary might be the solution, or that a dysfunctional intermediary is the cause of the problem (Akerlof, 1970, pp. 492-500).
A possible extension to the model that allows for the possibility of an intermediary for this problem would include the level of effort spent by firms to report thoroughly or accurately. So that the value of a news report (its approximation of θ) is given by: 𝑦𝑖 = 𝜃 + 𝑦̃− 𝜃𝑒 ; 𝑒 ≥ 1
19 Where e is some costly effort and 𝑦̃ is a signal indicating approximate value of θ, as also mentioned above. In this extension, firms thus expend effort in order to produce more accurate news items, thus lowering the variance of the population. The fact that variance of the population influences the speed of convergence towards equilibrium, gives room for intermediary organizations that counter fake news firms. They could influence 𝑛𝑦 𝑡+1
directly, by investigating possible malicious practices on the market and removing fake news firms. Or it could be an intermediary that reports on the effort levels of all firms on the market. Lastly, the intermediary, if it is a collective body, could subsidize effort levels of firms so as to influence the rate of convergence towards equilibrium. Various combinations and variations are possible.
The final possibility for intermediary functions, which Akerlof (1970) mentions, are licenses (p. 500). Some have proposed a network of fact-checkers (Brandenburg - van de Ven, 2018; Kshetri & Voas, 2018), which could serve as a licensing agency. Firms that are aware of their own effort level, and can thus deduce, by observing 𝑦𝑖 and 𝑦̃, the effort level
of others. As all news firms are distributed along a political spectrum, any licensing network could increase its legitimacy by including all sides of the spectrum. As a result, the licence for any particular firm or news item does not suffer from the question of whether it is (in part) inaccurate because of unintentional mistakes or an honest lack of information, or because it is politically biased. This is a major possible extension to the model which takes into account political bias, that is the subject of most literature related to this thesis.
However, the exact organisational or institutional way this intermediary licensing should be organised, is subject to far reaching political debates, which might be even more difficult to solve than an attempt at modelling such an intermediary.
Conclusion
It would seem, therefore, that the market for news under the influence of fake news can indeed be modelled in the style of Akerlof’s Market for “Lemons”, but not with the same results. Although the model developed in this thesis omits prices and valuations of the products, foregoing many possibilities for modelling firms strategies, the market for (fake) news can be modelled like the Market for “Lemons” in terms of consumer behaviour. In both models, consumers act on the basis of what little information they have access to: the distribution of the relevant variable. Also, in both models the consumers infer about the distribution of the relevant variable. Contrary to Market for “Lemons”, the market for (fake) news does not lead to a state where there are no products of quality available on the market.
For the market for (fake) news, it follows from the way the news items are selected by the consumer, that the proportion of news firms slowly converges towards 1. This result is highly dependent on the assumed distribution of fake news. In case of a different distribution, there are a lot of possible additions to the model. Such as firms attempting to influence the speed of convergence towards equilibrium. Possible additions are intermediary organisations, which might influence the convergence towards equilibrium.
20 However, possible remedies fighting the prevalence of fake news, if applicable, are subject to numerous political problems and objections. In any case, the market for (fake) news does not seem to be a market for “lemons”, but can be modelled as such.
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