Competition, Politics, & Social Media
aa r X i v : . [ ec on . GN ] D ec Competition, Politics, & Social Media
Benson Tsz Kin Leung ∗ Pinar Yildirim †‡ Abstract
An increasing number of politicians are relying on cheaper, easier to access technolo-gies such as online social media platforms to communicate with their constituency. Theseplatforms present a cheap and low–barrier channel of communication to politicians, po-tentially intensifying political competition by allowing many to enter political races. Inthis study, we demonstrate that lowering costs of communication, which allows many en-trants to come into a competitive market, can strengthen an incumbent’s position whenthe newcomers compete by providing more information to the voters. We show an asym-metric bad-news-good-news effect where early negative news hurts the challengers morethan the positive news benefit them, such that in aggregate, an incumbent politician’schances of winning is higher with more entrants in the market. Our findings indicate thatcommunication through social media and other platforms can intensify competition, how-ever incumbency advantage may be strengthened rather than weakened as an outcome ofhigher number of entrants into a political market.
Keywords: Elections, market turf, social media, communication technology, incumbencyadvantage, online platforms ∗ Faculty of Economics, University of Cambridge. Email: [email protected] . † Corresponding Author: Assistant Professor, The Wharton School, University of Pennsylvania. 3730 WalnutSt. Philadelphia PA 19104. Phone: (215) 746 2369. Email: [email protected] ‡ Authors are in alphabethical order. We thank Christophe Van den Bulte, Yi Liu, and Vladimir Pavlov fortheir valuable feedback on the paper. Introduction
When does lowering barriers to entering a competition strengthen, as opposed to weaken, in-cumbency advantage? We study a special case of this broad question in the context of politicalraces with barriers of marketing and communication. Historically, to reach out to large au-diences, public personalities and brands mainly relied on expensive and limited formats ofcommunication such as TV and newspaper advertising, or print and mailed pamphlets. Digitaltechnologies altered much of that by exponentially reducing communication costs, reducing bar-riers for entrants into politics, and thus widening access. Online platforms became the preferredchannel of communication for politicians to reach out to their constituency for campaigning andofficial communication. President Obama ran most of his 2012 campaign on Facebook (Borah,2014), President Trump used Twitter as his official channel of communicating with the publicwhile in office (Kreis, 2017), and more than half of campaign advertising dollars in the 2020Presidential race was spent on digital platforms (Gibson, 2020). On the one hand, social me-dia communication enables more politicians, particularly newcomers who lack funds, to finda platform to make their voices heard. Via these cheaper technologies they can reach out tovoters en masse and inform them about their candidacy, values, policies, and campaign activi-ties (Petrova et al., 2020). On the other hand, lowering barriers to entry can introduce a highnumber of challengers to an election who run not only against the incumbent, but also against each other . It is not ex ante clear whether lowering communication barriers to entering politicalraces or higher intensity of information about newcomers helps or hurts incumbents.Incumbency advantage in politics is well-documented (e.g., Ansolabehere et al., 2006; Levitt and Wolfram,1997). 75% of senate races between 1980 and 2012 had a participating incumbent (Garcia-Jimeno and Yildirim,2015). These incumbents held a 1-2% point advantage over their opponents in the 1940s, even-tually widening up to a 8-10% point advantage in early 2000s (Petrova et al., 2020). A hostof factors contribute to the presence of a competitive advantage. Scholars suggest that in-cumbents hold a repeated advantage over their opponents (Jacobson and Kernell, 1982), likelybecause they are higher quality candidates (Ashworth and Bueno de Mesquita, 2008), have ac-cess to resources of the offices they held, have more funding to run campaigns (Cox and Katz,1996; Fouirnaies and Hall, 2014), and receive more media coverage compared to challengers(Goldenberg and Traugott, 1980; Ansolabehere et al., 2006; Schaffner, 2006). The familiarityof voters with incumbent politicians and greater media coverage, combined with lack of fundsto run costly advertising campaigns, erected tall barriers for newcomers who want to enter into2olitical races, resulting in less competitive elections. Less competitive elections, in turn, re-sult in worse economic, social, and democratic outcomes (Myerson, 1993; Persson et al., 2003;Ferraz and Finan, 2011; Galasso and Nannicini, 2011).We build a model to study if lower informational barriers to enter into a race intensifycompetition, and in turn, reduce an incumbent’s advantage. Focusing on a two-party politicalcompetition between differentiated candidates where informing and persuading voters is the endgoal, we investigate if lower barriers of communication can alter the probability of winning foran incumbent politician. In our model, one party produces an incumbent as its candidate andthe other produces one or more challengers on the opposing side. We incorporate incumbencyadvantage by allowing the incumbents to reach constituents more widely than the challengersand by allowing the constituents to have more positive prior beliefs about the incumbents, inline with Ashworth and Bueno de Mesquita (2008). In this environment, we first investigatehow additional challengers’ entry to a race alters the incumbent’s probability of winning theelection and how this competitive advantage varies with the characteristics of the environmentof competition, such as the effectiveness of the communication and advertising channels.We find that, while lowering barriers of marketing and communications intensifies politicalcompetition, it does not necessarily reduce the re-election probability of an incumbent. Low-ering barriers to accessing communication channels can, in fact, strengthen the probability ofwinning for an incumbent when more challengers enter a race and when informational cam-paigns intensify. This is because, as more challengers enter a race, there is more media coverage,more communication via social media, and more campaign events by challenger candidates inthe primaries and these campaigns target or are followed by individuals who vote in the pri-maries. We show that more information arriving during the primary period has an asymmetriceffect. Voters who receive a negative information (e.g., attack ads, negative press coverage orsocial media buzz) about a challenger update their beliefs about the match of the candidatedownward, which reduces the likelihood of voting for the candidate in the primary and in thegeneral election against the incumbent. Similarly, a positive information received via the samemeans results in an upward update. The upside of a positive update, however, is small: whileit increases the probability that a voter would support the candidate in the primary, it makeslittle difference in the chances of the candidate winning against the incumbent, since the voter The observations from the 2020 Presidential Election in the US provide evidence for these statements. Arecord 29 challengers entered the race on the Democratic side (Jacobson, 2019). These candidates heavilyutilized communication channels such as Facebook and Twitter, spent a record amount on advertising (Fischer,2020), and held a high number of campaign events (Schwartz, 2020).
