Single-Issue Campaigns and Multidimensional
Politics
Georgy Egorov
Northwestern University
February 2014
Abstract
In most elections, voters care about several issues, but candidates may have to choose only
a few to build their campaign on. The information that voters will get about the politician
depends on this choice, and it is therefore a strategic one. In this paper, I study a model of
elections where voters care about the candidates’ competences (or positions) over two issues,
e.g., economy and foreign policy, but each candidate may only credibly signal his competence
or announce his position over at most one issue. Voters are assumed to get (weakly) better
information if the candidates campaign on the same issue rather than on di¤erent ones. I
show that the …rst mover will in equilibrium set the agenda for both himself and the opponent
if campaigning on a di¤erent issue is uninformative, but otherwise the other candidate may
actually be more likely to choose the other issue. The social (voters’) welfare is a non-monotone
function of the informativeness of di¤erent-issue campaigns, but in any case the voters are better
o¤ if candidates are free to pick an issue rather than if an issue is set by exogenous events or by
voters. If the …rst mover is able to reconsider his choice if the follower picked a di¤erent issue,
then politicians who are very competent on both issues will do so. If the decision to move …rst
or second is endogenized, choosing to move …rst signals incompetence in one of the issues and
thus politicians may wait until one has to make a move; however, the one moving …rst is more
likely to be overall more competent and more likely to be elected. The model and these results
may help understand endogenous selection of issues in political campaigns and the dynamics of
these decisions.
Keywords: Elections, campaigns, issues, salience, competence, probabilistic voting.
JEL Classi…cation: D72, D82 .
I am grateful to David Austen-Smith, V. Bhaskar, César Martinelli, Mattias Polborn, Maria Socorro Puy, Konstantin Sonin, and participants of the PECA 2012 conference, NES 20th Anniversary conference, the Econometric
Society meeting in Los Angeles in 2013, Wallis 2013 Conference in Rochester, and seminars at the University
of Pennsylvania, New York University, MIT, University of British Columbia, University College London, and
Queen’s University at Belfast for valuable comments.
1
Introduction
Political competition in one-dimensional policy space with two candidates is one of the most
well-studied areas in political economy, and formal analysis goes at least as far as Downs (1957).
The assumption of two candidates is not unrealistic; moreover, Duverger’s Law (Riker 1982,
see also Lizzeri and Persico 2005) predicts emergence of two major candidates or parties in
majoritarian (winner-take-all) elections, as is evident from the USA. Thus, we can expect, under
certain conditions, that political systems with multiple political actors endogenously evolve into
ones with only two. This observation prompts a natural question: Is it true, in some sense,
that political preferences over multiple dimensions nevertheless endogenously result in political
competition over a single issue, and if so, how is it chosen?
In this paper, I study endogenous selection of issues that make political campaigns. I assume that voter preferences are two-dimensional (and separable), and two politicians decide the
issue they want to run on. The following friction is key: I assume that a politician can only
credibly announce his position on (at most) one issue, but not on both.1 The motivation is that
persuading voters about his/her position or about his/her talents in a particular sphere is hard,
and there is only a limited amount of time. Many voters make their decision based on a single
debate performance, or a single rally they attend, so losing focus may be costly for candidates.
Even though occasionally candidates talk about several issues, the broad idea of the campaign
is typically clear: e.g., Bill Clinton ran on economy in 1992 while George H.W. Bush on foreign
policy; in 1972, George McGovern ran against the Vietnam war; in 2012, the Romney campaign
attacked the economic record of Barack Obama, and the Obama campaign eventually switched
from focusing on social issues to defending his economic record of his …rst term. Of course, in
the course of long campaigns, the focuses of the campaign, and even issue salience itself might
change, but it is probably reasonably to assume that at each stage (e.g., in a given debate) the
candidate has to adopt a particular line of attack or defense, which may or may not coincide
with the one chosen by his opponent.
The second key assumption I make is that voters get better information about a candidate’s
position or competence in a given issue if both of them choose to talk about it. In other words,
there is a certain complementarity: the more likely a politician’s opponent to talk about, say,
economy, the more credibly the politician himself will sound if he chooses to talk about economy
as well. Indeed, the other politician may criticize his positions, check whether the facts are
1
Polborn and Yi (2006) consider politicians who can either run positive (revealing information about themselves) or negative (revealing information about their opponents) informative campaigns, but not both. Alternatively, one can assume that politicians have a limited resource (time) that they need to spend on di¤erent issues.
Whenever this last problem has corner solutions, the two approaches yield identical results.
1
correct, and thus indirectly add credibility to the politician’s statements.2 On the other hand,
if the other politician talks on another topic, then one’s own credibility is undermined. I do
not model the process of making statements and having the other party check their veracity
explicitly; but the assumption that voters learn more if politicians have a sensible debate on the
same issue seems reasonable.3
These two assumptions lead to a simple and tractable model of issue selection. I …rst study a
basic case where one politician (dubbed “Incumbent”) commits to build his campaign on one of
the two dimensions. The other politician (“Challenger”) observes this, and decides whether to
reciprocate or to talk about the other dimension instead. Voters are Bayesian and they update on
the politician’s decisions and the information that they observe, and vote, in a probabilistic way,
for one of the two candidates. The model gives the following predictions. First, if campaigning
on di¤erent issues gives politicians very little credibility, there is a strong unraveling e¤ect,
which forces Challenger to respond to Incumbent and talk about the same issue. However, if a
politician is quite likely to be able to credibly announce his position or establish his competence
even when talking on a di¤erent issue, then divergence is possible, and in fact it is slightly more
likely that Challenger will choose a di¤erent issue. Second, the social welfare (as measured by
the expected competence of the elected politician) does not necessarily increase in the ability
of politicians to make credible announcements when the opponent campaigns on a di¤erent
issue. The reason is that when this ability is low, politicians will endogenously choose the same
issue, and this will help voters rather than hurt them. Third, if the …rst-mover (Incumbent)
is allowed to reconsider his initial choice of issue (e.g., because the opponent picked a di¤erent
issue, or because the campaign is long enough), then switching focus is not a signal of weakness;
rather, it signals competence in both issues. Fourth, if voters are more informed on Incumbent’s
competence in one of the issues, say, economy, then he is more likely to campaign on the other
issue that voters are more uncertain about. This implies, in particular, that Incumbent is likely
to run for reelection on an issue di¤erent than the one where he had a chance to demonstrate his
competence (or incompetence) during the …rst term; doing the opposite would be interpreted as
lack of competence in the other dimension. Finally, if politicians have some discretion whether
2
Bhattacharya (2011) models the process of information revelation through positive and negative ads.
One way to microfound this assumption is by introducing a third-party fact-checker which is active only some
of the time; however, if both candidates campaign on the same issue, their opponents serve this role automatically.
An alternative way is to assume that political positions of politicians are correlated, and when one candidate states
his position, voters update on the other candidate’s position as well. For example, Barack Obama’s open support
of gay marriage might be interpreted as a belief shared by all politicians of his generation and thus have little
impact for voters’decision to support or oppose him — unless his opponent makes a clear stance, too. In this case,
again, voters will get more precise signals about the di¤ erence in the candidate’s positions when they campaign
on the same issue. I do not model either mechanism explicitly in order to focus on consequences of single-issue
political competition.
3
2
to be the …rst or second to start a campaign, they would prefer to postpone a campaign as
this would signal their relative indi¤ence between the two issues; ultimately, those who choose
to move …rst are likely to be more competent (and also disproportionately competent in one
issue relative to the other). As such, there is a …rst-mover advantage, even though ex-ante, the
expected competences of the two politicians are equal.
Strategic choice of campaign issues by politicians has been the topic of a number of descriptive
and formal studies. Riker (1996, see also 1993), in an extensive study of the U.S. Constitution
rati…cation, observes that politicians were likely to abandon an issue where they could not
beat their opponents; this means that debating the same issue should be rarely observed, and
attributed to overcon…dence or lack of information. This observation predicts “issue ownership”
(as in Petrocik 1996, see also Petrocik et al., 2003), but it relies on the assumption of nonBayesian and somewhat “automatic” voters. An in‡uential body of the literature focuses on
priming, i.e., that campaigning on an issue attracts voters’ attention, and in the end of the
day, the voters begin to put a higher weight on this issue, i.e., their preferences change. In this
spirit, Aragonès, Castanheira, and Giani (2012) study a model where parties invest in generating
alternative proposals to the status quo, and then advertise this proposal to increase the salience
of this issue; they show that parties are most likely to invest in issues where they hold ex-ante
advantage (own), although sometimes “issue stealing” is also possible. Amorós and Puy (2007)
consider a model where campaigns allocate an advertisement budget to increase the salience of
an issue (see also Colomer and Llavador, 2011, for a related model). Other papers emphasize
signaling considerations; e.g., Morelli and van Weelden (2011) predict that a politician will spend
a disproportionate amount of e¤ort on a divisive issue for the purpose of credible signaling.
The paper most closely related to this one is Polborn and Yi (2006). There, politicians
also campaign on their qualities, and there may be positive and negative information available
about each of them (four dimensions totally). Politicians choose the campaign issue among two
options: each politician can reveal positive information about himself or negative information
about his opponent. The authors characterize a unique equilibrium, in which running a negative
campaign reveals lack of positive information about oneself. This setting corresponds to the case
where politicians have perfect ability to reveal their competence even if they talk about di¤erent
issues. This paper generalizes Polborn and Yi (2006) for the case of generic campaign issues
where campaigning on di¤erent issues may undermine voters’ability to learn the truth.4
The model in this paper assumes that the relevant characteristic of candidates is their competence in each of the two issues, and all voters have the same preferences (more competence
4
Duggan and Martinelli (2011) study a model of media slant, where media are assumed to collapse a multidimensional policy position to a one-dimensional one.
