Media Proliferation with Quasi-Rational Consumers
Jimmy Chan∗
Daniel F. Stone†
February 2010
Abstract
The number of Internet news media outlets has grown rapidly in recent years. This paper analyzes the effects
of media proliferation on electoral outcomes and social welfare. We assume voters are information-seeking but
choose outlets that are excessively partisan given the voters’ own ideological preferences, due to quasi-rationality.
We find that if voters who think (correctly or not) that they prefer extreme outlets may influence the election,
then media proliferation strictly improves welfare. Otherwise, welfare cycles as the number of outlets increases.
In general, welfare is lowest when there are only one or two media outlets, for all levels of rationality.
JEL Classification Numbers: D72, D81, D83, L82
Keywords: Media Bias; Media Competition; Bounded Rationality; Blogs; Elections
∗ Shanghai
† Oregon
University of Finance and Economics; jimmy.hing.chan@gmail.com.
State University, Department of Economics; dan.stone@oregonstate.edu.
1
Introduction
The news media landscape has changed dramatically in recent years. The main trend is well
known; the newspaper industry is undergoing major decline, while the readership of online news
has grown sharply. From 1996 to 2008, the percentage of Americans who “get most of their news
about politics and the election” from the Internet increased from 3% to 26%. The percentage for
hard-copy newspapers decreased from 60% to 28%. The percentage of U.S. adults who get some
political news from the Internet went up from 4% to 44% over the same period.1
The advent of the Internet has not only affected where people get their news, but also had a
tremendous impact on both media market structure and media politics. Because of high fixed costs,
traditional print and network television media markets are dominated by large firms considered
to have mass appeal to an audience across the political spectrum.2 By contrast, the online news
market, with much lower fixed costs, contains a far greater number of independent news outlets
with a variety of political views.3 Moreover, online news consumers are increasingly likely to select
news sources that share their political views.4
A number of journalists and social scientists have expressed concerns that this proliferation of
media choice and assortive matching between news consumers and news media may be socially
harmful.5 But the view that more choice could be bad is hard to reconcile with standard economic
1 The
figures are from the Pew Research Center’s report, “The Internet’s Role in Campaign 2008”, see
<<http://www.pewinternet.org/Reports/2009/6–The-Internets-Role-in-Campaign-2008/1–Summary-of-Findings/Summary-ofFindings.aspx?r=1>>, last accessed 8-2009. See also the Pew Research Center Biennial News Consumption Survey, available
at <<http://people−press.org/reports/pdf/444.pdf>>, which says from 1998 to 2008 the percentage of Americans age 18 or older who
“read a newspaper yesterday” (“regularly” watch nightly network news) declined from 48% to 34% (38% to 29%), while the percentage
who went online for news at least three times per week grew from 13% to 37%. The percentage “regularly” watching cable news grew
from 33% (in 2002, no data available prior to that year) to 39%. See De Waal et al. (2005) for more direct evidence of young consumers
switching from print to online newspapers.
2 For example, see Poniewozik (2009).
3 Seventeen distinct websites were “most frequented” by at least two percent of readers of online news in 2008
(<<http://people−press.org/reports/pdf/444.pdf>>), and according to Drezner and Farrell (2008) there were an estimated 70 million
blogs (worldwide) in May 2007. Although the majority are likely personal and not focused on news, there almost certainly hundreds,
and likely thousands, that mainly discuss politics.
4 From the Pew’s Internet 2008 report, cited above: “Fully 44% of Democratic online political users (up from 34% in 2004) and 35%
of Republican online political users (up from 26% in 2006) now say that they mostly visit sites that share their political point of view.”
5 See Sunstein (2001), Manjoo (2008), Kristof (2009), and Drezner and Farrell (2008). Sunstein (2001) refers to consumers exploiting
expanded choice to personalize the information they consume as “filtering” and says, “From the standpoint of democracy, filtering is
a mixed blessing... Unanticipated encounters, involving topics and points of view that people have not sought out and perhaps find
2
theory. Indeed, in his seminal work on democracy, Downs (1957) argues that a rational voter should
delegate the task of evaluating political platforms to a well-informed third party that shares his
view. According to this theory, the more third parties there are with different views, the better
off voters will be. However, this implication of the Downsian theory may require the assumption
that voters make optimal choices about news sources–i.e., that voters are fully rational. This
assumption has of course been increasingly questioned in the economics literature. Full rationality
is perhaps especially suspect in the context of news information consumption for a few reasons.
First, there is strong evidence that individuals are flawed information processors.6 A vast literature
shows individuals use heuristics, or mental short-cuts, to interpret new information due to cognitive
limitations. Second, individuals often prefer to obtain information consistent with previously held
views.7 Third, political information has no practical use in daily life, and individuals have almost
no influence on political outcomes. Like shoppers who are inattentive to shipping costs because
they are not incurred often, news consumers may not fully understand how to interpret news from
different sources since there is little incentive to develop this understanding. All of these reasons
may cause consumers to end up choosing media outlets that are too partisan relative to their true
political preferences.
Thus, we cannot take it for granted that more competition in the news market must make
consumers better informed and society better off. This point has been made in previous literature.
For example, Mullainathan and Shleifer (2005) show that if consumers have heterogenous tastes for
biased news, while a monopolist media outlet produces unbiased news for all consumers, duopolist
media outlets bias their news in opposite directions. We depart from the previous literature in two
ways. First, we examine markets in which the number of media outlets becomes unboundedly large,
irritating, are central to democracy and even to freedom itself.”
6 Tversky and Kahneman (1974) and Nisbett and Ross (1980) are classic references. See Stone (2009) for a recent empirical study
and discussion of the literature.
7 A great references is Nickerson (1998).
3
instead of focusing on monopoly and duopoly. Second, we explicitly analyze the welfare effects via
electoral outcomes resulting from consumers having a cognitive limitation they are unaware of.
We formally analyze media proliferation by adapting the model of Chan and Suen (2008). Voters with heterogenous ideological preferences consume news to learn which of the two political
parties better serves their interests. Consumers find news from outlets that share their political
preferences more informative, but have a cognitive or attention limitation that causes them to
underestimate how extreme media outlets really are. On the supply side, media outlets enter the
market and select ideological positions to compete for market share. In equilibrium the number of
media outlets and their positions are determined by the entry cost, the distribution of consumer
preferences, and the consumers’ cognitive limitation. A monopoly media outlet locates at the
ideological center. As the entry cost decreases, the number of outlets increases and their positions become more spread out. In the limit there will be a media outlet arbitrarily close to any
ideological position as the entry cost goes to zero.
We find the welfare consequence of having 20 media outlets–or even 100 or 1,000 outlets–instead
of 10 is quite different from that of having two instead of one. When two partisan outlets replace
a centrist outlet in the middle, consumers have no choice but to consume more partisan news. But
with 10 outlets, news consumers already have a rich array of ideological choices, including extreme
ones. Having another 10 (or more) merely fills the gaps between existing outlets. While some
moderate consumers will switch to more partisan outlets, others with more extreme preferences
will do the opposite. The overall welfare effect depends crucially on the relative importance of
consumers who think they prefer more extreme news. When these consumers may influence the
election outcome, they may also sometimes get propaganda, or the equivalent to no news. In section
4.2.1 we refer to this case as “the median voter sometimes gets propaganda”, and show welfare
4
is strictly increasing in the number of outlets. Intuitively, consumers with extreme preferences
never consume news from a centrist outlet. Having a greater number of media outlets, including
some fairly extreme ones, prevents these consumer from getting no news. Since in our model lower
rationality increases the electoral importance of consumers who think they prefer extreme news,
our main result is quite surprising: lower rationality makes it more likely that proliferating choice
is strictly socially beneficial.
The situation is different when the election is determined exclusively by the votes of consumers
who think they prefer more moderate news, who, unlike the more extreme ones, would consume
centrist news when there is no other choice (section 4.2.2; “the median voter never gets propaganda”). But even then, the harm caused by too many choices seems to be limited, and the benefit
possibly large. First, we show that in this case welfare will cycle rather than monotonically decrease in the number of outlets. Furthermore, while welfare with infinite choice is sometimes lower
than welfare under monopoly, it is always higher than the minimum welfare under duopoly. This
is because the duopolistic outlets may adopt fairly extreme position to cater to relatively extremist
consumers, who never determine the election outcome. This problem never arises as the number
of media outlets becomes large.
The main conclusion from our model is thus that media proliferation generally improves electoral
choices, despite quasi-rationality. We want to be clear though that we do not claim our work “closes
the book” on the relationship between media choice and welfare. We merely hope to clarify a few
points we believe to be subtle but important. At the end of the paper we provide remarks on other
issues neglected by our model that certainly merit future research.
5
1.1
Related Literature
This paper builds on the recent surge of economics literature on media markets. As mentioned
above, one major difference between our paper and the previous literature is our analysis of media
markets with an arbitrarily large number of firms.8 Papers that allow for many firms include
Nagler (2007) and Anand, Di Tella, and Galetovic (2007). The latter do not explicitly analyze
welfare; the former does, and also focuses on Internet media competition in particular. However,
he does not focus on political issues, and also makes a very different behavioral assumption, in
which consumers are more likely to get news from known brands as the number of choices grows
large.
We also depart from the previous literature in our method of modeling quasi-rationality. We
effectively model quasi-rationality as causing consumers to erroneously think they prefer more
partisan media. This consumer behavior is similar to that of the strand of the media literature
in which consumers are rational information processors, but are not solely information-seeking as
they also obtain utility from having their prior beliefs confirmed. In some models of this type, such
as Mullainathan and Shleifer (2005), consumers are capable of “backing out” the true news, which
they cannot do in our model. Bernhardt et al. (2008) also model consumers as gaining utility from
belief confirmation. In their model bias and increasing competition can adversely affect electoral
results. A key difference between our paper and theirs is that they assume all voters always
consume news, while we allow for the possibility that increased competition and bias expand the
market for news.
8 Other papers that focus on markets with a few firms are Gentzkow and Shapiro (2006), Burke (2008), Chan and Suen (2008), and
Duggan and Martinelli (2009).
6
2
The Model
2.1
Voter Behavior and the Electoral Outcome
The model is based on that of Chan and Suen (2008) (henceforth CS). There are two political
parties, liberal (L) and conservative (R), which are competing for electoral office. Each party is
committed to a fixed policy. The electoral outcome is determined by majority vote. There is a
continuum of voters of unit mass. The utility function of voter i is
ui (y, θ) =




θ − βi
if y = L,



βi − θ
if y = R,
where θ is a random variable distributed uniformly on [0, 1] that denotes the unobserved politicaleconomic state of the world, and y is the election winner. The parameter βi represents voter
i’s preference for the conservative party; voter i prefers L to R if and only if θ ≥ βi . We call
βi voter i’s ideal cutoff. Voter preferences are subject to a random aggregate shock ∆ that is
uniformly distributed on [−δ, δ] where 0 < δ ≤ 0.5. Given ∆, the fraction of voters who prefers R
when the state is x (i.e., with ideal cutoffs less than x, which is a realization of θ) is F (x − ∆).9
We assume that the distribution function F has a support on [−0.5, 1.5] and a density f that is
weakly single-peaked and symmetric over 0.5. We sometimes assume that F is uniform to obtain
closed-form solutions. Given ∆, the political preference of the median voter, denoted by βm , is
0.5 + ∆. Unconditional on ∆, βm is distributed uniformly on [0.5 − δ, 0.5 + δ]. Thus, the parameter
δ measures the stability of voter preferences. When δ = 0.5, the βm could be any value between 0
and 1.
