Why Primaries? The Strategic Choice of a
Candidate Selection Method∗
Gilles Serra†
May 5, 2008
Abstract
We elaborate a theory to explain why and when political parties
choose to hold primary elections. Party leaders face a trade-off between primary elections and elite-centered selections. The benefit of a
primary is revealing the campaigning skills of candidates. Its cost is
the ideological extremism that primary voters might induce on candidates. We find that primary elections are more likely when the party
leadership and the potential primary voters are ideologically similar
(which is consistent with the recent empirical research by Meinke, Staton and Wuhs (2006)). Intriguingly, our model predicts that parties
with extremist ideologies are more likely to be internally democratic.
∗
This paper benefitted from being presented at Nuffield College in October 2007, the
annual meetings of the American Political Science Association in September 2007 and the
Midwest Political Science Associatin in April 2008, and Harvard University in April 2007.
I thank James Adams, Christopher Avery, Robert Bates, Indridi Indridason, John Patty,
Brian Richardson, Kenneth Shepsle and James Snyder for their insightful comments. I also
thank the Institute of Quantitative Social Sciences for its financial support. All remaining
errors are my own.
†
Research Fellow, Nuffield College at Oxford University, gilles.serra@nuffield.ox.ac.uk
.
1
[T]he major political party is the creature of the politicians,
the ambitious office seeker and officeholder. They have created
and maintained, used or abused, reformed or ignored the political
party when doing so has furthered their goals and ambitions. The
political party is thus an "endogenous" institution.
John H. Aldrich (1995, p. 4)
1
Introduction
An institution whose origin needs better understanding is the primary election. Indeed, political parties can set different rules for the nomination of
the person who will run as their candidate in a given election. Broadly
speaking, a candidate-selection method can fall in two categories: open (or
democratic), where the party’s rank-and-file members and sympathizers take
a vote to nominate their candidate; and closed (or undemocratic), consisting
of a closed-door decision at the elite level of the party.1
This paper studies the endogenous adoption by political parties of one
type of candidate-selection method (CSM) over another. Of all the selection
methods that parties can use, the most salient one is the primary election,
not least because it has been increasingly frequent around the world.2 Indeed, primary elections are gaining popularity among reformers and party
leaders, as testified by recurrent journalistic editorials and public speeches.
As a result, the frequency of open CSMs is on the rise. They are already
prevalent in the U.S., where they have seen a startling increase in recent
decades. In 1968, only sixteen states and the District of Columbia were
holding primaries; whereas by 1996 the Democratic party was holding primaries in thirty-six states and the Republican party in forty-three (Morton
(2006)). Primary elections are seeing a surge in Latin America as well: Carey
and Polga-Hecimovich (2006) report that the percentage of candidates nominated by primaries increased from 3% in the 1980s to 4% in the 1990s to
12% in the 2000s, thus increasing fourfold in two decades. The number of
democratic candidate selections has also increased in Europe since 1960 with
the introduction of membership ballots (Bille (2001)); and parties in other
1
There is of course some variety within these two extremes: for example, CSMs can
further be classified as semi-open or semi-closed (as described in Kanthak and Morton
(2001)). But, as our results will show, a division in two broad categories is sufficient for
the points this paper wishes to make.
2
By primary election we refer to any organized competition among aspiring candidates
within the same party that culminates in the democratic vote of all party members.
2
countries like Taiwan have also experimented with primaries in the recent
past (see Scarrow (2005) for a survey).
Party leaders themselves are for the most part responsible for this trend
whereby parties are voluntarily adopting primary elections. In most party
systems around the world, political parties have leeway in choosing their
CSM, and it is therefore not the case that primaries are exogenously imposed
on parties. As a matter of fact, there is abundant evidence that political
parties have serious deliberation on what CSM to adopt before they discuss
which candidate to select. Recently, the UMP in France debated for several
months whether to adopt its first primary in 2007; so did a coalition of leftwing parties in Italy in 2005. The PAN in Mexico held a heated convention
before deciding to adopt an American style primary election in 2006.3
Even in the United States, where the usual claim is that primaries are
exogenously imposed on parties by state law, parties have actually had a
large degree of freedom in designing their CSM. Meinke, Staton and Wuhs
(2006) document that the official CSM is not irrevocably fixed, but rather
parties can re-interpret or change the law to make their CSM more or less
open, which led those authors to characterize the American CSMs as rather
fluid. They claim that "state parties can and do change their rules repeatedly over time, often moving from a more open to a more closed rule (or
vice versa) and then back again" (Meinke et al. (2006), p. 183). And even
without changing any rule or charter, even when primary elections are irrevocably imposed on parties by state law, party leaders still have means to
impose their favorite candidate for example by endorsing her and providing
her with enough resources to prevent any challenge in the primary. It is
no coincidence that most primary elections in America have been uncontested in the last hundred years (Ansolabehere, Hansen, Hirano and Snyder
(2005)). So at the very least, the Democratic and the Republican leaders
can influence whether a primary is competitive or non-competitive.
A more explicit illustration of a party deciding to open its candidate
selection is the McGovern-Fraser reforms in the 1970s. In those years, the
Democratic Party had a period of introspection and laborious internal discussions which culminated in its adoption of primary elections for most
offices in most states (Abramowitz and Stone (1984), Bartels (1988), Geer
(1989), Aldrich (1980b)).
Further back in time, it can be argued that the introduction of the
American direct primary in the early twentieth century was also the result
3
In all three cases, the winner of the primary election went on to win the general
election.
3
of party leaders’ strategies. The dominant view among political scientists is
that primary elections arouse out of the struggle for openness of active party
outsiders: the Progressive Era reformers (see for example Ranney (1975)).
We do not deny that such an exogenous pressure to open the candidate
selection was preeminent. But another force, more in line with this paper’s
argument, might have been at play as well: the self-interest of party leaders.
Indeed Alan Ware (2002) claims that, at least in some states, it was the
old party bosses, at the county level first and then in state legislatures and
in governors’ mansions, who were the main actors bringing about primary
elections in their own parties. To underscore the party leaders’ interest in
adopting primaries he documents that "Especially in much of the eastern
United States, they could have prevented the introduction of the direct
primary —had they wanted to do so. But, for the most part, they did not."
(Ware (2002, p. 15)).
Thus, any theory of the adoption of competitive primary elections must
put party leaders at the center of the decision process. This presents a
paradox however: why would party bosses decentralize what could be a
closed process where they could have their pick of candidates? Why does
the establishment delegate the nomination to the party’s foot soldiers which
might have different preferences?
Our answer begins by claiming that primary elections have an advantage
over elite anointments: they reveal information about the candidates’ appeal
to voters. Our theoretical model describes this information revelation feature
in detail, and studies the conditions under which such a feature is attractive
enough for party leaders to adopt a primary election.
Party leaders can see a drawback in primary elections, however: primary
voters might push candidates to adopt policies far from the leaders’ preferences.4 Indeed, several empirical studies have found that primary voters
have extreme ideological views (Key (1956), Lengle (1981), Polsby (1983)),
and that primary elections make candidates adopt extremist platforms (King
(1999), Burden (2001, 2004)). It should be noted that several other empirical studies have found the opposite results: that primary voters do not have
extremist ideological views (Ranney (1968), Geer (1988), Kaufmann, Gim4
Party leaders could see additional drawbacks that this paper does not focus on. Ezra
(2001) argues that primaries might harm candidates in the general election by draining
their energy and resources. Ansolabehere, Hirano and Snyder (2004) argue that the loyalty
of legislators in the U.S. Congress toward their parties decreased following the adoption
of the American direct primary. Serra (2007b) argues that party leaders in Latin America
may lose the patronage and cronyism that result from directly handpicking a candidate
from their inner circle.
4
pel and Hoffman (2003)), and that primary elections do not make candidates
adopt extremist platforms (Gerber and Morton (1998), Ansolabehere, Snyder and Stewart (2001)). Given that the empirical literature is mixed, in this
paper we do not take a position on whether primary voters are extremist or
moderate, but rather we study both situations by treating the preferences
of primary voters as a continuous parameter that can be located anywhere
in the political spectrum. We can thus make comparative statics on that
parameter.
The main point is that party leaders face a trade-off between the costs
and benefits of a primary election. The results in this paper reveal that the
party leaders’ decision is not trivial and relies heavily on the decision made
by the leaders of the other party. The emergence of primary elections must
therefore be understood as a game-theoretic rather than a decision-theoretic
phenomenon.
The paper adds to the formal literature on endogenous institutions.
There has been virtually no formal theory studying the origins of primary
elections. But there is a growing literature studying their consequences.5
Some of that literature compares primaries to other CSMs (Chen and Yang
(2002), Jackson, Mathevet and Mattes (2007), Kang (2007)). But to the
best of our knowledge, our work is the first to model the decision of which
CSM to adopt.
As we elaborate later, this paper also draws on the literature on valence
(Stokes (1963)). This model is closest to the theoretical models that postulate a valence parameter, i.e. a parameter that adds utility on top of policy
to all voters.6
There is another paper that combines both literatures, the one on valence
and the one on primaries: Adams and Merrill (2006), which focuses on the
policies adopted by candidates and whether they diverge from the median
voter. The authors believe, like we do, that primaries tend to select the
candidates with highest valence; and they derive some important results,
most notably that "small" or "weak" parties benefit the most from having
primaries. But their paper has several modelling differences with ours, which
we will point out in footnotes subsequently.