3s ex ante more likely to support the challenger against the incumbent, anyway. The potentialharm from a negative news update, relative to the small upside of a positive news update,implies that more information during the primaries may hurt, rather than help a challengerwin against an incumbent.We also analyze the impact of some recent policies in political communication space byonline platforms such as Facebook, Google, and Twitter for narrowing options for politicalcommunication. Twitter eliminated political advertising on its platform entirely during 2020(Yaraghi, 2020), Facebook reduced the ability to micro-target political ads and stopped polit-ical advertising a week before the 2020 U.S. Presidential Election (Overly, 2020), and Googlesimilarly reduced micro-targeting for political advertisers (Lee, 2019). These bans were takenin an effort to reduce political tension and spread of misinformation in ads. They neverthe-less reduce access to political information, and their consequences on electoral races have notbeen studied, to the best of our knowledge. We investigate the impact of these policies on thechallengers’ and the incumbents’ ability to communicate with voters and argue that these bansmay strengthen incumbency advantage and make elections less competitive.Our study broadly contributes to the literature studying competition, entry barriers, strate-gic entry (e.g., Shen and Villas-Boas, 2010; Igami and Yang, 2014; Chen and Turut, 2018; Joshi et al.,2009) and incumbency advantage. Demsetz (1982) notes that information costs are a funda-mental barrier to entry, as they “constitute hurdles to all who would (and have) enter(ed) theindustry.” Industrial organization literature, more specifically, recognizes advertising and pro-motions as informational barriers to entry (Demsetz, 1982; Schmalensee, 1983). Informationcan alter consumer tastes towards the advertised product and may erect additional challengesfor the entrants (Cubbin, 1981; Dixit and Norman, 1978; Bagwell, 2007). Advertising can alsoresult in brand loyalty and consumer goodwill, thus new firms entering a market have to adver-tise more than the existing levels of advertising by incumbents to gain market share. Loweringinformational barriers is particularly useful in markets with imperfect information, where thereis sufficient uncertainty about the match value of products, and consumers resolve it throughinformation received via advertising. This is also the case for political markets where newpoliticians with little-known policy positions frequently appear.Our paper contributes to this literature by explicitly focusing on the growing use of socialand digital information channels, which lowers communication costs. While empirical marketingliterature on social media has been growing (e.g. Godes and Mayzlin, 2009; Schweidel and Moe,2014), there have been fewer theoretical examinations on the impact of social media channels4e.g., Joshi, 2015; Bart, 2017). To our knowledge, little focus is paid to easy access to mediaas an entry barrier. Recently Petrova et al. (2020) studied if access to cheaper communicationchannels such as social media could earn politicians fundraising benefits. Authors find that,upon opening a social media account, an average politician’s donations go up, but this increaseis mainly observed for political newcomers rather than experienced politicians. Authors con-clude that cheaper communication channels such as social media may mitigate incumbencyadvantage by allowing more politicians to enter into races and communicate with their con-stituency. Question is whether more communication facilitated by social media and cheaperdigital communications, particularly that among the entrants which take place earlier in anelection during the primaries, turn into a competitive advantage which can reverse the outcomeof an election. Our study complements this paper by focusing on the voting outcome, using atheoretical model which incorporates an incumbent’s informational advantage. This gives us achance to address whether the incumbency advantage can be reversed. We demonstrate that,counter-intuitively, lowering communication barriers via the use of social media and digital ad-vertising need not reduce incumbency advantage. An incumbent may preserve and increase hisadvantage with increasing number of challengers on the opposite side of the political spectrum.Finally, we contribute to the literature documenting the long-standing incumbency advan-tage in the United States. Prior literature focused on the sources of incumbency advantage(Levitt and Wolfram, 1997), listing structural advantages of being an incumbent as greater in-terest from media, fundraising, access to key individuals (Ansolabehere and Snyder, 2000; Prat,2002; Strömberg, 2004; Prior, 2006; Petrova et al., 2020). Incumbency advantage bars entryand reduces electoral competition, which in turn reduces accountability of politicians towardsconstituents (Carson et al., 2007). More competitive elections result in better political and eco-nomic outcomes (Myerson, 1993; Persson et al., 2003). Therefore understanding how loweringcommunication barriers can alter electoral competition is crucial.In the rest of the paper, we first introduce the model in Section 2 and follow with theanalysis in Section 3. Section 3.2 generalizes the model and offers extensions. We conclude inSection 4.
We develop a model of electoral competition, considering the race between an incumbent andone or more challengers on the opposite side of the political spectrum. Let the political ideology5f voters be represented on a Hotelling line as illustrated in Figure 1. Voters are represented by n and their political ideology is denoted by x n and is uniformly distributed on a horizontal line,i.e., x n ∼ U [0 , i and without loss of generality, their ideology orpolitical positions are assumed to be located at either end of this spectrum, on 0 or 1, labelledas “left” and “right” respectively. The political ideology of a candidate i is assumed to beexogenous, and is denoted as x i ∈ { , } . Through the rest of the paper, for ease of exposition,we will refer to voters whose ideology is on the lower half of this spectrum ( x n < /
2) as left-wing and voters whose ideology is on the upper half of this spectrum ( x n > /
2) as right-wingvoters.Without loss of generality, we assume that the incumbent politician is a right-wing politicianwith x i = 1. Moreover, we assume that there can be at most two challenger candidates on theleft hand side, taking an opposing left-wing position such that x i = 0. We will refer to theincumbent candidate as candidate 3 ( i = 3) and the challengers as candidates 1 and 2 ( i = 1 , x n ∈ [0 , i is represented by V ( n, i ) and depends on twofactors: the ideological match between a candidate’s and own political position and the indi-vidual evaluation of the candidate. Formally, V ( n, i ) = Q ni − t ( x n − x i ) + ǫ i =3 (1)where Q ni is the personal evaluation of candidate i ’s quality by voter n , − t ( x n − x i ) is thedistance between the voter’s and candidate’s political ideology, and ǫ ∼ N (0 , σ ǫ ) is a globaltaste shock of ideology that favors the incumbent politician iff ǫ > Here, t measures theimportance of the ideological match between the voter and a candidate. This modeling choiceallows us to capture the similarities and differences between voters when they evaluate the samecandidate. The ideological match component of the valuation ( − t ( x n − x i ) ) allows a candidateto be valued similarly by voters of similar ideologies, while the idiosyncratic component Q ni We generalize the model to more than 2 challengers in Section 3.2 in the paper. Alternatively, ǫ can also be interpreted as a piece of information received by all voters that could favor the leftor right-wing candidate. This element is commonly used in probabilistic voting model (Lindbeck and Weibull,1987). Q n , Q n , Q n , but know their prior distribu-tion. They receive signals about the quality of the candidates during the primary and generalelection stages and update their beliefs, as will be described shortly. We assume that voters’individual assessments about the quality of the incumbent is realized as a random draw from Q n ∼ N ( q, σ Q ) for some q and that for a new candidate as a random draw from an unbiaseddistribution Q ni ∼ N (0 , σ Q ) for i = 1 , n . Here q captures the dif-ference in the expected quality between the incumbent and the entrants, which could be drivenby the performance of the incumbent in his previous term. When q >
0, the difference describesa form of incumbency advantage such that, on average, the incumbent candidate is assessedmore positively. As discussed in the introduction, due to running political campaigns in thepast or having been elected to an office, incumbents are known to hold a competitive advantagein elections over challengers (Ansolabehere et al., 2006). This incumbency advantage may becaptured in the prior beliefs held.The game timeline is illustrated in Figure 2 and lasts two periods, with a possible primarystage and a general election stage. In period 1, the challengers (candidates 1 and 2) decidewhether to enter the election at some fixed cost C . This cost represents, in our framework,the barrier to entering a market. While there can be a number of such barriers, in line withthe focus of our paper, we will treat this cost as the cost of communication. If there are nochallengers or if only one challenger enters the race, a primary election is not necessary todetermine the candidate for the general election from the left, and the game goes straight tothe general election. If challengers 1 and 2 both decide to enter, however, they compete in aprimary. The assumption of independence implies that Q measures only the horizontal differentiation among thechallengers but not the vertical differentiation. The model can be easily generalized to incorporate an elementof vertical differentiation among the challengers, e.g., by adding a random variable U that is common to allvoter n . Our results hold as long as the upper bound of U is not too large, or when the distribution of U is nottoo dispersed. The election setting we consider resembles a senate election in the U.S., where the race has (historically)been between two party candidates, there is a primary period leading to the elections, and the winner is electedby plurality of votes. Notice that, technically, there is also a primary on the side of the incumbent. We abstract away frommodeling a challenger on the side of the incumbent for simplicity, but the solutions which involve a challengeron the incumbent’s side can be obtained from the authors. The key qualitative insights of the paper are notaltered by this modification. The model with a single incumbent captures the competitive advantage of theincumbent in a parsimonious model. andidates 1 and 2decide whether to enterthe race at a cost C .Both do not enter.Only candidate 3participates and winsin the general election. Only candidate i ∈ { , } enters. Both candidates i = 1 , x n < acquire information aboutcandidates 1 and 2 and votein the primary election.Candidate i ∈ { , } wins the primary election.Voters acquire informationabout candidates i and 3and vote in the generalelection after observing ǫ. Candidate with thehighest support wins.