It is important to note that while θ is modeled as a real number, it denotes complicated policy
9 To
be clear, βi is voter i’s preference after the realization of the shock ∆. βi can be thought of as the sum of a latent variable bi
and ∆, in which the distribution of b across voters is F ().
7
issues that are costly to learn. Few individual voters, for example, are willing to spend the time to
read enough about the science, economics, and international politics of global warming to decide
whether a cap-and-trade program is optimal policy. Instead of reporting the “whole unbiased
truth” that no consumer will read, a media outlet must select the news that is most relevant to
its readers. To capture the limited attention span of news consumers, we follow CS in assuming
all media outlets observe θ but only report whether θ is greater than some strategically chosen
threshold that, once chosen, is fixed and publicly known. We refer to θj , the threshold of media
outlet j, as the outlet’s editorial position, and we say the outlet reports l when θ ≥ θj and r when
θ < θj . Since a higher realization of θ favors party L, we can also interpret an l report as the
outlet’s endorsement of party L. Thus, outlets with editorial positions below 0.5 are liberal in the
sense that they are more likely to report l than r (or more likely to endorse L than R), and outlets
with θ > 0.5 are conservative in an analogous sense. In addition to the news media, each party
also produces “news” that always supports its own policy; we refer to this party-produced news
as propaganda. Parties L and R, therefore, are like news outlets with editorial positions 0 and
1, respectively. Instead of consuming news, voters can visit a party’s website and consume party
propaganda.
Since there is a continuum of voters, there is zero probability that an individual vote is pivotal.
To get around the “paradox” of voting, we follow CS in assuming that each voter cares about
voting “correctly.” This means each voter maximizes his psychological utility from voting, which is
the utility the voter would obtain if he were indeed the median voter. It is also assumed although
the explicit price of news is zero, a voter consumes news if and only if doing so weakly improves
his/her voting decision. There must at least be some psychological benefit to becoming more
informed to compensate for the time and effort cost of consuming news. This means that if a voter
8
consumes news from an outlet, then he must vote according to its recommendation.
Prior to reading news, a voter who consumes news from outlet j has expected psychological
utility:
E[ui (y, θ|θj )] = E[ui (R, θ|θ < θj )]P r(θ < θj ) + E[ui (L, θ|θ ≥ θj )]P r(θ ≥ θj )
=
¡
¢
βi2 − βi + 0.5 − (βi − θj )2 .
(1)
Since the first term in (1) is a constant, voter i’s objective is to choose outlet j to minimize
the second term. Intuitively, voter i wants to vote for party L when θ ≥ βi . But if he follows the
recommendation of outlet j, he votes for party L when θ ≥ θj instead. The loss in psychological
utility, therefore, increases in the distance between βi and θj .
It is consequently straightforward to derive the voting and news consumption behavior of rational voters. Suppose there are n distinct editorial positions θ1 < θ2 < ... < θn . Extreme voters with
ideal cutoffs strictly less than θ1 /2 or strictly greater than (1 + θn ) /2 do not consume any news,
since these voters are unwilling to vote according to any outlet’s recommendation. The first group
always votes for L while the second always votes R. Moderate voters with ideal cutoffs between
θ1 /2 and (1 + θn ) /2 consume news from the outlet whose editorial position is closest to their political preference.10 This can be seen as a way of formalizing the idea of Downs that rational voters
delegate the evaluation of policies to experts with goals similar to their own (Downs (1957)). Note
that since the voting decision is binary, a voter would not gain from consuming news from more
than one outlet.
If voters were fully rational, a standard revealed-preference argument would suggest voters are
better off when they can choose from a bigger set of news outlets. In the limit, if there is a news
10 The news consumption rule for moderate voters is linear, in the sense of being independent of the value of the particular political
preference and whether the closest outlet is located to a voter’s right or left, due to the linearity of the utility function.
9
outlet for every editorial position between zero and one, then each voter would consume news from
his ideal outlet and vote as if he knows θ. But, as we argue above, concerns over a proliferation
of media choice stem in part from worries that voters are less than fully rational. To address
these concerns, we incorporate quasi-rationality into the model by assuming voters do not fully
appreciate how partisan the media outlets are. In particular, we assume that voters mis-perceive
the editorial position of each outlet j as
1
θ j ≡ ρθj + (1 − ρ) ,
2
∼
in which ρ is a constant between 0 and 1. When ρ is 1, voters are fully rational. When ρ is 0,
voters treat all media outlets as neutral (they are equally likely to report l or r). Quasi-rational
voters with ρ ∈ (0, 1) perceive the media editorial positions in a way that is systematically biased
towards 0.5, causing the voters to underestimate the partisanship of non-perfectly centrist media
outlets (those not located at exactly 0.5). If voters mis-perceive the positions of news outlets, it is
natural that they mis-perceive the positions of the parties as well. Hence, we assume that voters
perceive party L’s position as 0.5(1 − ρ) and party R’s position as 0.5 (ρ + 1). We discuss our
quasi-rationality assumption further in Section 5.
Except for their misperception of the parties and news media, quasi-rational voters are just
like fully rational voters in that they only consume news from an outlet if they are willing to
vote according to its recommendation, and they choose the outlet that offers highest (perceived)
expected utility. Liberals with ideal cutoffs less than 0.25(1 − ρ) do not consume any news; those
³
∼ ´
with ideal cutoffs between 0.25(1 − ρ) and 0.5 0.5(1 − ρ) + θ 1 consume liberal propaganda.
Both groups always vote for L.11 Conservative voters with ideal cutoffs greater than 0.25 (ρ + 1)
11 Some of these voters are misled by the propaganda of party L. They expect to vote R when party L recommends party R, which
it never does.
10
³∼
´
do not consume news; those with ideal cutoffs between 0.5 θ n + 0.5 (ρ + 1) and 0.25 (ρ + 1)
consume party R’s propaganda. These two groups always vote for party R. Moderate voters in
between consume news from the outlet whose perceived editorial position is closest to their own.12
Compared to fully rational voters, quasi-rational voters are less likely to consume news as they
are susceptible to party propaganda, and when they do consume news, they tend to choose outlets
that are more partisan due to their misperception of the media positions.
Voters’ news consumption behavior satisfies a monotone sorting property: more liberal voters
consume news from more liberal news outlets. Since media reports are also monotone, if the
median voter votes L, voters more liberal than the median voter must also vote L. Likewise, if
the median voter votes R, more conservative voters must also vote R. The electoral outcome,
therefore, is indeed determined by the vote of the median voter.13
2.2
Media Competition and Equilibrium
We assume that media outlets set their editorial positions before the realization of ∆ is known, as
establishing a reputation for editorial politics takes time. As a result, each outlet’s readership size
is a stochastic function of ∆. We also assume each media outlet’s profits are linearly increasing
in the size of its readership (profits come only from advertising; as mentioned above the price of
news is zero). Define
1
F (x) ≡
2δ
∗
Z
δ
F (x − ∆) d∆
−δ
as the expected number of voters with ideal cutoff less than x. Let f ∗ denote the density of F ∗ . It
is straightforward to show that f ∗ , like f , is also symmetric and weakly single-peaked across 0.5.
12 We
assume if a voter is indifferent between two media options, the voter chooses the more partisan option.
the median voter’s outlet is the most conservative outlet that reports l, and the median voter is the most conservative reader
of that outlet, at least some voters who are more conservative than the median voter will vote L when the median voters does. Hence,
it is almost surely that a strict majority votes L when the median voter does.
13 Unless
11
Let θ = (θ1 , ..., θn ), 0 ≤ θ1 ≤ ... ≤ θn ≤ 1, denote the editorial positions of n news outlets. Write
∼
∼
θ 0 for the perceived position of party L, and θ n+1 for the perceived position of party R. Since
∼
∼
∼
∼
voters with ideal cutoffs between 0.5( θ i + θ i−1 ) and 0.5( θ i + θ i+1 ) consume news from an outlet
with editorial position θi , the expected market size of outlet i is

³



´
∼
∼
∼
∼
F ∗ (0.5( θ i + θ i+1 )) − F ∗ (0.5( θ i + θ i−1 )) N (θi )−1
when θi 6= 0, 1,
Si (θ) =
³
´
∼
∼
∼
∼


 F ∗ (0.5( θ i + θ i+1 )) − F ∗ (0.5( θ i + θ i−1 )) (N (θi ) + 1)−1 otherwise,
where N (θi ) is the number of outlets with position θi .14
Each outlet has a fixed cost of S and no marginal costs, so the minimum readership size for an
outlet to break even is S (appropriately scaled). We say that entry is deterred at θ = (θ1 , ..., θn )
if given these positions no media outlet can enter the market and adopt an editorial position that
will earn it an expected market share strictly greater than S. We use the notion of entry-deterring
equilibrium to analyze the effect of declining S on media competition and voter welfare.
Definition 2.1. The editorial positions θ = (θ1 , ..., θn ) constitute an entry-deterring equilibrium
if (1) entry is deterred; (2) Sk (θ) ≥ S for each incumbent outlet k; (3) for any incumbent outlet
k, there does not exist some θ0 = (θ1 , ..., θk−1 , θk0 , θk−1 , ..., θn ) such that Sk (θ0 ) > Sk (θ) and entry is
deterred at θ0 .
The notion of entry-deterring equilibrium is introduced for the rational-voter case by CS to
analyze how media outlets compete by choosing editorial positions in a media market where free
entry is permitted when the price of news is fixed. It requires that in equilibrium each active
outlet earns a non-negative profit, no further entry is strictly profitable, and no outlet can increase
its profit by altering its position without inviting entry. Anand, Di Tella, and Galetovic (2007)
14 Outlets with the same editorial position share readers equally, and outlets choosing position 0 or 1 have to share the readership
with the parties as well.
12
derived a similar concept in a multiple-stage entry game. They show that in any subgame perfect
Nash equilibrium, entry occurs only in the first period, and the positions chosen by the entrants
constitute an entry-deterring equilibrium.15
To summarize, the timing of the model is as follows:
1. Given S, n media outlets choose and announce their editorial positions, (θ1 , ..., θn ).
2. The random variables θ and ∆ are realized; the former is observed only by the media outlets
and the latter is observed by all voters (and the outlets, but this does not matter).
3. Given (θ1 , ..., θn ) and ρ, voters choose which outlet or party, if any, to consume news from,
and observe the outlets’ news reports.
4. All of the voters vote; one party is elected and voters obtain their payoffs from the elected
party and state.