3
is better), but similar forces would be in e¤ect if candidates were competing on more divisive
issues; in this case, the counterpart of competence would be proximity of candidate’s ideal point
(in a given dimension) to the median voter’s position. The model would predict that politicians
would have an incentive to campaign on an issue where their position is close to the median
voter’s one, and if politician turns out to be far from the median voter on the issue that he
chose, voters would suspect that he is even more radical on the other issue. The results would
be valid under the following assumptions: that the candidates cannot commit to any policy
position other than their ideal one in the course of the campaign, and also that they cannot lie
about their position (or, more precisely, that they cannot lie if the other party is campaigning
on the same issue and can reveal this lie to voters).5 In the current model, voters’preferences
are aligned and competence is unambiguously good, so pandering must take the form of exaggerating one’s competence. The results are driven by the assumption that doing so is easier if
the opponent talks about a di¤erent issue; the assumption that exaggeration is either in…nitely
costly (or at least that competence is fully revealed) or totally uncontrolled to the point that
the candidate has zero credibility makes the model tractable, but hardly drives the results.
The rest of the paper is organized as follows. Section 2 introduces the basic model. Section
3 studies the equilibria of the basic model, with sequential or simultaneous moves, and obtains
implications for social welfare and for election probabilities. In Section 4, I study a dynamic
version, where both politicians get a chance to respond to each other’s choice of issue. Section 5
discusses several extensions of the basic model. Section 6 concludes. Appendix A contains the
main proofs; Appendix B (not-for-publication) contains auxiliary proofs and out-of-equilibrium
cases.
2
Model
Consider a two-dimensional policy space, one dimension being economy (E) and the other being
foreign policy (F ). There are two politicians, which we will refer to as Incumbent, indexed by
i, and Challenger, indexed by c; this is mainly done for brevity, and otherwise they will be
symmetric until we relax this assumption in Section 5. Each politician has a two-dimensional
type a = (e; f ), which corresponds to his ability in economic and foreign policy questions,
respectively (in what follows, ajs will denote the projection of a on s 2 fE; F g). Consider an
5
There is a large literature on pandering to voters by partisan politicians as well as obscuring one’s positions,
both on campaign trail and in o¢ ce, starting with Shepsle (1972). Alesina and Cukierman (1990) suggest that
incumbents have an incentive to be ambiguous (see also Heidhues and Lagerlof, 2003). Callander and Wilkie (2007)
talk about lying on the campaign trail, as does Bhattacharya (2011). Kartik and McAfee (2007) consider signaling
motive in policy choices; Acemoglu, Egorov, and Sonin (2012) suggest that signaling may make politicians choose
policies further from the median voter rather than closer to his position.
4
electorate with perfectly aligned preferences: there is a continuum of voters, the utilities of each
of which if politician of type (e; f ) is elected is
U (e; f ) = e + f ;
(1)
in doing so, I assume that voters weigh both issues equally (this is done for tractability). The
politician’s type is his private information and is not known to the other politician and voters.
At the time of voting, all voters have the same information on both Incumbent and Challenger:
they know the history of the candidates’moves as well as the moves of the Nature; we capture
this information (or, more precisely, the distribution of (ei ; fi ; ec ; fc ) conditional on the history
by I for brevity). Voting is probabilistic: voter j votes for Incumbent if and only if
E (U (ei ; fi )
where
that
A<
is a common shock and
j
is distributed uniformly on
1
2
and B <
U (ec ; fc ) j I) > +
j,
(2)
is voter j’s individual shock. As is standard, we assume
1
1
2A ; 2A
and
j
distributed uniformly on
1
1
2B ; 2B
, where
A
2A+1 .
During the campaign, each politician can only talk about one issue, economy or foreign policy.
If both talk about the same issue, they have a reasonable conversation or debates, during which
the voters perfectly learn their competences on this dimension (ei and ec or fi and fc ). However,
if they end up talking about di¤erent issues, it is much harder for them to do it credibly (e.g.,
the other side is not actively engaged in responding or fact-checking). In this case, the chance
that the voters will actually …nd out his competence on this issue is
corresponds to zero credibility, and
2 [0; 1], where
= 0
= 1 is the other extreme where a politician’s ability to
make credible statements of his competence does not depend on the issue his opponent chose.
In a sense, a lower
corresponds to a higher noise in communication between politician and
voters if there is nobody around to limit exaggerations or blu¢ ng.6
More precisely, the types of Incumbent and Challenger are given, respectively, by ai =
(ei ; fi ) 2
i
= [0; 1]2 and ac = (ec ; fc ) 2
c
= [0; 1]2 . Incumbent moves …rst and chooses the
issue of his campaign, di 2 fE; F g; Challenger observes this and chooses his issue dc 2 fE; F g.
In other words, the set of Incumbent’s strategies is Si :
strategies is Sc :
c
fE; F g !
fE; F g.7
i
! fE; F g and the set of Challenger’s
Nature then decides whether each of the candidates
6
One may assume that a politician may announce any competence, but is heavily penalized if he is found to
have exaggerated. When politicians talk about the same issue, there is only chance that some third party is
willing to check the politician’s announcements. However, when they talk about the same issue, there is always
someone to do this, and in this case the politicians have to be credible, so voters learn the true competences on
this issue. On the other hand, if nobody puts a check on politicians, then all announce that they are the most
competent, and Bayesian voters learn nothing. I do not model this explicitly to simplify the exposition.
7
It should be emphasized that the two candidates do not have private information about the types of their
5
is successful in announcing his competence, and voters get signals
i;
issues that each politician is talking about. Thus, for Incumbent,
i
if dc 6= di , then
Challenger,
c
i
= ai jdi with probability
and
c
c
2 [0; 1] [ f?g about the
= ai jdi if di = dc ; whereas
= ? with probability 1
= ac jdc if dc = di , and if dc 6= di , then
c
. Similarly, for
= ac jdc with probability
otherwise. Each voter observes the history of moves (di ; dc ) and signals ( i ;
c ),
and
c
=?
and updates
accordingly, thus getting conditional distribution I, which allows him to compute the di¤erence
in competence between Incumbent and Challenger,
V (di ; dc ;
i;
c)
= E (U (ei ; fi )
After that, the common shock
U (ec ; fc ) j di ; dc ;
i;
c)
and idiosyncratic shocks
= E (U (ei ; fi )
j
U (ec ; fc ) j I) .
are realized for each voter j, and
voter j votes for Incumbent if and only if (2) holds.
Both politicians are expected utility maximizers. The utility of each is normalized to 0 if he
is not elected and to 1 if he is elected, and therefore each maximizes the chance of being elected.
Incumbent, therefore, solves the problem
max E (V (di ; dc ;
di
i;
c)
j ei ; fi ) ,
(3)
j ec ; fc ; di )
(4)
and Challenger maximizes
max E (V (di ; dc ;
dc
i;
c)
where both in (3) and (4) the expectations are taken over the opponent’s type (ac and ai ,
respectively), as well as moves by Nature.
The equilibrium concept is the following re…nement of the standard Perfect Bayesian equilibrium (PBE) in pure strategies: monotone strategies.8 This means that if Incumbent’s strategy
satis…es di (ei ; fi ) = E, then di (e0i ; fi ) = E for e0i > ei and di (ei ; fi0 ) = E for fi0 < fi , and similar
requirements are satis…ed for dc (ec ; fc ; di = E) and dc (ec ; fc ; di = F ).9 This re…nement means
that if some type chooses to discuss economy, then increasing his competence in this dimension
without changing the other does not make him switch to foreign policy, and vice versa. In
addition, I do not distinguish between equilibria which di¤er on a subset of types of measure
zero; for example, if all incumbents with ei = fi are indi¤erent between economy and foreign
policy and can choose either way, this is treated as a single equilibrium.
opponent, and in particular Incumbent makes his decision knowing the issue that Challenger chose, but not
Challenger’s competence in that issue. This is a simpli…cation of reality, but from a technical standpoint, it
prevents politicians from strategically jamming the opponent’s signal if they know it to be very high.
8
Polborn and Yi (2006) introduce a similar re…nement in a game where politicians choose to run a positive
campaign or a negative campaign.
9
Suppose that e0i > ei and the candidate with (ei ; fi ) strictly prefers E while (e0i ; fi ) strictly prefers F . In that
case, the latter would be strictly better o¤ if he could fake a lower competence, i.e., choose E and pretend to be
(ei ; fi ). We do not allow candidates to pretend to have lower competence to simplify the exposition.
6
3
Analysis
In this section, I start by analyzing the game introduced in Section 2. I then consider an
alternative story where both candidates choose issues simultaneously. I conclude this Section by
studying social welfare and compare the results for di¤erent values of the credibility parameter
and for both timings.
3.1
Sequential game
Let us …rst compute the probability of each politician to be elected for any possible posterior
distribution I. For a given , citizen j votes for Incumbent with probability
Pr (
j
< E (U (ei ; fi )
U (ec ; fc ) j I)
)=
1
+ B (E (U (ei ; fi )
2
U (ec ; fc ) j I)
),
(5)
which is also the share of votes Incumbent gets. Incumbent wins if and only if (5) exceeds 12 ,
which happens with probability
Pr ( < E (U (ei ; fi )
U (ec ; fc ) j I)) =
1
+ AE (U (ei ; fi )
2
U (ec ; fc ) j I) .