The rest of our paper is developed as follows: Section 2 describes the
5
For example see Aranson and Ordeshook (1972), Coleman (1972), Aldrich (1980a),
Owen and Grofman (2004), Morton and Williams (2001), Buchler (2005), Meirowitz
(2005), Adams and Merrill (2006), Owen and Grofman (2006), Snyder and Ting (2006).
6
Such a parameter is found in Enelow and Hinich (1982), Cukierman (1991), Londregan
and Romer (1993), Adams (1999), Ansolabehere and Snyder (2000), Groseclose (2001),
Schofield (2004).
5
information revelation mechanism by which primaries can reveal candidates’
appeal to voters. Sections 3 sets up the model of electoral competition.
Section 4 solves the last stages of the election, after parties have nominated
their candidates. Sections 5 and 6 solve the first stages of the election, that
is the nomination of candidates, under two different assumptions about the
technology available to parties: in Section 5 only one of the parties has
the technology to adopt a primary, and in Section 6 both parties have the
technology to adopt a primary. And Section 7 offers some conclusions. The
Appendix of this paper (available as a separate document) is divided in two
sections. The first section develops the information-revelation mechanism
in detail, and the second section contains the proofs of all the results in the
paper.
2
The benefits of a primary election
One of the concerns of a political party is choosing among the individuals seeking to become its candidate, often called pre-candidates, the most
electable one. The premise in this paper is that primaries help a party nominate the candidate with best campaigning skills.7 Indeed primaries have
two attractive features. First, they reveal some information about the personal characteristics of candidates that would be useful (or detrimental) in
the general election. In effect, a primary election is a full blown election
7
The premise that primaries may be strategically adopted by party leaders to nominate
electable candidates has empirical support. In a survey of party democratization around
the world, Scarrow (2005) argues that there are self-interested reasons for parties to adopt
primaries, namely that such procedures "may help parties win elections, recruit and select good candidates, and retain popular support". Several students of Mexican politics
have noted this phenomenon as well: in field interviews, Poiré (2002) found that all the
relevant leaders deciding the PRI’s CSM shared the view that "the winner of a primary
has a better chance of winning the election than whoever turns out to be a loser"; according to Wuhs (2006), all major parties make the calculation that "more open selection
processes should result in the nomination of more ’electable’ candidates who can perform better in the general election. Across Latin America, Carey and Polga-Hecimovich
(2006) suggest that "primaries may be more effective than elite-driven search processes
in identifying candidates with broad popular appeal". They find that primary-selected
candidates fare better in the general election than those selected by other procedures by
at leat six percent points. In studying presidential nominations in the Unites States, Geer
(1989) concludes that "in principle, primaries may provide more reliable estimates of a
candidate’s ability to win votes than can party leaders". Even Nelson Polsby, generally a
critic of the American primary system, observed that presidential primaries can serve "as
means by which politicians inform themselves about the relative popularity of presidential
aspirants" (Polsby (1960 p. 617)).
6
within a party which shares many of the features of the subsequent general
election between the parties: for example the pre-candidates need to run a
campaign, debate other candidates, run advertisements on television, and
be scrutinized by journalists. Primaries are therefore a good testing ground
for the performance that aspiring candidates would have during the general
election.
A second feature of primaries is opening the door to unknown or marginalized party members who might potentially have a large appeal to voters
but would not be identified through an inside-track nomination. Illustrations
of such surprise candidates abound. John F. Kennedy in the U.S., Carlos
Menem in Argentina and Ségolène Royal in France are examples of nonmainstream politicians who were able to prove their appeal to voters thanks
to a primary. All three obtained their party’ nomination over some powerful insiders who were also competing in the primary, and went on to have a
strong showing in the general election. Barack Obama is the latest example
of an outsider who was able to prove his formidable eloquence in the primary
election: it is easy to speculate that Obama would not have been "discovered" if the Democratic Party still chose its candidates in smoke-filled rooms
instead of primary elections.
To be precise, let us assume that a given party, which we will call party
B, is considering whether to hold a primary election or not. All the possible candidates are characterized by a parameter v denoting how appealing
their non-policy attributes would be to voters in that election. Parameters
such as v have been called "valence parameters" and can be given many
interpretations like charisma, honesty, competence and the ability to raise
campaign funds (for an overview see Ansolabehere and Snyder (2000) and
Groseclose (2001)). In the context of this paper, v is best interpreted as the
candidate’s campaigning skill. The parameter v can take a maximum value
of 1, and we say that a candidate with skill equal to 1 is high-skilled. At the
time that B needs to nominate a candidate, however, it is uncertain whether
B’s pre-candidates are high-skilled or not. We call vB the campaigning skill
of the candidate ultimately nominated by B.
Before selecting a candidate, the authorities of party B need to select
a candidate-selection method (CSM). There exist two CSMs: The default
CSM would be for those authorities to directly nominate an insider candidate
in a closed negotiation at the elite level (such as a back-room deal among
delegates in a national convention). Alternatively they could implement
a primary election where an outsider candidate has a chance to run, and
the decision to choose the nominee is delegated to the party’s rank and file
7
(RAF). So B must choose its candidate-selection method mB with mB ∈
{primary, leadership} , where primary refers to the adoption of a primary
election and leadership refers to a selection by the party leaders. We define
π B as the probability that the nominee will be high-skilled if B uses a
leadership selection. In other words, π B ≡ P (vB = 1|leadership) .
Our claim is that a primary election increases B’s probability of nominating a high-skilled candidate above π B in the amount SB . We define
SB ≡ P (vB = 1|primary) − P (vB = 1|leadership) , and we call SB the skill
bonus of a primary. Another way of expressing this definition is
P (vB = 1|primary) = π B + SB
We devote a section in the appendix to develop an information-revelation
model from which SB is derived. In formal-theoretic parlance, we derive
SB from "microfoundations". The driving force behind the primary skill
bonus is the information revealed during the primary campaigns about the
campaigning skills of the pre-candidates: Primary voters interpret the performance of a candidate in the primary election as a forecast of how well she
would perform in the general election.
Those forecasts are imperfect however, meaning that the true skills of
candidates are not fully revealed during the primary, but they are only partially revealed.8 To be concrete we assume that a candidate’s performance
in the primary has a probability q of "being correct",
¢ is, of accurately
¡ that
reflecting the true skill of that candidate, with q ∈ 12 , 1 . In other words,
q measures the quality of the information revealed during the primary. As
we describe in the appendix, primary voters take the performances they observe in the primary to update their prior beliefs about the candidates’ skills
(using Bayes rule).
Deriving SB in such a way leads to some intuitive properties that will
buttress the rest of the paper. The main properties of the primary skill
bonus, which are proved in the appendix, are stated below. In all cases, π
refers to a constant larger that one-half that depends on the parameters of
the model.
Property 1 The primary skill bonus SB is strictly positive for π B ∈ (0, π)
and zero for π B ∈ [π, 1).
8
Adams and Merrill (2006) also postulate that primaries reveal information about the
pre-candidates’ true valence, but in their model information is fully revealed in the primary
election, and there is no additional information in the general election. In contrast, in
our model the information is only partially revealed in the primary, and there is some
additional information in the general election.
8
This result is a formal statement of the claim that primaries bring an
electoral advantage to the parties that use them, at least when the leadership
selection is not expected to be particularly effective in delivering a highskilled candidate.
Property 2 The primary skill bonus SB is strictly decreasing with π B for
π B ∈ (0, π) .
It makes intuitive sense that SB would decrease with π B , because the
whole advantage of primaries is to improve upon the skill of the candidate
that would be nominated through a leadership selection, i.e. the insider
candidate. As the skill of the insider candidate is expected to be higher, it
becomes less likely that a primary will improve upon it. The message of this
result is that the electoral advantage brought by primaries is larger the less
appealing the insider candidate is to begin with.
Property 3 For any given quality of information q there exists a q 0 > q
such that when q increases to q 0 , SB will increase.
An increase in q can be interpreted as an improvement in the informationrevelation feature of primaries. To illustrate this point we can think of the
presidential primary in the United States. Such an improvement could arise
because the primary campaigns became longer, or the media paid more attention, or the number of debates on television increased, etc. It is thus
ironic that many members of the Democratic Party have complained that
the current race between Hillary Clinton and Barack Obama has been too
long and too testing. According to the result above, the more competitive
and challenging a primary, the higher the probability that the most skilled
candidate will be identified. Every extra debate, every additional town-hall
meeting, every new revelation, has a positive effect on SB .
In sum, the benefit to party leaders of adopting a primary election is a
larger probability of nominating a candidate with a high campaigning skill
—we called that extra probability the primary skill bonus. Primaries might
carry a cost however, in terms of the policy that candidates are induced
to adopt. That cost is described in detail in the following section. As a
consequence, party leaders need to carry out a cost-benefit analysis when
choosing whether to hold primary election or not.
9
3
Set-up of the electoral competition
In this section we lay out a model of competition between two parties without
any reference to primary elections.9 In essence, it should be thought of as
the "general election" that occurs after all parties have already resolved their
nomination cycle.
The election occurs in a unidimensional space where x is the policy
implemented, thus x ∈ R.
3.1
Parties
There are two parties competing in this election, labeled party A and party
B. Parties are policy-motivated and have ideal policy points XA and XB
respectively. A has a distinct ideology from B so that XA 6= XB . We
normalize the ideal point of the median voter in the general election to zero,
and without much loss of generality we assume XA < 0 < XB . The utility
functions of A and B are
UA (x) = − |XA − x|
UB (x) = − |XB − x|
(1)
(2)
Parties formulate policy platforms to compete in the election, and they
do so strategically in order to maximize their expected utility. We call those
platforms xA and xB respectively, with xA , xB ∈ R.
