# Strategic flip-flopping in political competition††thanks: The authors wish
to thank the participants of the Game Theory seminar in IHP, the eco-lunch
seminar in AMSE, the Workshop on Dynamic Games and Applications in Paris 2 and
the Workshop on Stochastic Methods in Erice for their highly valuable
comments. YT acknowledges the support of the Israel Science Foundation grants
#2566/20, #1626/18 & #448/22. GF thanks the “France 2030” investment plan
managed by the French National Research Agency (reference :ANR-17-EURE-0020)
and from Excellence Initiative of Aix-Marseille University – A*MIDEX.

Gaëtan Fournier, Alberto Grillo, Yevgeny Tsodikovich Aix Marseille Univ, CNRS,
AMSE, Marseille, France<EMAIL_ADDRESS>Paris-Panthéon-
Assas, LEMMA, Paris, France<EMAIL_ADDRESS>of
Economics, Bar Ilan University, Israel<EMAIL_ADDRESS>
###### Abstract

We study candidates’ positioning when adjustments are possible in response to
new information about voters’ preferences. Re-positioning allows candidates to
get closer to the median voter but is costly both financially and electorally.
We examine the occurrence and the direction of the adjustments depending on
the ex-ante positions and the new information. In the unique subgame perfect
equilibrium, candidates anticipate the possibility to adjust in response to
future information and diverge ex-ante in order to secure a cost-less victory
when the new information is favorable.

_JEL Classification_ : C72, D72, D82

_Keywords_ : Spatial voting, Imperfect information.

##  1 Introduction

In the run-up to an election, candidates largely rely on polls to learn about
voting intentions. Uncovering the electorate’s leanings, what is known as the
policy mood [Stimson, 1991; Stevenson, 2001], help them shape their campaign
message but may also pose a difficult choice: should they change their
positions to get closer to voters? In the long run, it is well documented that
politicians adapt their stances to reflect the evolving preferences of their
constituencies [Glazer and Robbins, 1985; Stratmann, 2000; Miler, 2016]. Yet,
policy changes that are too sudden involve substantial costs, in terms of not
only communication efforts but also electoral appeal. In the history of the
U.S. presidential elections, for example, the defeats of John Kerry in 2004
and Matt Romney in 2012 are often linked to their shifting views on salient
issues111Specifically concerning Kerry’s opposition to the war in Iraq after
his previous support and Romney’s multiple shifts, notably on abortion [Croco,
2016]..

This paper studies re-positioning choices as a strategic game between
candidates. We assume that voters prefer candidates with platforms aligned
with their ideal policies, but dislike flip-floppers, i.e. candidates who
strategically change their positions during the campaign. We examine which
candidates are more likely to adjust their positions following new
information, in which direction, and how successfully. We also investigate how
the anticipation of possible changes affects candidates’ positions ex-ante,
before the information is revealed.

A theory of policy adjustments is useful in light of the bias of the empirical
evidence. As pointed out by Tomz and Van Houweling [2012], “historical data
[…] reveal the consequences of re-positioning only in the specific
circumstances when politicians thought re-positioning would be optimal”. Our
analysis of the involved trade-offs aims to clarify what these specific
circumstances are.

Model. We enrich the Downs-Hotelling framework by supposing an information
shock creating a two-stage game. The shock reveals the location of the median
voter. This captures the idea that voters’ aggregate preferences fluctuate
over time and that their current leanings are disclosed during the electoral
campaign. In the model, two office-motivated candidates first select their
positions before the shock, knowing only the distribution of the median
voter’s location. They can then revise their positions after the shock (after
learning the actual location). We assume that such a change involves both an
_electoral cost_ – voters’ discount their evaluation of a candidate who
changed position – and an _organizational cost_ – the candidate’s payoff is
reduced due to the policy change.

On the one hand, the electoral cost represents voters’ negative feeling toward
a flip-flop. This aversion may be interpreted as a higher uncertainty about
what the candidate would do if elected, i.e. to an undermined credibility of
the position [Enelow and Munger, 1993]. Alternatively, voters may value
consistency on policy issues as a cue for character [Kartik and McAfee, 2007],
or as a signal for quality of implementation of the ex-post policy. We model
voters’ dislike for flip-flops in a reduced form, through a penalty in their
utility if a candidate changes position.

On the other hand, the organizational cost represents all other costs involved
by a change of position which are not related to voters’ reaction. The most
prominent is the organizational and financial cost for candidates of
communicating the change to the voters. While justifying a policy change to
the public may reduce its electoral harm, such communication requires costly
advertising.

Results. We study the subgame-perfect equilibria of the game, using backward
induction. In the second stage, if the revealed information is not clearly in
favor of one candidate, the adjustment choices determine the winner of the
election. The advantaged candidate would like to adjust only if the other
candidate threatens the advantage by also moving, while the disadvantaged
candidate would like to adjust only if his opponent does not. The only
equilibrium is therefore in mixed strategies, with both candidates flip-
flopping with positive probability and having a chance to win the election. If
instead the revealed information strongly favors one candidate, there are no
incentives to flip-flop as the election cannot be disputed anymore.

In the first stage, candidates anticipate their strategic responses to the
information shock. The game has at most one subgame-perfect equilibrium, which
exists if both the electoral and the organizational costs are sufficiently
high. In such equilibrium, candidates contradict the median voter theorem:
they choose differentiated platforms to secure a cost-less victory when the
information is in their favor.

Contribution. Assuming sufficiently high adjustment costs, our model yields
several implications concerning the occurrence of flip-flops. First, along the
equilibrium path, re-positioning happens only toward the center, while
candidates never adjust toward more extreme positions. This highlights a
moderation effect, according to which candidates cultivate separate
electorates when elections are far in time and then soften their positions
during the campaign. In the U.S., such an effect is often attributed to the
presence of primary elections, in which candidates need to convince a more
extreme median voter [Agranov, 2016]. Our framework provides a different
rationale for a similar dynamic, which can play out even in the absence of
primary elections.

A second prediction is that flip-flops consist mostly of small adjustments
made by an advantaged candidate in order to secure his victory. Only a
minority of flip-flops are large adjustments made by a disadvantaged candidate
who seeks to reverse the likely outcome of the election. Indeed, on the one
hand, when the favorite candidate adjusts his position, the magnitude of the
adjustment is smaller than when his opponent adjusts. On the other hand, we
find that a candidate favored by the new information is more likely to adjust
his position than a disadvantaged candidate. Hence, an adjustment by the
advantaged candidate guarantees his victory but an adjustment by the
challenger is more likely to be unsuccessful.

Finally, we provide comparative statics results with respect to changes in the
adjustment costs. We find that an increase in the electoral cost decreases the
polarization of candidates but increases their equilibrium payoffs. Indeed,
such an increase makes flip-flopping less likely, because it increases the
probability that the election is secured after the information shock, which
guarantees the favorite candidate a cost-less victory.

Extension. We look at how the results are modified by an asymmetry between
candidates. Asymmetry in organizational costs has no impact on the results.
Asymmetry in electoral costs makes the more flexible candidate choose a more
central platform, while the less flexible candidate offers a more polarized
platform. We show that a candidate who is significantly less flexible in
adjusting his position loses the election in equilibrium, even if he is
favored by the information on voters. In the general asymmetric game, payoffs
are decreasing in candidates’ own electoral cost, but increasing in the
opponent’s electoral cost.

Literature. Following the seminal work by Hotelling [1929] and Downs [1957],
most models of political competition assume that candidates can freely commit
to any electoral platform. At the extreme opposite, some papers take campaign
announcements as cheap-talk and assume that, once elected, candidates act
according to their own preferences [Alesina, 1988; Osborne and Slivinski,
1996; Besley and Coate, 1997].

We lay down a more realistic framework, in which platforms are binding
although not immutable, and candidates can costly adjust them over time. The
evidence that politicians respond in an adaptive way to changes in voters’
preferences is substantial in political science, see Stimson et al. [1995],
Adams et al. [2004], and Kousser et al. [2007]. Karol [2009] argues that elite
replacement is not necessary for parties’ policy changes, which are often
driven by incumbent politicians. Adams et al. [2006] find that mainstream
parties engage in re-positioning more than niche parties, while Tavits [2007]
relates the likelihood of success to whether the change concerned pragmatic or
principled issues. This literature has a strong empirical focus and recently
turned also to experimental settings [Tomz and Van Houweling, 2012; Doherty et
al., 2016; Robison, 2017]. We see our theoretical investigation as a useful
complement to this existing research.