(6)
Since A is a constant, Incumbent chooses di to maximize
E (E (U (ei ; fi )
U (ec ; fc ) j I))
and Challenger chooses dc to minimize it. Notice that at the time either politician makes a
decision, he knows that he can a¤ect the voters’posterior about his opponent (e.g., Challenger
can make a very competent Incumbent appear worse if he chooses the opposite …eld, because
the latter could then fail to make a credible announcement). However, if he takes expectation conditional only on the information he knows at the time of decision-making, the expected
voters’ posterior is given and he cannot change it. This simpli…es the problem a lot by e¤ectively separating the problems of Incumbent and Challenger (Appendix A contains the details).
From now on, Incumbent and Challenger maximize the expectations of E (U (ei ; fi ) j I) and
E (U (ec ; fc ) j I), respectively; more precisely, Incumbent maximizes
E (E (U (ei ; fi ) j di ;
i)
j ei ; fi ) ,
and Challenger maximizes
E (E (U (ec ; fc ) j dc ;
c ; di )
j ec ; fc ; di ) .
The properties of equilibrium critically depend on whether
exceeds
Proposition gives characterization of equilibrium for a relatively high .
7
1
2
or not. The next
Proposition 1 If
> 12 , there is a unique equilibrium. In this equilibrium, Incumbent chooses
the issue that he is more competent in: economy if ei > fi and foreign policy if fi > ei (and he
is indi¤ erent otherwise). If di = E, then Challenger chooses E if and only if
p
4 5 +
(8 7 )
2
1
fc + C, where C =
,
ec >
2
4 (2
)
(7)
and if di = F ; then a symmetric formula applies.
Not surprisingly, Incumbent always chooses the issue that he is most competent in. The
Challenger’s response depends on the chance that he will be heard if he chooses a di¤erent
issue. If
= 1, then his credibility is the same regardless of the issue, and then his strategy is
independent from the choice of Incumbent: he will always choose the issue his is most competent
in. If
1
2
<
< 1, then he may not be as credible when choosing a di¤erent issue. For a very
competent (in both dimensions) Challenger, this gives a reason to choose the same issue as
Incumbent, in order to signal his competence in either issue and avoid being pooled with the
less competent mass of potential challengers. Conversely, if Challenger lacks competence in both
dimensions, he is better o¤ choosing an issue di¤erent from Incumbent’s choice, because he will
have a better chance to be viewed as an average rather than a very bad type. In equilibrium,
if Incumbent chose Economy, then the set of Challengers who are indi¤erent between the two
alternatives forms a straight line which is steeper than the diagonal (see Figure 1). Intuitively, in
this case, Challenger will have a higher chance to reveal ec if he chooses E than he has to reveal
fc if he chooses F ; thus, it is not surprising that the choice of Challenger is more “sensitive”to ec
than to fc : for example, if
is close to 21 , then Challenger’s decision depends almost exclusively
on ec .
We therefore have the following equilibrium strategies of the politicians. Incumbent always
chooses the issue he is best at. Challengers who are good at one issue and bad at the other
also choose their preferred issue, regardless of the choice of Incumbent. However, Challengers
which have roughly equal abilities in both dimensions and who would otherwise be relatively
indi¤erent respond to Incumbent’s pick of issue in a non-trivial way: those who excel in both
dimensions pick the same issue, whereas very incompetent ones choose a di¤erent dimension.
The equilibrium strategies are depicted on Figure 2. Notice that for
close to 1, the lines
separating the four regions converge to the diagonal, and Challenger’s decision becomes largely
independent from that of Incumbent. Conversely, if
is close to 21 , the four regions have (almost)
equal size, and half of Challengers will condition their choice of issue on that of Incumbent (the
northeastern corner will always pick the same issue and the southwestern one will always pick
the other issue).
8
Figure 1: Challenger’s equilibrium response if Incumbent picked Economy;
Figure 2: Equilibrium strategies of Incumbent and Challenger if
9
1
2
<
1
2
<
< 1.
1.
The formal proof of Proposition 1 is in the Appendix, but the idea is relatively straightforward. The Incumbent’s problem is symmetric, and thus it is natural to expect symmetric
equilibrium strategies. At the same time, picking the issue where one has a disadvantage cannot
happen in equilibrium: then deviating and choosing the other issue would send the voters a
better signal on both issues. The Challenger’s problem is more complicated. For simplicity,
suppose that both issues E and F are picked with a positive probability, and suppose that
Challenger with type (ec ; fc ) = (x; y) is indi¤erent, whereas those with a higher ec or lower
fc choose E and those with lower ec or higher fc choose F (Appendix …lls in the details on
why such equilibrium exists and why there are no other equilibria). Suppose that Incumbent
chose E; then, if Challenger chooses E, the voters will perceive him as having total competence
x + E (fc j di = E; d (ec ; fc ) = E; ec = x) = x + y2 . If he chooses F , then with probability , sim-
ilarly, voters will perceive his total competence as y + E (ec j di = E; dc (ec ; fc ) = F; fc = y) =
y + x2 . With complementary probability 1
, however, they will only know that he chose F ,
and will think of him as E (ec + fc j di = E; dc (ec ; fc ) = F ), which we denote by ZF . Then the
indi¤erence condition for Challenger of this type may be written as
x+
y
=
2
y+
x
+ (1
2
) ZF ,
(8)
and thus we get an upward-sloping boundary between E and F with a slope
only if
2
2
1
(this is positive
> 12 ). Following the intuition above, conjecture that Challenger of type (1; 1) chooses
E and one of type (0; 0) chooses F ; then (8) intersects the boundary of
and (b; 1). If so, one can compute ZF directly: ZF =
a2 +ab+a+b2 +2b
3(a+b)
c
at some points (a; 0)
. Substituting this into (8),
we have an equation onp(x; y) which should
p hold for (a; 0) and (b; 1). It is then straightforward
(8 7 )
(8 7 )
3 +
4 5 +
,b=
, which establishes (7).
to …nd that a =
4(2 )
4(2 )
Are the two politicians ex ante more likely to campaign on the same issues or on di¤erent
issues? Surprisingly, the answer is that campainging on di¤erent issues is more likely (and this
is captured on Figure 2): Challenger if more likely to choose economy if Incumbent chose foreign
policy, and vice versa. More precisely, we have the following result.
Proposition 2 If
> 12 , then the probability of having politicians talk about di¤ erent issues is
higher than 12 : it increases in
at
=
4
5.
on
1 4
2; 5
and decreases on
4
5; 1
, thus reaching its maximum
Nevertheless, the probability that either politician successfully communicates his
competence on the chosen dimension is strictly increasing in .
It is true that politicians who choose the opposite issue have worse expected quality, and thus
candidates have an incentive to pool with those who choose the same one. However, the result
will be more intuitive if we recall that a campaign where one can reveal one’s competence on one
10
dimension but not the other punishes politicians who are equally competent (or incompetent) on
both issues disproportionately harshly, and these types are key to determine whether Challenger
is more likely to choose the same or a di¤erent issue. In particular, suppose that type
1 1
2; 2
is indi¤erent, which would be true if both cases are equally likely. If Challenger of this type
reveals his competence in either issue, the voters’posterior belief about his competence will be
1
2
+
1
4
= 34 . However, it is easy to see that the average competence of politicians choosing the
opposite issue is (weakly) larger than this, and therefore, as long as
< 1, the type
1 1
2; 2
would
choose the opposite issue, too. Intuitively, politicians who excel in only one dimension are more
likely to be interested in revealing their competence in this dimension, whereas more symmetric
politicians are relatively less interested in communicating credibly and thus are more likely to
campaign on a di¤erent issue. The di¤erence reaches its maximum at
on di¤erent issues are almost eight percentage points (precisely,
p
3
3
= 45 , where campaigns
1
2
0:077) more likely
than campaigns on the same issue.
If
1
2,
the equation (8) no longer de…nes an upward-sloping boundary, and thus equilibrium
must be di¤erent. As it turns out, there are multiple equilibria.
Proposition 3 If
1
2,
then there are multiple equilibria, and Incumbent’s strategy is the
same (di = E i¤ ei > fi ) in every symmetric equilibrium. The strategy of Challenger may fall
into one of the two classes:
(i) Challenger always conforms to Incumbent’s choice of issue: dc = di ;
(ii) If Incumbent chose di = E, then Challenger chooses E if ec > a and chooses F if ec < a;
similarly, if di = F , then Challenger chooses F if fc > a and E if fc < a. There is an equilibrium
with such equilibrium strategies if and only if a 2 [ ; 1
].
The equilibrium strategy of Challenger may, therefore, be either to reciprocate Incumbent or
to play a strategy which only depends on his competence in the issue chosen by Incumbent. If
Challenger is expected to play a similar strategy (mutatis mutandis) for the two possible choices
of Incumbent, the latter expects that he would be successful in communicating his competence
with the same probability regardless of the issue he chooses. In this case, Incumbent will
use a symmetric strategy. In principle, there exist equilibria where Challenger plays di¤erent
strategies if Incumbent chose E and F , and then Incumbent must play asymmetrically as well:
for example, if Challenger always picks the same issue as Incumbent if di = E but not if
di = F , then Incumbent of type (1; 1) should strictly prefer E and Incumbent of type (0; 0)
should strictly prefer F . To avoid these complications, from now on I focus on symmetric
equilibria, i.e., equilibria where the equilibrium play if Incumbent and Challenger have types
11
(ei ; fi ; ec ; fc ) = (a; b; c; d) and (ei ; fi ; ec ; fc ) = (b; a; d; c) will be the opposite.10
To illustrate the equilibria of type (i), suppose Incumbent chose E.