3.2
Candidates
Each candidates is characterized by a parameter v denoting the candidate’s
campaigning skill. A candidate’s campaigning skill can take two values: a
low value normalized to zero corresponding to a low-skilled candidate, and
a high value normalized to one corresponding to a high-skilled candidate.
So v ∈ {0, 1} .10
9
This section replicates, with a few modifications, the valence-policy model developed
in Serra (2007c). Therein can be found a more detailed exposition of the set-up.
10
Assuming that campaigning skills can take only two values is mostly done for presentation purposes. Assuming instead that the campaigning skill could take a continuum
of values would unnecessarily complicate the Bayesian updating that we study in the appendix, and would not lead to different conclusions than the ones we obtain. In related
research we did study the case where skill can take a continuum of values, and found very
similar results (Serra 2007c).
10
Each party has only one candidate to choose from: an insider.11 We call
AI and BI the insider pre-candidates of A and B respectively, and we call
vAI and vBI their campaigning skills.
The exact values of the candidates’ campaigning skills are uncertain before the election. However the parties have some prior information about the
skill of their insider candidate. That information could come from previous
performance in office, from past elections, or from polls. According to that
information, the probabilities that AI and BI are high-skilled are π AI and
π BI respectively, with π AI , π BI ∈ (0, 1) . All this information is common
knowledge.
The true campaigning skills of A and B’s candidates are revealed when
they have to campaign to win the general election, and thus vAI and vBI
are fully known by the time the general electorate votes between A and B.
Candidates adopt the policy platform that their party has designed, and
will implement such policy if they win. In other words, a candidate will
behave as a perfect agent of those who nominated her. The interpretation is
that in striving to win the nomination, the pre-candidates are forced to cater
to the wishes of those selecting them, and accept any policy commitment.
3.3
The general electorate
The electorate is policy motivated. To simplify the analysis we will assume
that there is a "median voter", which we call M, whose preferences are
decisive in the election. We normalize her ideal point to zero.
In addition to the policy implemented x, the electorate also cares about
the skill v of the winning candidate. Voters place a weight γ on skill relative
to policy. The utility function of M is given by
UM (x, v) = − |x| + γv
M will vote for the party whose candidate maximizes her utility. If
the candidates of both parties give her the same utility, we will make the
following indifference assumption:12
A1: If M is indifferent between the two parties, she will vote for
the one whose candidate has the highest skill. If both candidates
11
We can think of that "only" insider as the most skilled among all the available insiders.
The model is robust to changes in this assumption, which we only make to ensure the
existence of equilibria between parties. If we changed it, an equilibrium might not exist
but the behavior of parties would still converge ever so closely to the equilibria described
in the text.
12
11
have the same skill, she will randomize equally between the two.
3.4
Timing and solution concept
The timing of this election is the following:
1. The general-election campaigns: (no decision)
Candidates’ campaigning skills vAI and vBI are fully revealed.
2. The platform design:
Both parties design their platforms xA and xB simultaneously.
Candidates adopt those platforms.
3. The general election vote:
The median voter elects A or B.
The game must be solved by backward induction and the solution concept is subgame-perfect equilibrium (SPE) in pure strategies.
4
The general election
Stage 1 does not involve any decision; the skills of both candidates are fully
revealed to everyone. The first decision is made in Stage 2, when each
party must design its platform in the unidimensional political spectrum.
The candidates then adopt and announce those platforms. In Stage 3, once
candidates have announced their platforms xA and xB , the median voter
elects A or B to office.
Before stating the main results of this section we must define some important variables. We call δ the difference in skill between B’s candidate and
A’s candidate, adjusted by the weight voters’ place on that skill. To be concrete, δ ≡ γvBI −γvAI . Note that δ can take three values: δ ∈ {−γ, 0, γ} .We
call x∗A and x∗B the equilibrium strategies of parties A and B, and x∗ the
winning platform.
Theorem 1 The equilibrium strategies and equilibrium outcomes of this
election for given values of vAI and vBI are given in the following table,
where δ ≡ γvBI − γvAI
12
Value of δ
Nash equilibrium/
equilibria
x∗A ∈ R
x∗B = XB
Winning platform
x∗
XB
Winning candidate
0 < δ < XB
x∗A = 0
x∗B = δ
δ
B
δ=0
x∗A = 0
x∗B = 0
0
A or B with
equal probability
XA < δ < 0
x∗A = δ
x∗B = 0
δ
A
XB ≤ δ
B
x∗A = XA
XA
A
x∗B ∈ R
There are several comments to make about this table. First note the
results when δ = 0, that is, when there is no skill difference between the
candidates: both parties converge completely to the median voter’s ideal
point. As soon as δ 6= 0 however, the candidate with highest skill is able to
diverge from the median voter toward the ideal point of her party, and still
win the election based on her superior skill. So the policy implemented is
biased toward B when δ > 0, biased toward A when δ < 0, and unbiased
when δ = 0. And that bias is in proportion to δ: the larger the advantage in
skill of one party over the other, the more the advantaged party is able to
move policy toward its ideal point. Actually, if δ is large enough the party
with the highest skilled candidate is able to pull policy exactly to its ideal
point.
Given that the size of δ is proportional to γ, the last two statements
can be reinterpreted in an interesting way: the larger the weight that voters
place on skill, the more the advantaged party is able to move policy toward
its ideal point; and if that weight is large enough, the party with the highest
skilled candidate is able to pull policy exactly to its ideal point.
δ ≤ XA
5
The nomination
Now imagine that one of the two parties is pondering whether to hold a
primary election or not. We must assume that this party has the necessary
13
infrastructure to hold a primary, which is not a given.13
As we suggested in the introduction, it is increasingly common for political parties around the world to face that choice. And yet, such a decision
is exceptionally understudied, this paper being the first one that we are
aware of analyzing it formally. We do not study, however, the case where
both parties are simultaneously deciding whether to hold primaries: our
analysis is "decision-theoretic", meaning that only one party has the ability
to choose a primary election, rather than "game-theoretic", meaning that
both parties have the ability to make that decision. We believe that such
a decision-theoretic set-up is a good starting point for the analysis, in addition to being a realistic scenario.14 The game-theoretic scenario remains
important to study however, and we are doing so extensively in a related
research (Serra (2007a)). Given the space limitations, we cannot show all
that analysis here, but we summarize some of the most relevant results in
the discussion.
5.1
Set up
Party B consists of a leadership (or "establishment") and a rank and file
(or "membership"). The leaders of B will be referred to as BL. They are
policy-motivated and have an ideal policy point XBL , with XBL > 0. The
utility function of B’s leadership is
UBL (x) = − |XBL − x|
The rank and file (RAF) of B is also policy-motivated. To simplify
the analysis we will assume that the RAF has a "median member" whose
preferences are decisive in the primary election. We call BR the median
member of B’s RAF, with XBR > 0. The utility function of BR is
UBR (x) = − |XBR − x|
In general we will have XBL 6= XBR so there is a tension between the
policy preferences of a party’s leadership and its RAF. It will be useful to
13
Indeed running a primary requires material resources like booths and ballots; an
effective organization including well trained election officers; a developed party structure
which includes an accurate membership roster; and a credible and authoritative arbiter
to ensure an orderly process. So not every party might have the ability to run a primary.
14
Indeed, the case where some parties have the ability to adopt a primary election while
other parties cannot is common, especially in countries where old institutionalized parties
coexist with new undeveloped parties, as is the case in much of Africa and Latin America.
14
have a measure of the ideological distance between a party’s establishment
and its primary voters, so we define λB ≡ XBR − XBL as the "ideological
distance in party B".
The leadership of the party needs to choose a candidate-selection method.
There are two CSMs to choose from: a leadership selection or a primary election. We call mB the CSM that B adopts, with mB ∈ {primary, leadership} .
Following standard language in the party-politics literature, we will call selectorate the group in charge of selecting a party’s candidate. If the CSM is
a leadership selection, the selectorate is the party’s leadership. If the CSM
is a primary election, the selectorate is the party’s RAF.
Parties are also responsible for formulating policy platforms to compete
in the election. If party B uses a leadership selection, then its leaders formulate the policy platform. If, instead, party B uses a primary election, then
its RAF formulates the policy platform. Note that both the leadership and
the RAF would behave strategically, i.e. they would not passively adopt
their ideal points without rational calculation.
If the CSM is an elite selection, the party has only one candidate to
choose from: an insider. In contrast, if the CSM is a primary election, the
party has two candidates (or rather, "pre-candidates") to choose from: an
insider and an outsider.15 We call BI and BO the insider and outsider precandidates of B, and we call vBI and vBO their campaigning skills. We call
vB the skill of the candidate that is finally nominated by B.
The exact values of the candidates’ campaigning skills are uncertain
before the election. As we mentioned previously, the probability that BI is
high-skilled is π BI . On the other hand, there is no prior information about
the outsider candidate, so she is believed to have a probability of one-half
of being high-skilled.16 All this information is common knowledge.
As we mentioned, the true campaigning skills of A and B’s candidates
are fully revealed when they have to campaign to win the general election.
In addition, if B adopts a primary election the campaigning skills of its
precandidates are partially revealed, as described in Section 2.
Candidates adopt the policy platform that their selectorate has designed
for them. In other words they behave as a perfect agents of whichever group
inside their party nominated them, thus acquiring the preferences of either
15
The model could easily be extended to having more than one outsider. The main
effect would be a larger primary skill bonus.
16
Alternatively, we could assume that there is some prior information about the pool of
outsider candidates. The results would all hold, with only notational adjustments, if we
assumed that BO had a probability πBO 6= 12 of being high-skilled.