Figure 2: Timeline of the gameNote that, in a primary, only the voters with left-wing views, x n < , vote and they learnabout the quality of the challengers by gathering noisy information about them. They thenvote for the candidate with the higher expected value. More specifically, we assume that thecommunications during the primary are aimed at the voters who participate in the primaries,such as advertising or social media messages of the candidates. Each voter x n < receivesa private signal s pni about candidate i = 1 , p stands for‘primary.’ Signals s pn and s pn are independently distributed according to N ( Q ni , σ s ).After receiving signals s pn , s pn , voters n with political ideology x n < update their expectedvaluation of candidates 1 and 2 and vote for the one with the highest expected value in theprimary election. More specifically, they update their expected value according to the Bayes’8ule, as follows: E ( V ( n, | s pn ) = σ s σ Q + σ s E ( V ( n, σ Q σ Q + σ s s pn − tx n = σ Q σ Q + σ s s pn − tx n ,E ( V ( n, | s pn ) = σ s σ Q + σ s E ( V ( n, σ Q σ Q + σ s s pn − tx n = σ Q σ Q + σ s s pn − tx n . (2)Note that, during the primary stage, voters with x n < receive a signal about each chal-lenger candidate. Importantly, voters with x n > do not vote in the left-wing primary, andtherefore receive less information, which we normalize to no signals in the model for simplicity.This asymmetry reflects the fact that voters with aligned political ideology pay more attentionto the primary compared to voters whose ideology is misaligned with the party whose primaryis held.Next, suppose candidate i ∈ { , } participates in the general election, either via winningthe primary election or due to being the only candidate on the left. In the general election,voters receive extra information about the candidate i ∈ { , } on the left and candidate 3 onthe right. In general election, all voters receive a signal s gn about the incumbent that follows N ( Q n , βσ s ), where g stands for general election. On the other hand, voters with x n < receivea signal s gni about the challenger candidate i ∈ { , } which follows N ( Q ni , βσ s ), while voterswith x n > receive a signal s gni about the challenger candidate i which follows N ( Q ni , λβσ s ).Here, two parameters allow us to study the differentiation between the effectiveness ofpolitical marketing campaigns during the primary and general election stages, and that betweenthe incumbent and the challengers. They are not, however, necessary for driving our generalqualitative insights.First, we introduce β ∈ (0 , ∞ ), which measures the informativeness of the signals in a generalelection compared to the primary election for all candidates. Specifically, this parameter allowsthe general election marketing campaigns to yield more or less precise signals in informing votersrelative to the primary stage. It is possible, for instance, that politicians use different formatsof communication or advertising, or media or voters pay more attention to the election duringthe primary stage. Or as we will show later, more competition could lead to more activitiesand information provision in the primary. All of these differences between the primary andgeneral election environment that are common across the politicians would be captured bythis parameter, and a larger (smaller) β indicates a more (less) informative signal about thecandidates in the primary period compared to that in the general election.9econd, the parameter λ ≥ λ = 1, there are no disadvantages. When λ >
1, the challengers’ signalsare less effective in informing the supporters of the incumbent ( x n > / x n > / λ allowsus to capture the difference between the incumbent and challengers in the ability to reach outto incumbent’s base (voters with x n > /
2) using communication tools. Restrictions specific topaid political communication, for instance, narrows a challenger’s opportunity to reach voterswith x n > /
2. As reported in the literature, for incumbents, informing voters with x n < / λ ).To summarize the information provision in a general election, upon receiving s gni , i ∈ { , } and s gn , voters with x n > update their expected value from voting for each candidate accord-ing to: E ( V ( n, i ) | s gni ) = σ Q σ Q + λβσ s s gni − tx n ; E ( V ( n, | s gn ) = βσ s σ Q + βσ s q + σ Q σ Q + βσ s s gn − t ( x n − . (3)On the other hand, if there was no primary preceding the general election, upon receiving s gni , i ∈ { , } , and s gn in the general election, voters with x n < update their expected valuation10ccording to: E ( V ( n, i ) | s gni ) = σ Q σ Q + βσ s s gni − tx n ; E ( V ( n, | s gn ) = βσ s σ Q + βσ s q + σ Q σ Q + βσ s s gn − t ( x n − . (4)If there was a primary election preceding the general election, upon receiving s gni , i ∈ { , } ,and s gn in the general election, voters with x n < update their expected valuation accordingto: E ( V ( n, i ) | s pni , s gni ) = βσ s σ Q + βσ s + σ s s gni + σ s σ Q + βσ s + σ s s pni − tx n ; E ( V ( n, | s gn ) = βσ s σ Q + βσ s q + σ Q σ Q + βσ s s gn − t ( x n − . (5)After both stages, and having observed the realization of ǫ , voters vote for the candidatewhich generates higher expected value for them in the general election. The candidate whowins the general election is assumed to receive a utility or a prize of 1. If no candidates on theleft enter the election, candidate 3 wins with probability 1. We next solve the game described in the previous section using backward induction. To thisend, we will consider each sub-game of a general election where there can be zero, one, ortwo challenger candidates entering the race and then we will compare these cases to assess ifmore challengers reduce incumbency advantage in a general election. We will also draw generalinsights about the informational disadvantages of the challengers and how they contribute tothe presence of the incumbency advantage.