3
Equilibrium Editorial Positions
There are two types of media outlets in equilibrium. We call outlets with editorial positions strictly
between 0 and 1 interior outlets and those with editorial positions at 0 or 1 boundary outlets. Let
θ0∗ = 0. For i ≥ 1, define θi∗ recursively by the equation
 ∼∗ ∼∗ 
µ ∗¶
∼
θ + θ i−1 
= S,
F ∗ θi − F ∗  i
2
(2)
∼∗
where θ i = ρθi∗ + (1 − ρ) 0.5 is voters’ perception of θi∗ .16 Let q (S) ≡ max {i|θi∗ < 0.5} .The
following proposition characterizes the equilibrium positions.17
15 The set up of the model of Anand, Di Tella, and Galetovic (2007) is slightly different from ours. But with minor modifications,
a similar result could be obtained for our model. We do not repeat the proof here as it is not crucial for the subsequent analysis.
Interested readers should consult their paper for details.
16 Given
∼∗
∼∗
θ i−1 , θ i is uniquely defined since f (θ) is increasing when θ is less than 0.5.
results are also obtained in CS.
17 Similar
13
Proposition 3.1. Suppose f ∗ (x) is strictly increasing when x ∈ (0, 0.5) and strictly decreasing
when x ∈ (0.5, 1). Then there is a unique entry-deterring equilibrium for almost all S, and the
equilibrium positions of the interior outlets are as follows:
µ
1. When S ≥ 1 − 2F
∼∗
0.5+ θ 0
2
∗
¶
, there is no informative media outlet.
µ
µ
·
¶
¶¶
∼∗
∼∗
0.5+ θ 0
0.5+ θ 0
∗
∗
2. When S ∈ 0.5 − F
, 1 − 2F
, there is one interior outlet with editorial
2
2
position 0.5.
µ
3. When S < 0.5 − F
∗
∼∗
¶
0.5+ θ 0
2
, for all i ≤ q (S) there is a pair of interior outlets with editorial
µ
¶
∼∗
0.5+ θ q(S)
∗
∗
∗
positions θi and 1 − θi , respectively. If, in addition, S < 1 − 2F
, there is an
2
outlet with editorial position 0.5 .
To understand Proposition 3.1, note that the marginal effect on an outlet’s expected market
size from an increase in its editorial position (the partial derivative of Si (θ) with respect to θi ) is
³
´
∼
∼
∼
∼
∂Si
= 0.5ρ f ∗ (0.5( θ i + θ i+1 )) − f ∗ (0.5( θ i + θ i−1 )) .
∂θi
When outlet i shifts its editorial position to the right, it attracts the readers indifferent between
outlets i and i + 1 (the first term in the right-hand side), while losing those indifferent between
outlets i and i−1 (the second term). Hence, outlet i would gain from such a move if there are more
indifferent readers on the right than on the left. Suppose i = n = 1 (there is only one media outlet,
plus the parties). Then outlet i would gain from a shift right if θi < 0.5, given the symmetry and
strict single-peakedness of f ∗ . The exact opposite would occur if θi > 0.5, so i would gain readers
by shifting left in that case. This implies that any non-perfectly centrist outlet has an incentive
to move towards the middle, and ultimately must take a position of 0.5. Part 1 of Proposition 3.1
identifies the condition under which a single outlet with position 0.5 would not break even, and
14
part 2 identifies the condition under which a second interior outlet is deterred from entry.
Now suppose n > 1 and θi+1 ≤ 0.5 (so that θi−1 , θi , and θi+1 are all liberal). Since f ∗ (θ) is
strictly increasing when θ < 0.5,
∂Si
∂θi
> 0 and outlet i would gain readers by moving towards the
center. So outlet i will move as close to 0.5 as possible without inviting entry by a more liberal
outlet on its left. If such an outlet were to enter, then the previous argument would suggest that
it would be best for the entrant to take a position marginally to the left of outlet i and gain an
expected readership of
 ∼∗ ∼∗ 
µ ∗¶
∼
θ + θ i−1 
.
F ∗ θi − F ∗  i
2
Thus, (2) must be satisfied in equilibrium to deter entry. Since θ0 = 0, (2) can be used to
∗
characterize the positions of all liberal outlets, and θq(S)
is the position of the liberal outlet closest
to 0.5. Since f ∗ is symmetric across 0.5, conservative outlets are the mirror image of liberal outlets.
∗
Now suppose θi+1 > 0.5 and θi < 0.5. By symmetry, θi∗ = 1 − θi+1
, and there still must be more
mass of f () to i’s right than left, and so i will still move as close to 0.5 as possible. Finally, part 3 of
Proposition 3.1 gives the condition for there to be an outlet with a perfectly centrist position 0.5;
the condition ensures that the neutral outlet could break even. The positions of interior outlets
are unique except for a measure zero set of S where the expected readership of the neutral outlet
at 0.5 is exactly equal to S.
µ ∼∗
∼∗
θ 1+θ 0
2
¶
−
The total expected readership of party L and other liberal boundary outlets is SL = F ∗
µ ∼∗ ¶
µ ∼∗ ¶
θ0
θ0
∗
∗
F
if there is an interior outlet, otherwise, SL = 0.5 − F
. The number of boundary
2
2
outlets with editorial position 0 is equal to greatest integer less than (SL /S) − 1. Since the model
is symmetric, there is an equal number of boundary outlets with editorial position 1.
The comparative statics are straightforward to derive and can be seen in Figure 1, which
illustrates the equilibria for different values of ρ. When S is largest there are 0-2 informative
15
outlets. Declining S causes entry and the outlets to shift to more partisan positions to deter
entry, by (2). When voters are less rational (ρ is lower), the boundary outlets are perceived to
∼∗
∼∗
∼∗
be more centrist as θ 0 becomes greater. Since θ i is increasing in θ i−1 by (2), the perception of
each liberal outlet also becomes more centrist as ρ decreases. This causes more liberal voters
consume propaganda, and the market for informative news to be smaller. As a result, the number
of interior media outlets decreases, while that of the boundary outlets increases. As S goes to zero,
the numbers of both interior and boundary outlets become unboundedly high.
θ(S)
ρ = 0.25
ρ = 0.75
1
1
0.5
0.5
0.5
0
Boundary Outlets
ρ = 0.5
1
0
0.2
0.1
0
0.2
0.1
0
0
40
40
40
30
30
30
20
20
20
10
10
10
0
0
0
0.2
0.1
0
0.2
0.1
0
0.2
0.1
0
0.2
0.1
0
S decreasing
Figure 1: Illustration of equilibrium outlet positions and number of boundary outlets as functions of S, ρ; f ()
truncated normal distribution with σ = 0.2, δ = 0.2
The following corollaries describe properties of the equilibria that are useful for the subsequent
welfare analysis.
Corollary 3.2. For almost all S, for all interior outlets i, if f () is strictly single-peaked, then
each outlet is closer to its more moderate neighbor than its more partisan neighbor: θi∗ > (<)(=
∗
∗
)/2 if θi∗ < (>)(=)0.5.
+ θi−1
)(θi+1
This corollary is very intuitive and follows directly from Proposition 3.1 and (2) (the proof is
16
omitted); it says politically moderate outlets are closer together than those that are politically
extreme. Since there is more consumer mass “in the middle”, each outlet in the middle needs to
be relatively close to its more partisan neighbor to deter entry.18
Corollary 3.3. For all ² > 0 and x ∈ [0, 1], ∃S 0 > 0 such that ∀S < S 0 , there exists i such that
|θi∗ − x| < ².
The proof is in the appendix. This corollary says that in the limit as S approaches zero, there
is an outlet with editorial position that is arbitrarily close to any value θ may take. The media
completely cover the political spectrum. This is useful for our analysis of welfare in the limiting
case.
4
Welfare
4.1
Social Welfare Loss
Since each voter i’s utility function is linear in βi , the average utility of the voters is equal to the
utility of the average voter. Since the distribution βi is symmetric across βm , the average voter
and the median voter are the same. As a result, social voter welfare in our model can be proxied
by the welfare of the median voter, and so we focus on the median voter’s welfare in this section.
We established above that the electoral outcome is determined by the vote of the median
voter, and that unconditional on ∆, βm ∼ U [0.5 − δ, 0.5 + δ]. Also, a median voter with βm ∈
h
i
∼
∼
∼
∼
0.5( θ i−1 + θ i ), 0.5( θ i + θ i+1 ) consumes news from media outlet i and votes L when θ ≥ θi , and
that if the median voter does not consume news from an interior outlet, he votes L if liberal
(βm ≤ 0.5) and R if conservative (βm > 0.5). Define social welfare loss as the difference between
the median voter’s utility when he directly observes θ and his utility when he does not. It follows
18 The corollary only holds for “almost all” S as there are measure-zero sets of S where some of the strict inequalities are weak, which
does not affect subsequent results.
17
from (1) that the expected welfare loss (over θ) is (βm − θi )2 when the median voter follows the
recommendation of outlet i. Suppose there are n interior news outlets with positions θ1 ≤ ... ≤ θn .19
We continue to let θ0 = 0 and θn+1 = 1 be the positions of the liberal and conservative parties,
respectively. Define
n
o
α ≡ min i | 0.5(θei + θei+1 ) ≥ 0.5 − δ ;
n
o
e
e
α ≡ max i | 0.5(θi + θi−1 ) ≤ 0.5 + δ .
Among the outlets (including parties) that a median voter may choose, outlet α is the most liberal
and outlet α the most conservative. Then expected welfare loss (with the expectation taken over
θ and βm , and normalized with respect to δ) is
Z
e θeα+1 )
0.5(θα+
n−α Z
X
2
(βm − θα ) dβm +
W L (θ1 , ..., θn ) =
0.5−δ
Z
+
j=α+1
0.5+δ
0.5(θeα +θeα−1 )
0.5(θej +θej+1 )
0.5(θej−1 +θej )
(βm − θj )2 dβm
(βm − θα )2 dβm .
(3)
The welfare effect of a marginal change in the editorial position of outlet i = α, ..., α, is
³
³
h ³
´
³
´i´´
∂W L
= ρ (θi+1 − θi−1 ) θi − E βm |βm ∈ 0.5 θei−1 + θei , 0.5 θei + θei+1
∂θi
|
{z
}
Direct Effect
+ 0.5ρ (1 − ρ) (θi+1 − θi ) (1 − (θi+1 + θi )) Ii6=α + 0.5ρ (1 − ρ) (θi − θi−1 ) (1 − (θi + θi−1 )) Ii6=α , (4)
{z
}
|
{z
} |
Readership Effect (Left)
Readership Effect (Right)
in which Ii6=k is an indicator function that equals 1 when i 6= k, and defining θ−1 = θ0 and
θn+2 = θn+1 .20
19 The number of boundary outlets does not affect welfare since these outlets are just sharing with the parties the fraction of consumers
who get propaganda. The number of boundary outlets does not affect the size of this fraction.
20 The welfare effect of marginal changes in outlet with index i strictly less than α or strictly greater than a is zero as it is never read
by the median voter.