Challenger chooses E, then E (fc j di = E; d (ec ; fc ) = E; ec = x) =
1
2
If for all (ec ; fc ),
for all x; at the same
time, E (ec j di = E; dc (ec ; fc ) = F; fc = y) = zy may be chosen arbitrarily, as may ZF =
E (ec + fc j di = E; dc (ec ; fc ) = F ). If Challenger of type (x; y) chooses E, voters believe his
competence is x + 12 , and if he chooses F , they believe it is
(y + zy ) + (1
) ZF . Clearly, the
type most likely to deviate is (x; y) = (0; 1); Challenger of this type does not deviate if and only
if
1
2
Since z1 ; ZF
(1 + z1 ) + (1
0, such equilibrium is possible only if
) ZF :
1
2.
At the same time, for such values
of , it is indeed an equilibrium; it su¢ ces to set zy = 0 for all y and ZF = 0.
Equilibria of type (ii) are also possible only if
1
2.
Indeed, …x a, and suppose that
Incumbent chose E and Challenger chooses E if ec > a and chooses F if ec < a. Then
1
2
E (fc j di = E; d (ec ; fc ) = E; ec = x) equals
if x > a and may be chosen arbitrarily if x < a
(denote it zx ). At the same time, E (ec j di = E; dc (ec ; fc ) = F; fc = y) =
have E (ec + fc j di = E; dc (ec ; fc ) = F ) =
a
2
a
2
for all y; we also
+ 12 . To verify whether threshold a constitutes an
equilibrium, it su¢ ces to concentrate on the types most likely to deviate. In equilibrium, type
(a + "; 1) must prefer E and type (a
a
a+"+
1
2
" + za
"
"; 0) must prefer F ; this yields two conditions:
a
+ (1
2
a
0+
+ (1
2
1+
)
)
a 1
+
2 2
a 1
+
2 2
;
:
Since these conditions need to hold for " arbitrarily close to 0, the …rst condition implies a
whereas the second one (setting za
"
= 0) implies a
1
,
. It is now straightforward to
show that Challengers with ec 6= a do not have incentives to deviate. Technically, it remains to
de…ne strategies for ec = a (this may not be done arbitrarily) and show that there are still no
deviations, but this is simple and is done in the Appendix.
In Subsection 3.3, I compute welfare of voters under di¤erent equilibria and parameter values,
and it turns out that if
1
2,
the equilibrium of type (i) dominates every equilibrium of type
(ii) in terms of welfare. The reason for this is that the two politicians always campaign on the
same issue, and a low
does not result to loss of information due to their campaigns lacking
credibility. The equilibrium of type (i) is unique for any , and thus this equibrium re…nement
10
There are many alternative ways to de…ne the same restriction. E.g., it would be su¢ cient to assume that
the chances of Incumbents (Challengers) with type (x; y) and with type (y; x) to be elected are the same. It is
remarkable that for > 21 , such symmetry need not be assumed, but may rather be proved (see Proposition 1).
12
extends the uniqueness result from Proposition 1 to the case
1
2.
This is the equilibrium we
focus from now on.
3.2
Simultaneous game
In this subsection, consider an alternative game, where the two politicians must choose the issue
simultaneously (or, which is equivalent, Challenger starts his campaign before he gets a chance
to observe the choice of Incumbent). Since there is no way to know the exact sequence of moves,
this makes it a priori an equally interesting case to consider. Thus, we consider exactly the same
game as the one introduced in Section 2, except that Challenger, when deciding on dc , does not
observe the choice of Incumbent, di .
Given the symmetry of the game, it is not surprising that there is a symmetric equilibrium,
where each politician j picks dj (ej ; fj ) = E if ej > fj and dj (ej ; fj ) = F if ej < fj . Indeed, for
either politician, the probability of ending up campaigning on the same issue is exactly
1
2
and
this does not depend on the issue; consequently, each politician is able to send a credible signal
with probability
1
2
+
2
1
2,
regardless of the issue he chooses. Consequently, if one politician
follows this symmetric strategy, then the other politician also must do so in equilibrium (see
Appendix B for a comprehensive analysis of all alternatives). Hence, symmetric strategies by
the politicians are “best responses” to one another, and thus such an equilibrium exists for all
.
This argument does not preclude existence of equilibria which are not symmetric in the two
issues: for example if
1
2,
then there is an equilibrium where di (ei ; fi ) = dc (ec ; fc ) = E for
both politicians and all types; indeed, in Subsection 3.1, it was shown that if Incumbent plays E,
then there is an equilibrium where Challenger picks E regardless of his type as well.11 However,
for
> 21 , only symmetric equilibria exist. Formally, we have the following result.
Proposition 4 For any
there exists a symmetric equilibrium where each politician j 2 fi; cg
chooses dj (ej ; fj ) = E if ej > fj and dj (ej ; fj ) = F if ej < fj . Moreover, if
> 21 , then this is
the only equilibrium.
3.3
Social welfare
In this Subsection, we study the consequences of the issue selection game on social welfare and
provide comparative statics results. For the society, the relevant variable is the expected competence of the elected politician. The following lemma shows that in the probabilistic voting
11
There are other, more exotic equilibria. For example, the following is an equilibrium for
1
E whenever ej
, campaign on F otherwise.
4
13
= 15 : campaign on
model as above, there is a simple formula for this expected competence. This formula applies to sequential and simultaneous games and as well as some other situations, and we use it
throughout.
Lemma 1 Let I (ei ; fi ; ec ; fc ) be the distribution of the skills of the two politicians,
(ei ; fi ; ec ; fc ) 2
i
c,
that voters will get if these politicians follow the equilibrium play.
The expected quality of the elected politician equals
0
Z
B
E (U (ei ; fi ))2 + (U (ec ; fc ))2 j I (ei ; fi ; ec ; fc ) d
1 + A@
(ei ;fi ;ec ;fc )2
where
1
C
2A ,
c
i
is the uniform measure on
i
c.
(9)
Proof. The expected competence of the elected politician equals (dropping the argument at I
for brevity):
Z
1
2
+
= 1+A
Z
+ AE (U (ei ; fi ) U (ec ; fc ) j I) E (U (ei ; fi ) j I)
AE (U (ei ; fi ) U (ec ; fc ) j I) E (U (ec ; fc ) j I)
1
2
E (U (ec ; fc )
d
U (ei ; fi ))2 j I d .
The result would follow immediately if the expected utilities inside the integral were independent.
They are not; for example, a high realization of E (U (ei ; fi ) j I) means that most likely the
politicians are talking about the same issue, and therefore Challenger is likely to have high
average competence. However, conditional on the choices of issues by both politicians, these
variables are independent. In addition, the choice of issue by one politician does not depend on
the competence of the other one, except through the choice of issue, even if the case of sequential
moves. This implies that social welfare may be computed using (9). Appendix A …lls in the
details.
Lemma 1 simpli…es the computations considerably. In particular, it shows that the expected
competence of the elected politician only depends on the sum of variances of posterior beliefs
that politicians’equilibrium play generates. This allows to do the computations for Incumbent
and Challenger separately, only taking into account the equilibrium strategies.
In evaluating the welfare consequences, the following benchmarks are useful. First, if the
winner were picked at random, the average competence would be the unconditional expectation,
i.e., 1. At the opposite extreme is the full information case: if both candidates revealed their
competences to voters on both dimensions, then the expected quality of the winner would be
Z 1Z 1
1
1+A 2
(x + y)2 dxdy 2 = 1 + A.
(10)
3
0
0
14
Thus, 31 A is the extra bene…t of having elections as opposed to picking the candidate randomly;
this is the maximum one can achieve with probabilistic voting where a less competent candidate
has a chance due to a shock to preferences . As the variance of common shock decreases (A
becomes higher), the expected competence of the elected politician would increase.
In the case of sequential voting, we have the following result.
Proposition 5 The expected competence of the elected politician is increasing in A. It is non1
2
5
A if
monotone in ; more precisely, it equals 1+ 24
(if type (i) equilibrium from Proposition
> 21 , it monotonically increases
3, which yields the highest social welfare, is played), and for
from 1 +
3
16 A
<1+
5
24 A
to 1 + 14 A > 1 +
5
24 A.
The nonmonotonicity result is not surprising if one takes into account that for
< 21 , the two
politicians are guaranteed to discuss the same issue (in the equilibrium that maximizes social
welfare at least), whereas for
slightly exceeding
1
2
they will talk about di¤erent issues half of
the time, and thus there is a chance of one-fourth that they will fail to announce their respective
competences. In fact, the expected competence exceeds 1 +
this falls short of the maximal possible gain of
1
3 A,
5
24 A
although if
only if
> 0:7. In all cases,
is close to 1, then 75% of this
9
gain is realized, and even in the worst-case scenario this chance exceeds 56% ( 16
).
Consider now the welfare implications of a game where strategies are chosen simultaneously.
Proposition 4 established that a symmetric equilibrium exists for all , and arguably it is the
most plausible one (and unique if
> 21 ). We get the following comparison in this case.