No challengers enter the election.
We first consider the case when there are no challengers.The analysis of this case is straightforward since candidate 3 wins the election and gets payoff1, by default, and candidates 1 and 2 receive 0 payoff.
Only one challenger enters the election.
Next, we consider the case when there is onlyone challenger entering the race. Without loss of generality, we will assume that candidate 111nters the race as the challenger on the left. The expected valuation of voter x n of candidates1 and 3, conditional on receiving s gn and s gn , are now E ( V ( n, | s gn ) = σ Q σ Q + λ ′ βσ s s gn − tx n ,E ( V ( n, | s gn ) = βσ s σ Q + βσ s q + σ Q σ Q + βσ s s gn − t ( x n − + ǫ (6)where λ ′ = 1 for voters with x n < and λ ′ = λ for voters with x n > . In the following, wecharacterize the vote received by candidate 3 given ǫ . Fixing ǫ , a voter with political ideology x n votes for candidate 3 if and only if upon updating the beliefs about the candidate, he has ahigher expected valuation, which holds if: βσ s σ Q + βσ s q + σ Q σ Q + βσ s s gn − t ( x n − + ǫ > σ Q σ Q + λ ′ βσ s s gn − tx n or, equivalently σ Q σ Q + βσ s s gn − σ Q σ Q + λ ′ βσ s s gn > t (1 − x n ) − ǫ − βσ s σ Q + βσ s q. For voters with x n < , σ Q σ Q + βσ s s gn − σ Q σ Q + λ ′ βσ s s gn ∼ N (cid:18) σ Q σ Q + βσ s q, σ Q σ Q + βσ s (cid:19) . Therefore for agiven ǫ , the vote share of candidate 3 among the voters with x n < equals to: Z − G ( t (1 − x n ) − ǫ − q ) q σ Q + βσ s √ σ Q dx n , where G is the c.d.f. of the standard normal distribution. On the other hand, for voters with x n > , σ Q σ Q + βσ s s gn − σ Q σ Q + λ ′ βσ s s gn ∼ N (cid:18) σ Q σ Q + σ g q, (cid:18) σ Q σ Q + λβσ s + σ Q σ Q + βσ s (cid:19)(cid:19) . For the given ǫ , the votereceived by candidate 3 from voters with x n > equals: Z − G ( t (1 − x n ) − ǫ − q ) σ Q σ Q + λβσ s + σ Q σ Q + βσ s ! − dx n . Using the above distributions and vote shares, we can now characterize in Lemma 1 theprobability that the incumbent politician will be elected when there is only one challengerentering the primary.
Lemma 1. (Incumbent’s chance of winning with one challenger)
When only one can-didate on the left enters the race, the incumbent is elected with probability G (cid:16) − ǫ ∗ σ ǫ (cid:17) , where ∗ ≤ − q is the solution of Z G ( t (1 − x n ) − ǫ ∗ − q ) q σ Q + βσ s √ σ Q dx n + Z G ( t (1 − x n ) − ǫ ∗ − q ) σ Q σ Q + λβσ s + σ Q σ Q + βσ s ! − dx n = 12 (7) Lemma 1 shows that the incumbent is elected when the global taste shock of ideology favorshim such that voters evaluate him highly, i.e., when ǫ is large. More specifically, the incumbentis elected when ǫ is larger than the threshold ǫ ∗ and with probability 1 − G (cid:16) ǫ ∗ σ ǫ (cid:17) = G (cid:16) − ǫ ∗ σ ǫ (cid:17) .The threshold ǫ ∗ thus measures the incumbent’s advantage when only one challenger entersthe race: the lower ǫ ∗ is, the larger is the incumbency advantage. We will use Lemma 1 alongwith the next lemmas in driving the results with regards to how the number of challengersentering the race influences the chances of the incumbent to get re-elected. Two challengers enter the election.
Now suppose that both challengers, candidates 1 and2, enter the election. We start by analyzing the outcome of the primary election and then moveon to the analysis of the general election. The analysis resembles what we have carried out untilnow. Given the communication by politicians in the primary stage, s pn and s pn , the expectedvaluation of the voters with x n < of candidates 1 and 2 are E ( V ( n, | s pn ) = σ Q σ Q + σ s s pn − tx n E ( V ( n, | s pn ) = σ Q σ Q + σ s s pn − tx n Since the signals received during the primary election for both candidates, s pn and s pn , followthe same distribution ( N (0 , σ Q + σ p )), candidates 1 and 2 each receive half the support fromvoters with x n < during the primary stage and end up with a probability of winning theprimary. Without loss of generality, let’s assume candidate 2 wins the primary election andbecomes the candidate for the general election on the left.To understand how a more competitive primary influences the outcome of a general election,let’s first describe how these candidates are seen by the voters. During the general electionperiod, voters receive additional information about candidates 2 and 3. Let voters with x n > receive signals s gn and s gn . In the following, as in the case with only one entrant, we characterize13he vote share of the incumbent given ǫ . As voters with x n > do not receive additionalinformation about candidate 2 in the primary period, similar to the case with only one entrant,their expected valuation of candidates 2 and 3 becomes: E ( V ( n, | s gn ) = σ Q σ Q + λβσ s s gn − tx n ,E ( V ( n, | s gn ) = βσ s σ Q + βσ s q + σ Q σ Q + βσ s s gn − t ( x n − + ǫ. (8)Fixing ǫ , the vote received by candidate 3 from voters with right-wing views is the same asin the case with one challenger, i.e., Z − G ( t (1 − x n ) − ǫ − q ) σ Q σ Q + λβσ s + σ Q σ Q + βσ s ! − dx n . Now, for voters with x n < , their expected evaluation towards candidate 2 and 3 given s pn , s gn and s gn are E ( V ( n, | s pn , s gn ) = βσ s σ Q + βσ s + σ s s gn + σ s σ Q + βσ s + σ s s pn − tx n E ( V ( n, | s gn ) = βσ s σ Q + βσ s q + σ Q σ Q + βσ s s gn − t ( x n − + ǫ. (9)Their evaluation of candidate 3 follows the same expression as in the case where only onechallenger enters the election. However, the expression of their evaluation of candidate 2 isdifferent as equation (6) because they have received information about candidate 2 during theprimary election. Fixing ǫ , a voter with political ideology x n < votes for candidate 3 if andonly if βσ s σ Q + βσ s q + σ Q σ Q + βσ s s gn − t ( x n − + ǫ > βσ s σ Q + βσ s + σ s s gn + σ s σ Q + βσ s + σ s s pn − tx n σ Q σ Q + βσ s s gn − βσ s σ Q + βσ s − σ s s gn + σ s σ Q + βσ s + σ s s pn > t (1 − x n ) − ǫ − βσ s σ Q + βσ s q. σ Q σ Q + βσ s s gn − βσ s σ Q + βσ s + σ s s gn − σ s σ Q + βσ s + σ s s pn ∼ N σ Q σ Q + βσ s q, σ Q σ Q + βσ s + σ Q + βσ s β σ s ( σ Q + βσ s + σ s ) + σ Q + σ s σ s ( σ Q + βσ s + σ s ) . Thus, the vote received by candidate 3 from voters with x n < is equal to R − G ( t (1 − x n ) − ǫ − q ) σ Q σ Q + βσ s + σ Q + βσ sβ σ s ( σ Q + βσ s + σ s ) + σ Q + σ sσ s ( σ Q + βσ s + σ s ) − dx n . Similar to Lemma 1, the vote received by candidate 3 is increasing in ǫ , thus he is elected ifand only if ǫ is bigger than some threshold which gives us the following result. Lemma 2.