18
A marginal increase in θi directly affects the voting behavior of the voters who consume news
from outlet i—they will vote R instead of L when the state is θ. We refer to this effect as the
direct effect. It is captured by the first term on the right-hand side of (4). In addition, a marginal
increase in θi will cause voters who are indifferent between outlets i and i + 1 to switch to i, and
those who are indifferent between i and i − 1 to switch to i − 1. These effects are captured by the
second and third terms on the right-hand side of (4). We refer to them as the readership effects.
The readership effects are zero when ρ = 1. Since a rational voter must obtain the same utility
from two outlets if he is indifferent between them, his welfare is unaffected by a switch from one
to another.21 Quasi-rational voters, however, tend to select outlets that are too partisan. If a
quasi-rational voter is indifferent between two outlets, he would be strictly better off choosing the
more moderate one. Hence, when an outlet becomes more moderate, it creates two readership
effects that both increase welfare loss, as the outlet loses readers to its more partisan neighbor
and gains from its more moderate one. In (4), both readership effects are positive for liberal
outlets with θi < 0.5 and negative for conservative outlets. These effects are strict except for
the right readership effect for the centrist liberal outlet, and left readership effect for the centrist
conservative outlet, which equal zero when there is no perfectly centrist outlet.
For all ρ > 0, the direct effect of a marginal increase of θi leads to a higher welfare loss if
and only if θi is greater than the average ideal cutoff of the median voter who consumes news
from i. The welfare loss due to the direct effect is minimized when the position of each outlet is
equal to the average ideal cutoff of the median voter who consumes news from outlet i. Note that
determining the signs of both the readership and direct effects does not rely on the assumption
that the distribution of the median voter is uniform.
Thus, if an outlet’s position is less partisan than its average reader, welfare loss is reduced
21 When
ρ = 0, voters perceive every outlet as neutral regardless of its actual position. In that case, both the direct and readership
effects are zero.
19
by the outlet becoming more partisan. Both the readership and direct effects of an increase in
partisanship are socially beneficial. If an outlet’s position is equal to the ideal cutoff of its average
reader, welfare loss is minimized–only if consumers are perfectly rational. The direct effect is
zero, and the readership effects are always zero in this case. Perhaps counter-intuitively, however,
welfare loss would be reduced further by the outlet becoming more partisan if consumers were
quasi-rational. Although the direct effect would still be zero, the readership effect of an increase
in partisanship would be socially beneficial. Consequently, in the quasi-rational case it is socially
optimal for outlets to be more partisan than their consumers.
It might be natural to think that it would be optimal for the position of each outlet to be
unbiased relative to the preferences of its readers. But such thinking ignores the fact that the
composition of the readership of an outlet is endogenous to the position of that outlet and its
neighbors. While many commentators worry about less-than-fully-rational voters being misled by
partisan outlets, this argument shows it is precisely when voters are not fully rational that media
outlets should be partisan. This is so as to prevent news consumers from consuming even more
partisan news or not consuming any news at all.22 It is helpful to elucidate these points for the
subsequent analysis of the effects of decreasing S on welfare.
4.2
Decreasing S
In this subsection we look at whether decreasing S causes the outlets to move towards or away
from their optimal positions. There are two cases to consider: 1) α = 0 and α = n + 1, meaning
that when the realization of ∆ is very high or low and the median voter is most partisan he gets
propaganda; 2) α > 0 and α < n + 1, so the median voter always get informative news/never gets
22 This point provides a different rationale for a similar result in CS. That paper shows that media outlets should be “biased” relative
to the preferences of their readers in order to induce political parties to choose more moderate policies. Here we show that media outlet
should be biased when the party policies are fixed when voters are quasi-rational.
20
propaganda. Case 1 occurs if and only if the average of the perceived liberal party and outlet 1
∼∗
∼∗
positions, 0.5( θ 0 + θ 1 ), is greater than the most liberal median voter ideal cutoff, 0.5−δ. Conditions
for the cases to occur are characterized as follows.
Lemma 4.1.
1. A sufficient condition for the median voter to sometimes get propaganda (case
1) for all S > 0 is 2δ ≥ ρ.
2. A sufficient condition for the median voter to never get propaganda (case 2) for all S > 0
such that there is at least one informative outlet is ρ > 4δ.
3. If 4δ ≥ ρ > 2δ, there exists S 0 > 0 such that case 1 occurs if S ≥ S 0 , and case 2 occurs if
S < S 0.
The proof is in the appendix. Part 1 says if ρ is sufficiently low relative to δ, rationality is
sufficiently low and preferences sufficiently unstable that for large realizations of ∆, the median
voter is sufficiently partisan to think propaganda is more informative than any interior outlet.
Consequently, α = 0 for all S. Part 2 says if ρ is sufficiently high relative to δ, the median voter
voter never gets propaganda, as he always finds the centrist outlet more informative than the
parties. And for intermediate cases (4δ ≥ ρ > 2δ), the median voter will get propaganda when ∆
is near its upper or lower bounds only when S is sufficiently high and there are no highly partisan
informative outlets. We analyze the two cases in turn.
4.2.1
Case 1: The median voter sometimes gets propaganda
A decline in S will have either one or two effects. It will always cause all non-perfectly centrist
interior outlets to shift positions to continue to deter entry. And for some S, a new outlet will
enter in spite of the shifts by the incumbent outlets. If there is only the first effect, the marginal
effect of S on W L∗ can be decomposed into the sum of partial effects of marginal effects of outlet
21
locations, times the marginal effects of S on the outlet locations:23
¯
α
dW L∗ X ∂W L ¯¯
dθi∗
=
¯ ∗ dS .
dS
∂θ
i
θ=θ (S)
i=α
(5)
Using (4), the partial effect of outlet i for α < i < α, which has both left and right readership
effects, can be rearranged as
∂W L
= 0.5ρ(2 − ρ)(θi+1 − θi−1 )(θi − 0.5(θi+1 + θi−1 )).
∂θi
(6)
This expression takes the sign of its last term, (θi − 0.5(θi+1 + θi−1 )). Corollary 3.2 states that
if f () is single-peaked, then this term, evaluated at the equilibrium positions, is positive when
i is liberal, negative when i is conservative, and equal to zero when i is perfectly centrist. The
equilibrium positions are bunched in the middle–so as S declines and the outlets spread out, this
improves welfare. The intuition follows directly from the discussion of 4.1. Since the outlets are
optimally at least as partisan as their audiences, but in equilibrium they are less partisan, welfare
is improved by the outlets spreading out.
The signs of the marginal effects of S on the equilibrium outlet positions, the
dθi∗
’s,
dS
are very
intuitive: positive if i is liberal, negative if i is conservative, and zero if i is centrist or one of the
parties. If i is liberal (conservative), we know the outlet becomes more partisan as S declines to
continue to deter entry, so θi decreases (increases) as S decreases. Finally, because outlets α and α
are the political parties in this case,
dθi∗
dS
= 0 if i ∈ {α, α}. Thus we can ignore the partial effects for
these outlets, as they are multiplied by zeros, and the marginal effect of S on W L can be signed:
23 We
now take into account the fact that the equilibrium positions depend on S, so we can write the vector of equilibrium positions
∗ (S)). We sometimes continue to sometimes suppress S for convenience, and write W L(θ ∗ (S), ..., θ ∗ (S)) as just W L∗ .
as (θ1∗ (S), ..., θn
n
1
22
¯
¯
¯
∗
X
dW L
∂W L ¯¯
=
dS
∂θi ¯
i=1:q(S) | {z }¯¯
+
This implies
dW ∗ (S)
dS
dθi∗
dS
|{z}
θ=θ ∗ (S)
+
¯
¯
¯
X
∂W L ¯¯
+
∂θi ¯
i=n+1−q(S):n | {z }¯¯
−
< 0 (assuming n > 1, otherwise
dW ∗ (S)
dS
θ=θ∗ (S)
dθi∗
> 0.
dS
|{z}
(7)
−
= 0), and welfare improves as S
declines, as long as the number of outlets stays fixed, since this causes the outlets to spread out.
If a decline in S triggers entry, this also weakly improves welfare. To see this, recall that since
outlets set their positions so that an entrant outlet cannot recover its fixed cost by taking a just
partisan position, entry always occurs in the middle. If the number of incumbent outlets is odd,
entry will not have an immediate, or discontinuous, effect on welfare, as the difference between
the entrant’s and incumbent centrist’s positions has measure zero. If the number of incumbent
outlets is even, there is no perfectly centrist incumbent outlet, and entry will cause welfare to jump
upward. The most moderate consumers will switch from the most moderate partisan outlets to
the entrant outlet, and the incumbent firm locations do not change discontinuously. This strictly
improves welfare for all ρ, since we know quasi-rational consumers tend to choose outlets that are
excessively partisan, and rational voters are never worse off when they switch outlets.
Thus, contrary to what one might expect, we have shown that when consumers are sufficiently
irrational they always benefit from the proliferation of news sources. The intuition follows from the
discussion of 4.1 and the fact that single-peaked preferences cause outlets to cluster in the political
center when fixed costs are high. The results are summarized in the following proposition.
Proposition 4.2. Monotone welfare improvement: if the median voter sometimes gets propaganda
(4δ > ρ with sufficiently high S), then welfare increases (strictly if n > 1) as S decreases.
See Figure 2 for examples of media outlet positions and corresponding social welfare levels for
this case. It shows how W ∗ (S) increases discontinuously when a new outlet enters the market only
23
when the number of incumbent outlets is even.
ρ = 2/3
ρ = 1/3
θ*(S)
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.2
0.15
0.1
0
0.2
0.05
0.15
0.1
0.05
0.15
0.1
0.05
0.162
0.148
W*(S)
0.16
0.146
0.158
0.144
0.156
0.154
0.142
0.2
0.15
0.1
0.05
0.152
0.2
S decreasing
Figure 2: Examples of outlet positions (θ∗ (S)) and welfare (W ∗ (S)) for case 1 (the median voter sometimes gets
propaganda); f () truncated normal distribution with σ = 0.2, δ = 0.5
4.2.2
Case 2: The median voter never gets propaganda
In this case the outlets patronized by the median voter when he is most partisan (the shock to
preferences is near its upper bound), outlets α and α, are interior media outlets: α > 0 and
α < n + 1. Since interior outlets provide real news rather than propaganda, this means the median
voter never gets propaganda. This has a major effect on the qualitative nature of the results.
We can see why by examining the simplest case in which declining S affects the outlet positions:
24
n = 2, α = 1 and α = 2. Using (4), we can write
∂W L
= ρθ2 (θ1 − E (βm |βm ∈ [0.5 − δ, 0.5])) + 0.5ρ (1 − ρ) (θ2 − θ1 ) (1 − (θ2 + θ1 )),
{z
} |
{z
}
|
∂θ1
Direct Effect=DE1
Readership Effect (Right)
∂W L
= ρ (1 − θ1 ) (θ2 − E (βm |βm ∈ [0.5, 0.5 + δ])) + 0.5ρ (1 − ρ) (θ2 − θ1 ) (1 − (θ2 + θ1 )) .