15
BL or BR.
The timing of this election is the following:
1. The selection of the candidate-selection method:
The leaders of party B choose its CSM.
2. The primary campaign in party B: (no decision)
If the CSM in party B is a primary election, the skills of the precandidates in that party are partially revealed.
If the CSM in party B is a leadership selection, no information is
revealed.
3. The nomination in party B:
Party B selects its candidate, BI or BO.
4. The general-election campaigns: (no decision)
Candidates’ campaigning skills vAI and vBI are fully revealed.
5. The platform design:
Both parties design their platforms xA and xB simultaneously.
Candidates adopt those platforms.
6. The general election vote:
The median voter elects A or B.
The game must be solved by backward induction and the solution concept is subgame-perfect equilibrium (SPE) in pure strategies.
5.2
Payoffs
The case where B adopts a leadership selection would lead to a situation
in which both parties nominate their insider candidates. Thus the probabilities that A’s candidate and B’s candidate are high-skilled are given by
π AI and π BI , respectively. A primary election has two differences: first, the
probability that B’s nominee is high-skilled increases from π BI to π BI + SB ,
where SB is the skill bonus of adopting a primary in B. As elaborated in Section 2, that skill bonus comes from the fact that an outsider candidate joins
the primary race, and that the primary campaigns reveal information about
pre-candidates’ skills. And second, it would be the RAF’s preferences, and
16
not the leadership’s preferences, that would determine B’s policy platform;
and thus it would be XBR and not XBL that B’s candidate would adopt if
she had skill advantage over A’s candidate.
That trade-off (a higher skilled candidate but a worse policy platform)
will be clearly reflected in the expressions we find for the expected utility
of B’s leadership from adopting a primary election. Those expressions will
crucially depend, it turns out, on the value of γ, the weight that voters
place on skill. We must divide the possible values of γ in three intervals
corresponding to a "large" weight where XBL , XBR , −XA < γ, a "small"
weight where γ < XBL , XBR , −XA , and an intermediate weight, where γ
takes any value in between the other two intervals.
These remarks are summarized in the following results, where we call
EUBL (mB ) the expected utility of B’s leadership from adopting mB as its
CSM. We start with the results for a large γ.
Lemma 1 If XBL , XBR , −XA < γ, the expected utility of B’s leadership for
each value of mB is
EUBL (leadership) = −XBL [π AI π BI + (1 − π AI ) (1 − π BI )]
− (XBL − XAL ) π AI (1 − π BI )
(3)
EUBL (primary) = − |XBL − XBR | (1 − π AI ) (π BI + SB )
−XBL [π AI (π BI + SB ) + (1 − π AI ) (1 − (π BI + SB ))]
− (XBL − XAL ) π AI (1 − (π BI + SB ))
(4)
We now turn to the results for small γ.
Lemma 2 If γ < XBL , XBR , −XA , the expected utility of B’s leadership for
each value of mB is
EUBL (leadership) = − (XBL − γ) (1 − π AI ) π BI
−XBL [π AI π BI + (1 − π AI ) (1 − π BI )]
− (XBL + γ) π AI (1 − π BI )
EUBL (primary) = − (XBL − γ) (1 − π AI ) (π BI + SB )
−XBL [π AI (π BI + SB ) + (1 − π AI ) (1 − (π BI + SB ))]
− (XBL + γ) π AI (1 − (π BI + SB ))
17
The results for an intermediate γ are easy to compute but too long to
report in full.17 They also offer little surprise, since they all fall in between
the results reported in the previous two lemmas. So we just summarize
the most important features: First, EUBL (leadership) is increasing (or
rather, non-decreasing) with γ. The intuition is that the larger γ is, the
more BL is able move policy toward its ideal point when it has the highest
skilled candidate. In contrast, EUBL (primary) is decreasing (or rather,
non-increasing) with γ. The intuition is that the larger γ is, the more BR is
able move policy away from BL’s ideal point when it has the highest skilled
candidate.
Armed with these results, the leadership in party B can measure the
consequences of choosing one CSM over the other.
5.3
The optimal selection of a CSM
The leadership in party B will choose the optimal rule mB by comparing
EUBL (leadership) and EUBL (primary) and choosing the CSM that yields
the highest expected utility. In case both expressions are equal we will make
the following indifference assumption for convenience:18
A2: If a party’s leadership is indifferent between a primary election and a leadership selection, it will choose a primary election.
Once again we need to consider three different intervals for γ, large,
small and intermediate. We start with the analysis of a large γ, which is
the most illuminating case. Indeed, it is when voters care deeply about the
skill of candidates that searching for the highest skilled candidate becomes
the most crucial.
By glancing at equations 3 and 4 we can readily see the trade-off that B’s
leadership faces in choosing a primary election over a leadership selection:
the benefit of a primary is that the probability of nominating a low-skilled
candidate decreases (due to the primary skill bonus SB ). The cost is that the
payoff from having a skill advantage also decreases (due to the ideological
distance XBL − XBR ). Put differently, a primary makes losing less likely
but makes winning less attractive.
A primary will therefore be adopted if and only if EUBL (leadership) ≤
EUBL (primary) . That condition leads to the following instructive result,
recalling that λB ≡ XBR − XBL :
17
Indeed, there are 12 intermediate cases, corresponding to the remaining permutations
of γ, XBL , XBR and −XA .
18
Eliminating or reversing this assumption would not change the essence of the results.
18
Theorem 2 If XBL , XBR , −XA < γ, the leadership of party B will adopt a
primary election if and only if
−ΛB ≤ λB ≤ ΛB
with ΛB ≡
SB [XBL (1−π AI )−XAL π AI ]
.
(1−π AI )(π BI +SB )
The intuition behind this result is that B’s leadership will delegate the
nomination to its RAF only when the RAF’s ideology is close enough to its
own. The same intuition can be obtained from Figure 1. If B’s RAF is so
ideologically extreme that ΛB < λB , then the leadership will not adopt a
primary election.
[Figure 1 about here]
The theorem also describes a striking flip-side to this result: that the
party’s leadership might find its RAF too centrist to adopt a primary. For
certain values of the parameters it is possible to have λB < −ΛB , in which
case a primary election will not be adopted because the primary voters are
considered too moderate.19
Given that the region of RAF ideologies for which a primary is adopted
is determined by the threshold ΛB , we can interpret ΛB as indicating the
likelihood that B will adopt a primary. Indeed for a larger ΛB it is more
likely that the ideological distance between B’s establishment and RAF will
be acceptable to adopt a primary. Then a way of phrasing this theorem
is that the likelihood of opening the CSM decreases with the ideological
distance between the party’s leadership and the primary voters.
It should be noted that this result is compatible with the empirical results found in Meinke et al. (2006). In that creative paper the authors
measure the likelihood that parties will open the delegate selection rules
in the United States as a function of the ideological distance between the
party establishment and the potential primary voters. Their strong and statistically significant results indicate that "as the preferences of state party
leaders and the voting public diverge, party leaders do become less willing
to open their selection processes."
We now study the case of a small γ, whose results are simple and striking.
Intuitively we would imagine that party leaders would be less inclined to
adopt a primary election when the skill of their candidate has little impact
on voters —after all, the benefit of a primary in this model is to reveal the
19
Such an interval will exist as long as XBL is large enough, specifically XBL >
. Otherwise, there will not exist an interval where the RAF is "too moderate".
−XAL SB π AI
π BI (1−π AI )
19
skill of precandidates, so the appeal of a primary would seem to decrease
with a small γ. And yet, the following result says the opposite.
Theorem 3 If γ < XBL , XBR , −XA , the leadership of party B will always
adopt a primary election.
The reason behind this counterintuitive result is that a small γ prevents
the RAF to move policy anywhere different than the leadership would. In
other words, when B wins the election, the effect of having a high-skilled
candidate is so small that policy can barely be moved way from the median
voter; both the leadership and the RAF would move it in the same direction
and exactly in the same (small) amount. BR would love to move the policy
to its ideal point but it cannot; BL therefore loses nothing by delegating the
nomination to BR. So choosing a primary does not carry any cost in policy
to party leaders, but it still has a benefit. The benefit of increasing the
chances of winning the election remains: a primary election, irrespective of
γ, still increases the probability that a high-skilled candidate is nominated.
The effect is therefore that choosing a primary election, as compared to a
leadership selection, has a benefit but does not have a cost. Consequently
leaders always choosing primaries.
The results for intermediate values of γ fall in between these two theorems. Once again, the full analysis is too long to display here, so we only
provide a summary of the main finding: all things equal, the likelihood of
adopting a primary is larger than it is for large γ, and lower than it is for
small γ.
5.4
Comparative statics
We would like to gain insight on what makes the adoption of primary elections more likely. According to Theorem 2, the value of relevance is ΛB ,
which determines that likelihood when γ is sufficiently large. So we now
study how ΛB changes with the parameters in the model. In the following
q2
result, π is a constant equal to 1−2q+2q
2 as calculated in the appendix.
Theorem 4 For π BI ∈ (0, π) the threshold ΛB is:
1. Strictly positive
2. Strictly increasing with SB
3. Strictly decreasing with π BI
20
4. Strictly decreasing with π AI
5. Strictly decreasing with XAL
6. Strictly increasing with XBL
For π BI ∈ [π, 1) the threshold ΛB is always equal to zero.