When two challengers from the left enter the race, the incumbent is elected withprobability G (cid:16) − ǫ ∗ σ ǫ (cid:17) , where ǫ ∗ is the solution to Z G ( t (1 − x n ) − ǫ ∗ − q ) σ Q σ Q + βσ s + σ Q + βσ s β σ s ( σ Q + βσ s + σ s ) + σ Q + σ s σ s ( σ Q + βσ s + σ s ) − dx n + Z G ( t (1 − x n ) − ǫ ∗ − q ) σ Q σ Q + λβσ s + σ Q σ Q + βσ s ! − dx n = 12 . (10) Similar to Lemma 1, the threshold ǫ ∗ in Lemma 2 measures the incumbency advantagewhen two challengers enter the race: the lower ǫ ∗ is, the greater is the incumbency advantageand the incumbent is elected with a higher probability. Next, using this lemma, we can comparethe winning probability of the incumbent with one or two challengers on the left and show thata higher number of challengers, or more competition, could benefit the incumbent.Note that, as the left hand side of Equation (10) is decreasing in ǫ ∗ , candidate 3 wins witha higher probability when both challengers on the left enter, i.e., ǫ ∗ < ǫ ∗ , iff Z G ( t (1 − x n ) − ǫ ∗ − q ) σ Q σ Q + βσ s + σ Q + βσ s β σ s ( σ Q + βσ s + σ s ) + σ Q + σ s σ s ( σ Q + βσ s + σ s ) − dx n < Z G ( t (1 − x n ) − ǫ ∗ − q ) q σ Q + βσ s √ σ Q dx n . (11) x n < given ǫ = ǫ ∗ . Now note that by Lemma 1, ǫ ∗ ≤ − q and thus t (1 − x n ) − ǫ ∗ − q > x n ∈ [0 , ]. It implies that without any information, all voters with x n < would vote for the challenger candidate 2. Thus, inequality (11) is equivalent to σ Q + βσ s β σ s ( σ Q + βσ s + σ s ) + σ Q + σ s σ s ( σ Q + βσ s + σ s ) > σ Q σ Q + βσ s . (12)Put differently, incumbent candidate 3 is elected with a higher probability when both, insteadof only one, challengers enter the race if and only if the evaluation of the challenger candidate2 from voters with left-wing views, i.e., x n < , is noisier. Proposition 1 demonstrates this keyfinding. Proposition 1. (Number of Challengers and Incumbency Advantage)
Candidate , orthe incumbent, wins the general election with a higher probability when there are two challengersinstead of one if σ s σ Q or β is sufficiently high. Formally, ǫ ∗ < ǫ ∗ holds iff: σ s σ Q > q ( β − + 8 − ( β − β . (13)Proposition 1 shows that, counter-intuitively, the incumbent could win with a higher proba-bility when there are more challengers entering an electoral race. In this case, more competitionstrengthens the incumbency advantage. To see the intuition for this result, consider how a sig-nal received in the primary stage influences the likelihood of a challenger winning in the generalelection, depending on the number of challengers entering the race. In terms of purely theirideological match, the voters with x n < prefer the challenger candidates over the incumbentbefore receiving any signals. However, importantly, these voters are targeted during the pri-maries with additional information received through various media. When some voters drawnegative signals about a challenger candidate, some may update their valuation of a candidatedownward, and those at the ideological margin may switch to vote for the incumbent in thegeneral election. The noisier the evaluation of the challenger is, the higher is the probabilitythat they will get a negative signal and switch to the incumbent. When σ s σ Q is high, the extrainformation that left-wing voters receive in the primary is noisier, which makes their evaluationof the challengers noisier. As a result, there is a higher probability that voters with x n < x n < ) is noisier.When β is high, the information received in the primary period is more informative than thatin the general election and therefore voters put more weight on them in their belief updating.A negative signal received during the primary stage thus has a heavier weight in the voter’sevaluation. On the other hand, if they receive a positive signal in the primary stage, althoughit protects the challenger from negative signals in the general election, the incremental benefitis small, because the information in the general election weighs less in voters’ evaluation andis unlikely to affect voters’ decision. Thus, there is an asymmetric effect of good news and badnews in the primary election for the incumbent and the challenger.This result also highlights the potential adverse effects of noisier political communicationarriving at later stages of an election. Such noise may arrive from misinformation campaigns oradvertisements with false information launched closer to the election date. Such efforts are morelikely to hurt the challengers rather than the incumbents. Similarly, political communicationrestrictions brought on at later stages of elections to reduce political information may strengthenan incumbent’s chances of winning. In a related recent development, Facebook announced thatit will ban political advertisements during the one week period before the 2020 U.S. Presidentialelection. This restriction, which comes late in the election period, is an example of a policy whichreduces information received during the general election stage compared to the primary stage,and based on the predictions from our model, may disproportionately harm the challengersrather than the incumbent politician. Now we are ready to characterize the subgame perfect Nash equilibrium. We focus on purestrategy equilibria. We also assume a candidate chooses to enter the race in case of indiffer-ence. Proposition 2 investigates how the changes in the fixed cost of entering a market altersthe competitiveness of political races and the resulting probability of the incumbent’s chancesof winning in an election. Such fixed costs may be associated with the initial fundraising,procedural challenges, or the difficulty of initiating marketing and political communication. In the mixed strategy equilibria, there might be a coordination failure among candidates on the left, inwhich case no candidates on the left enter the election with strictly positive probability. This type of equilibriais unrealistic and is therefore ignored. roposition 2. (Cost of Entry and Competitiveness of Races) When G (cid:16) ǫ ∗ σ ǫ (cid:17)
As the cost of entering a race( C ) decreases, incumbency advantage, or candidate ’s probability of winning the race is higherif G (cid:16) ǫ ∗ σ ǫ (cid:17) < G (cid:16) ǫ ∗ σ ǫ (cid:17) and C < G (cid:16) ǫ ∗ σ ǫ (cid:17) , and G (cid:16) ǫ ∗ σ ǫ (cid:17) < G (cid:16) ǫ ∗ σ ǫ (cid:17) holds if: σ s σ Q > q ( β − + 8 − ( β − β . C ), which increases thenumber of challengers from 1 to 2, increases the probability of the incumbent’s re-election.Proposition 2 and Corollary 1 deliver the main message of the paper: although a lower entrycost induces more competition, i.e., more challengers to enter the race, this does not necessarilyweaken incumbency advantage. In fact, reducing barriers to entry could benefit the incumbentand boost his probability of being re-elected.While our main findings are in the context of cost of marketing as an entry barrier topolitics, parallel arguments can be brought up if new and similarly positioned firms first facea competition among themselves. Such a competition may be for startup funding, regionaldistribution competition, or competition for shelf-space of a retailer before facing a nationallyknown and well-advertised incumbent. Our analysis suggests that, availability of social mediaand cheap digital advertising may facilitate entry of new brands which are differentiated fromthe incumbent, but intensified competition may not be sufficient to weaken the market shareof the incumbent.In the following, we analyze how the perceived quality of the incumbent and other parame-ters about the informativeness of signals in primary and general election affect the incumbencyadvantage. Proposition 3. (Perceived Quality and Informational Advantage of Incumbent) (i)