Key to our analysis is the assumption that policy changes are costly. Models
by Bernhardt and Ingberman [1985], Ingberman [1989] and Enelow and Munger
[1993] consider a voters’ utility function which is decreasing in the size of
a change in candidates’ policies. In these studies, the loss is derived from
the increased uncertainty concerning the policy that will be effectively
implemented. Evidence of an intrinsic preference of voters for consistency is
also discussed as a “waffle effect” in political psychology [Carlson and
Dolan, 1985]. Hoffman and Carver [1984] found that even proposing a policy in
agreement with the voters’ preferences may not be rewarded if it follows an
inconsistent track-record. Our specification is consistent with DeBacker
[2015], who finds empirically that the electoral cost is increasing in the
size of the change. The organizational cost, instead, captures in a reduced
form the idea that informative advertising directed towards voters is costly
for candidates [Coate, 2004; Ashworth, 2006]. Our model abstracts away the
role of interest groups in financing political campaigns, and assumes for
simplicity that candidates bear communications costs themselves.222See Prato
and Wolton [2019] for a similar assumption.

Given the result of ex-ante divergence, our paper belongs to a class of models
in which policy differentiation is adopted to soften competition in a second
dimension. The baseline argument is familiar in industrial organization
models, in which firms differentiate their products in order to reduce the
subsequent competition in prices [Tirole, 1988; Shaked and Sutton, 1982]. In
political competition, Ashworth and Bueno De Mesquita [2009] and Zakharov
[2009] examine the reduced need for differentiated candidates to invest in
valence. Eyster and Kittsteiner [2007] consider parties as collections of
candidates. Parties choose general platforms first and candidates can deviate
from their party’s platform at a cost, in order to increase their chances in
their specific constituency. Divergence of parties’ platforms results from the
incentive to minimize the aggregate cost of candidates’ re-positioning for the
party. In Balart et al. [2022], candidates diverge to minimize future costly
advertising, which would be needed to impress voters if the ideological divide
were low. We focus instead on candidates’ choice of changing their own
position over time, when doing this is not only financially costly but also
penalized electorally. While ex-ante divergence arises analogously to these
previous models in the literature, our framework allows us to highlight the
emerging patterns of flip-flopping along the (subgame-perfect) equilibrium.

The paper is more loosely related to three other research lines. First, flip-
flopping has been investigated in two-stage models with primary elections
[Hummel, 2010; Agranov, 2016]. Although we do not have primary elections, our
model exhibits a similar dynamic of moderation during the campaign stage.
Second, the paper shares with Kamada and Sugaya [2020] an interest in the
dynamic aspect of policy announcements during the campaign stage. In their
paper, however, the opportunity to revise a position arises stochastically in
a revision game à la Kamada and Kandori [2020], and the focus is on
candidates’ reactions to each others’ announcements. Finally, policy
adjustments in our model can be seen as a pandering phenomenon. However, such
pandering only describes a spatial movement and ignores considerations of
optimality from a common-value perspective, addressed in Maskin and Tirole
[2004] and Andreottola [2021].

The rest of the paper is organized as follows. Section 2 presents the model
and Section 3 the equilibrium analysis and the results. We study the case of
asymmetric candidates in Section 4. Technical proofs and lemmas are postponed
to the appendix.

##  2 The Model

There are two office seeking candidates (namely, candidate $1$ and $2$) and a
continuum of voters that we identify with their ideal policy $t\in\mathbb{R}$.
The location of the median voter $m$ is drawn according to a uniform
distribution on a compact interval, w.l.o.g. fixed to $[0,1]$, and is revealed
during the campaign. Candidates choose their platforms twice, both before and
after the reveal of the median voter’s position. We denote the ex-ante
platforms $x_{1}$ and $x_{2}$, and the ex-post platforms $y_{1}$ and $y_{2}$.

We interpret the choice of the ex-post platforms as the adjustment that
candidates may make with respect to their previous positions after learning
the electorate’s preferences more precisely. The choice of the ex-ante
platforms reflects instead the positions in which candidates invest over the
long term, knowing that future revisions are allowed but costly.

Voter $t$’s utility from a victory of candidate $i$ depends on both the
candidate’s ex-ante and the ex-post platform according to the following
functional form

$u_{t}(x_{i},y_{i})=-(t-y_{i})^{2}-a(y_{i}-x_{i})^{2}.$ (1)

The first term represents voters’ preference for a candidate whose final
policy is closer to their ideal one. The second term represents how voters
penalize candidates for changing their platform with respect to their previous
position. In our model, voters trust that policy $y_{i}$ will be implemented
if candidate $i$ wins, hence do not care about the distance between $t$ and
$x_{i}$. Yet, they have an intrinsic preference for candidates who are
consistent. The parameter $a>0$ measures the relative effect of this electoral
penalty and thus how voters trade off policy considerations with their dislike
for a candidate changing position.

The election is decided by majority rule. Because voters’ preferences are
single-peaked and their utilities have the same functional form, the candidate
that attracts the median voter wins the election. The payoff of the candidates
depends on both the outcome of the election and the possible organizational
cost of a policy change. The benefit from winning the election is set to $1$
and that from losing to $0$. In addition, a candidate that changed his policy,
by selecting $y_{i}\neq x_{i}$, pays a fixed cost
$\phi\in(0,\frac{1}{2})$.333We assume that the cost $\phi$ is smaller than
$\frac{1}{2}$, which is the expected gain from competing in the election since
players are symmetric. If $\phi>0.5$, saving the organizational costs is more
important than competing for the election. We interpret $\phi$ as the cost
induced by the need to communicate the change of policy to the public.
Although greater changes may require more substantial communication
strategies, we abstract from the dependency of the organizational cost on the
magnitude of the change for technical simplicity. Our assumption of a fixed
$\phi$ captures the presence of some fixed cost, which is likely present
independently of the magnitude of the change. The payoff of candidate $i$ is
then

$g_{i}(\textbf{x},\textbf{y})=\mathbb{E}({1\hskip-2.5pt{\rm l}}_{i\text{
wins}}-\phi{1\hskip-2.5pt{\rm l}}_{y_{i}\neq x_{i}}),$

where ${1\hskip-2.5pt{\rm l}}_{i\text{ wins}}$ is the indicator that candidate
$i$ wins the election, and the expectation is taken with respect to the
position of the median voter $m$ and the possibly mixed actions of the
players. Ties are broken by a toss of a fair coin, so in case of a tie,
$\mathbb{E}({1\hskip-2.5pt{\rm l}}_{i\text{ wins}})=\frac{1}{2}$.

To summarize, the timing of the game is as follows.

  1. 1.

First Stage: The two candidates choose ex-ante platforms $x_{1}$ and $x_{2}$.

  2. 2.

Information Shock: The position of the median voter $m$ is revealed.

  3. 3.

Second Stage: The two candidates choose ex-post platforms $y_{1}$ and $y_{2}$.

  4. 4.

The Voting: The candidate’s preferred by the median voter is elected and
candidates’ payoffs are realized.

##  3 Equilibrium analysis

We are interested in the subgame-perfect equilibrium of the game. Hence, we
solve the game using backward induction and start by studying the strategic
flip-flopping of candidates after their observation of the ex-ante platforms
$x_{1},x_{2}$ and the location of the median voter $m$ (Propositions 1 and 2).
The analysis shows that two scenarios are possible in the second stage: either
the advantage of one candidate is too large and he is sure to win, or the
election is still open and the outcome depends on candidates’ reactions. In
the first case, it is optimal for both candidates not to change platforms and
remain at their ex-ante positions. In the second case, both candidates mix
between moving to an ex-post optimal position and not moving.

Next, given the strategies in the second stage, we analyze the optimal choice
of ex-ante platforms. We show that for sufficiently high values of the costs
$(a,\phi)$, the unique subgame-perfect equilibrium requires candidates to
invest in divergent ex-ante positions (Proposition 3). Candidates
differentiate in order to maximize the chances of having a sufficient
advantage in the second stage, for which they save on the organizational cost
of changing policy. In the remaining region of the parameter space (see Figure
3), a subgame-perfect equilibrium does not exist but an $\epsilon$-equilibrium
exists in which candidates take centrist positions one $\epsilon$ away from
each other (Proposition 4). Finally, we summarize the implications in terms of
flip-flopping behavior that emerge from our equilibrium analysis (Proposition
5 and 6).

###  3.1 Second Stage: Choice of Ex-Post Platforms $y_{1}$, $y_{2}$

We take $x_{1}$, $x_{2}$ and $m$ as given and consider separately the cases
where the ex-ante platforms are the same or different, as the analysis in
these two cases differs significantly.

####  3.1.1 Different Ex-Ante Platforms $x_{1}\neq x_{2}$

Let us assume first $x_{1}\neq x_{2}$ and consider, without loss of
generality, the case $x_{1}<x_{2}$. Given $m$, we refer to the candidate with
ex-ante platform closer to $m$ as the _favorite_ candidate.

######  Definition 1.

Candidate $i$ is the _favorite_ and candidate $j$ is the _challenger_ if
$|x_{i}-m|<|x_{j}-m|$.

The intuition for the term _favorite_ is that, if candidates do not change
their platforms, the favorite candidate $i$ wins the election as
$u_{m}(x_{i},x_{i})>u_{m}(x_{j},x_{j})$. We ignore the case where
$m=\frac{x_{1}+x_{2}}{2}$ which occurs with null probability.