Proposition 6 In the symmetric equilibrium of the game with simultaneous moves, the expected
quality of the elected politician equals 1 +
game of sequential moves if
1
2,
1+
8
A; this is lower than the expected quality in the
but it is higher in the case
1
2
<
< 1.
This intuition for this result is simple: all things equal, voters make a more informed choice,
and therefore get a higher utility, if politicians reveal more information about their competences.
This is more likely to happen (again, all things equal) if
is high, and given that strategies are
the same, the expected quality is increasing in . The probability of campaigning on the same
issue is in the game with simultaneous moves is
game with sequential moves, it is less than
for such . If
1
2,
1
2
for
1
2;
2
however, Proposition 2 states that in the
1
2; 1
. This explains higher voter welfare
then sequential moves allow the politicians to converge on the same issue
and have an informative discussion at least on that issue. With simultaneous moves, they lack
the ability to converge, and thus the welfare of voters is lower in this case.
If there is a concern that social welfare is lower if politicians fail to coordinate on the same
issue, then one alternative would be to choose an issue (say, at random), and then require that
15
both politicians campaign on that issue. The formula (9) applies to this case as well, and the
expected quality of the winner equals
1+A 2
Z
0
1Z 1
0
1
x+
2
2
dxdy
!
2
1
= 1 + A.
6
(11)
Comparing this result to Proposition 5 reveals that for all , the expected competence in the
case where politicians are free to choose their issue is higher than if they are not. This seems
surprising, because in this case, there is no loss due to possible failure to announce their competences credibly in any dimension. However, there is a di¤erent force at play: with endogenous
choice of issues, the announcement of a politician of his competence over one dimension carries
quite a bit of information about his competence on the other issue, which is not the case if they
were forced to talk on a given issue (this is not true for Challenger, who is the second-mover, if
< 12 , but for Incumbent this is true for any ). It turns out that the latter e¤ect dominates.
Proposition 7 If politicians were forced to campaign on the same dimension, then social welfare
would be strictly lower than in the case of sequential moves for all values of . It would be lower
than in the game with simultaneous moves as well, provided that
>
1
3;
if, however,
<
1
3,
then forcing politicians to campaign on an exogenously given dimension results in a higher social
welfare than a simultaneous-move game.
Figure 3 illustrates the expected qualities of elected politicians under di¤erent scenarios.
Proposition 4, in particular, suggests that campaigning on di¤erent issues may decrease social
welfare, if this prevents voters from getting precise information about candidates’ policies.12
This will not happen if politicians choose issues in a well-established sequence, or at least if
the one who moves second has a chance to observe the choice of the …rst politician and to
respond. This leads to the following nontrivial implication. When candidates choose an agenda
for an entire campaign, the decisions are unlikely to be made at once, and hence constraining
the candidates with an exogenously given agenda is not a good idea. However, when it comes
to some particular event, such as debates, where candidates are likely to prepare their strategy
without observing the opponent’s plan, …xing a particular issue or set of related issues may
make sense. Notice that for
2
1
2; 1
, it is better for voters if politicians made their campaign
decisions independently; intuitively, it is due to the fact that with sequential decisions, it is
more likely that they will campaign on di¤erent issues, while with independent decisions this
probability is …xed at 12 .
12
Caselli and Morelli (2004) and Mattozzi and Merlo (2007) consider very di¤erent models that lead to selection
of incompetent politicians. In this paper, incompetent politicians may get elected because voters do not necessarily
make a strong inference about a politician’s incompetence in the issue he is not campaigning on.
16
Figure 3: Social welfare under di¤erent scenarios and values of .
3.4
Probability of being elected
Another natural question is whether the timing of the game gives an advantage to Incumbent or
Challenger. At …rst glance, the timing allows Incumbent to pick the issue that he prefers, and
thus can guarantee that he will talk about the issue that he excels at. However, the voters are
aware of this incentive and behavior, and discount Incumbent’s competence on the other issue
accordingly. Challenger, on the other hand, does not have such ‡exibility, and if
<
1
2
he is
forced to choose the same issue as the challenger, which might not be his strong side. However,
voters understand this, and do not infer Challenger’s competence in the other dimension from his
announcement, and thus if Challenger turns out to be incompetent in the issue that Incumbent
picked, he is not penalized further. It turns out that in the probabilistic voting model, these
e¤ects cancel each other out; the expected probability of winning only depends on the expected
competence, which is equal in our case for the two politicians. This implies, for example, that
before knowing his type, a politician is indi¤erent between being a …rst-mover and a secondmover in the game.
Proposition 8 The probability of Incumbent winning and Challenger winning are equal.
At the same time, a given type of politician need not be indi¤erent. For example, if
< 1,
then all politicians who are equally competent on both dimensions, with ej = fj < 1, would
prefer to be second-movers rather than …rst-movers. In Section 5, Incumbent will be given an
option to postpone his announcement and allow Challenger to move …rst. It turns out that while
17
Figure 4: Timing of proposal-making by Incumbent and Challenger in the dynamic game.
not all types of Incumbent will use this option, those who do are likely to be competent, and
this in the end of the day will result in observable …rst-mover (incumbent’s) advantage.13
4
Dynamics of campaign
In this Section, I extend the baseline model by allowing the Incumbent (who moves …rst) to
reconsider his initial choice if the Challenger made a di¤erent suggestion. The idea here is to
capture the process of …nding a common theme for the campaign or debates and to make sure
that both players have an opportunity to react to each other’s suggestion. Speci…cally, consider
the following timing. First, Incumbent picks a (tentative) issue for his campaign, d~i . Challenger
observes this choice and responds with his own (…nal) decision dc . If the two issues coincide,
d~i = dc , then the parties proceed to campaigning on this issue, and so Incumbent’s …nal decision
is di = d~i . If, however, d~i =
6 dc , then with some probability , Incumbent has an opportunity
n
o
to revise his initial decision, and is free to pick any di 2 d~i ; dc = fE; F g.14 This timing (as
well as a possible conversation) is shown on Figure 4.15 I assume that voters observe the entire
sequence of moves d~i ; dc ; di ; in particular, they know whether Incumbent proposed the issue
of his campaign di himself, or he started with a di¤erent issue and then switched.
13
In Ashworth and Bueno de Mesquita (2008), incumbency advantage arises in a probabilistic voting model
due to his ex-ante higher competence (which in turn is present because he had won elections before). Since this
paper assumes, for simplicity, that the candidates are ex-ante symmetric, this e¤ect is not present.
14
An alternative interpretation for is the share of voters who will listen to Incumbent after he switches, and
thus 1
is the share fo voters who paid attention in the beginning of the campaign.
15
If Incumbent were allowed to revise his initial pick even if Challenger chose the same issue, then the …rst-stage
announcement by Incumbent would have to be “babbling”. The reason is that in a candidate equilibrium where
…rst announcement is informative, switching to another issue should be interpreted as competence in both issues.
Thus, it would make sense to announce the weaker issue for at least some types, and this cannot be true in
equilibrium. This is discussed later in more detail. I am indebted to V. Bhaskar for the suggestion to explore the
possibility that Incumbent must stick to his original choice if Challenger approves the choice of issue.
18
We are interested in symmetric equilibria, so the …rst proposal by Incumbent is d~i = E if
ei > fi and d~i = F otherwise. If Challenger responds with dc = d~i , then Incumbent does not
have another move; thus, consider the case where d~i = E and dc = F . In this case, Incumbent
may either agree to discuss foreign policy or insist on talking about economy. Suppose …rst
that Incumbent is very competent in economy but not in foreign policy, for example, his type is
(1; 0). For such Incumbent, switching to foreign policy is unlikely to do him any good: indeed, he
would have a (weakly) higher chance to signal his competence credibly, but in the foreign policy
dimension, this is precisely what he prefers to avoid. In contrast, suppose that Incumbent is very
competent, say, type (1; 1
"). In this case, it makes perfect sense to comply with Challenger’s
proposal and switch to foreign policy. By doing so, not only he would be able to signal that
he has a very high competence in foreign policy, in addition, voters will take into account that
foreign policy is his weaker issue, and therefore he must be even more competent in economy
(which he truly is). For such Incumbent, therefore, switching allows to signal competence in both
dimensions, which is otherwise di¢ cult to achieve. If he insisted in talking about economy, he
would, with some probability, reveal his highest competence in this issue, but voters would view
his foreign policy credentials to be average at best (and, in fact, worse than average, because they
expect those who excel in foreign policy to switch). Therefore, one can expect an equilibrium
where, having chosen an issue in the beginning, Incumbent will insist on campaigning on this
issue if his competence in the other issue is low, but will be open to switching if he is competent
on the other issue as well.
What is the choice of Challenger who received a proposal to talk about, say, economy, and
anticipates that if he proposes foreign policy instead, then Incumbent will follow the strategy
above? Such Challenger knows that if he agrees on E, then both will campaign on economy, and
his signal will be credible with probability 1. At the same time, if he chooses F instead, he will
talk about F , but will only be able to send a credible signal with probability
where p is the probability of Incumbent switching to F . Assuming
avoid dealing with multiple equilibria), we are guaranteed to have
>
0
1
2
0
= p + (1
p) ,
(which we do here to
> 21 , and therefore the
characterization of Challenger’s strategies from Proposition 1 applies. It remains to verify that
Incumbent’s strategy to start with the issue he is most competent in is indeed an equilibrium;
p
one can verify that this is true whenever < 2 2 3 3
0:93.16 We therefore have the
following result, which is illustrated by Figure 5.