Fixing the number of challengers, the incumbent is re-elected with a higher probability ifhe has a higher perceived quality or a higher informational advantage, i.e., ǫ ∗ and ǫ ∗ bothincrease in q and λ . (ii) When more competition reduces incumbency advantage, then the incumbent is re-electedwith a higher probability when q and λ increase. In contrast, when more competition strengthensincumbency advantage, the effects of q and λ are non-monotonic. The proposition demonstrates how the quality advantage (i.e., the expected match value q ) and the informational advantage ( λ ) of the incumbent impact the re-election probabilityof the incumbent. Part (i) indicates that a higher expected match value of the incumbent( q ) implies a higher winning probability of an incumbent. As a candidate increases in hisappeal to voters, naturally, he gains more support. A higher informational advantage relative19o the challengers, for instance, the recent bans on political advertisements which limit politicaloutreach to voters, is similarly likely to hurt the challenger’s ability to convince the voters toswitch and vote for him and increase the probability that the incumbent will win the generalelection. The effect of an increase in the two variables, when the number of entrants to the raceis exogenous, is straightforward and is hurtful to the chances of a newcomer. Part (ii) of theproposition demonstrates how, when the number of entrants to the race is also endogenouslydetermined by the changes to these parameters, the incumbency advantage is altered. Since anincrease in either parameter discourages entry of challengers, when more competition reducesthe incumbency advantage, a higher q or λ implies reduced number of challengers entering a race,and an increase in the incumbency advantage for two reasons: first, there are fewer numberof challengers entering the race, and second the challengers who enter are more limited incompeting against the incumbent. If more competition strengthens the incumbency advantage,however, the first effect reverses. A higher q or λ implies fewer number of challengers entering arace and a decrease in the incumbency advantage while still indicating a more limited ability ofthe challengers competing against the incumbent, resulting in an ambiguous outcome in terms ofthe change in incumbency advantage. The key point is that, in both scenarios, when politicaladvertisement bans of online platforms reduce the ability of the challengers to disseminateinformation about their candidacy more than they reduce that of the incumbent, this may alterthe results of elections in a way to favor known, career-politicians. We now generalize the key finding that more competition could strengthen incumbency advan-tage to settings with more than two challengers. We capture the increase in competition inthe primary through its effect on the information structure in this stage. More specifically, weassume that more challengers lead to a more informative signal structure in the primary as theyhave to fight harder for attention, for which we provide a micro-foundation in Appendix B. Inthe appendix, we show without unnecessarily complicating the derivations that, the challengershave more incentives to engage in communication (e.g., more media coverage, debates, discus-sions or activities on social media) in a more competitive primary. Formally, let’s denote thenumber of challenger candidates as e . We generalize the baseline model such that the parame-ters of the signal structures σ s ( e ) and β ( e ) are functions of e . Based on our discussion above,more challengers entering an electoral race induces more communication during the primarystage, but does not affect the communication in the general election. To capture this relation-20hip, we assume that the precision of the signal in the primary stage increases in e , i.e., σ s ( e )decreases in e , while the precision of signals in the general stage remains a constant. That is, β ( e ) σ s ( e ) is invariant in e . In this scenario, we have the following result. Corollary 2. (More Competition in the Primary and Incumbency Advantage)
Sup-pose the competition in primary period intensifies such that the signals obtained in the primaryare more precise relative to the information provision in the general election. Suppose that thenumber of challengers increases from e to e + 1 , the incumbent wins with a higher probability if β ( e ) σ s ( e ) σ Q > q ( β ( e ) − + 8 − ( β ( e ) − . In words, the incumbency advantage is stronger if either σ s ( e ) is sufficiently small or if β ( e ) is sufficiently large. Moreover, if the incumbency advantage is stronger when the number ofchallengers increases from e to e +1 , it is also stronger when the number of challengers increasesfrom e + 1 to e + 2 . The results from the corollary are illustrated in Figure 3. In Figure 3a, when σ s ( e ) σ Q > √ ( β ( e ) − +8 − ( β ( e ) − β ( e ) for all e ≥ e ≥
2. The entry of eachadditional challenger strengthens the incumbent’s chance of re-election. In Figure 3b, when σ s ( e ) σ Q ≤ √ ( β ( e ) − +8 − ( β ( e ) − β ( e ) for e ≥ e , i.e., in this example when e ≥ σ s ( e ) decreases and β ( e ) increases). As aresult, the information from the primary weighs more on the evaluation of voters with left-wingviews. Negative information has a larger adverse effect on the challenger’s chances of beingelected. On the other hand, although a positive signal in the primary protects the challenger21 o. ofchallengers1 2 3 4 5 61Incumbent’s likelihoodof re-election (a) σ e ( e ) σ Q > √ ( β ( e ) − +8 − ( β ( e ) − β ( e ) for e ≥ (b) σ e ( e ) σ Q > √ ( β ( e ) − +8 − ( β ( e ) − β ( e ) for e ≥ Figure 3:
Illustration of how the number of challengers affects the incumbent’s probability of being re-elected. from negative news in the general election, this positive effect is small as the information inthe general election weighs less in voters’ evaluation and thus is unlikely to persuade left-wingvoters to switch. In this case, the negative news in the primary hurts the challengers more thanthe positive news benefit them, and as a result more competition strengthens the incumbencyadvantage.