In the case where $m=x_{1}$, candidate $1$ can win the election without
changing platform and incurring the organizational cost $\phi$, as the median
voter has the highest possible utility from his victory. Candidate $2$, on the
other hand, loses the election regardless of his ex-post platform, so it is
optimal for both candidates to not move and save on the organizational cost.
This argument remains true when $m\neq x_{1}$ but is close enough to $x_{1}$:
candidate $1$ still wins the election without changing his platform, and
candidate $2$ also does not change his platform as he loses the election
anyway. In this case, we say that the favorite candidate has _secured_ the
election.

######  Definition 2.

The election is _secured_ for the favorite candidate $i$ if
$u_{m}(x_{i},x_{i})>\max\limits_{y\in[0,1]}u_{m}(x_{j},y)$. Otherwise, the
election is _still open_.

A favorite candidate who has secured the election has a strictly dominant
strategy $y_{i}=x_{i}$, by which he wins without paying the organizational
cost $\phi$. The challenger also has a strictly dominant strategy
$y_{j}=x_{j}$, as he cannot win the election and should save the
organizational cost. Hence, if the the election is secured, the strategies
$y_{1}=x_{1}$, $y_{2}=x_{2}$ constitute the unique equilibrium of the second
stage subgame. The equilibrium payoffs are $1$ for the favorite candidate and
$0$ for the challenger. In Lemma 1 in the appendix, we solve the inequality
that appears in Definition 2 and show that the election is secured for
candidate $1$ whenever $m\in(\underline{m},\overline{m})$, where

$\underline{m}:=\tfrac{\alpha x_{1}-x_{2}}{\alpha-1}\vee
0,\qquad\overline{m}:=\tfrac{\alpha x_{1}+x_{2}}{\alpha+1},$

and where

$\alpha=\sqrt{\tfrac{1+a}{a}}$ (2)

is a useful parameter. Similarly, the election is secured for candidate $2$
whenever $m\in(\underline{n},\overline{n})$ where

$\underline{n}:=\tfrac{\alpha
x_{2}+x_{1}}{\alpha+1},\qquad\overline{n}:=\tfrac{\alpha
x_{2}-x_{1}}{\alpha-1}\wedge 1.$

Figure 1 summarizes the regions where each candidate has secured the election,
and those in which the election is still open. We observe that the election is
secured if $m$ realizes in a neighborhood of each candidate’s ex-ante platform
$x_{i}$.

$0$$\underline{m}$$x_{1}$$\overline{m}$$\tfrac{x_{1}+x_{2}}{2}$$\underline{n}$$x_{2}$$\overline{n}$$1$1FS2FS1FO1FO2FO2FO
Figure 1: An illustration of the status of each player being the favorite
while the election is secured (FS) or still open (FO) before choosing the ex-
post platform, with respect to the position of the median voter $m\in[0,1]$.

If the election is still open, the favorite candidate wins the election if
neither candidate changes his platform. However, he can lose the election if
he keeps the platform while the challenger moves closer to the median voter.
Hence, the challenger wants to move if the favorite does not change his
platform. Analogously, the favorite also wants to move if the challenger
moves, in order to secure his victory. In Lemma 2 in the appendix, we show
that in this case both candidates have an optimal platform $\hat{y_{i}}$ which
they want to adopt if they decide to change their ex-ante platform. The
optimal platform is given by

$\hat{y_{i}}=\frac{m+ax_{i}}{1+a}.$ (3)

This optimal platform is a weighted average between the ex-ante platform
$x_{i}$ and the realization of the median voter location $m$. For each
candidate, the platform $\hat{y_{i}}$ is optimal in the sense that all other
platforms different from $x_{i}$ are either dominated or redundant, as proved
also in Lemma 2. We can represent the ex-post game between the two candidates
by the following $2\times 2$ one-shot game, assuming that each candidate
chooses between moving to $\hat{y_{i}}$ and not moving. Without loss of
generality, we consider candidate $1$ as the favorite.

22[Favorite candidate $1$][Challenger candidate $2$] & $\hat{y_{2}}$ $x_{2}$

$\hat{y_{1}}$ $(1-\phi,-\phi)$ $(1-\phi,0)$

$x_{1}$ $(0,1-\phi)$ $(1,0)$

Table 1: The normal form game that candidates face if the election is still
open.

If both candidates move to their optimal platforms, candidate $1$ wins the
election: his ex-ante platform is closer to the median voter, so his ex-post
platform is both closer to the median voter and requires a smaller adjustment
$|\hat{y_{1}}-x_{1}|<|\hat{y_{2}}-x_{2}|$. We observe that the favorite
candidate wants to take the same action as the challenger, while the
challenger wants to take the opposite action as the favorite. Hence, such a
game has no equilibrium in pure strategies but has a unique equilibrium in
mixed strategies. At this equilibrium, the favorite changes his platform with
probability $(1-\phi)$ and the challenger changes his platform with
probability $\phi$. The expected equilibrium payoffs are $1-\phi$ for the
favorite candidate and $0$ for the challenger.

This discussion is summarized in the following proposition.

######  Proposition 1 (Equilibrium in the second stage subgame for $x_{1}\neq
x_{2}$).

Suppose that different ex-ante platforms $x_{1}\neq x_{2}$ were chosen. The
reaction of the candidates to the revelation of $m$ is:

  * •

If the favorite candidate has secured the election, the unique subgame
equilibrium is $(y_{1},y_{2})=(x_{1},x_{2})$. The equilibrium payoffs are $1$
for the favorite and $0$ for the challenger.

  * •

If the election is still open, the unique subgame equilibrium is in mixed
strategies:

$y_{i}=\begin{cases}\hat{y_{i}}&\text{ with probability }1-\phi,\\\
x_{i}&\text{ with probability }\phi,\end{cases}$

for the favorite candidate $i$ and

$y_{j}=\begin{cases}\hat{y_{j}}&\text{ with probability }\phi,\\\ x_{j}&\text{
with probability }1-\phi,\end{cases}$

for the challenger $j$. The equilibrium payoffs are $(1-\phi)$ for the
favorite and $0$ for the challenger.

###### Proof.

See Appendix A.2. ∎

If the the election is secured, the uniqueness of the equilibrium is clear, as
$(x_{1},x_{2})$ are strictly dominant strategies. Otherwise, the uniqueness
concerns the equilibrium payoff but not the equilibrium strategies, as in some
configurations other strategies might be redundant to the strategies
$\hat{y_{i}}$ and constitute an equilibrium with the same payoffs. As shown in
Lemma 2, disregarding these strategies is without loss of generality because
it does not impact the payoffs nor the analysis.

The comparison of the subgame equilibrium payoffs when the election is secured
or open illustrates the inefficiency of the second scenario. When the election
is still open, the favorite gets $(1-\phi)$ and the challenger $0$ (in
expectation), which is the same payoffs as if the challenger were to concede
but the favorite candidate still paid the organizational cost to change his
policy.

####  3.1.2 Identical Ex-Ante Platforms $x_{1}=x_{2}$

In the case of identical ex-ante platforms, the election is always open and
there is neither a favorite nor a challenger, which breaks down the analysis
from Proposition 1. As before, both candidates have an optimal platform
$\hat{y_{i}}$ where they want to move if they do. This optimal position is
again equal to $\hat{y_{i}}=\tfrac{m+ax_{i}}{1+a}$ and is now identical for
both candidates. Hence, if both candidates move or both do not move, each has
a probability of $\tfrac{1}{2}$ to win the election. We can then restrict our
attention to the ex-post game given by the $2\times 2$ matrix in Table 2.

22[Candidate $1$][Candidate $2$] & $\hat{y_{2}}$ $x_{2}$

$\hat{y_{1}}$ $(\frac{1}{2}-\phi,\frac{1}{2}-\phi)$ $(1-\phi,0)$

$x_{1}$ $(0,1-\phi)$ $(\frac{1}{2},\frac{1}{2})$

Table 2: The normal form game that candidates face if the ex-ante platforms
are identical.

Since $\phi<\tfrac{1}{2}$, we have $\tfrac{1}{2}-\phi>0$ and
$1-\phi>\tfrac{1}{2}$. It follows that keeping the ex-ante platform is a
strictly dominated strategy and in the unique equilibrium both candidates
choose platforms $\hat{y_{1}}=\hat{y_{2}}$. The above discussion proves the
following Proposition:

######  Proposition 2 (Equilibrium in the second stage of the game for
$x_{1}=x_{2}$).

Suppose that ex-ante, the identical platforms $x_{1}=x_{2}$ were chosen. The
unique subgame equilibrium in the second stage is $(\hat{y_{1}},\hat{y_{2}})$
(given by Eq. (3)). The equilibrium payoffs are
$(\frac{1}{2}-\phi,\frac{1}{2}-\phi)$.