The first four results of this theorem are quite intuitive. As long as
the party considers its insider candidate to be relatively weak (meaning
that π BI < π), it has a strictly positive likelihood of adopting a primary.20
When the primary skill bonus SB increases, indicating that primaries have
become better at revealing the skills of candidates, primaries will be more
attractive. When the probability that the insider candidate is high-skilled
π BI is larger, perhaps because some polls indicate her surge in popularity,
then the party leadership is more inclined to appoint her directly through a
leadership selection. And when the probability that A’s candidate is highskilled increases, then party A becomes more of a threat to party B having a
higher probability of winning the election and implementing its leftist ideal
policy, which gives B an incentive to adopt a primary election.
The fifth result is more surprising. All things equal, an increase in XAL
indicates that the A has become more centrist. According to the theorem,
B then becomes less inclined to adopt a primary. The reason is that a
moderation of party A makes its victory less painful for B. So when A’s
ideal point becomes more moderate, B can better afford to lose the election.
The most important result is the sixth one: that primaries are more likely
when XBL increases meaning that the leadership of party B becomes more
extremist. The reason is that the more extremist the preferences of a party’s
establishment, the more painful a defeat would be for that establishment;
and therefore the party establishment is more inclined to adopt a primary
to improve its probability of winning. To illustrate this result consider two
cases: the first one where XBL and XBR are moderate with a certain ideological distance λB , and the second where XBL and XBR are extremist with
the same ideological distance λB . The theorem says that λB is more likely to
produce a primary election in the latter case where the preferences of leaders
and RAF are extremist.21 A clear empirical prediction of this result is that
extremist parties should, all things equal, be internally more democratic.
20
This result has a similar flavor to the main result in Adams and Merrill (2006) that
"weak" parties have the most to gain from a primary election.
21
If party B becomes so extremist that γ < XBR , XBL , then the use of a primary
actually becomes a certainty.
21
The qualifier "internally" is important in this claim. Communist and fascist
parties, which are at the extreme left and extreme right, might very well
stand for autocracy at the country level. But the prediction of this model is
that they will use more democratic ways of selecting candidates within their
parties, such as primary elections.
Those are the most relevant comparative statics to analyze. When γ
is "small", meaning that γ < XBL , XBR , −XA , there are no comparative
statics at all: according to Theorem 3, the party will always adopt a primary
for any value of the parameters. The case of an "intermediate" γ does not
offer surprises: the comparative statics go in the same direction as the case
where γ is "large", but might be mitigated.
This analysis does offer one surprising comparative static with respect
to γ. Overall, the larger the weight voters place on skill, the smaller the likelihood that a primary election will be adopted. This result was unforeseen
given that the stated benefit of primary elections is to increase the expected
skill of the party’s nominee. It is thus remarkable that the party leaders are
perfectly certain to adopt a primary when γ is small, somewhat certain when
γ is intermediate, and least certain when γ is large. The reason is that γ
constrains how far the selectorates can move policy toward their ideal point:
the larger γ, the riskier is it for party leaders to let their rank and file loose.
6
Significance and extensions
Primary elections are a frequent method used by political parties around the
world to select their candidates —and increasingly so. Some benefit, private
or public, needs to be pointed out if we are to explain the increasing popularity of primary elections. In particular, some benefit needs to be identified
for the party leaders who are responsible for choosing whether to adopt a
primary or not. The premise in this paper is that primary elections can serve
as a mechanism to reveal information about the candidates’ personal appeal
to voters. In particular, by forcing candidates to run a primary campaign
before the general election campaign, the candidates reveal their campaigning skills and the primary voters can select them accordingly. A second
feature of primary elections is opening the door to previously unknown or
marginalized candidates, thus expanding the pool of pre-candidates that the
party can choose from.
An implication of those two features is that a primary election will increase the expected valence of the party’s nominee. However, this benefit
can be offset by the possibility that primaries will push candidates to adopt
22
policies that the party leaders disapprove of. Therefore party leaders face a
trade-off between choosing a primary election versus a selection by the party
establishment.
According to the results in this paper, that trade-off depends on parameters in systematic, but sometimes surprising ways. Quite intuitively, party
leaders are more likely to adopt primary elections when the candidates they
have otherwise available are expected to have low campaigning skills; when
primaries are highly effective at revealing accurate information about the
candidates’ appeal to voters; when the potential primary voters are not too
extremist; and when the opposing party has a strong candidate. More surprisingly, party leaders are also more likely to adopt primaries when the
potential primary voters are not too moderate; when the opposing party is
centrist; and when those leaders have extremist policy preferences. An empirical prediction of this last property is that parties at the extreme left and
the extreme right of the political spectrum are more likely to adopt primary
elections. And counterintuitively, primaries are more likely when voters are
less sensitive to the skill of candidates.
An important extension that was not presented here, is the case where
both parties, and not only one, have the ability to hold a primary election. There are many countries, indeed, where more than one party has
the necessary infrastructure and organization to stage a mass vote among
its membership. It is thus interesting to study the game-theoretic situation
where all parties are simultaneously deciding, taking each others decision
into account, whether to use a primary election or not. In a separate research are studying such a game-theoretic interaction in detail, and we have
found that most of the results in this paper still hold. We have also found
two additional and interesting results. First, many reasonable values in the
parameters lead to multiple equilibria. And second, parties are expected to
display a significant degree of contagion, meaning that a party’s adoption of
a primary election will influence the other party to adopt a primary election
as well (see Serra 2007a).
More broadly, this paper adds to the literature on the origin of institutions that regards the rules governing political action as the choice of
political actors themselves. Such is the case of candidate-selection rules,
whose origin lies in the strategic calculations of the selectors themselves. To
paraphrase what John Aldrich (1995) said of parties in general, the nomination method is the creature of the politicians, the ambitious office seeker and
officeholder. They have created and maintained, used or abused, reformed
or ignored the candidate selection when doing so has furthered their goals
and ambitions. The CSM is thus an "endogenous" institution.
23
7
Figures
Figure 1
24
8
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9
9.1
Appendix A: Primaries as an information-revelation
mechanism
The model
Assume that a given party, which we will generically call party K, needs
to nominate a candidate for an upcoming election. Candidates are characterized by a parameter v denoting the candidate’s campaigning skill. A
candidate’s campaigning skill can take two values: a low value of zero corresponding to a low-skilled candidate, and a high value of one corresponding
to a high-skilled candidate. So v ∈ {0, 1} .
29
Before choosing its candidate, K’s leaders must choose its candidateselection method mK with mK ∈ {primary, leadership} , where primary
refers to the adoption of a primary election and leadership refers to a
leadership selection. If mK = leadership, party K has only one precandidate to choose from called the insider candidate and denoted by KI.
If mK = primary party K has two pre-candidates to choose from, which
are the insider, KI, and an outsider denoted by KO. We call vKI and vKO
the campaigning skills of KI and KO respectively, and we call vK the skill
of the candidate that is finally nominated by K. The exact values of KI’s
and KO’s campaigning skills are uncertain ex-ante, but the party has some
prior information about the skill of its insider candidate KI. According to
that information, candidate KI has a probability π KI of being high-skilled,
with π KI ∈ (0, 1) . On the other hand, there is no prior information about
candidate KO, so she is believed to have a probability of one-half of being
high-skilled. All this information is common knowledge.
Party leaders and party members want to maximize the expected skill of
the party’s nominee, E (vK ). Note that the expected value of K’s nominee
is proportional to her probability of being high-skilled, given that
E (vK ) = P (vK = 1) · 1
And therefore it is equivalent for party K to maximize E (vK ) or to maximize
P (vK = 1) , which is what we will focus on in the rest of this section.
If party leaders choose to select the candidate themselves they will directly nominate KI and therefore P (vK = 1|leadership) = π KI . If however
they choose to adopt a primary election the candidate KO will join the race,
and the nomination is delegated to the party’s RAF who will vote between
KI and KO.
If there is a primary election, party members interpret the performance
of a candidate in the primary-election campaign as a forecast of how well
they would perform in the general-election campaign.
A candidate’s performance in the primary can itself reflect high skill or
low skill. We assume that a high-skilled candidate has a larger probability
of having a high-skilled performance in the primary. To be concrete, we
denote by sj the performance of candidate j, with j = KI, KO; and we
say that sj = high if j’s performance in the primary was high skilled, and
sj = low if j’s performance in the primary was low skilled. We assume that
a candidate’s performance in the primary has a probability q of accurately
forecasting
the performance she would have in the general election, with q ∈
¡1 ¢
,
1
.
In
other
words, sKI and sKO have probability q of "being correct".
2
30
Essentially, sKI and sKO are (independently distributed) random variables
whose distribution depend on vKI and vKO in the following way:
P (sj = high|vj = 1) = P (sj = low|vj = 0) = q
P (sj = high|vj = 0) = P (sj = low|vj = 1) = 1 − q
j = KI, KO
Once the party members observe the candidates’ performances, they
can update their beliefs about KI’s and KO’s skills using Bayes rule. The
candidates’ performances are public, and therefore all the RAF members
observe the same sKI and sKO . Thus all the RAF members of K update
their prior beliefs with the same information.22
Given its interest in winning the general election, the RAF will vote for
the candidate that is believed to have the highest skill. If both candidates
are believed to have the same skill thus making the RAF indifferent, we will
make the following indifference assumption for convenience.
A3: When a party member is indifferent between KI and KO she
will vote for the one whose prior probability of being high-skilled
was largest. If both had the same prior, she will randomize
equally.
9.2
Results
We define SK ≡ P (vK = 1|primary) − P (vK = 1|leadership) , which we
call the skill bonus of a primary. An important implication of this definition
is that
P (vK = 1|primary) = π KI + SK
which means that SK can also be interpreted as the extra probability of
having a high-skilled candidate that a primary brings above a leadership
selection.