{z
} |
{z
}
|
∂θ2
Direct Effect=DE2
(8)
Readership Effect (Left)
Using these results together with (5), we obtain
dW L∗
∂W L
dθ∗ ∂W L
dθ∗
|θ1 =θ1∗ (S) 1 +
|θ2 =θ2∗ (S) 2
=
dS
∂θ1
dS
∂θ2
dS
∗
dθ
= 2DE1 1 ,
dS
since
dθ1∗
dS
(9)
dθ ∗
= − dS2 , DE1 = −DE2 by symmetry, and the readership effects in the middle exactly
cancel, since the consumer who is indifferent between the two outlets always has ideal cutoff 0.5.
The following proposition is immediate.
Proposition 4.3. If n = 2, α = 1 and α = 2, then a marginal decline in S improves (worsens)
welfare iff θ1 is greater (less) than the average ideal cutoff of liberal median voters:
dW ∗
dS
> (=)(<)0
iff θ1 < (=)(>)E (βm |βm ∈ [0.5 − δ, 0.5]).
We know that welfare improves from an outlet becoming more partisan, so long as it is less
partisan than the average preference of the median voter, conditional on the median voter being
one of its consumers. When α = 0, the condition that the outlet is less partisan than its median
voter consumers is assured; when α > 0, the condition may not hold. Since there are no readership
effects, and the sign of the direct effect may change as S declines and the outlets spreads out, the
sign of the marginal effect of S on welfare may change–i.e., welfare is possibly a non-monotone
function of S. Decreasing S is socially beneficial at first, optimal when the outlets are positioned
at the mean preference of their consumers, and negative if and when the outlets become too spread
25
out. Put differently, the lack of a readership effect means when outlet 1 becomes more partisan, it
does not attract readers (who may be the median voter) from the liberal party. So an increase in
partisanship simply causes the median voter to get more partisan news, which may very well be
less informative.
Note that this result also does not rely on βm being uniform, and may hold for all ρ (even ρ = 1).
In fact, ρ needs to be sufficiently high for this case to occur. The median voter may be worse off
when the cost of entry declines when ρ is higher because this causes the partisan consumers to be
more interested in informative news. The outlets respond to the demand of partisan consumers
who may never be the median voter, causing the median voter to be limited to highly partisan
news, which appears biased to him.
To get closed form results characterizing the welfare function in duopoly, we assume f () is
uniform. This allows us to solve for the equilibrium positions, but they are not unique since f () is
not strictly single-peaked. Any outlet located between two other outlets (or the parties) is indifferent between its current position and any other position between the same two outlets/parties.
However, a uniform f () can be thought of as the limit of a sequence of strictly single-peaked fˆ()’s.
Since each single-peaked fˆ() has a unique equilibrium, there will be a unique limit to the sequence
of equilibria. This limit will clearly be an equilibrium when f () is uniform–the equilibrium in
which each outlet is located as close to 0.5 as possible. It is natural to focus on this particular
equilibrium when f () is indeed uniform, and we do so subsequently.
Proposition 4.4. If f () is uniform, and α = 1 for some S such that n = 2, then welfare is
maximized over S such that n = 2 at the value of S such that θ1∗ = E (βm |βm ∈ [0.5 − δ, 0.5]).
Welfare is minimized when the outlets are most spread out if δ is sufficiently small relative to ρ;
otherwise, welfare is minimized when the outlets are closest together. A sufficient condition for
26
welfare to be minimized when the outlets are most spread out is min{ρ/4, 1/6} > δ.
The proof is in the appendix. It follows directly from Proposition 4.3 that welfare increases as S
decreases so long as θ1∗ > E (βm |βm ∈ [0.5 − δ, 0.5]). The interesting question really is does θ1∗ ever
in fact equal, or become less than, E (βm |βm ∈ [0.5 − δ, 0.5]), when n = 2. That is, is welfare in
fact hill-shaped when n = 2, or monotonic. This proposition shows that welfare is non-monotonic
(hill-shaped), and the lowest part of the hill–minimal welfare–may occur when S is lowest and the
outlets are most spread out. This is more likely to happen when δ is low and ρ high. If δ is low
the median voter’s ideal cutoff is limited to a small centrist interval, and consequently the median
voter will have high welfare loss when the outlets are most partisan. If ρ is high, the median
voter will be more likely to get informative news when the two outlets are closest together, making
welfare loss relatively low as compared to when the outlets are most spread out. The proposition
does not state this explicitly, but the degree to which welfare can decrease is limited since a third
outlet eventually enters the market for low enough S.
Figure 3 illustrates the relationship between welfare and S, for the case that the median voter
never gets propaganda, for duopoly and more general settings. It shows that welfare can indeed
decline when n = 2, but that welfare jumps up again when the third firm enters the market.
Welfare continues to be non-monotonic as S decreases and more outlets enter. While the patterns
appear unclear, the intuition is the same as in the duopoly case. The effects of outlets α and α
spreading out are socially negative, since they are more partisan than the average median voter
who gets news from them. The readership effects no longer disappear when n > 2, but they are still
dominated by the direct effects, since the readership effects of outlets α and α are only one-sided,
in contrast to the two-sided readership effects for the other interior outlets, which dominate their
direct effects.
27
The other key point to note from Figure 3, besides the non-monotonicity, is that most of the
welfare “action” occurs when S is relatively high, and the number of outlets relatively low. Social
welfare sometimes becomes very low in duopoly (and possibly with three or even four outlets) as
S declines because the two partisan outlets cater to extremist consumers who have no potential to
be the median voter. The median voter does not have the option of getting news from moderate
outlets since they do not exist, or they appear too moderate. This problem never occurs as S
declines and n becomes large, since this ensures the existence of a diverse set of moderate media
choices.
In the supplementary appendix, we analyze a special case of the model (uniform f () and ρ = 4δ)
in detail for arbitrarily small S, and show that welfare actually cycles, i.e., goes through a welldefined repeated pattern of decline followed by improvement, as S goes to zero. We conjecture, but
do not prove, that welfare cycles in a similar way for all parameter values satisfying the condition
for case 2. The cycles are too complex to be seen clearly in Figure 3, but can be seen in additional
examples presented in the supplementary appendix. The fact that welfare cycles provides some
intuition into why the overall changes in welfare are small as S declines when the number of outlets
is somewhat large; since welfare both improves and worsens, the changes mostly cancel each other
out. To examine whether the overall effect of the changes in welfare are socially beneficial or not,
we compare welfare with a continuum of media choices to welfare with very limited choice in the
following subsection.
4.3
Welfare with a Continuum of Media Choices
In the limit, as S approaches zero, Corollary 3.3 implies there is a continuum of media choices and
all consumers get news from the outlets they perceive to be optimal, so long as their locations are
in [0,1]. A consumer with ideal cutoff β will perceive her optimal outlet location to be the one
28
θ*(S)
ρ = 1/3
ρ = 2/3
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.3
0.2
0.0126
0.3
0.2
0.1
0.3
0.2
0.1
0.0126
0.0124
W*(S)
0
0.1
0.0124
0.0122
0.0122
0.012
0.012
0.0118
0.0116
0.0118
0.3
0.2
0.1
S decreasing
Figure 3: Examples of outlet positions (θ∗ (S)) and welfare (W ∗ (S)) for case 2 (the median voter never gets
propaganda); f () truncated normal distribution with σ = 0.2, δ = 0.05
∼∗
such that θ = β, i.e. ρθ∗ + (1 − ρ)/2 = β, so in the limit the consumer will choose an outlet
with position θ∗ =
β−(1−ρ)/2
.
ρ
Suppose the median voter gets informative news in monopoly. That
is, if there is only one outlet with location 0.5, the median voter prefers news from this outlet to
news from either of the parties. Then, due to the linearity of consumer utility, if the median voter
is liberal (βm < 0.5) he will be worse off getting news from his perceived optimal outlet than in
monopoly if and only if
βm −
βm − (1 − ρ)/2
> 0.5 − βm .
ρ
(10)
This inequality holds if and only if ρ < 0.5. Since the condition does not depend on the particular
29
value of βm , we can conclude welfare is greater with infinite media choice than monopoly if and
only if ρ > 0.5 independent of the distribution of βm , assuming the median voter always gets
informative news in monopoly (Lemma 4.1 implies ρ > 4δ is necessary and sufficient for this).
This is because consumers perceive that their news is coming from an outlet with location 0.5 with
probability (1 − ρ). If ρ is less than 0.5, consumers compensate for their misperception by choosing
an outlet that is even further from their tastes than 0.5 (the monopolist’s location).
Proposition 4.5. If ρ > 4δ, then monopoly welfare is greater than welfare in the limit as S goes
to zero (W ∗ (S|n = 1) > lim W ∗ (S)) iff ρ < 0.5.
+
S →0
This is a robust result in that it is independent of f () and the distribution of ∆, and is interesting
in that it illustrates the downside of media competition and choice with consumer quasi-rationality.
That is, if rationality is low (ρ < 0.5), but preferences are stable (δ < ρ/4), society is better off
with one perfectly centered media choice than a continuum of choices. This result suggests there
may be social benefits to severely restricting media choice; however, the result should be treated
with a great deal of caution, since we know there are reasons outside our model that a monopolist
may not in fact take a perfectly centrist position.24
We also compare duopoly to the case with infinite media choice; obtaining a very strong result
for the uniform f () case, summarized as follows.
Proposition 4.6. If f () is uniform, minimal duopoly welfare is less than welfare with a continuum
of media choices (min{W ∗ (S|n = 2)} < lim W ∗ (S)) for all ρ and δ.
S
+
S →0
The proof is in the appendix. This result says that duopoly welfare is potentially worse than
welfare in the limit, for all levels of rationality and preference stability. Consequently, a sufficiently
24 For example, a monopolist with a political agenda, or captured by the government, would likely take a partisan position although
this would cause it to sacrifice some of its audience. We do not analyze these issues in detail since, as described in the introduction, they
have been the focus of previous literature, and the changing media landscape has motivated our focus on markets with many outlets.
Our results regarding markets with more than one firm are likely more robust to these types of issues, since (as shown by previous
research) competition tends to reduce their impacts.
30
risk averse social planner would prefer to have very many media choices to just two choices,
assuming the risk was with respect to S. Although consumers make quite poor media choices
when ρ is low, consumers still may be better off with infinite choice than with two media options.
This is surprising since Proposition 4.5 says monopoly welfare is greater than welfare in the limit
when ρ and δ are both very low. Since duopoly is better than monopoly for some S for all ρ and
δ by Propositions 4.2 and 4.3, it would be natural to think there are cases of the model in which
duopoly welfare is generally greater than welfare with infinite choice. Duopoly welfare is always
worse than welfare in the limit for some S though, because whenever monopoly welfare is higher
than welfare in the limit, ρ must be relatively large compared to δ. And when this occurs, welfare
always decreases substantially as the two outlets become sufficiently spread out. If ρ is absolutely
small, δ must be very small, so the outlets spreading out is socially very harmful. The median
voter’s ideal cutoff is limited to a very small interval, and so his welfare is very low when forced
to get news from a very partisan outlet. This can be considered a formalization of the observation
above that welfare sometimes becomes very low in duopoly as the outlets spread out when δ is
very small (see Figure 3). If ρ is large (both absolutely and relative to δ), then monopoly welfare
is less than welfare in the limit anyway by Proposition 4.5. This is sufficient for minimal duopoly
welfare being less than welfare in the limit, since the former is less than or equal to monopoly
welfare. And when ρ is not so large compared to δ, the benefit from the outlets spreading out
decreasing the chance the median voter gets propaganda is the dominant effect.