Note that $\frac{1}{2}-\phi$ is the worst possible equilibrium payoff in this
game. Indeed, no matter the strategy of his opponent, candidate $i$ can select
$x_{i}=\frac{1}{2}$ and $y_{i}=\hat{y_{i}}$. By doing so he wins the election
with probability of at least $\frac{1}{2}$, while paying the organizational
cost $\phi$.

###  3.2 First Stage: Choice of Ex-Ante Platforms $x_{1}$, $x_{2}$

By backward induction, candidates choose their ex-ante platforms by
considering that the subgame equilibrium given by Proposition 1 or 2 is played
in the second stage. In the first stage, we restrict attention to pure
strategies $x_{i}$ for both candidates.

A first result is that there cannot exist a subgame-perfect equilibrium in
which candidates select the same position in the first stage. If the ex-ante
positions were identical, each candidate would have a profitable deviation by
playing an infinitesimally different ex-ante position, since such a deviation
induces a better equilibrium payoff after the second stage. We prove this
claim within the proof of Proposition 4 below but anticipating this result
allows us to focus on ex-ante platforms that properly define a favorite and a
challenger, as in Section 3.1.1.

The next proposition shows that if $\phi$ and $a$ are sufficiently big, then
the ex-ante platforms diverge from the center in the unique subgame-perfect
equilibrium. The intuition behind this result is that, conditional on being
the favorite, candidates prefer the election to be secured rather than still
open, when the position of $m$ is revealed. Divergence occurs because the
probability of a secured election is proportional to the distance between the
ex-ante platforms $|x_{2}-x_{1}|$. More precisely, given the unique subgame
equilibrium described in Proposition 1, each candidate $i$’s expected payoff
is:

$1\times\mathbb{P}(i\text{ is the favorite and the election is secured})+\\\
+(1-\phi)\times\mathbb{P}(i\text{ is the favorite and the election is still
open})+0$ (4)

where the probability $\mathbb{P}$ represents the randomization of the median
voter’s location and the $0$ is the expected payoff if $i$ is the challenger.
On the one hand, player prefer to be the favorite, which creates an incentive
to move towards the opposite candidate in order to increase the likelihood of
being the closest candidate to $m$. On the other hand, being the favorite is
not the only concern of the candidates, because they also want to maximize the
probability of the election being secured conditional on being the favorite.
The fact that the probability that the election is secured for the favorite is
proportional to the distance between ex-ante platforms $|x_{2}-x_{1}|$ creates
an incentive to diverge. Indeed, by differentiating their platforms,
candidates create “secure electorates” – regions of the policy space which
guarantee a victory of a candidate if the median voter is revealed in such a
region without the need of a costly adjustment of platform. This centrifugal
force is to be traded off with the centripetal force and prevails in
equilibrium if the condition on $\phi$ in Proposition 3 holds.

######  Proposition 3 (Differentiation of ex-ante platforms).

Suppose that $\phi>\frac{1}{1+4\sqrt{a(1+a)}}$ and without loss of generality
that $x_{1}\leq x_{2}$. In the unique subgame-perfect equilibrium, candidates’
ex-ante platforms are

$(x_{1}^{*},x_{2}^{*})=\left(\frac{1}{\alpha+1},\frac{\alpha}{\alpha+1}\right)$
(5)

where $\alpha=\sqrt{\frac{1+a}{a}}$, and the ex-post behavior is according to
Propositions 1 and 2.
The equilibrium expected payoffs are

$g_{1}^{*}=g_{2}^{*}=\frac{1}{2}-\frac{\phi}{2}\left(\frac{\alpha-1}{\alpha+1}\right)^{2}$
(6)

###### Proof.

See Appendix A.3. ∎

In these equilibrium payoffs, the first term represents the expected gain of
the election and the second term represents the expected organizational cost.
Indeed, in equilibrium the election is still open with probability
$\left(\frac{\alpha-1}{\alpha+1}\right)^{2}$ and in this case each player
deviates with probability $1-\phi$ or $\phi$ whether he is the favorite or
not, so he pays the cost $\phi$ with probability
$\frac{1}{2}\phi+\frac{1}{2}(1-\phi)=\frac{1}{2}$.

Figure 2 shows the intervals on the policy space in which the election is
secured for the favorite candidate or still open at the subgame-perfect
equilibrium, which depend only on the electoral cost $a$ through the parameter
$\alpha$.

$0$$x_{1}=\frac{1}{\alpha+1}$$\frac{2\alpha}{(\alpha+1)^{2}}$$\tfrac{1}{2}$$\frac{\alpha^{2}+1}{(\alpha+1)^{2}}$$x_{2}=\frac{\alpha}{\alpha+1}$$1$1FS2FS1FO2FO
Figure 2: The status of each player being the favorite when the election is
secured (FS) or still open (FO) at the subgame-perfect equilibrium described
by Proposition 3, with respect to the position of the median voter $m$.

The following proposition shows that if the condition on $a$ and $\phi$ in
Proposition 3 is reversed, a subgame-perfect equilibrium does not exist.
Instead, we prove the existence of an $\epsilon$-equilibrium, at which neither
candidate can unilaterally improve his payoff by more than $\epsilon$, for
every $\epsilon>0$.

######  Proposition 4.

If $0<\phi<\tfrac{1}{1+4\sqrt{a(1+a)}}$, there does not exist a subgame-
perfect equilibrium. There exists instead an $\epsilon$-equilibrium, given by
$(x_{1},x_{2})=(\tfrac{1}{2}-\epsilon,\tfrac{1}{2}+\epsilon)$ for every
$\epsilon>0$ small enough.

###### Proof.

See Appendix A.4. ∎

The intuition for the result is that if the organizational cost of changing
platform is sufficiently small, candidates prefer to converge towards the
center to increase the chances of being the favorite candidate, even if this
decreases the probability of a secured election. However, the centripetal
incentive stops when candidates take the same position because of the
discontinuity in the payoff when $x_{1}=x_{2}$. Indeed, if candidates have
minimally differentiated platforms, each has an advantage in the second stage
if the median voter is located on his side of the policy space. Formally, by
converging fully to $x_{1}=x_{2}=\frac{1}{2}$ each candidate obtains an
expected payoff equal to $\frac{1}{2}-\phi$. Instead, by minimally
differentiating from the center and choosing $x_{1}=\frac{1}{2}-\epsilon$,
$x_{2}=\frac{1}{2}+\epsilon$ they both obtain a higher expected payoff, which
converges to $\frac{1}{2}-\frac{\phi}{2}$ as $\epsilon$ goes to $0$.

The condition on the parameters for the existence of a subgame-perfect
equilibrium is drawn in Figure 3 in the $(a,\phi)$ plane. The region dealt
with in Proposition 3 is designated by $R_{0}$ and formally defined as
$\\{(a,\phi)|\Psi(a)<\phi<\tfrac{1}{2}\\}$ where

$\Psi(a)=\frac{1}{1+4\sqrt{a(1+a)}}.$ (7)

In the region named $R_{1}$, only an $\epsilon$-equilibrium exists.

$0.5$$1$$1.5$$2$$2.5$$3$$0.1$$0.2$$0.3$$0.4$$0.5$$R_{1}$: No SPNE$R_{0}$: SPNE
$x_{1}\neq x_{2}$$\Psi(a)$$a$$\phi$ Figure 3: The two regions $R_{0}$ and
$R_{1}$ in the $a-\phi$ space. The colored region represents the area $R_{0}$
concerned with Proposition 3.

###  3.3 Adjustments along the equilibrium path

In light of our analysis, we can now put forward a few implications concerning
the occurrence of candidates’ policy changes. We focus on the parameter region
in which the electoral and organizational costs are high enough, and thus a
subgame-perfect equilibrium with divergent ex-ante positions exists.

######  Proposition 5.

Suppose that $\phi>\frac{1}{1+4\sqrt{a(1+a)}}$. At the subgame-perfect
equilibrium,

  1. (i)

candidates flip-flop to voters’ preferences only towards the center;

  2. (ii)

the favorite candidate is more likely to flip-flop than the challenger;

  3. (iii)

when the favorite candidate flip-flops, the magnitude of the adjustment is
smaller than when the challenger flip-flops;

  4. (iv)

flip-flopping is always successful when done by a favorite candidate, while it
is more likely to be unsuccessful than successful for a challenger.

###### Proof.

See Appendix A.5. ∎

Property (i) describes a dynamic of moderation along the electoral campaign,
according to which candidates start out with more extreme positions and
converge to the center if and once the median voter realizes in the center.
The property also implies that candidates never cross over the position of the
opponent. As such, even flip-flopping candidates always keep their relative
ideological stands, despite the fact that they are office seekers and have no
preferred policy.

The intuition behind property (ii) relies on the mixed strategies used by
candidates at equilibrium: the favorite is indifferent between flip-flopping
or not only when the probability to lose the election is relatively small,
that is when his opponent is less likely to flip-flop. Instead, the
disadvantaged candidate is indifferent only when the victory is relatively
unlikely, that is when the favorite secures his advantage with high
probability.