The exact values of P (vK = 1|primary) , P (vK = 1|leadership) and SK
are calculated in Section 2 of this appendix. Their most salient feature is
the following result. In all results below, the two constants π and π are
(1−q)2
q2
given by π = 1−2q+2q
2 and π = 1−2q+2q 2 .
22
In related research we have studied the case where the estimates are not public but
private, and thus each primary voter has a different sKI and sKO . We found that the
effects of holding a primary are the same as here, but slightly weaker.
31
Proposition 1 The primary skill bonus SK is strictly positive for π KI ∈
(0, π) and zero for π KI ∈ [π, 1).
9.3
Comparative statics
We now study how SK changes with respect to its two main determinants:
the prior about the insider candidate’s skill, π KI , and the accuracy of the
candidates’ performances q. We first describe the comparative statics with
respect to π KI .
Proposition 2 The primary skill bonus SK is strictly decreasing with π KI
for π KI ∈ (0, π) , and constant (equal to zero) for π KI ∈ [π, 1).
Now we describe the comparative statics with respect to q, which are
still intuitive but less straightforward. As the following result shows, for
intermediate values of π KI the primary skill bonus is increasing with q. But
surprisingly, it can be insensitive to small changes in q when π KI is very
large or very small.
Proposition 3 The effect on the primary skill bonus SK of a marginal increase in q is strictly positive for π KI ∈ [π, π], but is null for π KI ∈ (0, π)
and π KI ∈ (π, 1) .
There is a sense, however, in which q has a positive effect on the primary
skill bonus for any value of π KI if large enough improvements in q can be
achieved. This is underlined by the result below.
¢
¡
Proposition 4 For every q ∈ 12 , 1 there exists a q 0 > q such that for any
π KI ∈ (0, 1) , when q increases to q 0 , SK will increase.
10
10.1
Appendix B: Proofs
Theorem 1
Proof. We study each of the five possible regions for δ in turn.
• Case XB ≤ δ :
First note that in this case the valence advantage of candidate B is
so large that B could propose its ideal point and still win the election,
irrespective of the policy announced by candidate A. That is because
32
XB ≤ δ ⇒ γvAI ≤ −XB + γvBI ⇒ UM (0, vAI ) ≤ UM (XB , vBI ) ⇒
maxxA UM (xA , vAI ) ≤ UM (XB , vBI ) which implies that M will vote for
B for any platform xA (these equations hold with equality because δ > 0
and due to the indifference assumption A1). So if B announced xB = XB ,
A could announce any platform xA ∈ R and still lose the election; and thus
neither of the two candidates would have an incentive to deviate. This NE
is depicted in the figure below.
Note that no other platform xB can be sustained in equilibrium because
B can always deviate to XB and obtain its maximum payoff. There is
one other way for candidate B to obtain its maximum payoff, which is for
candidate A to adopt xA = XB , and for B to adopt a very extreme platform
that would allow A to win; but in that case A could benefit from deviating
closer to its ideal point, so this cannot be a NE.
• Case 0 < δ < XB :
Let us study all the possible locations of xA to see which ones can be
sustained in a NE. If xA is too extreme, namely |xA | ≥ XB − δ, then B can
propose xB = XB and win the election. That is because |xA | ≥ XB − δ ⇒
− |xA | +γvAI ≤ −XB +γvBI ⇒ UM (xA , vAI ) ≤ UM (XB , vBI ) . But then A
could deviate unilaterally to a more moderate platform and win the election,
and therefore this cannot be an equilibrium.
On the other hand, if xA is moderate enough, namely |xA | < XB − δ,
then B’s best response is to propose the rightmost platform that allows it
to win the election, which is xB = |xA | + δ. That is because UM (xA , vAI ) =
UM (|xA | + δ, vBI ) , so B could adopt xB = |xA |+δ and win the election (due
to the indifference assumption A1 and the fact that δ > 0). Note the crucial
role that A1 plays in this last statement: without that assumption there
would not exist an optimal xB for B. Our results are robust to eliminating
that assumption, however: An equilibrium would not exist but the optimal
behavior of B would be nearly identical to the one described in the theorem
—namely to increase xB ever so closely to |xA | + δ without reaching it.
If |xA | > 0 and xB = |xA | + δ then B is best responding to xA , but A
could adopt a more centrist platform and win the election, so this cannot
be an equilibrium. Only if xA = 0 and xB = δ we have B best-responding
to xA and A best-responding to xB , and that is because A cannot adopt a
more centrist platform. This is therefore the only NE, as illustrated in the
figure below.
33
• Case δ = 0 :
In this case none of the candidates has a valence advantage. M will
therefore vote for the candidate whose platform is closest to zero, or will
randomize equally between the two if both platforms are equidistant from
zero (as established by the indifference assumption A1). It is well known
in this setting that the unique Nash equilibrium is for both candidates to
converge to the median voter’s ideal point (see for example Calvert (1985) or
Roemer (2001)). Such an equilibrium is depicted in the figure below, where
x∗A = x∗B = 0.
• Case XA < δ < 0 :
This is the mirror image of the case 0 < δ < XB . Hence the unique NE is
for candidate B to converge to the median voter’s ideal point, x∗B = 0, and
for candidate A to adopt the leftmost platform that can win the election,
i.e. x∗A = δ.
• Case δ ≤ XA :
This is the mirror image of the case XB ≤ δ. Hence candidate A is able
to adopt its ideal point, x∗A = XA , and win the election irrespective of the
platform adopted by candidate B, x∗B ∈ R.
10.2
Lemma 1
Proof. First we calculate EUBL (leadership) based on the prior beliefs
about AI and BI’s skills. The distribution of possible values of δ, with
δ ≡ γvBI − γvAI , is
⎧
⎨ γ with probability (1 − π AI ) π BI
δ=
0 with probability π AI π BI + (1 − π AI ) (1 − π BI )
⎩
−γ with probability π AI (1 − π BI )
which, according to Theorem 1 and given that XBL , XBR , −XA < γ, leads
to the following distribution of equilibrium platforms:
⎧
⎨ XBL with probability (1 − π AI ) π BI
0 with probability π AI π BI + (1 − π AI ) (1 − π BI )
x∗ =
⎩
XAL with probability π AI (1 − π BI )
34
Accordingly, the expected utilities of B’s leaders are given by
E (UBL (x∗ ) |leadership) = − |XBL − XBL | (1 − π AI ) π BI
− |XBL − 0| [π AI π BI + (1 − π AI ) (1 − π BI )]
− |XBL − XAL | π AI (1 − π BI )
which is equivalent to the result stated in the lemma.
We now calculate EUBL (primary). We first calculate the new distribution of δ, noting that P (vB = 1|primary) = π BI + SB as stated in Section
2. This gives
⎧
⎨ γ with probability (1 − π AI ) (π BI + SB )
δ=
0 with probability π AI (π BI + SB ) + (1 − π AI ) (1 − (π BI + SB ))
⎩
−γ with probability π AI (1 − (π BI + SB ))
which, according to Theorem 1 and given that XBL , XBR , −XA < γ, leads
to the following distribution of equilibrium platforms (remembering that it
is XBR and not XBL that is implemented when B has the highest skilled
candidate).
⎧
⎨ XBR with probability (1 − π AI ) (π BI + SB )
∗
0 with probability π AI (π BI + SB ) + (1 − π AI ) (1 − (π BI + SB ))
x =
⎩
XAL with probability π AI (1 − (π BI + SB ))
Accordingly, the expected utility of B’s leadership is given by
E (UBL (x∗ )) = − |XBL − XBR | (1 − π AI ) (π BI + SB )
− |XBL − 0| [π AI (π BI + SB ) + (1 − π AI ) (1 − (π BI + SB ))]
− |XBL − XAL | π AI (1 − (π BI + SB ))
which is equivalent to the expression of EUBL (primary) given in the lemma.
10.3
Lemma 2
Proof. First we calculate EUBL (leadership) based on the prior beliefs
about AI and BI’s skills. The distribution of possible values of δ, with
35
δ ≡ γvBI − γvAI , is
⎧
⎨ γ with probability (1 − π AI ) π BI
δ=
0 with probability π AI π BI + (1 − π AI ) (1 − π BI )
⎩
−γ with probability π AI (1 − π BI )
which, according to Theorem 1 and given that γ < XBL , XBR , −XA , leads
to the following distribution of equilibrium platforms:
⎧
⎨ γ with probability (1 − π AI ) π BI
0 with probability π AI π BI + (1 − π AI ) (1 − π BI )
x∗ =
⎩
−γ with probability π AI (1 − π BI )
Accordingly, the expected utilities of B’s leaders are given by
E (UBL (x∗ ) |leadership) = − |XBL − γ| (1 − π AI ) π BI
− |XBL − 0| [π AI π BI + (1 − π AI ) (1 − π BI )]
− |XBL + γ| π AI (1 − π BI )
which is equivalent to the result stated in the lemma.
We now calculate EUBL (primary). We first calculate the new distribution of δ, noting that P (vB = 1|primary) = π BI + SB as stated in Section
2. This gives
⎧
⎨ γ with probability (1 − π AI ) (π BI + SB )
δ=
0 with probability π AI (π BI + SB ) + (1 − π AI ) (1 − (π BI + SB ))
⎩
−γ with probability π AI (1 − (π BI + SB ))
which, according to Theorem 1 and given that γ < XBL , XBR , −XA , leads
to the following distribution of equilibrium platforms
⎧
⎨ γ with probability (1 − π AI ) (π BI + SB )
x∗ =
0 with probability π AI (π BI + SB ) + (1 − π AI ) (1 − (π BI + SB ))
⎩
−γ with probability π AI (1 − (π BI + SB ))
Accordingly, the expected utility of B’s leadership is given by
E (UBL (x∗ )) = − |XBL − γ| (1 − π AI ) (π BI + SB )
− |XBL − 0| [π AI (π BI + SB ) + (1 − π AI ) (1 − (π BI + SB ))]
− |XBL + γ| π AI (1 − (π BI + SB ))
36
which is equivalent to the expression of EUBL (primary) given in the lemma.