5
Discussion
There are several assumptions of our model that warrant further discussion. The quasi-rationality
assumption stands out in particular. The nature of this assumption is not as important as its
31
behavioral implication: that it causes consumers to choose excessively partisan outlets. This is
the behavior of interest for the analysis of this paper. While we believe the specific assumption
we use is reasonable, other reasonable assumptions would yield similar behavioral implications.
Namely, if we assumed consumers simply liked being told the party they were planning to vote for
was “correct”, or if consumers were subject to other information-processing biases, such as conservative belief updating or overconfidence in the precision of their priors, consumers would choose
excessively partisan outlets, and thus there would be similar results regarding the relationship
between welfare and S.25
Other papers in this literature, such as Gentzkow and Shapiro (2006) and Burke (2008), focus
on contexts in which consumers make non-electoral decisions based on media news. Our model’s
results can also be interpreted as applying to these situations. If δ = 0.5 and f () is the uniform
distribution, out model becomes equivalent to one with a unit interval of consumers with preferences ∼ U [0, 1], who each make binary decisions based on media news. Clearly the monotone
welfare improvement result (Proposition 4.2) would hold in this case–thus the model would predict media consumer, and not just voter, welfare would improve as media choice expands. The
assumption that δ may be as high as 0.5–although it allows political preferences to be potentially
very unstable–is useful for this alternative interpretation of the model.
The assumption that ∆ is uniform is not qualitatively important; for example, Proposition 2
does not depend on it. We do make a number of assumptions that, if weakened, might affect the
main results. One is that the quality of news reporting is constant. If smaller outlets reported
news less accurately, this could cause increased media competition to reduce welfare. Blogs in
particular are considered likely to report unsubstantiated news and increase the propensity of false
25 If consumers liked having their prior vote confirmed, they would choose an outlet with an editorial position more partisan than
their ideal cut-off. If consumers were conservative belief updaters or overconfident, they would have to get news from more partisan
outlets to be convinced to change their votes.
32
rumors to gain traction (Munger (2008)). The quality of media outlets also affects their ability to
perform the “watchdog” role. Media outlets act as watchdogs of parties forming policy platforms
before elections, the performance and corruption of elected officials–and of other media outlets. It
is unclear how increased competition will affect media behavior in this respect. There is a positive
effect of having more outlets looking out for problems, but a negative effect of the outlets being
smaller and having fewer resources to conduct in-depth investigations.26
Our model also abstracts from issues such as voter interest in news and voting (turn-out), and
the social cost of political polarization. Sunstein (2001) claims that increased media competition
will enable consumers to limit their news consumption to niche interests and avoid more socially
important issues.27 If voter turn-out were endogenous the effects of increased media competition
might be more ambiguous.28 Sunstein (2001) and others also suggest that increased societal
polarization or fragmentation (the left moving left and right moving right) could result from a
proliferation of partisan news sources, which may be socially harmful.29 We think this is plausible,
but would have to involve a mechanism outside our model. Finally, other important issues we
abstract from are the manipulability of political preferences by the media, heterogeneity of ρ and
correlation with partisanship, and the interaction of media competition with behavior (pricing,
etc.) of other industries that advertise in news media. These are all topics worthy of future
research.
While our model does not suggest a need for greater regulation of media markets–in particular, the results do not support the revival of the Fairness Doctrine, which is receiving some
26 Bloggers
have already contributed to breaking several major news stories, such as the firing of US attorneys by the Bush administration in 2007 and the use of forged documents relating to Bush’s national guard service by the TV show 60 Minutes. See Massing
(2009) for a good discussion of these issues.
27 Prior (2005) finds evidence that having more choices does indeed allow some people to avoid political news who otherwise might
not have been able to, and that this phenomenon is causing an “information gap” in the electorate to increase.
28 If increased competition increases turn-out by stimulating interest in politics, this would make electoral results more representative,
but also perhaps more likely to be affected by uninformed voters.
29 Stroud (2008) finds empirical evidence of this phenomenon, specifically, that partisanship increased over the course of the 2004
presidential campaign. See also Rosenblat and Mobius (2004) and Sunstein (2002).
33
consideration–we do not claim to have the last word on this topic.30 Empirically, perhaps the ultimate question is whether the changes in media market structure are causing the electorate to make
more or less informed decisions. Our model predicts that the potential for an uninformed electorate
is highest by far when the number of outlets is very small. A related, but merely instrumental
question, is what are the media consumption patterns that have been arising during this period
of change; what types of news are being consumed by people with different political ideologies.
Progress on these empirical issues will facilitate analysis of the validity of our theoretical model,
our understanding of new media markets in general, and formation of optimal policy in this area.
References
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Product Differentiation,”Journal of Economics & Management Strategy, vol. 16(3), 635-682.
Bernhardt, D. and Krasa, S. and Polborn, M. (2008). “Political Polarization and the Electoral
Effects of Media Bias,” Journal of Public Economics, vol. 92(5-6), 1092–1104.
Burke, J. (2008). “Primetime Spin: Media Bias and Belief Confirming Information,” Journal of
Economics & Management Strategy, vol. 17(3), 633–665.
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De Waal, E. and Schönbach, K. and Lauf, E. (2005). “Online newspapers: A substitute or complement for print newspapers and other information channels?” Communications, vol. 30(1),
55–72.
30 See,
e.g., Eggerton (2009) for a discussion of reinstating the Fairness Doctrine, which required holders of broadcast licenses to present
balanced views on political issues, and was abolished in 1987. In our model, consumers always have news options with symmetrical
political editorial positions, so there is no need for government to require the supply of opposing views.
34
Downs, A. (1957). An economic theory of democracy, Addison-Wesley Publishing, Inc.
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Balance’ in Media,” Broadcasting & Cable.
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Manjoo, F. (2008). True Enough: Learning to Live in a Post-Fact Society . (Wiley).
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2009.
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95(4), 1031–1053.
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vol. 134(1), 125–138.
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7(1), art. 29.
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Nisbett, R. and Ross, L. (1980). “Human Inference: Strategies and Shortcomings of Social Judgment.” Prentice Hall, Inc.
Poniewozik, J. (2009). “Walter Cronkite: The Man with America’s Trust,” Time, July 17, 2009.
Prior, M. (2005). “News vs. entertainment: How increasing media choice widens gaps in political
knowledge and turnout,” American Journal of Political Science, 577–592.
Rosenblat, T.S. and Mobius, M.M. (2004). “Getting Closer or Drifting Apart?” Quarterly Journal
of Economics, vol. 119(3), 971–1009.
Stone, D.F. (2009). “Testing Bayesian Updating with the AP Top 25,” working paper.
Stroud, N.J. (2008). “Media Use and Political Predispositions: Revisiting the Concept of Selective
Exposure,” Political Behavior, vol. 30(2), 341–366.
Sunstein, C.R. (2001). Republic.com, Princeton University Press.
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10(2), 175–195.
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Science, vol. 185(4157), 1124–1131.
36
A
Proofs
A.1
Corollary 3.3
Proof. For all ² > 0, by (2) it is clear that there exists S 0 such that θ1∗ < ²/2. By Corollary 3.2,
∗
∗
this implies for each interior outlet i, |θi∗ − θi−1
| < ²/2 and |θi∗ − θi+1
| < ²/2. Thus, if S = S 0 , for
all x ∈ [0, 1] there is an i such that |θi∗ − x| < ². And for all S < S 0 , θ1∗ (S) < θ1∗ (S 0 ) (by (2) and
abusing notation by writing the equilibrium outlet positions as functions of the fixed cost), and so
again each outlet is within ²/2 of each neighboring outlet, and again there will be an i such that
|θi∗ − x| < ².
A.2
Lemma 4.1
Proof. It is clear that the median voter gets news from the liberal party or not at all if βm is less
∼
∼
than 0.5( θ 0 + θ 1 ), and since the support of βm is [0.5−δ, 0.5+δ], the requirement for case 1 to occur
∼
∼
∼
∼
∼
∼
∼
is 0.5( θ 0 + θ 1 ) ≥ 0.5 − δ. And, since θ 0 = 0.5(1 − ρ) and lim θ 1 = θ 0 , we know 0.5( θ 0 + θ 1 ) ≥ 0.5 − δ
S→0
for all S > 0 iff 0.5(1 − ρ) ≥ 0.5 − δ, i.e. 2δ ≥ ρ. The median voter will always prefer an outlet with
∼
position 0.5 to either party iff 0.5( θ 0 + 0.5) < 0.5 − δ, which is equivalent to ρ > 4δ. So under this
condition, case 1 never occurs when there is at least one informative outlet. If 4δ ≥ ρ > 2δ, then
∼
∼
case 1 occurs (0.5( θ 0 + θ 1 ) ≥ 0.5 − δ) only when S (and consequently θ1 ) is sufficiently high.
A.3
Proposition 4.4
Proof. It follows from Proposition 4.3 that the maximum level of welfare in duopoly will occur
either at the S such that θ1∗ (S) = E (βm |βm ∈ [0.5 − δ, 0.5]) = 0.5 − δ/2, or the minimum S such
that n = 2. Let S 0 denote the latter value. We need to show θ1∗ (S 0 ) ≤ 0.5 − δ/2.
First note if f () is uniform, then f (x) = 0.5 = f ∗ (x) and so F (x) = 0.5x + 0.25 = F ∗ (x).
37
Equation (2) thus implies
∼∗
∼∗
θ + θ i−1
0.5( θ i − i
) = S
2
∼∗
∗
+ 4S/ρ.
θi∗ = θi−1
(11)
This, together with Proposition 3.1, can be used to easily show that if f () is uniform, the minimum
S such that n = 1 is ρ/8, the minimum S such that n = 2 is ρ/12 (so this is S 0 ), and when n = 2,
θ1∗ (S) = 4S/ρ = 1 − θ2∗ (S). Thus, θ1∗ (S 0 ) = 1/3, for all ρ and δ.
Next, since by supposition α = 1 for some S such that n = 2, it is implied that α = 1 when
∼
∼∗
S = S 0 . Consequently, 0.5( θ 0 + θ 1 (S 0 )) = 0.5(0.5(1 − ρ) + ρ/3 + 0.5(1 − ρ)) < 0.5 − δ, implying
3δ < ρ and δ < 1/3. This implies 0.5 − δ/2 > 1/3, and thus 0.5 − δ/2 > θ1∗ (S 0 ).