16
If Incumbent always had the possilibity to switch, then the types who are very competent in both dimensions
would have incentives to pick the weaker issue. Indeed, they plan to switch if Challenger proposes a di¤erent
issue. However, we saw, in Proposition 2, that this will happen with a probability exceeding 12 , therefore, an
incumbent who, on the margin, would prefer to talk about economy, has incentives to start with proposing foreign
policy. The assumption that with some small (less than 10% ) probability Incumbent will not have a chance to
switch ensures that this e¤ect is not strong enough.
19
Figure 5: Equilibrium strategies by Incumbent and Challenger in the dynamic game if
Proposition 9 Suppose that
p
<2 2 3
3
0:93. If
> 12 .
> 12 , there is a unique symmetric
equilibrium. In this equilibrium, Incumbent initially chooses the issue that he is more competent
at, and if Challenger picks a di¤ erent issue, then a positive share of incumbents switch. The
boundary between those willing to switch and those that are not is linear; for
is ‡exible if min (ei ; fi ) >
1
2,
1
2
= 1, Incumbent
max (ei ; fi ), i.e., if Incumbent is relatively symmetric; at
Incumbent is ‡exible if min (ei ; fi ) >
4
p
10
close to
0:28. Challenger follows the strategy described
3
in Proposition 1.
5
Extensions
In this section, I consider several extensions of the baseline model of Section 2.
5.1
Asymmetric uncertainty
The baseline model assumed that the ex-ante distributions of Incumbent and Challenger’s abilities are the same. This was obviously a simpli…cation; …rst, the voters are likely to be better
informed about Incumbent’s competence, and also Incumbent is more likely to be more competent on the grounds that he was selected into the o¢ ce earlier. To study this possibility, suppose
that, from voters’ perpective, Incumbent’s two-dimensional type is taken not from a uniform
distribution on [0; 1]
[0; 1], but instead from a uniform distribution on
0
i
= [e1 ; e2 ]
[f1 ; f2 ];
in particular, e1 = e2 would imply that the voters are perfectly informed about Incumbent’s
20
Figure 6: Information structure and equilibrium strategies.
ability on economy, and f1 = f2 would imply the same on foreign policy. The new distribution
of Incumbent’s type is shown on Figure 6.
The Challenger’s strategies, for either choice made by Incumbent, are the same as speci…ed
in Proposition 1 for
>
1
2
out, critically depends on the shape of the set
than
i
= [0; 1]
1
2.
and Proposition 3 for
0.
i
The strategy of Incumbent, as it turns
In particular, if it is a square, albeit smaller
[0; 1], which means that the residual uncertainty of Incumbent’s competence
in the two issues is the same, then Incumbent will have equal probabilities of campaigning
on both issues. Interestingly, this does not depend on whether he is known to be competent or
incompetent in either dimension; all that matters is residual uncertainty. If, however, uncertainty
about Incumbent’s competence in one of the dimensions is less, then Incumbent is more likely to
campaign in the other dimension (and in particular, the most competent and the least competent
of Incumbent’s types will choose the other dimension, provided that
is not equal to exactly 1.
More formally, we have the following result.
Proposition 10 Suppose that Incumbent’s type is distributed on
(i) If e2
choose E if ei
(ii) If e2
e1 = f2
then if
e1 > fi
e1 < f2
applies if e2
1
e1 > f2
= [e1 ; e2 ]
[f1 ; f2 ]. Then:
f1 , then Incumbent is equally likely to choose either dimension, and will
f1 and will choose F if ei
e1 < fi
f1 .
f1 , then in any equilibrium, Incumbent is more likely to choose F than
E. More precisely, let
e2 e1
f2 f1
0
i
= p+
(1
p) be the probability of being credible in communication;
, then all types of Incumbent will choose F . A similar characterization
f1 .
In other words, Incumbent is always more likely to campaign on the issue where voters are
more uncertain about his competence. At the extreme, if voters perfectly know his competence
21
on either of the dimensions, Incumbent must campaign on a di¤erent issue. Interestingly, this
perfect knowledge is not required for this result. If
6= 1, so there is a slightest chance of
miscommunication, then Incumbent will never choose economy if
will never choose foreign policy if
f2
f 2 f1
e2 e1
e2 e1
f2 f1
is small enough, and
is small enough. Intuitively, if e2
e1 is small and
f1 is not, then even Incumbent with ei = e2 will not want to waste his campaign on the
issue of economy where the campaign cannot make a big di¤erence, and will talk about foreign
policy instead. Thus, one can expect that an Incumbent who had a chance to demonstrate
his true competence (or true incompetence) in one of the issues will build his campaign on a
di¤erent issue.
5.2
Asymmetric issues
We have so far assumed that voters care about both issues equally. However, more generally,
the weights voters put on economy and foreign policy may be di¤erent, i.e., it is possible that
voter’s utility is we e + wf f with we 6= wf .
The case of di¤erent weights is similar to the previous one in Subsection 5.1, with issue
that has a higher weight corresponding to an issue where there is more uncertainty about both
politicians types. (The di¤erence is that both the incumbent and the challenger would have the
same ex-ante distribution, with higher variance on one dimension.) The easiest way to see this is
to renormalize the policitians’types to the scale that voters use to evaluate them. Namely, one
can de…ne e~j = we ej , f~j = wf fj , for j 2 fi; cg (so that both a
~i and a
~c are distributed uniformly
on [0; we ]
[0; wf ]), and from then on assume that the weights are equal: w
~e = w
~f = 1.
It will follow immediately that both the challenger and the incumbent are more likely to
campaign on the issue that voters care about more. Indeed, the e¤ective uncertainty about the
politicians’ types would be greater over that dimension rather than over the issue that voters
care little about. This is realistic: there is little reason for politicians to campaign on the issue
that does not interest voters, and even more, doing so would be interpreted negatively as lack
of competence (or occupying an extreme position) on the issue that concerns voters more. If
the issues in the model were ones where voters had disagreement about, then campaigning on
more divisive issues would sway fewer voters; and so the model predicts that campaigns would
focus on candidates’qualities on issues where preferences are the same, rather than on the issues
where they are di¤erent.17
One less trivial insight is the impact of di¤erent weight that voters put on issues on social
17
The di¤erence with Morelli and van Weelden (2011) is that there, taking positions on divisive issues serves
as signaling, while the issues where voters’preferences are similar does not have the competence component that
voters care about a lot.
22
welfare. In addition to the trade-o¤ between losing information on the other important dimension
and the chance of learning nothing at all, a new problem is that politicians will end up talking
about the less important issue. It turns out that this is not a concern: in equilibrium, politicians
are more likely to choose the issue that voters care about more, and this still makes it optimal
to give politicians the freedom to choose the issue for their campaign rather than force them to
focus on a single (even ex-ante more important) issue. This result, of course, need not hold if
politicians made choices simultaneously.
5.3
Delaying campaign
So far, Incumbent was assumed to be the …rst mover and Challenger was assumed to follow.
Let us augment this game by assuming that Incumbent may instead postpone starting a campaign and wait for Challenger’s choice of issue. More precisely, suppose that with probability
p1 , Incumbent must move …rst, with probability p2 he must wait and move second, and with
probability p0 = 1
p1
p2 he has discretion. The question we study here is under which
conditions he will decide to move …rst and when he will prefer to wait.
The voters observe whether Incumbent moves …rst or second, but are not aware about the
reason (whether he had the discretion or had to for exogenous reasons). The following two
parameters are key:
=
p0
p0 +p1
and
=
p0
p0 +p2 ;
here,
is the probability that moving …rst was a
strategic decision rather than imposed by Nature, and similarly,
captures the probability that
Incumbent who waited had an option to move. As we will see, voters will also update on the
competence that the politician announces; in particular, a …rst move, coupled with a declaration
of a very low competence, is likely to be attributed to the politician being forced to move …rst,
rather than to a conscious decision.
The equilibrium is easy to characterize in the extreme situations, with
and
equal (or
close) to 0 or
1.18
…rst is
= 0 (equivalently, p0 = 0), where voters attribute decision to move …rst or second
=
We therefore study the two limit cases which are tractable analytically. The
to exogenous factors. In this case, if Incumbent gets a chance to decide, he will e¤ectively
compare the costs and bene…ts of being the …rst-mover and the second-mover. The following
result shows that, the types of Incumbent who are very asymmetric will decide to act …rst, and
those who are equally competent in both issues will pass.
18
For Bayesian voters who observe that Incumbent did not make the …rst move, information on whether Incumbent did not have a chance to move …rst or had this chance but decided not to is payo¤-relevant information,
because this tells them something about his competence on the other issue. However, the probability that he
had this chance will depend on the competence over the issue he campaigns on. This complicates the problem
considerably; in particular, the boundaries of incumbent types choosing whether to move …rst or to pass, and
later whether to campaign on economy or foreign policy need not be linear.
23
Proposition 11 Suppose that
=
= 0, and also
<
1
2,
so politicians are guaranteed to
campaign on the same issue in equilibrium. In this case:
(i) Incumbent will move …rst if
j1
j1
1
ei j
2
=
;2
fi j
2
(12)
and will pass otherwise. If he moves …rst, he will choose the issue he is more competent in;
(ii) conditional on moving …rst, the chance that Incumbent wins is
1
2
+ 16 A > 12 ;
(iii) the expected competence of the elected politician is higher if Incumbent chose to move
…rst than if he chose to move second or did not have a chance to choose strategically.