10.4
Theorem 2
Proof. Given our indifference assumption A2, B will adopt a primary
election if and only if EUBL (leadership) ≤ EUBL (primary) . Recalling
that λB ≡ XBR − XBL , it is just a matter of algebra to prove that
EUBL (leadership) ≤ EUBL (primary) ⇔
0 ≤ − |λB | (1 − π AI ) (π BI + SB )
−XBL [π AI SB + (1 − π AI ) (−SB )]
− (XBL − XAL ) π AI (−SB ) ⇔
0 ≤ − |λB | (1 − π AI ) (π BI + SB ) + SB [XBL (1 − π AI ) − XAL π AI ]
Solving for |λB | in the expression above tells us that the leadership of
BL (1−π AI )−XAL π AI ]
party B will adopt a primary election if and only if |λB | ≤ SB [X(1−π
.
AI )(π BI +SB )
By defining ΛB ≡
−ΛB ≤ λB ≤ ΛB .
10.5
SB [XBL (1−π AI )−XAL πAI ]
,
(1−πAI )(π BI +SB )
that inequality is equivalent to
Theorem 3
Proof. Given our indifference assumption A2, B will adopt a primary
election if and only if EUBL (leadership) ≤ EUBL (primary) . It is just a
matter of algebra to prove that
EUBL (leadership) ≤ EUBL (primary) ⇔
0 ≤ − (XBL − γ) (1 − π AI ) (π BI + SB )
−XBL [π AI SB + (1 − π AI ) (−SB )]
− (XBL + γ) π AI (−SB ) ⇔
0 ≤ γSB
which is always true because γ ≥ 0 and SB ≥ 0.
10.6
Theorem 4
Proof. First note that ΛB = 0 whenever SB = 0. And we know from
Proposition 1 that SB = 0 for π BI ∈ [π, 1). Therefore ΛB equals zero in that
interval, and will not change with any of the parameters.
37
We also know from Proposition 1 that SB > 0 for π BI ∈ (0, π) , and thus
ΛB > 0 in that interval. To derive the comparative statics in the theorem
we differentiate ΛB with respect to the parameters.
[XBL (1−π AI )−XAL π AI ]π BI
B
To study the effect of SB we note that ∂Λ
∂SB =
(1−π AI )(π BI +SB )2
which is strictly positive, and therefore ΛB is strictly increasing with SB .
To study π BI we must note that it has two effects on ΛB : a direct effect,
and an indirect effect through its effect on SB . In total, we have that dπdΛBI =
∂ΛB
∂π BI
+
∂ΛB ∂SB
∂SB ∂π BI .
−SB [XBL (1−π AI )−XAL πAI ]
(1−π AI )(π BI +SB )2
B
the other hand we just calculated that ∂Λ
∂SB is
∂SB
from Theorem 2 that ∂πBI is strictly negative.
We can calculate that
∂ΛB
∂πBI
=
which is strictly negative. On
strictly positive, and we know
We therefore have that dπdΛBI < 0 and ΛB is strictly decreasing with π BI .
SB (πBI +SB )(−XAL )
∂ΛB
= ((1−π
To study the effect of π AI we note that ∂π
2 which
AI
AI )(π BI +SB ))
is strictly positive, and therefore ΛB is strictly increasing with π AI .
∂ΛB
B [−π AI ]
To study the effect of XAL we note that ∂X
= (1−πSAI
)(π BI +SB ) which
AL
is strictly negative, and therefore ΛB is strictly decreasing with XAL .
∂ΛB
B (1−π AI )
= (1−πSAI
To study the effect of XBL we note that ∂X
)(π BI +SB ) which
BL
is strictly positive, and therefore ΛB is strictly increasing with XBL .
10.7
Proposition 1
We start by stating and proving a claim regarding the RAF’s behavior.
We need one more definition: When sKI 6= sKO , we say that the RAF
"votes according to the performances" if it votes for the candidate whose
performance was best.
Claim 1 Party K’s RAF will
if π KI ∈ (0,
¡ π],
¢ always vote for KO
1
if π KI ∈ π, 2 , vote according to the performances if sKI 6= sKO , and
vote for KO if sKI = sKO
if π KI = 12 , vote according to the performances if sKI 6= sKO , and
randomize between
¡
¢ KI and KO if sKI = sKO
if π KI ∈ 12 , π , vote according to the performances if sKI 6= sKO , and
vote for KI if sKI = sKO
if π KI ∈ [π, 1), always vote for KI
with
(1 − q)2
q2
and
π
=
π=
1 − 2q + 2q 2
1 − 2q + 2q 2
38
Proof of Claim 1. Party K’s RAF will vote for the candidate that she believes to have highest probability of being high-skilled. The beliefs she holds
about each candidate’s skill depend on two pieces of information: her prior
beliefs, and the information acquired throughout the primary campaign.
Given that the RAF is rational, it will update her prior beliefs based on the
performances sKI and sKO to form a couple of posterior beliefs about the
probabilities that KI and KO are high-skilled. If the RAF uses Bayes Rule
to update her prior beliefs after receiving a given estimate, her posterior
beliefs will be given by
(1 − q) π KI
(1 − q) π KI + q (1 − π KI )
qπ KI
P (vKI = 1|sKI = high) =
qπ KI + (1 − q) (1 − π KI )
P (vKO = 1|sKO = low) = 1 − q
P (vKI = 1|sKI = low) =
P (vKO = 1|sKO = high) = q
There are four couple of performances (sKI , sKO ) that the RAF could
receive, which are (0, 0) , (1, 1) , (0, 1) and (1, 0) , We study each of them
in turn, along with the decision that the RAF makes upon receiving those
couples of estimates.
• If the RAF observes estimates sKI = low and sKO = low :
The RAF will vote for KI if P (vKO = 1|sKO = low) < P (vKI = 1|sKI = low)
which is equivalent (after some algebra) to 12 < π KI . Then, given our indifference assumption A1, the RAF will vote for KO if π KI < 12 , will vote for
KI if 12 < π KI , and will randomize equally if π KI = 12 .
• If the RAF observes estimates sKI = high and sKO = high :
The RAF will vote for KI if P (vKO = 1|sKO = high) < P (vKI = 1|sKI = high)
which is equivalent (after some algebra) to 12 < π KI . Then, given our indifference assumption A1, the RAF will vote for KO if π KI < 12 , will vote for
KI if 12 < π KI , and will randomize equally if π KI = 12 .
• If the RAF observes estimates sKI = low and sKO = high :
The RAF will vote for KI (in other words, disregard the candidates’
performance) if P (vKO = 1|sKO = high) < P (vKI = 1|sKI = low) which
is equivalent (after some algebra, and noting that 1 − 2q + 2q 2 > 0) to
39
q2
< π KI . Then, given
1−2q+2q2
q2
1
that 2 < 1−2q+2q
2 ), the RAF
2
q
π ≡ 1−2q+2q
2.
our indifference assumption A1 (and noting
will vote for KI if and only π ≤ π KI , with
• If the RAF observes estimates sKI = high and sKO = low :
The RAF will vote for KO (in other words, disregard the candidates’
performance) if P (vKO = 1|sKO = low) < P (vKI = 1|sKI = high) which
is equivalent (after some algebra, and noting that 1 − 2q + 2q 2 > 0) to
(1−q)2
1−2q+2q 2 . Then,
(1−q)2
that 1−2q+2q2 < 12 ), the
(1−q)2
π ≡ 1−2q+2q
2
π KI <
given our indifference assumption A1 (and noting
RAF will vote for KO if and only π KI ≤ π, with
The following table summarizes these results:
sKI = low
sKI = high
sKI = low
sKI = high
sKO = low
sKO = high sKO = high sKO = low
if π KI ∈ (0,
¡ π]¢ Vote for KO Vote for KO Vote for KO Vote for KO
if π KI ∈ π, 12
Vote for KO Vote for KO Vote for KO Vote for KI
1
if π KI = ¡2 ¢ Randomize
Randomize
Vote for KO Vote for KI
if π KI ∈ 12 , π
Vote for KI Vote for KI Vote for KO Vote for KI
if π KI ∈ [π, 1) Vote for KI Vote for KI Vote for KI Vote for KI
Which is what the claim states.
We can now use the RAF’s behavior described in the claim to prove the
proposition.
Proof of Proposition 1. We first endeavor to calculate P (vK = 1|primary) .
We do so by noting that
X
X
P (vK = 1|sKI , sKO ; vKI , vKO )·P (sKI , sKO |vKI , vKO )·P (vKI , vKO )
P (vK = 1) =
vKI ,vKO sKI ,sKO
which uses the definition of conditional probability twice.