Proposition 4.3 implies welfare is minimized when the outlets are closest together or most
spread out. Lemma 4.1 implies if ρ ≤ 4δ, then the median voter does not always get informative
news in monopoly. Again by supposition, the median voter does always get informative news in
duopoly when the outlets are most spread out. Using (3), we can show welfare loss in monopoly is
(2/3)(δ 3 −(3/2)δ 2 +(3/4)δ +(3/32)ρ2 −(3/16)ρ) and in duopoly, (2/3)(δ 3 −δ 2 /2+δ/12). The latter
is greater when (3/16)ρ(1 − ρ/2) > δ(2/3 − δ), whose left (right)-hand side is increasing as ρ (δ)
increases, given as shown above δ < 1/3. If ρ > 4δ, the median voter always gets informative news
in monopoly and duopoly. Thus a liberal median voter gets news from an outlet with position 0.5
in monopoly, and position 1/3 in duopoly. Due to the linearity of utility and uniform distribution
of βm , welfare is higher in monopoly iff the mean ideal cutoff of liberal median voters, 0.5 − δ/2,
is closer to 0.5 than to 1/3, i.e., if δ < 1/6. This proves the last part of the proposition.
38
A.4
Proposition 4.6
Proof. We need to prove that if f () is uniform, there exists S 0 such that W L(S 0 |n = 2) >
lim W L∗ (S). Proposition 4.4 implies welfare loss with n = 2 is maximized when S is highest
+
S →0
or lowest. If the former, the upper bound of welfare loss with n = 2 is monopoly welfare loss,
which we denote W L1 . If the latter, the maximal welfare loss with n = 2 occurs when S = ρ/12
and θ1 = 1/3 = 1 − θ2 (see proof of Proposition 4.4). Denote welfare loss when S = ρ/12 as W L2 .
So we need to show max{W L1 , W L2 } > lim W L∗ (S).
+
S →0
First note if ρ ≤ 2δ, then the result is straightforward since by Proposition 4.2 welfare always
monotonically increases as S declines for sufficiently low S. If ρ > 2δ, then all consumers get news
from their perceived optimal outlets in the limit as S goes to zero (see beginning of Section 4.3),
and
Z
∗
0.5+δ
lim W L (S) =
+
(βm −
0.5−δ
S →0
= (2/3)(
βm − (1 − ρ)/2 2
) dβm
ρ
1−ρ 2 3
)δ .
ρ
(12)
From the proof of Proposition 4.4, we know if ρ ≤ 4δ, then W L1 = (2/3)(δ 3 − (3/2)δ 2 + (3/4)δ +
(3/32)ρ2 − (3/16)ρ). Write ρ = aδ. Then W L1 > lim W L∗ (S) iff
+
S →0
((3/32)a4 − (3/2)a2 + 2a)δ > (3/16)a3 − (3/4)a2 + 1.
(13)
The right-hand side is negative if a ∈ (2, 3.585), and the left-hand side is weakly positive if a > 2.96.
If a ≤ 2.96, then (13) will hold if it holds with δ = 0.5, so a sufficient condition for it to hold is
(3/64)a4 + a > 1 + (3/16)a3 , which is true for all a > 2.
Thus, if a ≤ 3.585, W L1 > lim W L∗ (S). If a > 3.585, then for sufficiently small δ, (13)
+
S →0
39
will not hold, so we need to show W L2 > lim W L∗ (S). From the proof of Proposition 4.4,
+
S →0
W L2 = (2/3)(δ 3 − δ 2 /2 + δ/12). This is greater than lim W L∗ (S) iff
+
S →0
a2 /12 − 1 > aδ(a/2 − 2).
(14)
If a ≤ 4, the right-hand side is weakly negative, so a sufficient condition is a > 120.5 = 3.46, which
clearly holds.
If a > 4, then the median voter always gets informative news in monopoly. Then, according
to Proposition 4.4, W L2 > W L1 iff δ < 1/6. Under this condition, it is easily shown (14) holds,
given a > 4, and so W L2 > lim W L∗ (S). If δ ≥ 1/6, then ρ > 0.5, and so by Proposition 4.5
+
S →0
W L1 > lim W L∗ (S).
+
S →0
B
Supplementary Appendix: The special case of ρ = 4δ
To make further progress analyzing the welfare effects of decreasing S for case 2 (“the median
voter never gets propaganda”), we focus on the special case in which ρ = 4δ. While seemingly very
restrictive, this case does not prohibit ρ from taking any value in (0, 1) and, more importantly, has
properties which are qualitatively very general, which we illustrate numerically. We also assume
f () is uniform, and restrict attention to the equilibrium that is the limit of equilibria for strictly
single-peaked f ()’s converging to uniform f () (as discussed in the body text).
Recall that case 2 is defined by α, the index of the media outlet used by the median voter for
the largest realizations of ∆, taking a value greater than zero. We first informally characterize the
relationship between α and S.
Remark B.1. α takes increasingly high integer values, and never skips a value, as S goes to 0. α
can be thought of as a decreasing step-function of S.
40
n
o
To understand this, recall that α ≡ min i | 0.5(θei + θei+1 ) ≥ 0.5 − δ and the condition for
∼∗
case 2 to occur is
θ 1 + 1−ρ
2
2
< 0.5 − δ. Since θe1∗ >
But for each liberal outlet i, θei∗ converges to
1−ρ
2
1−ρ
2
for all S > 0, it follows that
1−ρ
+ 1−ρ
2
2
2
< 0.5 − δ.
as S goes to zero for all i, so 0.5(θei + θei+1 ) must
also be less than 0.5 − δ for all i, for sufficiently low S. It follows directly that, whenever case 2
occurs for some S > 0, α becomes unboundedly high as S goes to zero.
This remark is useful because it makes clear that each value α takes corresponds to a unique
interval of S.31 Within any interval, as S declines new outlets enter the market when the existing
outlets can no longer deter entry. In order to analyze the welfare effects of declining S for a given
value of α, we need to understand the relationship between the α “function” and entry.
Using (11), it is easily shown the nth outlet enters the market when S declines marginally from
ρ
.
4n
If α ≥ 1, α is defined by
θeα +θeα+1
2
≥ 0.5 − δ and
θeα +θeα−1
2
< 0.5 − δ. Consequently, again using
(11), it can be shown the upper bound of S for a given value of α is
ρ/2−δ
.
4α−2
In general, these facts are not sufficient for parsimoniously characterizing the relationship between entry behavior and α. This is because different numbers of outlets may enter the market for
different values of α. However, when ρ = 4δ exactly three new outlets enter the market for each
value of α. Moreover, α “jumps” at precisely the same S at which a fourth new outlet enters the
market.32
This makes the model relatively tractable, allowing us to show that, holding the value of α
fixed, for sufficiently small S, as S declines, welfare first gets worse and then gets better. Since
the value of α increases periodically as S declines (Remark B.1), welfare decreases and increases
periodically as S declines. That is, welfare cycles as S vanishes, and each value of α corresponds
to a cycle. Figure 4 illustrates this phenomenon. At the start of the interval of S for which α = 2,
31 We will sometimes refer to α as a function and sometimes to refer to it as a particular value of the function; the context should
make clear the sense in which we are using the notation.
ρ/2−δ
ρ
ρ
32 This is shown using the equations ρ/2−δ =
,
= 4n
, and substituting x = 2.
4α+2
4(n+2x) 4α−2
41
welfare declines, then jumps when the first new outlet enters the market, then welfare continues to
decline, then increases monotonically after the second new outlet enters (with another at the value
of S at which the third new outlet enters); the same pattern occurs for cycles 3-5. The intuition is,
for a given cycle, the socially harmful effect of outlet α becoming more partisan is strongest when
outlet α is most moderate, i.e. when S is high. This is because the direct effect is largest then, as
the probability the median voter gets news from outlet α is highest when S is highest. The socially
harmful effect of α becoming more partisan dominates the socially positive effect of the centrist
outlets spreading out before the second new outlet enters. The beneficial effects of the centrist
outlets spreading out go to zero just before this entry, as the middle outlets become evenly spaced.
However, the effect of the centrist outlets spreading out jumps–becoming dominant–as soon as the
second new outlet enters, because the magnitude of the partial effects of the shifts in the centrist
outlets jumps at that point. Immediately after the second outlet enters the two centrist outlets
are very close to each other, and so the social benefit of them spreading out is substantial, since
almost all of their consumers benefit from those outlets’ editorial positions moving closer to the
consumers’ tastes. While the effect of the centrist outlets spreading out again goes to zero just
before the third new outlet enters, the effect of outlet α’s movement has become socially positive
by that point (the readership effect has begun to dominate), so the net marginal effect of a decline
in S is positive.
The following formal results provide a characterization of the welfare cycles. The reader may
have also noticed in Figure 4 that welfare at what appears to be the peak of the cycles does not
change across cycles, while welfare at the troughs increases, as S declines. We also show formally
that the first property is not general–in fact the peaks increase (decrease) if and only if ρ is strictly
greater (less) than 0.5. The second property is general (true for all ρ).
42
ρ = 0.5; δ = 0.125
0.0312
0.0312
W(S)
0.0311
0.0311
0.031
0.02
0.018
0.016
0.014
0.012
α=2
0.01
α=3
0.008
α=4
0.006
α=5
S decreasing
Figure 4: Illustration of four welfare cycles; dashed vertical lines denote cycle start/end points
It is helpful to first define a few more pieces of notation. Let S̄α (Sα ) denote the upper (lower)
¯
bound to S for a fixed α, and Sαmid denote the entry mid-point for that α, in the sense that it is
the value at which the second (out of three) new outlets enter for the given α.
Lemma B.2. There exists S 0 such that ∀S < S 0 , dW (S)/dS > 0 for all S ∈ (Sαmid , S̄α ) such that
dW (S)/dS is defined, and W (S) > W (S 0 ) for all S, S 0 ∈ (Sα , Sαmid ) such that S < S 0 .
¯
Proof. Remark B.1 implies that the condition ∀S < S 0 is equivalent to ∀α > α0 . Thus, we prove
that the result holds for all sufficiently high α.
ρ
ρ
It is easily shown using results referred to above that [Sα , S̄α ) = [ 8(2α+1)
, 8(2α−1)
). Call this
¯
interval α. It is also easily shown there are 4α + 1, 4α, 4α − 1 and 4α − 2 outlets when S is in the
ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
sub-intervals [ 8(2α+1)
, 4(4α+1)
), [ 4(4α+1)
, 16α
), [ 16α
, 4(4α−1)
) and [ 4(4α−1)
, 8(2α−1)
), respectively.
We refer to these four sub-intervals as the four “legs” of interval α, and since we focus on
ρ
ρ
, 8(2α−1)
), is the first leg, the next highest sub-interval
declining S, the upper sub-interval, [ 4(4α−1)
the second leg, etc. Let S̄αi and Sαi denote the upper and lower bounds, respectively, of the ith leg of
¯
43
interval α. Using this notation, to prove the lemma we need to prove dW/dS > 0 for S ∈ (S 2 , S̄ 2 )
¯
and S(S 2 , S̄ 2 ), and dW/dS < 0 for S ∈ (S 4 , S̄ 4 ) and S ∈ (S 3 , S̄ 3 ) (omitting the subscripts), since
¯
¯
¯
dW/dS < 0 is only undefined at points of entry, which are the boundaries of the interval legs (we
know that welfare strictly increases discontinuously–jumps up–at all of the entry points).