Property (iv) means that a flip-flopping favorite always wins the election in
equilibrium, while a flip-flopping challenger loses whenever the favorite also
flip-flops, i.e. with probability $1-\phi>\frac{1}{2}$. Taken together, these
properties suggest that campaign flip-flopping consists most often in a minor
adjustment by the favorite candidate in order to consolidate his victory, and
only less likely it is a major and risky move by the challenger who tries to
reverse the election outcome.

The next proposition focuses on the comparative statics properties with
respect to the electoral cost parameter $a$.

######  Proposition 6.

When the electorate is more tolerant towards candidates changing positions,
that is when $a$ decreases,

  1. (i)

the candidates’ ex-ante policies are more polarized;

  2. (ii)

the election is more likely to be still open after the reveal of $m$, hence
each candidate is more likely to flip-flop;

  3. (iii)

the candidates’ equilibrium payoffs are lower.

###### Proof.

See Appendix A.6. ∎

The comparative statics analysis highlights two main phenomena. First,
candidates prefer an intransigent electorate that is less tolerant of flip-
flopping. The underlying mechanism is described in claim (ii): more tolerant
electorates lead to increased competition, with more elections to be still
open after the revelation of the median. As a result, winning elections
without flip-flopping becomes less likely, leading to smaller equilibrium
payoffs.
Additionally, while the electoral cost is a necessary ingredient in
Proposition 5 to obtain differentiation, the degree of polarization decreases
as the electoral cost increases. Indeed, in equilibrium, candidates push their
secured intervals to the limits of the policy space, as $\underline{m}=0$ and
$\overline{n}=1$, and they position in the middle of these intervals. However,
when $a$ is small, these intervals shrink due to increased competition (the
challenger can more easily flip-flop to reverse the election), causing the
center of these intervals to shift closer to the extremities, and the degree
of candidates polarization is thus larger.

##  4 Asymmetric candidates

In this section, we relax the symmetry between candidates by supposing that
voters penalize differently each candidate for changing platform, via
different values of the parameter $a$.444A similar analysis can be made to
consider different organizational costs $\phi$, without affecting the results
significantly. Thus, the utility of voter $t$ from voting for candidate $i$
with ex-ante and ex-post platforms $x_{i}$ and $y_{i}$ is

$u_{t}^{i}(x_{i},y_{i}):=-(t-y_{i})^{2}-a_{i}(y_{i}-x_{i})^{2}$ (8)

as opposed to Eq. (1).

An interesting phenomenon results from this heterogeneity between candidates:
for a range of possible realizations of the median voter’s location, the
favorite candidate is not guaranteed to win the election even if he adjusts
his position. Indeed, the optimal adjustments are still given by weighted
averages between the ex-ante platforms and the realization of the median
voter, namely $\hat{y_{i}}=\frac{m+a_{i}x_{i}}{1+a_{i}}$, but because the
weights are different for each candidate, the challenger might attract the
median voter after both candidates adjust. Suppose, for example, that
$a_{1}<a_{2}$ and that platforms $x_{1}<x_{2}$ were chosen ex-ante. If $m$
realizes sufficiently close to $\frac{x_{1}+x_{2}}{2}$ then candidate $1$ can
adjust his position closer to $m$ than candidate $2$, thanks to his smaller
electoral cost, and win the election. This is true even if $m$ is larger than
$\frac{x_{1}+x_{2}}{2}$, i.e. if candidate $2$ is the favorite according to
Definition 1. This justifies the following definition:

######  Definition 3.

Candidate $i$ is a weak favorite and candidate $j$ is a strong challenger if
$|x_{i}-m|<|x_{j}-m|$ but
$u^{i}_{m}(x_{i},\hat{y}_{i})<u^{j}_{m}(x_{j},\hat{y}_{j})$.

In other terms, a weak favorite candidate wins the election when no candidate
adjusts, but looses the election when both candidates adjust their platforms.
More precisely, the (second-stage) game played by a weak favorite and a strong
challenger is as follows:

22[Strong challenger candidate $1$][Weak favorite candidate $2$] &
$\hat{y_{2}}$ $x_{2}$

$\hat{y_{1}}$ $(1-\phi,-\phi)$ $(1-\phi,0)$

$x_{1}$ $(0,1-\phi)$ $(0,1)$

The game admits $(\hat{y_{1}},x_{2})$ as its unique equilibrium, since both
actions are strictly dominant. That is, in equilibrium, a strong challenger
adjusts his position and wins the election with certainty, while a weak
favorite does not move. The equilibrium payoffs are given by $(1-\phi,0)$ and
are identical to the case in which candidate $1$ is the favorite and the
election is still open in the symmetric game.

For $a_{1}<a_{2}$, by solving
$u^{2}_{m}(x_{2},\hat{y}_{2})<u^{1}_{m}(x_{1},\hat{y}_{1})$ with respect to
$m$, we find that candidate $2$ is a weak favorite when
$m\in(\frac{x_{1}+x_{2}}{2},\tilde{m})$, in which
$\tilde{m}=\frac{\alpha_{1}x_{2}+\alpha_{2}x_{1}}{\alpha_{1}+\alpha_{2}}$ and
$\alpha_{i}=\sqrt{\frac{1+a_{i}}{a_{i}}}$. This region is represented in light
blue in Figure 4.

$0$$x_{1}$$\overline{m}$$\frac{x_{1}+x_{2}}{2}$$\tilde{m}$$\underline{n}$$x_{2}$$1$1FS2FS1FO2FO2WF
Figure 4: The status of each player being the favorite when the election is
secured (FS) or still open (FO) at the subgame-perfect equilibrium described
by Proposition 7, with respect to the position of the median voter $m$, for
$a_{1}<a_{2}$. In the light blue region, candidate $2$ is the weak favorite
(WF)

The analysis of the second stage in all other regions (when the favorite is
not weak) is the same in the asymmetric game as in the symmetric game of the
previous sections. The next proposition describes the subgame-perfect
equilibrium considering both stages of the game.

######  Proposition 7 (Equilibrium with asymmetric electoral costs).

Assume that $a_{1}<a_{2}$ and that $\phi>\max(\Psi(a_{1}),\Psi(a_{2}))$. There
exists a unique subgame-perfect equilibrium, in which the ex-ante locations
are given by

$(x_{1}^{*},x_{2}^{*})=\left(\frac{\alpha_{1}-1}{\alpha_{1}\alpha_{2}-1},\frac{\alpha_{2}(\alpha_{1}-1)}{\alpha_{1}\alpha_{2}-1}\right)$

where $\alpha_{i}=\sqrt{\frac{1+a_{i}}{a_{i}}}$.

The ex-post behavior depends on $m$:

  * •

For $m\in\left[\tfrac{x^{*}_{1}+x^{*}_{2}}{2},\tilde{m}\right]$ where
$\tilde{m}=\tfrac{\alpha_{1}x_{2}^{*}+\alpha_{2}x_{1}^{*}}{\alpha_{1}+\alpha_{2}}=\tfrac{\alpha_{2}(\alpha_{1}^{2}-1)}{(\alpha_{1}\alpha_{2}-1)(\alpha_{1}+\alpha_{2})}$,
candidate $1$ adjusts to $\hat{y}_{1}$ and wins the election, whereas
candidate $2$ remains at $x_{2}^{*}$ and loses the election.

  * •

Otherwise, it follows Proposition 1.

If $\phi<\max(\Psi(a_{1}),\Psi(a_{2}))$, there is no subgame-perfect
equilibrium.

###### Proof.

See Appendix A.7∎

While the ex-ante positions were symmetric around $\tfrac{1}{2}$ in the case
of identical electoral costs, we now have that

$\frac{x_{1}^{*}+x_{2}^{*}}{2}=\frac{1}{2}+\frac{\alpha_{1}-\alpha_{2}}{2(\alpha_{1}\alpha_{2}-1)}$
(9)

hence, the candidate with a lower $a$ (a higher $\alpha$) takes a more
centrist ex-ante position and has a higher expected payoff. Expected payoffs
in equilibrium are modified with respect to the expressions in Eq. (6) by the
possibility for the favorite to be weak, as follows:

$\displaystyle
g_{1}=1\times\left[\tfrac{2\alpha_{2}(\alpha_{1}-1)}{(\alpha_{1}\alpha_{2}-1)(\alpha_{2}+1)}\right]+(1-\phi)\times\left[\tfrac{(\alpha_{1}-1)(\alpha_{1}\alpha_{2}+\alpha
2)}{(\alpha_{1}\alpha_{2}-1)(\alpha_{1}+\alpha_{2})}-\tfrac{2\alpha_{2}(\alpha_{1}-1)}{(\alpha_{1}\alpha_{2}-1)(\alpha_{2}+1)}\right]$
(10) $\displaystyle
g_{2}=1\times\left[1-\tfrac{(\alpha_{1}\alpha_{2}+1)(\alpha_{1}-1)}{(\alpha_{1}\alpha_{2}-1)(\alpha_{1}+1)}\right]+(1-\phi)\times\left[\tfrac{(\alpha_{1}\alpha_{2}+1)(\alpha_{1}-1)}{(\alpha_{1}\alpha_{2}-1)(\alpha_{1}+1)}-\tfrac{\alpha_{2}(\alpha_{1}^{2}-1)}{(\alpha_{1}\alpha_{2}-1)(\alpha_{1}+\alpha_{2})}\right]$

With respect to Proposition 5, if candidates are asymmetric, it is still true
that flip-flopping happens only towards the center. It is no longer true, in
general, that a favorite candidate is more likely to flip-flop than a
challenger or that the magnitude of his adjustment is smaller. The validity of
these statements depends now on whether the favorite is weak or not and on the
exact difference in the electoral costs $a_{i}$. Flip-flopping is now always
successful both for a favorite candidate (recall that a weak favorite never
flip-flops in equilibrium) and for a strong challenger, while it remains more
likely unsuccessful than successful for a challenger who is not strong.