Given the prior probabilities about the skills of KI and KO, we have
that
⎧
(1 − π KI ) 12 for vKI = 0 and vKO = 0
⎪
⎪
⎨
(1 − π KI ) 12 for vKI = 0 and vKO = 1
P (vKI , vKO ) =
π 1 for vKI = 1 and vKO = 0
⎪
⎪
⎩ KI 21
π KI 2 for vKI = 1 and vKO = 1
And given the probabilities that the estimates about the skills of KI and
KO are correct, the value of P (sKI , sKO |vKI , vKO ) is given in the following
40
table
Value of
P (sKI , sKO |vKI , vKO )
For sKI = low and sKO = low
For sKI = low and sKO = high
For sKI = high and sKO = low
For sKI = high and sKO = high
If vKI = 0
and vKO = 0
q2
q (1 − q)
(1 − q) q
(1 − q)2
If vKI = 1
and vKO = 0
(1 − q) q
(1 − q)2
q2
q (1 − q)
If vKI = 0
and vKO = 1
q (1 − q)
q2
(1 − q)2
(1 − q) q
If vKI = 1
and vKO = 1
(1 − q)2
(1 − q) q
q (1 − q)
q2
In order to calculate P (vK = 1|sKI , sKO ; vKI , vKO ) we use Lemma 1,
which tells us who the nominee is for every couple sKI , sKO . In accordance
with that lemma, we need to study several intervals for π KI separately.
• If π KI ∈ (0, π] :
The nominee is always KO, and therefore P (vK = 1|sKI , sKO ; vKI , vKO ) =
P (vKO = 1) = 12 . Plugging those values in the addition above leads to
P (vK = 1) =
1
2
¡
¢
• If π KI ∈ π, 12 :
The nominee and her probability of being high-skilled are given by the
following table
Value of P (vK = 1|sKI , sKO ; vKI , vKO ) :
Values of
(sKI , sKO )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
Nominee
KO
KI
KO
KO
If vKI = 0
and vKO = 0
0
0
0
0
If vKI = 1
and vKO = 0
0
1
0
0
If vKI = 0
and vKO = 1
1
0
1
1
Plugging those values in the addition above leads to
1
1
P (vK = 1) = π KI q 2 + q − q 2 − π KI q + π KI
2
2
• If π KI =
1
2
:
41
If vKI = 1
and vKO = 1
1
1
1
1
The nominee and her probability of being high-skilled are given by the
following table
Value of P (vK = 1|sKI , sKO ; vKI , vKO ) :
Values of
(sKI , sKO )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
Nominee
KO or KI
KI
KO
KO or KI
If vKI = 0
and vKO = 0
0
0
0
0
If vKI = 1
and vKO = 0
If vKI = 0
and vKO = 1
1
2
1
2
1
0
0
1
1
2
1
2
If vKI = 1
and vKO = 1
1
1
1
1
Plugging those values in the addition above leads to
1
1
P (vK = 1) = q +
2
4
• If π KI ∈
¡1
¢
,
π
:
2
The nominee and her probability of being high-skilled are given by the
following table
Value of P (vK = 1|sKI , sKO ; vKI , vKO ) :
Values of
(sKI , sKO )
(0, 0)
(0, 1)
(1, 0)
(1, 1)
Nominee
KI
KI
KO
KI
If vKI = 0
and vKO = 0
0
0
0
0
If vKI = 1
and vKO = 0
1
1
0
1
If vKI = 0
and vKO = 1
0
0
1
0
If vKI = 1
and vKO = 1
1
1
1
1
Plugging those values in the addition above leads to
1
1
P (vK = 1) = π KI q − π KI q 2 + q2 + π KI
2
2
• If π KI ∈ [π, 1) :
The nominee is always KI, and therefore P (vK = 1|sKI , sKO ; vKI , vKO ) =
P (vKI = 1) = π KI . Plugging those values in the addition above leads to
P (vK = 1) = π KI
We can now calculate our value of interest, SK for each value of π KI . We
use the values above to calculate SK ≡ P (vK = 1|primary)−P (vK = 1|leadership) ,
42
remembering that P (vK = 1|leadership) = π KI .
⎧ 1
⎪
2 − π KI if π KI ∈ (0, π]
¡
¢
⎪
⎪
⎪
⎨ π KI q 2 + q − 12 q 2 − π KI q + 12 π KI − π KI if π KI ∈ π, 12
1
1
1
SK =
2 q + 4 − π KI if π KI = 2
¡
¢
⎪
⎪
π KI q − π KI q 2 + 12 q2 + 12 π KI − π KI if π KI ∈ 12 , π
⎪
⎪
⎩
π KI − π KI if π KI ∈ [π, 1)
which gives, with some algebra and noting the continuity of SK at π KI = π,
π KI = 12 and π KI = π,
SK
⎧
⎪
⎪
⎨
1
2
− π KI if π KI ∈ (0, π]
£ 1¤
π,¤2
π KI q 2 − π KI q − 12 q 2 − 12 π KI + q if π KI ∈
£
=
1 2
1
1
2
−π KI q + π KI q + 2 q − 2 π KI if π KI ∈ 2 , π
⎪
⎪
⎩
0 if π KI ∈ [π, 1)
Now we need to analyze the sign of SK . If π KI ∈ (0, π] we have that
SK = 12 − π KI > 0 ⇔ π KI < 12 , but that is satisfied
π KI ≤
£ 1because
¤
1
π and we have already noted that π < 2 . If π KI ∈ π, 2 we have that
2q−q 2
SK = π KI q2 − π KI q − 12 q 2 − 12 π KI + q > 0 ⇔ π KI < 1+2q−2q
2 (noting that
1 + 2q − 2q 2 > 0) which is satisfied because
we have that SK = −π KI q 2 + π KI q + 12 q 2 −
2
2q−q 2
1
1
2 < 1+2q−2q2 . If π KI ∈ [ 2 , π)
q2
1
2 π KI > 0 ⇔ π KI < 1−2q+2q 2
q
which is satisfied because π = 1−2q+2q
2 . And finally if π KI ∈ [π, 1) we have
£
¤
SK = 0. So we have indeed SK > 0 for π KI ∈ (0, π] ∪ π, 12 ∪ [ 12 , π) and
SK = 0 for π KI ∈ [π, 1), as the theorem claims.
10.8
Proposition 2
Proof. We calculate the differential of SK with respect to π KI and check
¡ 1its
¢
∂SK
sign. If π KI ∈ (0, π), ∂πKI = −1 which is strictly negative. If π KI ∈ π, 2 ,
¡
¢
¡1 ¢
∂SK
2 − q − 1 which is strictly negative for q ∈ 1 , 1 . If π
=
q
∈
KI
∂π KI
2
2
2, π ,
¡1 ¢
∂SK
2
∂π KI = −q + 2q − 1 which is strictly negative for q ∈ 2 , 1 . So SK is
decreasing with π KI in all those intervals. SK is non-differentiable at π KI =
π and π KI = 12 , but is continuos at both points, and is therefore decreasing
just like their neighboring
SK decreases with π KI when π KI ∈
¡
¢
¢ © ªpoints.
¡ Hence
(0, π) ∪ {π} ∪ π, 12 ∪ 12 ∪ 12 , π .
If π KI ∈ [π, 1), SK is constant for all values of π KI (and equal to zero),
so an increase in π KI will not affect it.
43
10.9
Proposition 3
Proof. We calculate the differential of SK with respect to q and check
(1−q)2
its sign, remembering that the values of π and π are π = 1−2q+2q
2 and
π=
q2
.
1−2q+2q2
∂SK
∂q
According to the values of SK in Theorem 1, if π KI ∈ (0, π),
K
= 0; similarly if π KI ∈ (π, 1) , ∂S
∂q = 0. So in those intervals, SK is
unresponsive to marginal
in q.
¢
¡ changes
K
However, if π KI ∈ π, 12 , ∂S
∂q = 2π KI q − π KI + 1 − q which is strictly
¡
¢
1 ∂SK
1
positive; if π KI = 2 , ∂q = 2 which is strictly positive; if π KI ∈ 12 , π ,
∂SK
∂q = −2π KI q + π KI + q which is strictly positive.
SK is strictly increasing with marginal increases in q.
So in those intervals,
To analyze the cases where π KI = π and π KI = π, note that
³
´
(1−q)2
1−2q+2q2
i
h
(1−q)2
1
,
2
1−2q+2q 2
∂
∂q
0, so with a marginal increase in q, π remains in the interval
where
just
³ we
´ proved that SK is increasing with q. Similarly note that
q2
∂
> 0, so with a marginal increase in q, π remains in the interval
2
i
h∂q 1−2q+2q
2
q
1
2 , 1−2q+2q2 where we just proved that SK is increasing with q.
To summarize, SK is unresponsive to marginal changes in¡ q for
¢ π©KI
ª∈
1
1
1)
,
and
is
strictly
increasing
with
q
for
π
∈
{π}∪
π,
∪
(0,
π)∪(π,
KI
2
2 ∪
¡1 ¢
2 , π ∪ {π} .
10.10
Proposition 4
Proof. First assume that π KI ∈ [π, π] with π =
(1−q)2
1−2q+2q 2
and π =
q2
.
1−2q+2q 2
K
In that case we know from Lemma 2 that ∂S
∂q > 0. So SK is strictly
0
increasing when q increases to any q > q.
(1−q)2
Now assume that π KI ∈ (0, π) . Note that limq→1 1−2q+2q
2 = 0, so there
(1−q 00 )2
< π KI , in which case π KI ∈ [π 00 , π 00 ]
1−2q 00 +2q 002
(1−q 00 )2
q 002
00 =
. And given the result above,
π 00 = 1−2q
00 +2q 002 and π
1−2q 00 +2q 002
00
strictly increasing when q increases to any q 0 > q00 . And given that
exists q 00 > q such that
where
SK is
SK cannot have decreased when q increased to q 00 , we have that SK will
increase strictly when q increases to q 00 .
q2
A similar logic is used when π KI ∈ (π, 1) by noting that limq→1 1−2q+2q
2 =
1.
44
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