We use (5) to evaluate the sign of dW (S)/dS. By (6) and (11), the expressions for the partial
effects for liberal outlets i > α turn out to equal zero except for outlet q(S), the most centrist
liberal outlet (due to uniform f ()). Thus, by symmetry,
µ
sign
dW L∗
dS
¶
µ
= sign
¶
∗
dθq(S)
dθα∗
∂W L∗
∂W L∗
|θ=θ∗ (
)+
|θ=θ∗ (
).
∂θα
dS
∂θq(S)
dS
(15)
Using (4) and (11), for outlet α the partial effect is
¢¡
¢
¡
∂W L∗
|θ=θ∗ = − 2 δ − ρ/2 + 2(2α + 1)S 0.5(1 − ρ/2 − δ) + (2α + 1 − 4α/ρ)S
∂θα
|
{z
}
Direct Effect
+ 2S(1 − ρ)(1 − 4(2α + 1)S/ρ) .
{z
}
|
(16)
Readership Effect (Right)
(6) and (11) yield the partial effect for outlet q(S). The expression for the
dθi∗
’s
dS
is simple given
(11): 4i/ρ. And q(S) is equal to 2α − 1 in the first two legs and 2α in the second two legs of
interval α. Together these results can be used to show the following.
44
lim1−
S→S̄α
lim1+
S→Sα
¯
dW ∗
1 − 0.5ρ
>0↔α>
,
dS
1−ρ
dW ∗
α(1 + 2α(1 − ρ) − ρ/4)
=
> 0, ∀α, ρ,
dS
(4α − 1)2
dW ∗
1 − 5ρ/8
> 0 if α >
,
1−ρ
S→S̄α dS
dW ∗
2−ρ
lim2+
=
> 0, ∀α, ρ,
16α
S→Sα dS
¯
dW ∗
dW ∗
= − lim2+
< 0, ∀α, ρ,
lim3−
S→Sα dS
S→S̄α dS
lim2−
¯
dW ∗
0.5 − ρ/8
lim3+
< 0 if α >
> 0,
1−ρ
S→Sα dS
¯
dW ∗
2α(−1 − α(1 − ρ) + 5ρ/8)
lim4−
=
< 0, ∀α, ρ,
(4α − 1)2
S→S̄α dS
lim4+
S→Sα
¯
dW ∗
α(1 − ρ)
=−
< 0, ∀α, ρ.
dS
2(2α − 1)
(17)
These results imply that for sufficiently high α, dW/dS takes the signs needed to prove the
lemma as S approaches the boundaries of all four legs. To complete the proof then, we just need
to show dW/dS takes the same sign in the interior of the legs as it does at the boundaries. It
is difficult to do this explicitly since the algebra is very complicated. Another way to do this is
to show d2 W/dS 2 takes the same sign for all S in each leg. This takes some work, but is doable.
The easiest way seems to be to use the facts that d3 W/dS 3 > 0 for all S, and that d2 W/dS 2 > 0
evaluated at S = Sαi for each leg i, for sufficiently high α (specific results omitted but available
¯
upon request).
Lemma B.3. For each interval α,
mid
1. W (Sαmid ) > W (Sα−1
);
45
2. W (Sα ) > (<)W (S̄α ) = W (Sα−1 ) if and only if ρ > (<)0.5.
¯
¯
Proof. To start with part 2 of the lemma, notice first that the RHS equality, W (S̄α ) = W (Sα−1 ), is
¯
true because although entry occurs when S is marginally less than S̄α , the entry causes the number
of outlets to change from odd to even and the entrant’s position, in the (upper) limit with respect
to S, is the same as the incumbent centrist’s position, 0.5. Entry itself then has no immediate
effect on welfare, since the entrant is not offering news with a unique editorial position.
To prove the inequality part of part 2, we use the fact that W (S̄α1 ) − W (Sα1 ) =
¯
R S̄α1
S1
¯α
dW ∗
dS,
dS
which we can calculate using expressions derived in the proof of Lemma B.2. Entry occurs at
S = Sα1 which appears to complicate things. However, when S declines from S̄α to Sαmid , the
¯
number of outlets changes from n = 2(2α − 1) to n + 2. The n/2 liberal outlets move left, and
the n/2 conservative outlets move right, and two new outlets appear at position 0.5. Since the
most moderate of the (n initial) liberal/conservative outlets start at position 0.5 as well, we can
also calculate W (S̄α ) − W (Sαmid ) by integration of the marginal effects, in which the middle partial
effect is for outlet n/2, and its neighbor to the right, outlet 1 + n/2, has position 0.5.
Thus, W (S̄α ) − W (Sα ) =
¯
R S̄α
mid
Sα
dW ∗
dS
dS
+
R Sαmid
Sα
¯
dW ∗
dS,
dS
which can be shown to be
(1 − 2ρ)αρ
W (S̄α ) − W (Sα ) =
> 0 ↔ ρ < 0.5.
¯
18(4α2 − 1)2
(18)
This proves part 2 of this lemma.
We use the same method to prove part 1:
Z
mid
W (Sα−1
)
−
W (Sαmid )
S̄α
=
mid
Sα
dW ∗
dS +
dS
Z
mid
Sα−1
Sα−1
¯
dW ∗
dS.
dS
(19)
This expression is extremely complex and omitted, but it is unambiguously negative, which proves
46
the result.
The first part of this lemma says that, at the entry midpoint of each cycle, welfare unambiguously improves as the level of competition in the market increases, independent of the parameter
ρ. The second part says that whether welfare improves/worsens within a cycle, depends only on
whether ρ is above a cut-off, and is independent of the index of the cycle and all other parameters.
The key to the intuition is that the first part relates to changes in welfare across cycles, while
the second part relates to the change in welfare that results from S declining within a particular
cycle. Within a cycle, the proof discusses how the change in welfare can be expressed as the sum
of two integrals of boundary and middle effects. In both integrals the value of α is the same.
Although both (α and q(S)) partial effects are sensitive to ρ, the α effect is more so. This is
because a decrease in ρ has competing effects within the q(S) effects. A lower ρ causes the most
conservative liberal outlet to move further from 0.5 from the beginning to end of the cycle. The
downside of this greater movement is that consumers who stay with the outlet get more distorted
news. The upside is that the consumers who switch from the more liberal neighbor outlet get a
greater improvement in the distortion of their news, since the distortion of this neighbor was also
decreasing in ρ. These effects nullify each other, to some extent. On the other hand, a decrease
in ρ causing outlet α to become more extreme only worsens the α effect, since outlet α is never
attracting median voters from a more extreme outlet (since all median voters are already getting
media news). As a result there is a cut-off value of ρ that determines which effect dominates.
Part 1 of the lemma results from the value of α changing, since this part of the lemma refers to a
change in welfare across cycles. As α increases, the fraction of median voters who are affected by
the α and q(S) effects decreases substantially. This causes the amplitude of the cycles to decrease
47
mid
as S declines. Since when S declines from Sα−1
to S̄α , W (S) is increasing when S is in the cycle
with the larger amplitude (cycle α − 1) and decreasing when S is in the cycle with the smaller
amplitude (cycle α), it is intuitive that the increase dominates the decrease. We use these lemmas
to describe how the peaks and troughs of the cycles change as S declines.
Proposition B.4. There exists S 0 > 0 such that for all S < S 0 , the trough of cycle α is greater
than the trough of cycle α − 1 for all ρ, and the peak of cycle α is greater (less) than the peak of
cycle α − 1 if and only if ρ > (<)0.5.
2
Proof. Lemma B.3 says that for all α, W (Sα2 ) > W (Sα−1
), so if the trough of cycle α occurs at
¯
¯
S = Sα2 for all α we will have the part of the result relating to troughs. To show this, note that
¯
Lemma B.2 implies that for sufficiently high α, the trough of each cycle occurs at S = Sα1 or
¯
S = Sα2 , so we simply need to show W (S 2 ) < W (S 1 ) to show the trough indeed occurs at S 2
¯
¯
¯
¯
(subscripts omitted). One way to do this is to show the change in welfare resulting from entry at
S 1 is less than the change in welfare resulting from S going from S̄ 2 to S 2 (obtained by integrating
¯
¯
the partial effects as in the proof of Lemma B.3). Using Mathematica, this can be shown true
asymptotically (results omitted).
We prove that the peaks are increasing if ρ > 0.5 in a similar manner. Lemma B.3 says that
W (Sα ) > (<)W (Sα−1 ) if and only if ρ > (<)0.5, so we need to prove that the peak of cycle α
¯
¯
occurs at S = Sα . From Lemma B.2, for sufficiently high α, W (S) is monotone increasing for
¯
S < S mid = S 2 as S decreases, and monotone decreasing for S > S mid as S decreases, where
¯
W (S) is differentiable. The points where it is not differentiable are the leg boundaries, S 1 , S 2
¯ ¯
and S 3 . Welfare jumps at S equal to S 1 and S 3 , because at those points the number of outlets
¯
¯
¯
changes from even to odd, and W (S) is continuous at S = S 2 . It follows that the candidates for
¯
the S where the peak occurs are S 4 , S 1 (in the limit as S approaches from below) and S̄ 1 . Since
¯ ¯
48
4
W (S̄α1 ) = W (Sα−1
), W (S 4 ) will be greater than W (S̄ 1 ) if ρ > 0.5. So we just need to show that
¯
¯
lim W (S) is less than W (S̄ 1 ) or W (S 4 ). One way to do this to show the change in welfare from
¯
S→S 1−
¯
S increasing from S 2 to (marginally less than) S 1 is less than the change in welfare resulting from
¯
¯
S going from S 2 to S̄ 1 . Again, using Mathematica, this can be shown true asymptotically.
¯
Lemmas B.2 and B.3 together almost imply that asymptotically the trough of each cycle occurs
at the cycle’s entry midpoint, and the peak occurs either at the cycle start or end. The only
complication comes from the jump in welfare that comes at the point where the first new outlet
enters, S =
ρ
.
16α−4
But we show this jump is relatively small, and so neither the peak nor trough
occurs at that point. This proves the proposition. Figure 5 illustrates the asymptotic welfare
cycles for a range of ρ (three of which do not satisfy the ρ = 4δ condition). They are especially
irregular for the case ρ = 1. However, in all cases it is clear that: a) there are indeed cycles; b)
the troughs increase as S declines; c) the peaks increase when ρ > 0.5.
49
ρ = 0.3; δ = 0.1
ρ = 0.4; δ = 0.1
0.0235
0.0246
0.0235
0.0246
0.0246
W(S)
0.0235
0.0246
0.0235
0.0246
0.0235
0.0246
0.0246
0.0235
10
8
6
S
10
8
ρ = 0.7; δ = 0.1
0.0253
6
S
−4
x 10
−4
ρ = 1; δ = 0.1
x 10
0.0253
0.0253
0.0253
0.0253
0.0253
W(S)
0.0253
0.0253
0.0253
0.0253
0.0253
0.0253
0.0253
0.0253
0.0253
5.4
5.3
5.2
S
5.1
5.4
5.3
5.2
S
−4
x 10
5.1
−4
x 10
Figure 5: Asymptotic welfare cycles; all parameters correspond to case 2 (“the median voter never gets propaganda”), uniform f ()
50