Incumbency advantage has grown steadily since the 1940ies (Ansolabehere and Snyder, 2000).Candidates who have been elected to an office once hold continuing advantages over theiropponents, which remains an important barrier to making elections more competitive. A keysource of this advantage is the difference between the incumbents and challengers in their abilityto run marketing and communication campaigns. More specifically, the difference in ability toaccess media to inform and persuade voters, either because experienced politicians are morelikely to be covered in media or because newcomers lack funding to buy advertising or otherforms of paid messaging, resulted in persistent re-election success of incumbents.Internet, digital advertising, and social media relaxed this limitation by giving new politi-cians a platform to communicate with masses (Petrova et al., 2020) thereby reducing infor-mational barriers to enter politics. Many political newcomers now communicate with theirconstituency via social media such as Facebook and Twitter to inform and persuade them,which makes electoral races more competitive. But does more competition necessarily help toreverse incumbency advantage? We answer this question considering a specific informational22nvironment: marketing campaigns in political races. We develop a model where incumbencyadvantage can come into play through two channels. First, voters may hold a positive priorabout the match value of the incumbent. Second, the incumbent may hold structural advantagesin reaching out to voters who are ideologically different than their base relative to challengers.We find, first, that lowering cost of communication via digital advertising and social mediareduces the barriers to entering politics and will make races more competitive. But higher num-ber of challengers, and resulting higher levels of marketing and communication campaigns dur-ing the primary, do not necessarily mitigate incumbency advantage, and may in fact strengthenit. The incumbent can benefit from intensified marketing efforts of other candidates during theprimary. Specifically, compared to an election where there are fewer challengers, an electionwith more than two challengers may increase an incumbent’s chances of re-election. This is be-cause of the asymmetric effect of negative versus positive information during the primary thatvoters use to resolve uncertainty about the candidates. Challengers’ communication during theprimary targets the individuals who vote in the primaries, and these are the individuals whosepolitical opinions are aligned with that of the challengers. Those voters ex-ante prefer chal-lengers, but upon receiving negative information in the primary, they might switch to vote forthe incumbent. Thus positive information at the primary stage is unlikely to gain a positionaladvantage for the challengers. Negative news is more likely to dominate the impact of positivenews, especially when communication during the primary is more informative than the generalelection.Second, we find that restricting political advertising and micro-targeting could hurt thechallengers disproportionately more than the incumbent, when incumbents hold strategic ad-vantages in accessing the entire body of voters. This finding implies that recent political adver-tising and micro-targeting bans instituted by online platforms such as Twitter, Facebook, andGoogle may ultimately hurt the chances of challengers in electoral races. Managers of onlinesocial media platforms should be cognizant of the decisions they make regarding advertisingpolicies, as these policies will influence the outcome of elections, despite the intention to protectconsumers.Our findings demonstrate the key benefits and costs of cheaper communications technologyfacilitated by digital advertising and social media platforms. This topic is very timely andimportant, and our study is focusing on the outcome of these changes on political competition.There may be other effects of easily accessing such platforms which are peripheral to ourstudy. Future research can consider other factors. Researchers can also empirically test the23redictions of this study to test how incumbents fare in elections compared to newcomers, astools of communication got cheaper over the years or as political advertising bans have beenerected by online platforms, using a large sample size of politicians.
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Appendix A:
Proofs of Propositions, Lemmas, and CorollariesProof of Lemma 1:
Here we show that ǫ ∗ ≤ − q . As will be shown in Proposition 3, ǫ ∗ decreasesin λ . Moreover, note that when λ = 1, ǫ ∗ = − q satisfies Equation 7. Thus as λ ≥ ǫ ∗ ≤ − q . Proof of Proposition 1:
Inequality 12 can be rewritten as follows: σ Q + βσ s β σ s ( σ Q + βσ s + σ s ) + σ Q + σ s σ s ( σ Q + βσ s + σ s ) > σ Q σ Q + βσ s σ Q σ s ( σ Q + βσ s ) + σ Q β σ s ( σ Q + σ s )( σ Q βσ s + σ Q σ s + βσ s ) > σ Q σ Q + βσ s ( β + 1) σ Q + ( β + β ) σ s ( σ Q ( β + 1) + βσ s ) > σ Q + βσ s ( β + 1) σ Q + ( β + 1) βσ Q σ s + ( β + β ) σ s σ Q + ( β + β ) βσ s > ( σ Q ( β + 1) + βσ s ) β ( β − σ Q σ s + β σ s > βσ Q β σ s σ Q + β ( β − σ s σ Q > σ s σ Q + β − β ! > β − β σ s σ Q > p ( β − + 8 − ( β − β he second part of the proposition follows from: ∂∂β p ( β − + 8 − ( β − β ! = 12 β β p ( β − + 8 − β − ( q ( β − + 8 − ( β − ! = 14 β p ( β − + 8 (cid:18) β − (2( β − + 16 + 2 q ( β − + 8) (cid:19) = 14 β p ( β − + 8 (cid:18) − β + 3 β − − − q ( β − + 8 (cid:19) < β p ( β − + 8 (cid:16) − β + 3 β − − β − (cid:17) = 14 β p ( β − + 8 (cid:16) − β + β − (cid:17) = 14 β p ( β − + 8 ( − β (2 β − − < Proof of Proposition 3:
We first show that ǫ ∗ decreases in q . Note that the left hand side ofEquation (7), denoted as G , decreases in both q and ǫ ∗ because G decreases in both q and ǫ ∗ . Thus, ∂ G ∂ǫ ∗ ∂ǫ ∗ ∂q + ∂ G ∂q = 0 ∂ǫ ∗ ∂q = − ∂ G ∂q (cid:18) ∂ G ∂ǫ ∗ (cid:19) − < . Next, we show that ǫ ∗ increases in λ . To see that, first note that we must have t (1 − − ǫ ∗ − q < ǫ ∗ > − t − q. Otherwise Z G ( t (1 − x n ) − ǫ ∗ − q ) q σ Q + βσ s √ σ Q dx n > Z G ( t (1 − x n ) − ǫ ∗ − q ) σ Q σ Q + λβσ s + σ Q σ Q + βσ s ! − dx n ≥ ∂ G ∂λ <