The introduction of heterogeneous electoral costs also allows us to refine the
comparative statics properties in Proposition 6. We can deduce from Eq. (9)
and Eq. (10) that, when voters are more tolerant towards a change of position
by candidate $i$ (i.e. if $a_{i}$ decreases for a given $a_{j}$), the ex-ante
position of $i$ is more centrist and his expected payoff is higher, while the
ex-ante position of $j$ is more extreme and his expected payoff is lower.
Hence, while in the symmetric case, candidates are better off when the
electorate penalizes flip-flopping more, in the asymmetric case candidates are
better off only if a higher electoral cost concerns the opponent. In
principle, a higher electoral cost could be beneficial for a candidate by
reducing the likelihood of adjusting the position and paying the
organizational cost. We find instead that, in equilibrium, each candidate
would prefer voters to be more tolerant with himself. Indeed, the region in
which a candidate $i$ has a secured election is independent of $a_{i}$ and
only depends on the electoral cost $a_{j}$ for the opponent. Hence, any
increase in $a_{i}$ simply decreases the likelihood of $i$ being in a better
position (non-weak favorite or strong challenger) in a still-open election,
and it is therefore detrimental.

##  Appendix A Proofs

###  A.1 Useful Lemmas

In this section we present results which we use in the paper. Most of these
results are simple computation and are omitted from the main text for the ease
of reading.

######  Lemma 1.

Suppose $x_{1}\neq x_{2}$ and without loss of generality $x_{1}<x_{2}$.
Candidate $1$ has secured the election if and only if
$m\in(\underline{m},\overline{m})$ with

$\underline{m}:=\tfrac{\alpha x_{1}-x_{2}}{\alpha-1}\vee 0$
$\overline{m}:=\tfrac{\alpha x_{1}+x_{2}}{\alpha+1}$

candidate $2$ has secured the election if and only if
$m\in(\underline{n},\overline{n})$ with

$\underline{n}:=\tfrac{\alpha x_{2}+x_{1}}{\alpha+1}$
$\overline{n}:=\tfrac{\alpha x_{2}-x_{1}}{\alpha-1}\wedge 1$

###### Proof.

candidate $1$ has secured the election when
$u_{m}^{1}(x_{1},x_{1})>\max\limits_{y}u_{m}^{2}(x_{2},y)$. After
simplification, we find that it is the case when:

$(m-x_{2})^{2}>\frac{1+a}{a}(m-x_{1})^{2}$

A case by case analysis (whether $m<x_{1}$, $m\in[x_{1},x_{2}]$ or $m>x_{2}$
gives that candidate $1$ has secured the election if $m\in(\tfrac{\alpha
x_{1}-x_{2}}{\alpha-1},\tfrac{\alpha x_{1}+x_{2}}{\alpha+1})$. Because $m$ has
a support in $[0,1]$ that is $m\in(\underline{m},\overline{m})$. The
computations are symmetric for candidate $2$. We easily verify that
$\underline{m}<x_{1}<\overline{m}<\tfrac{x_{1}+x_{2}}{2}<\underline{n}<x_{2}<\overline{n}$.
∎

######  Lemma 2.

Consider the ex-post platform that maximizes the utility of the median voter
from candidate $i$:

$\hat{y_{i}}=\operatorname*{arg\,max}_{y_{i}}u_{m}(x_{i},y_{i})=\tfrac{m+ax_{i}}{1+a}$

We have:

  1. 1.

If candidate $i$ is the favorite, any platform
$y_{i}\notin\\{x_{i},\hat{y_{i}}\\}$ is either dominated by or redundant with
a strategy in $\\{x_{i},\hat{y_{i}}\\}$.

  2. 2.

If candidate $i$ is the challenger, any platform
$y_{i}\notin\\{x_{i},\hat{y_{i}}\\}$ is weakly-dominated by the platform
$\hat{y_{i}}$ .

  3. 3.

Weakly dominated actions are not played in equilibrium.

Hence, at equilibrium, candidates select $\hat{y_{i}}$ when they adjust their
platforms.

###### Proof.

Suppose w.l.o.g. that candidate $1$ is the favorite. For any pair of ex-ante
platforms $(x_{1},x_{2})$, we define
$W_{1}:=\\{y_{1}|u_{m}^{1}(x_{1},y_{1})>\displaystyle\max\limits_{y_{2}}u_{m}^{2}(x_{2},y_{2})\\}$,
the set of policies of candidate $1$ that guarantee him to win the election.

1\. By definition, $\hat{y_{1}}\in W_{1}$. In addition, If $x_{1}\in W_{1}$,
any action $y_{1}\neq x_{1}$ is strictly dominated by $x_{1}$ because $x_{1}$
yields a payoff of $1$ and any other actions yields at most $1-\phi$, hence it
is strictly dominated by $x_{1}$.

If $x_{1}\notin W_{1}$, any action $y_{1}\in W_{1},y_{1}\neq\hat{y_{1}}$ is
redundant with $\hat{y_{1}}$ and any action $y_{1}\notin W_{1},y_{1}\neq
x_{1}$ is weakly dominated by $\hat{y_{1}}$. Indeed, consider $y\in
W_{1}\setminus\\{\hat{y_{1}}\\}$. Both $y$ and $\hat{y_{1}}$ ensure a victory
with payoff $1-\phi$ regardless of the platform of candidate $2$. Hence, $y$
is redundant with $\hat{y_{1}}$.

Consider instead $y\notin W_{1},y\neq x_{1}$. Consider strategy $\sigma$ that
chooses $y$ with positive probability and strategy $\sigma^{\prime}$ which is
identical to $\sigma$, except that it plays $\hat{y_{1}}$ instead of $y$. Then
the payoff of candidate $1$ with $\sigma^{\prime}$ is weakly higher than with
$\sigma$ regardless of the strategy of candidate $2$. Indeed, the expected
payoff when using $y$ includes a term of the form $p-\phi$, where $p$ is the
probability to win when using $y$ against the strategy of candidate $2$, while
the corresponding term when using $\hat{y_{1}}$ is $1-\phi$, as $\hat{y_{1}}$
wins with probability 1. The other terms in the expected payoff remain the
same.

2\. Any strategy that chooses $y\notin\\{\hat{y_{2}},x_{2}\\}$ with some
positive probability is weakly dominated by a strategy that transfers this
probability to $\hat{y_{2}}$. In both cases, $\phi$ is paid, but
$u^{2}_{m}(x_{2},\hat{y_{2}})>u^{2}_{m}(x_{2},y)$ so $\hat{y_{2}}$ wins in all
the events in which $y$ wins (and possibly in other events), resulting in a
weakly higher payoff.

3\. Weakly dominated action cannot be played in equilibrium. Suppose $y\notin
W_{1},y\neq x_{1}$ and suppose that in equilibrium, candidate $1$ plays $y$
with positive probability. Then his expected payoff when choosing $y$ (which,
by indifference, is his equilibrium payoff) is $p-\phi$ where $p$ is the
probability of winning the election, $p\leq 1$. If $p<1$, then $\hat{y_{1}}$
is a profitable deviation of candidate $1$ as he obtains $1-\phi$. If $p=1$,
candidate $2$ always looses the election and his expected payoff is non-
positive. Then in equilibrium candidate $2$ must choose $y_{2}=x_{2}$ which
grants a payoff of $0$, and candidate $1$ again has a profitable deviation to
$y_{1}=x_{1}$ which yields a payoff of $1$. This contradicts the assumption
that $y$ is played with positive probability in equilibrium. ∎

###  A.2 Proof of Proposition 1

If the favorite candidate has secured the election, he has a strictly dominant
strategy $y_{i}=x_{i}$ whose best reply for the challenger is $y_{j}=x_{j}$.
Hence this is the unique subgame equilibrium.