Proposition 11 is intuitive, and the only role of the assumption
<
1
2
is to make the condition
(12) so simple to analyze. The most competent Incumbent, (ei ; fi ) = (1; 1), will be indi¤erent
between acting …rst and passing. However, if he is competent in one dimension but incompetent
in the other, he will …nd waiting too risky, and thus candidates close to (1; 0) and (0; 1) will
move …rst. On the other hand, candidates with ei = fi < 1 will strictly prefer to wait: they
know that if they move …rst, their competence in the dimension they choose not to campaign on
will be heavily discounted, and this is less of a problem if they are moving second. Indeed, in the
latter case, the voters will think that they were forced to campaign on a dimension chosen by the
challenger, and as such the penalty would not be so high. It is also intuitive that least competent
politicians (in particular, those with ei ; fi < 21 ) prefer to pass and become second-movers: for
them, moving …rst and campaigning on either issue is going to release a (justi…ed) negative
signal about their competence in both dimensions; at the same time, if they move second, this
would only be true for one dimension. These strategies are illustrated on Figure 7; notice that
in the case
< 12 , the candidates will eventually campaign on the same issue regardless of who
moves …rst.
According to Proposition 11, if Incumbent gets a chance to move …rst, it is equally likely
that he will use this chance and that he will not. However, there is another di¤erence in the
types of Incumbent who choose either strategy. Not only the ones who act …rst are more likely
to be competent in one dimension and challenged in the other, but they are more likely to be
more competent overall. Indeed, suppose, for example, that such Incumbent has ei > fi ; then
simple calculation shows that his expected competence in economy E is
competence in foreign policy F is
1
3,
5
6,
and his expected
which makes his overall expected competence
7
6.
Conse-
quently, incumbents who move …rst are also more competent than challengers, in expectation.
This insight leads to the other two results in the proposition; …rst, there is …rst-mover advantage,
in the sense that an incumbent who gets a chance to move …rst is more likely to win, and second,
when campaigns choose their issues earlier, the expected competence of the elected politician is
24
Figure 7: Incumbent’s strategy when he can postpone his campaign.
higher. The latter e¤ect arises in the equilibrium because more competent politicians are more
likely to move …rst, and incompetent ones prefer to wait.19
Consider now the opposite case,
=
= 1, corresponding to the case where the politician
almost always has discretion (p0 = 1). The trade-o¤ that the incumbent faces now looks somewhat di¤erently. Moving …rst allows them to reveal their competence on the issue they are good
at, but the cost is that they will be thought of as very incompetent on the other issue. On
the other hand, if they wait, voters will think that their skills on the two issues are relatively
close, and therefore announcing their competence on one issue will not hurt them on the other
dimension, or will hurt very little. Hence the types of Incumbent close to the boundary have an
incentive to pass, and this only increases the penalty for moving early. When
=
= 1, this
leads to full unraveling: no Incumbent will ever move …rst.
Proposition 12 If ;
= 1, then the unique equilibrium requires Incumbent to postpone choos-
ing the campaign issue until Challenger emerges and makes the choice.
This results suggests that when voters know that a politician is free to decide between
moving …rst and moving second, the best response for the politician is to wait. In this model,
this means that strategies on choosing issues do not change, and neither does social welfare.
More broadly, this result implies that while making a …rst announcement is an attractive option
for at least some politicians, not using this option is even more attractive, as it serves as a
positive signal of possessing balanced competence in the two issues, which is valuable in the
19
Another e¤ect, which is not modeled explicitly, is that deciding on an issue earlier gives politicians more time,
and allows voters to get more precise signals about politicians’ competences, which again raises the expected
competence of the winner.
25
environment where campaigns may be run on one issue only. As a result, the opportunity to
make an early commitment to the issue of the campaign will not be used. This insight may be
extended further, to a game where both politicians get, e.g., alternating opportunities to move
…rst; backward induction will immediately suggest that both politicians will wait until the very
last opportunity, and on the equilibrium path, waiting will be something expected rather than
a positive signal. (Interestingly, o¤ equilibrium path, once one politician makes the choice, the
other one has no incentive to wait further.)
The insights obtained in these extreme cases suggest the following implications. First, politicians are not likely to seize the very …rst opportunity to pick an issue, and the reason is not the
aggregate uncertainty (i.e., they might want to learn, which issues voters …nd most important),
but rather signaling considerations. Second, politicians will use opportunities which present a
good reason not to wait. For example, if some event or story makes it impossible or very hard
for a politician not to react, the politician might well make the …rst move (e.g., a stock market
crash creates a good reason to start campaigning on economy). A politician may also want to
use an opportunity which would otherwise go unnoticed (and therefore he would not get enough
credit for waiting). If none of these event types get realized, politicians are likely to wait until
the last moment, when the remaining time becomes a binding constraint.
6
Conclusion
The paper studies the incentives of politicians to choose issues to run their campaigns on. Voters
are Bayesian, and by campaigning, the candidates cannot change voters’ preferences, but can
a¤ect the information they possess. I assume that both candidates are more credible when they
run on the same issue, and this creates a non-trivial interplay between their incentives. The
…rst mover (Incumbent) has a disproportionate in‡uence on the course of the campaign, but
this does not necessarily help him win. Whenever a politician gets a chance to postpone the
announcement of his issue, he will, because this will signal his competence on the issues he will
not focus his campaign on. The politicians who nevertheless choose to act …rst are more likely to
be competent, and more likely to win. The model predicts that allowing politicians a free choice
of campaign issues reveals more information in the course of campaign, and ultimately raises the
chance of electing the most competent candidate. The model o¤ers non-trivial insights on the
nature of issue selection in campaigns, and is tractable enough to allow for di¤erent extensions.
In particular, building on the dynamic model of Section 4 and further studying dynamics of
political campaigns as sequential issue selection seems to be an interesting avenue for future
research.
26
Appendix A — Main Proofs
Proof of Proposition 1. Suppose that di = E. For Challenger, we have, in notation of Lemma
5 from Appendix B,
= 1 and
E
F
= , thus r =
1
2 [1; 2). By Lemma 5, there is unique
upward-sloping boundary with types who are indi¤erent, and Lemma 6 and Lemma 7 imply
that there is no other equilibrium response by Challenger in this case. We therefore need to …nd
the required boundary.
2
1 2
Since the slope of the boundary is given by
, there are several cases to consider. If types
(a; 0) and (1; b) are indi¤erent (and a > 0, b < 1), then we would get a contradiction with Lemma
8. Similarly, if types (0; a) and (b; 1) are indi¤erent, then we again get a contradiction with the
same Lemma 8, because
E
= 1 > 32 . Therefore, we must have that types (a; 0) and (b; 1) are
indi¤erent, and a < b. Then the expected quality of types choosing F is
a2 +ab+b2
3(a+b)
+
a+2b
3(a+b) .
The
conditions that types (a; 0) and (b; 1) are indi¤erent are indi¤erent are then
a =
b+
For
1
2
=
> 21 , this has solution a =
a2 + ab + a + b2 + 2b
,
3 (a + b)
a2 + ab + a + b2 + 2b
+ (1
)
.
3 (a + b)
p
(8 7 )
(8 7 )
3 +
,b=
.
4
8 4
a
+ (1
2
b
1+
2
p
4 5 +
8
)
The probability that p
Challenger chooses the same issue is found in the proof of Proposition
(8 7 )
3(2 )
. Given the symmetry of Challenger’s strategies, Incumbent
2; it equals p =
4(2 )
p
(1 ) 8 7 2
has E = F = p + (1 p) = +3
> 23 for all . Lemma 5 implies that for
4
4(2 )
Incumbent r = 1, and there is a unique equilibrium.
Proof of Proposition 2. The probability that the incumbent choosing issue F when the
challenger chose E (or vice versa) equals 0 if < 21 and
!
p
p
(8 7 ) 3 +
(8 7 )
2
1 4 5 +
+
=
2
8 4
8 4
if
> 21 Di¤erentiating this with respect to yields
!
p
+
(8 7 )
d 2
=
d
4 (2
)
2 (2
This is increasing for
<
4
5
and decreasing for
4
2
)
5
p
p
+
(8
4 (2
)
(8
7 )
7 )
.
> 54 , and the maximal value equals
1
4
+
p
3
6
0:539. The probability that a politician
communicates his competence on the chosen dimension
p
(1 ) 8 7 2
+3
successfully equals 4
, which is increasing on 12 ; 1 . This completes the proof.
4(2 )
27
Proof of Proposition 3. This result immediately follows from Lemma 6 and Lemma 7 in
the case of Challenger; the case of Incumbent is considered similarly to Proposition 1
Proof of Proposition 4. The proof of this result is trivial and is omitted.
Proof of
Lemma
The expected competence of the elected politician is
1.
EE (U (ei ; fi ) j I) = E (U (ei ; fi )) = 1 (and similarly
for Challenger).