0. First, if t (1 − x n ) − ǫ ∗ − q ≤ x n ∈ [ , (cid:18) σ Q σ Q + λβσ s + σ Q σ Q + βσ s (cid:19) − increases in λ . Next, if t (1 − x n ) − ǫ ∗ − q > x n ∈ [ , x ∈ [ , ] such that t (1 − x n ) − ǫ ∗ − q > n < ¯ x , then we have Z G ( t (1 − x n ) − ǫ ∗ − q ) σ Q σ Q + λβσ s + σ Q σ Q + βσ s ! − dx n = Z +2(¯ x − ) G ( t (1 − x n ) − ǫ ∗ − q ) σ Q σ Q + λβσ s + σ Q σ Q + βσ s ! − dx n + Z +2(¯ x − ) G ( t (1 − x n ) − ǫ ∗ − q ) σ Q σ Q + λβσ s + σ Q σ Q + βσ s ! − dx n = 12 + Z +2(¯ x − ) G ( t (1 − x n ) − ǫ ∗ − q ) σ Q σ Q + λβσ s + σ Q σ Q + βσ s ! − dx n which is increasing in λ as ( t (1 − x n ) − ǫ ∗ − q < x n ≥ + 2(¯ x − ). Thus, ∂ G ∂λ <
0, and ∂ G ∂ǫ ∗ ∂ǫ ∗ ∂λ + ∂ G ∂λ = 0 ∂ǫ ∗ ∂λ = − ∂ G ∂λ (cid:18) ∂ G ∂ǫ ∗ (cid:19) − < . The results regarding ǫ ∗ follow similar arguments.Now for the overall effect, as shown above, when q or λ increase, both ǫ ∗ and ǫ ∗ decreases.Thus G (cid:16) ǫ ∗ σ ǫ (cid:17) and G (cid:16) ǫ ∗ σ ǫ (cid:17) decreases, and the number of challengers decrease in q and λ . WhenEquation (13) does not hold, as shown in Proposition 1, a decrease in the number of challengersnecessarily imply that incumbent is re-elected with a higher probability. Thus, combined with the factthat G (cid:16) ǫ ∗ σ ǫ (cid:17) and G (cid:16) ǫ ∗ σ ǫ (cid:17) decreases in q and λ , the probability of the incumbent being re-electedalso decreases in q and λ .To show that the overall effect is non-monotonic when Equation (13) holds, consider the examplewhere C = G (cid:16) ǫ ∗ σ ǫ (cid:17) < G (cid:16) ǫ ∗ σ ǫ (cid:17) . In this scenario, two challengers enter the race. Now consider asmall increase in q or λ , it implies that G (cid:16) ǫ ∗ σ ǫ (cid:17) > C > G (cid:16) ǫ ∗ σ ǫ (cid:17) , and now only one challenger entersthe race. By Proposition 1, the incumbent is re-elected with a lower probability when the change in q and λ is small enough. Proof of Corollary 2:
We divide the proof into two parts. First, we prove that ∂ǫ ∗ ∂σ s (cid:12)(cid:12)(cid:12) βσ s ) is positiveif inequality (13) holds. It implies that keeping βσ s fixed, a marginal drop in σ s decreases ǫ ∗ , i.e.,increases incumbency advantage. Second, we prove that fixing βσ s , if ∂ǫ ∗ ∂σ s (cid:12)(cid:12)(cid:12) βσ s < σ s , wealso have ∂ǫ ∗ ∂σ s (cid:12)(cid:12)(cid:12) βσ s < σ s < σ s , which proves the corollary.Denote that the left hand side of equation (10) as G . Note that as ∂ǫ ∗ ∂σ s (cid:12)(cid:12)(cid:12) βσ s = − (cid:16) ∂ G ∂ǫ ∗ (cid:17) − (cid:18) ∂ G ∂σ s (cid:12)(cid:12)(cid:12) βσ s (cid:19) and (cid:16) ∂ G ∂ǫ ∗ (cid:17) − <
0, we prove that the left hand side of equation (10), denoted as G , increases in σ s when βσ s is fixed. First, when βσ s is fixed, the second item of G is unchanged. Second, as quation (10) implies that t (1 − ) − ǫ ∗ − q >
0, fixing βσ s , G increases in σ s if and only if ∂∂σ s σ Q σ Q + βσ s + σ Q + βσ s β σ s ( σ Q + βσ s + σ s ) + σ Q + σ s σ s ( σ Q + βσ s + σ s ) (cid:12)(cid:12)(cid:12) βσ s < . With simple algebra, ∂∂σ s σ Q σ Q + βσ s + σ Q + βσ s β σ s ( σ Q + βσ s + σ s ) + σ Q + σ s σ s ( σ Q + βσ s + σ s ) (cid:12)(cid:12)(cid:12) βσ s =2 σ Q + βσ s β σ s ( σ Q + βσ s + σ s ) ( 1 σ s ) + − σ Q σ s − σ s ( σ Q + βσ s + σ s ) + 2 σ Q + σ s σ s ( σ Q + βσ s + σ s ) ( 1 σ s )which is smaller than 0 if and only if2 σ Q + βσ s β σ s + 2 σ Q + σ s σ s < ( 2 σ Q σ s + 1)( 1 σ Q + 1 βσ s + 1 σ s )2( σ Q + βσ s ) + 2 β ( σ Q + σ s ) < (2 β σ Q σ s + β σ s )( 1 σ Q + 1 βσ s + 1 σ s )2( σ Q + βσ s ) + 2 β ( σ Q + σ s ) < β σ s + 2 βσ Q + 2 β σ Q + β σ s σ Q + βσ s + β σ s β (2 β + 2 β ) σ s σ Q < β σ s σ Q ! + (3 β + β ) σ s σ Q + 2 β + 2 β β σ s σ Q ! + β ( β − σ s σ Q > − β ) (14)which holds if inequality (13) holds because 2(1 − β ) <
2. We now prove that fixing βσ s , if ∂ǫ ∗ ∂σ s (cid:12)(cid:12)(cid:12) βσ s < σ s , we also have ∂ǫ ∗ ∂σ s (cid:12)(cid:12)(cid:12) βσ s < σ s < σ s . As βσ s is fixed, a smaller σ s implies a bigger β . First note that the inequality (14) holds when β ≥
1. Next, when β <
1, inequality (14) can berewritten as: β σ s σ Q ! + β ( β − σ s σ Q > − β ) βσ s σ Q + β − ! > − β ) βσ s σ Q > q − β ) − β − β . The result thus follows. PPENDIX B:
Micro-foundation for Challengers and Information Provision
In this section, we show a simple model to illustrate a channel that induces more information whenmore challengers enter the race.Suppose there are N candidates in the primary, and each decides how many signals to send tothe voters. Assume for simplicity that they get utility 1 if they win the primary, and that theirwinning probability increases in the number of signals they send compared to their competitors. Thisassumption is motivated by the fact that more signals make a candidate more visible and thus winwith a higher probability. More specifically, denoting the number of signals sent by candidate i with q i ,we assume that the winning probability of a candidate follows the Tullock contest function (Tullock,1980), which is equals to: q ri P Nj =1 q rj where r ≤
1. We also assume that the cost of the number of signals follows a quadratic form A q i forsome constant A . The Tullock contest function ensures that the winning probability is increasing andconcave in q i , such that the best response is characterized by the first order condition.In the following, we derive the equilibrium number of signals, denoted as q ∗ . In a symmetricequilibrium, given all candidates other than i send q ∗ signals, the best response of candidate i equalsto q ∗ and follows the following first order condition: (cid:16)P j = i ( q ∗ ) r (cid:17) r ( q ∗ ) r − (cid:16)P Nj =1 ( q ∗ ) r (cid:17) = Aq ∗ r ( N − q ∗ ) r − N ( q ∗ ) r = Aq ∗ ( q ∗ ) = r ( N − AN which clearly shows that q ∗ increases in N . The intuition of this result is reminiscent to the literatureof advertising. In particular, as there are more candidates, the aggregate signals in the whole marketincreases, and thus it is more difficult for a particular candidate to stand out, and in equilibrium eachcandidate invests more on information provision.. The intuition of this result is reminiscent to the literatureof advertising. In particular, as there are more candidates, the aggregate signals in the whole marketincreases, and thus it is more difficult for a particular candidate to stand out, and in equilibrium eachcandidate invests more on information provision.