If the election is still open, by Lemma 2, we can limit the analysis to the
$2\times 2$ game in Table 1. The game has no pure strategy equilibria. The
unique mixed strategy equilibrium is such that both candidates are indifferent
between the two pure strategies, given the mixed strategy of the opponent.
Hence the probability $p$ of playing $\hat{y_{i}}$ for the favorite candidate
solves

$-p\phi+(1-p)(1-\phi)=0\quad\Rightarrow\quad p=1-\phi,$

and the probability $q$ of playing $\hat{y_{i}}$ for the challenger candidate
solves

$(1-\phi)=(1-q)\quad\Rightarrow\quad q=\phi.$

###  A.3 Proof of Proposition 3

We perform a backward analysis by supposing that after choosing actions
$x_{1},x_{2}$, players play the second-stage equilibrium provided in
Proposition 1 and Proposition 2.

We first show that in equilibrium, the best response to a certain ex-ante
platform is never a more extreme platform to the same side. Formally and
without loss of generality, if $x_{2}<\frac{1}{2}$ then $BR_{1}(x_{2})\geq
x_{2}$, and if $x_{2}>\frac{1}{2}$ then $BR_{1}(x_{2})\leq x_{2}$, where
$BR_{1}$ is the best response ex-ante platform of Candidate $1$ to the ex-ante
platform $x_{2}$ of Candidate $2$.

To prove that, we show that if $x_{1}<x_{2}<\frac{1}{2}$ then
$g_{1}(x_{1},x_{2})<g_{1}(1-x_{1},x_{2})$. By the uniform distribution of $m$,
$g_{1}(x_{1},x_{2})=g_{1}(1-x_{1},1-x_{2})$. Hence, proving
$g_{1}(x_{1},x_{2})<g_{1}(1-x_{1},x_{2})$ is equivalent to proving that
$g_{1}(x_{1},x_{2})<g_{1}(x_{1},1-x_{2})$. When $x_{1}<x_{2}<\tfrac{1}{2}$ and
Candidate $2$ moves from $x_{2}$ to $1-x_{2}$:

  * •

candidate $1$ is the favorite more often as $\tfrac{x_{1}+x_{2}}{2}$ increases
with $x_{2}$.

  * •

candidate $1$ secures the election more often as both $\overline{m}$ increases
and $\underline{m}$ decreases with $x_{2}$.

We conclude that the expected payoff $g_{1}$ of candidate $1$ is larger, given
$g_{1}=1\times\mathbb{P}(1\text{ FS })+(1-\phi)\times\mathbb{P}(1\text{ FO
})$. We can therefore suppose that $x_{2}\geq\tfrac{1}{2}$ and conclude that
the best response of Candidate $1$ belongs to $[0,x_{2}]$.555In the border
case where $x_{2}=\frac{1}{2}$, both sides of $x_{2}$ are symmetric and for
every best response $x_{1}$ in $[x_{2},1]$, the ex-ante platform
$1-x_{1}\in[0,x_{2}]$ is also a best response. The case where
$x_{2}\leq\tfrac{1}{2}$ and $x_{2}\leq x_{1}$ is symmetric.

For $x_{1}<x_{2}$, in each region of the graph in Figure 1, the payoff is
determined by Proposition 1:666We can safely ignore null probability events
regarding the boundaries, such as $m=\overline{m}$.

  * •

If $m\in(\underline{m},\overline{m})$, candidate $1$ has secured the election
and his second-stage equilibrium payoff is $1$.

  * •

If $m\in[0,\underline{m})\cup(\overline{m},\frac{x_{1}+x_{2}}{2})$, candidate
$1$ has secured the election and his second-stage equilibrium payoff is
$1-\phi$.

  * •

If $m\in(\frac{x_{1}+x_{2}}{2},1]$, candidate $1$ is the challenger and his
second-stage equilibrium payoff is $0$.

Because $m$ is uniformly drawn on the unit interval, the ex-ante payoff of
candidate $1$ is

$g_{1}(x_{1},x_{2})=(\overline{m}-\underline{m})+(1-\phi)\left(\tfrac{x_{1}+x_{2}}{2}-\overline{m}+\underline{m}-0\right)$

Substituting the values of $\overline{m}$ and $\underline{m}$ from Lemma 1, we
have that for $x_{1}<x_{2}$

$g_{1}(r,x)=\begin{cases}x_{1}(\tfrac{1-\phi}{2}+\tfrac{\phi\alpha}{\alpha+1})+x_{2}(\tfrac{1-\phi}{2}+\tfrac{\phi}{\alpha+1})\mbox{
if }0\leq x_{1}\leq\tfrac{x_{2}}{\alpha}\\\
x_{1}(\tfrac{1-\phi}{2}-\tfrac{2\phi\alpha}{\alpha^{2}-1})+x_{2}(\tfrac{1-\phi}{2}+\tfrac{2\phi\alpha}{\alpha^{2}-1})\mbox{
if }\tfrac{x_{2}}{\alpha}<x_{1}<x_{2}\end{cases}$

The payoff function $g_{1}$ is therefore increasing with $x_{1}$ in the region
$[0,\frac{x_{2}}{\alpha}]$. In the region $[\tfrac{x_{2}}{\alpha},x_{2})$, the
function $g_{1}$ decreases with $x_{1}$ when
$\tfrac{1-\phi}{2}-\tfrac{2\phi\alpha}{\alpha^{2}-1}<0$, which is equivalent
to

$\phi>\tfrac{\alpha^{2}-1}{\alpha^{2}+4\alpha-1}=\tfrac{1}{1+4\sqrt{a(1+a)}}=\Psi(a)$

It follows that the best response to $x_{2}$ for candidate $1$ in the region
$[0,x_{2})$ is $x_{1}^{*}(x_{2})=\tfrac{x_{2}}{\alpha}$. If instead
$x_{1}=x_{2}$, then according to Proposition 2 the payoff $g_{1}$ is
$\tfrac{1}{2}-\phi$, regardless of $x_{2}$. Note that by choosing
$x_{1}=x_{2}-\epsilon$, the payoff $g_{1}$ would be $x_{2}(1-\phi)$, which is
always greater than $\tfrac{1}{2}-\phi$ for any $x_{2}\geq\tfrac{1}{2}$ and
$\phi>0$. Hence $x_{1}=x_{2}$ cannot be a best reply for candidate $1$.

The same arguments apply to candidate $2$ for any $x_{1}\leq\frac{1}{2}$. The
function $g_{2}$ is increasing with $x_{2}$ in the
$[x_{1},1-\tfrac{1-x_{1}}{\alpha}]$ region and decreasing with $x_{2}$ in the
$[1-\frac{1-x_{1}}{\alpha},1]$ region, when $\phi>\Psi(a).$ Hence, the best
response to $x_{1}$ for candidate $2$ in the region $(x_{1},1]$ is
$x_{2}^{*}(x_{1})=1-\tfrac{1-x_{1}}{\alpha}$. Choosing $x_{2}=x_{1}$ cannot be
optimal as it is dominated by $x_{2}=x_{1}+\epsilon$.

To conclude, when $\phi>\Psi(a)$ the function $g_{1}$ admits a global maximum
in $x_{1}^{*}(x_{2})=\tfrac{x_{2}}{\alpha}$ and the function $g_{2}$ admits a
global maximum in $x_{2}^{*}(x_{1})=1-\tfrac{1-x_{1}}{\alpha}$. In
equilibrium, both candidates best respond to each other, and these two
equations provide the unique profile: $x_{1}^{*}=\tfrac{1}{\alpha+1}$ and
$x_{2}^{*}=\tfrac{\alpha}{\alpha+1}$. The payoffs associated with these
locations are:

$g_{1}^{*}=g_{2}^{*}=\frac{1}{2}-\frac{\phi}{2}\left(\frac{\alpha-1}{\alpha+1}\right)^{2}$

###  A.4 Proof of Proposition 4

According to the proof of Proposition 3, since $\phi<\Psi(a)$, the payoff
$g_{1}$ is strictly increasing with $x_{1}$ in the $[0,x_{2})$ region. The
unique possible equilibrium is then $x_{1}=x_{2}$, where the payoff is
discontinuous. At the limit $x_{1}\rightarrow x_{2}$, both $\underline{m}$ and
$\overline{m}$ converge to $x_{2}$, so:

$\lim\limits_{x_{1}\to x_{2}}g_{1}(x_{1},x_{2})=\lim\limits_{x_{1}\to
x_{2}}\tfrac{x_{1}+x_{2}}{2}(1-\phi)+\phi(\overline{m}-\underline{m})=x_{2}(1-\phi)$

On the other hand, the second stage equilibrium when $x_{1}=x_{2}$ yields a
payoff of $g_{1}=\tfrac{1}{2}-\phi$. The condition for equilibrium can then be
written $x_{2}(1-\phi)\leq\tfrac{1}{2}-\phi$ or equivalently
$x_{2}\leq\tfrac{\tfrac{1}{2}-\phi}{1-\phi}$. Because $0<\phi<\tfrac{1}{2}$
the right hand side of the previous inequality is smaller than $\tfrac{1}{2}$,
so $x_{2}<\tfrac{1}{2}$. By repeating the same calculation for candidate $2$,
we obtain that $x_{1}=x_{2}$ is an equilibrium only when $x_{1}>\tfrac{1}{2}$,
which cannot hold with $x_{1}=x_{2}$ and $x_{2}<\tfrac{1}{2}$.

We now prove that if $\phi<\Psi(a)$,
$(x_{1},x_{2})=(\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon)$ is an
$\epsilon$-equilibrium. Based on the expression computed in subsection A.3,
candidate $1$’s payoff is
$g_{1}(\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon)=(\frac{1}{2}-\epsilon)(\tfrac{1-\phi}{2}-\tfrac{2\phi\alpha}{\alpha^{2}-1})+(\frac{1}{2}+\epsilon)(\tfrac{1-\phi}{2}+\tfrac{2\phi\alpha}{\alpha^{2}-1})=\frac{1-\phi}{4}+\epsilon\frac{4\alpha\phi}{\alpha^{2}-1}$.
On the other hand, because $\phi<\Psi(a)$, $g_{1}(x_{1},\frac{1}{2}+\epsilon)$
is increasing with $x_{1}$ in the region $[0,\frac{1}{2}+\epsilon)$, therefore
we can bound from above the possible payoff of candidate $1$ by
$\displaystyle\lim_{x_{1}\uparrow\frac{1}{2}+\epsilon}g(x_{1},\frac{1}{2}+\epsilon)=\frac{1-\Phi}{4}+\epsilon\frac{1-\phi}{2}$.
The loss of candidate $1$ when playing $\frac{1}{2}-\epsilon$ is

$\displaystyle\lim\limits_{x_{1}\rightarrow\frac{1}{2}-\epsilon}g_{1}(x_{1},\frac{1}{2}+\epsilon)-g_{1}(\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon)$
$\displaystyle=\frac{1-\phi}{4}+\epsilon\frac{1-\phi}{2}-\frac{1-\phi}{4}+\epsilon\frac{4\alpha\phi}{\alpha^{2}-1}$
$\displaystyle=\epsilon\left(\frac{1-\phi}{2}-\frac{4\alpha\phi}{\alpha^{2}-1}\right)\rightarrow
0\text{ as }\epsilon\rightarrow 0.$

###  A.5 Proof of Proposition 5

(i) Based on Proposition 1 and 3, the election is open if and only if
$m\in\left[\frac{2\alpha}{(\alpha+1)^{2}},\frac{\alpha^{2}+1}{(\alpha+1)^{2}}\right]$.
On the other hand, $x_{1}^{*}$ is on the left of this interval and $x_{2}^{*}$
on the right. Because candidate $i$ eventually flip-flops to
$\hat{y_{i}}=\frac{m+ax_{i}^{*}}{1+a}$, he flip-flops towards the center.

(ii) At the mixed equilibrium played in the second stage, if the election is
still open, the favorite candidate flip-flops with probability
$1-\phi>\frac{1}{2}$ and that the challenger flip-flops with probability
$\phi<\frac{1}{2}$.

(iii) Based on Eq. (3), the optimal platforms $\hat{y_{1}}$ and $\hat{y_{2}}$
are the same weighted average of $m$ and respectively $x_{1}$ and $x_{2}$. If
player $1$ is the favorite, $x_{1}$ is closer to $m$ than $x_{2}$. Therefore,
the distance between $\hat{y_{1}}$ and $x_{1}$ is smaller than the distance
between $\hat{y_{2}}$ and $x_{2}$.

(iv) By point (ii) and the fact that a favorite candidate $i$ always wins if
he moves to $\hat{y_{i}}$.

###  A.6 Proof of Proposition 6

(i) Based on Proposition 3, equilibrium location are given by
$\left(\frac{1}{1+\alpha},\frac{\alpha}{1+\alpha}\right)$. The distance
between ex-ante platforms is therefore equal to $\frac{\alpha-1}{\alpha+1}$
which is increasing with $\alpha$, thus decreasing with $a$.

(ii) Propositions 1 and 3 together prove that at equilibrium, the election is
still open with probability $\left(\frac{\alpha-1}{\alpha+1}\right)^{2}$ which
is also increasing with $\alpha$, thus decreasing with $a$. Conditionally on
the election to be still open, the likelihood of flip-flopping does not depend
on $a$ (it is $1-\phi$ and $\phi$ for the favorite and the challenger
respectively).

(iii) Proposition 3 gives that equilibrium payoffs are decreasing with the
probability of the election to be still open
$\left(\frac{\alpha-1}{\alpha+1}\right)^{2}$, so this claim is a corollary of
claim (ii).

###  A.7 Proof of Proposition 7

We follow the proof of the symmetric competition (Appendix A.2 and A.3) and
discuss the differences. First, the optimal ex-post positions (given by Eq.
(3) in the symmetric case) differs with $a_{i}$: we find
$\hat{y_{i}}=\tfrac{m+a_{i}x_{i}}{1+a_{i}}$. A candidate with a larger $a_{i}$
is more penalized by the voters and changes less his ex-ante platform than his
opponent. Second, the interval of $m$ for which each candidate has secured the
election generalizes to
$(\underline{m},\overline{m})=(\tfrac{\alpha_{2}x_{1}-x_{2}}{\alpha_{2}-1}\vee
0,\tfrac{\alpha_{2}x_{1}+x_{2}}{\alpha_{2}+1})$ for candidate $1$ and
$(\underline{n},\overline{n})=(\frac{\alpha_{1}x_{2}+x_{1}}{\alpha_{1}+1},\frac{\alpha_{1}x_{2}-x_{1}}{\alpha_{1}-1}\wedge
1)$ for candidate $2$, where $\alpha_{i}=\sqrt{\frac{1+a_{i}}{a_{i}}}$. Notice
that the region where each candidate has secured the election does not depend
on his own parameter $\alpha_{i}$ but on his opponent’s. The intuition is that
the region where candidate $i$ has secured the election is the region where he
wins the election without moving, so the penalty for his movement does not
play a role. Moreover, the larger $a_{j}$, the smaller movement of candidate
$j$ towards the median voter and therefore the region where candidate $i$ has
secured the election is on a broader region of possible $m$.

Next, we solve the inequality
$u_{m}^{1}(x_{1},\hat{y}_{1})>u_{m}^{2}(x_{2},\hat{y}_{2})$, which holds for
$m<\tfrac{\alpha_{1}x_{2}+\alpha_{2}x_{1}}{\alpha_{1}+\alpha_{2}}:=\tilde{m}$.
Note that $\tilde{m}>\tfrac{x_{1}+x_{2}}{2}$ for $a_{1}<a_{2}$ with equality
when $a_{1}=a_{2}$. Hence, in the region
$\left(\tfrac{x_{1}+x_{2}}{2},\tilde{m}\right)$ candidate $2$ is favorite, the
election is still open and candidate $2$ cannot defend his advantage: when
candidate $1$ moves to $\hat{y}_{1}$, he wins the election regardless of the
action of candidate $2$. Since candidate $1$ loses the election if he does not
move, moving is a dominating strategy and in this region the optimal strategy
for candidate $1$ is to move, for candidate $2$ not to move and the payoff is
$(1-\phi,0)$.

Note that the organizational costs are unchanged, so the second stage remains
strictly identical to Proposition 1 for
$m\notin\left(\tfrac{x_{1}+x_{2}}{2},\tilde{m}\right)$. Moreover, the payoff
in $\left(\tfrac{x_{1}+x_{2}}{2},\tilde{m}\right)$ is exactly the same as the
expected payoff in the region where candidate $1$ is favorite and the election
is still open, so, although the second stage strategy is different, in terms
of continuation payoff we can analyze the first stage as if candidate $1$ is
favorite and the election is still open in the region
$[\overline{m},\tilde{m}]$ instead of $[\overline{m},\tfrac{x_{1}+x_{2}}{2}]$.

Finally, repeating the argument in Appendix A.3, we find that it is necessary
and sufficient to have $\phi>\Psi(a_{1})$ to guarantee the existence of a best
response of candidate $1$, which is given by
$x_{1}(x_{2})=\tfrac{x_{2}}{\alpha_{2}}$, and analogously, we have
$x_{2}(x_{1})=1-\tfrac{1-x_{1}}{\alpha_{1}}$ when $\phi>\Psi(a_{2})$.
Together, we conclude that if $\phi>\max(\Psi(a_{1}),\Psi(a_{2}))$, then
$x_{1}^{*}=\frac{\alpha_{1}-1}{\alpha_{1}\alpha_{2}-1}$ and
$x_{2}^{*}=\frac{\alpha_{2}(\alpha_{1}-1)}{\alpha_{1}\alpha_{2}-1}$. If any of
the thresholds is higher than the organizational cost, there exists no
equilibrium.

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