Z
E ((U (ei ; fi ) U (ec ; fc )) j I) d = 1. We split the
Denote = i
c , Let us show that
entire space of types into four regions, (di ; dc ) 2 f(E; E) ; (E; F ) ; (F; E) ; (F; F )g according to
the dimension chosen in equilibrium (note that these need not be independent, as the second
player’s choice depends on the …rst player’s one). We have
Z
=
E ((U (ei ; fi ) U (ec ; fc )) j I) d
X
(di ;dc )
X
(di ;dc )
(di ;dc )2S
=
X
di 2fE;F g
X
=
di 2fE;F g
=
X
di 2fE;F g
= 1
E (U (ei ; fi ) U (ec ; fc )) j
Z
E U (ei ; fi ) j
(di ;dc ) ; ai jdi ; ac jdci
d
(di ;dc )
(di ;dc )2S
=
Z
(di ;dc )
Z
X
(di ;dc )
dc 2fE;F g
di
E (U (ei ; fi ) j
E (U (ei ; fi ) j
Z
E (U (ei ; fi ) j
di ; ai jdi )
di
di
di 2fE;F g
Z
di E (U
E U (ec ; fc ) j
di ; ai jdi ) E (U
dc (di )
Z
X
Z
(di ;dc ) ; ai jdi
di ; ai jdi )
(ei ; fi ) j
X
(di ;dc )
di E (U
(ec ; fc ) j
dc 2fE;F g
Z
(di ;dc ) ; ac jdc
d
(ec ; fc ) j
di ;
dc
(di ) ; ac jdc ) d
dc d di
E (U (ec ; fc ) j
di ;
dc
(di ) ; ac jdc ) d
dc d di
dc (di )
di ; ai jdi ) d di
di ) d di
= 1 1 = 1.
di
(where ai jdi is the projection of type ai on the dimension di ). We used that E (U (ec ; fc ) j
does not in fact depend on di and equals 1. Consequently,
0
Z
Z
2
@
1+A E (U (ei ; fi ) U (ec ; fc )) j I d = 1+A
E (U (ei ; fi ))2 + (U (ec ; fc ))2 j I d
di )
1
2A .
Within each region, E (U (ei ; fi ) j I) and E (U (ec ; fc ) j I) are independent, and thus
E ((U (ei ; fi ) U (ec ; fc )) j I) = E (U (ei ; fi ) j I) E (U (ec ; fc ) j I). This completes the proof.
28
1
2.
Proof of Proposition 5. First, suppose
Without loss of generality, assume that
the challenger has ec > fc and thus chooses E, and the incumbent does the same. Computing
the relevant integrals in the right-hand side of (9), we obtain
Z 1
Z 1
1 2
53
3 2
5
y+
dy =
x 2xdx +
=2+ .
2
2
24
24
0
0
1
2.
Now suppose that
We …rst calculate the integral for the challenger (again, as-
suming that
similarly). With probability
p ec > fc as the opposite case may be considered
R1 3 2
2 +
(8 7 )
1
, the incumbent chooses E, too. This gives 0 2 x 2xdx = 89 . With probabil4(2
p )
(8 7 )
2 +
9
ity
) 1. Totally,
, the incumbent chooses F , and the integral equals
8 + (1
4(2 )
the contribution of the challenger is
!
p
2
+
2
(8 7 )
9
1
+
8
4 (2
)
p
+
(8
4 (2
)
7 )
9
8
=
9
+ (1
8
1
32 (2
)
) 1
2
+
p
(8
7 )
.
Now, consider the contribution of the incumbent.
The part ofpthe integral coming from the
p
3 +
4 5 +
(8 7 )
(8 7 )
,b=
):
set where he chooses E is (as before, a =
8 4
8 4
Z
b
a
1x
x+
2b
2
a
a
x
b
Z 1
a
1
dx+
x+
a
2
b
2
dx =
13 3a + 9b + 4ab2 + 4a2 b + 8ab + 4a2 + 4a3 + 12b2 + 4b3
.
12
48
The part of the integral coming from the set where he chooses F may be found as follows: with
probability , it is
Z 1
1
y + (a + (b
2
0
2
a) y)
and with probability 1
(a + (b
a) y) dy =
1
4a + 12b + 3ab2 + 3a2 b + 8ab + 4a2 + 3a3 + 12b2 + 3b3 ,
48
, it is
a2 + ab + b2
a + 2b
+
3 (a + b)
3 (a + b)
2
a+b
1
=
a2 + ab + a + b2 + 2b
2
18 (a + b)
2
.
This means that the total integral for the challenger and the incumbent, after substituting for
the values of a and b, is
2+
19 + 6
2
68 (2
)2 + 29
192 (2
2
6
)
3
3
52 + 24
p
(8
7 )
.
It may be shown directly that this is an increasing function of , and that its value for
2+
3
16
and for
= 1 is 2 +
1
4.
=
1
2
is
This completes the proof.
Proof of Proposition 6. It was proved earlier that an equilibrium where both politicians
always choose E or always choose F is only possible if
29
> 12 . On the other hand, it is obvious
that picking one’s better dimension is an equilibrium for all values of . For this strategy, each
politician is able to announce his competence credibly with probability
1
2
do so with probability
(1
+
1
2
, and he fails to
). Consequently, the integral in the right-hand side of (9) equals
!
Z 1
3 2
1+
1
x 2xdx + (1
) 1 =2+
.
2
2
8
0
1 1
+
2 2
2
1
2
Therefore, the expected competence of the elected politician is monotonically increasing, and it
exceeds the competence in the case of an exogenously given issue (2 + 16 ) if and only if
> 13 .
This completes the proof.
Proof of Proposition 7. Proved in the text.
Proof of Proposition 8. The expected probability of challenger winning is obtained by
taking expectation of (6) over all possible realizations of (ec ; fc ), (ei ; fi ), as well as
The law of iterated expectations implies that this equals
incumbent winning also equals
1
2.
1
2,
c
and
i.
as thus the expected probability of
This completes the proof.
Proof of Proposition 9. In the symmetric equilibrium, Incumbent chooises d~i = E if
ei > fi and d~i = F if ei < fi . Consider the case where d~i = E and dc = F . Consider an
agent with competence (x; y); suppose that he is indi¤erent. If so, then if he chooses E, he
gets
x + y2 + (1
) ZE , where ZE = E ei + fi j d~i (ei ; fi ) = di (ei ; fi ) = E; dj = F . If he
chooses F instead, he gets y +
x+y
2 .
Consequently, he is indi¤erent if and only if
y=
2
3
This de…nes an upward-sloping line for
1
1
2
x+
<
2 (1
3
)
ZE .
(A1)
1. Moreover, for such , y > 0, provided that
x > 0. Consequently, if there are positive measures of Incuments who stay and who switch,
the line should connect some point (a; a) with some point (1; b); moreover, it must be that
b=a+
2
3
1
(1
a) and 0
a < b < 1.
Let us now …nd ZE as a function of a.
The area E consists of two triangles, ob-
tained from the bottom-right triangle by connecting (a; a) with (1; 0). One triangle has vertices (0; 0) ; (1; 0) ; (a; a); its area is
1
3 (1 + a +
(1 a) a+ 23
a) = 2a+1
3 .
1
(1 a)
2
2a+b+2
3
=
2a+(1 a)
a
2
and the sum of the coordinates of its mass center is
The other triangle has vertices (1; b) ; (1; 0) ; (a; a); its area is
and the sum of coordinates of its mass center is
a+ 23
1
(1 a) +2
3
ZE =
. Therefore,
1 2a2 b + 2a2 ab2 + a + b2 + 2b
.
3
a + b ab
30
1
3
(1 a)b
2
=
(1 + 1 + a + b + a) =
Thus, (A1), when evaluated at point (x; y) = (a; a), simpli…es to
If
H (a; ) = 3
3
+ 81
2
2
+ 23
84 + 64 a3 + 288
171 + 84 a + 30
108
2
20
2
192 a2
(A2)
10 = 0.
= 1, the equation (A2) has a unique root on [0; 1), a = 0, and in this case the indi¤erence
= 12 , we must have a = b, and thus a > 0;
line connects (0; 0) and 1; 12 . In the other extreme,
in this case, the only root is a =
Let us show that for all
(A2) equals 10 (2
4
p
10
0:28.
3
2 (0; 1), there is a unique root a 2 (0; 1). The left-hand side of
1) (1
) > 0 if a = 0 and it equals
3 (2
) (3
)2 < 0 if a = 1. Thus,
there exists a root a 2 (0; 1) for any such . Moreover, if = 1, the three roots of the cubic
p
p
2; 0; 1 + 2 , and so a = 0 is a simple root. Therefore, for close to 1,
equation are 1
the equation (A2) has a unique simple root on (0; 1). If for some
2
1
2; 1
there are two or
three roots on (0; 1), then at least one of the following three alternatives must be true: either
for some (perhaps other)
2
1
2; 1
, a = 0 is a root, or a = 1 is a root, or there is a double root
a0 2 (0; 1). The …rst two possibilities are ruled out, because we know that a = 0 may be a root
only for
2
1
:2 ; 1
, and a = 1 may be a root only for
2 f2; 3g. Suppose that some a
~ 2 (0; 1)
is a double root, then the second derivative of (A2) must vanish at some a between the single
root and the new double root:
6 3
Since
3
+ 23
2
84 + 64 a + 2 288
6= 43 , this is equivalent to
a=
which is decreasing from
8
9
to
to
as
)2
2 (3
(16
2
3
2 )2
9
12
9 +
2
increases from
10
4
261
3
108
2
192 = 0.
(A3)
16
,
16
1
2
(A4)
to 1. For a given by (A4), (A2) simpli…es
+ 1053
2
1536 + 768 .
It is straightforward to check that the last factor has no roots on
1
2; 1
; this proves that the
root is unique.
We can show that this root is decreasing in
@H(a; )
@a
. Indeed, the root 0 for
< 0; since we proved that there is no double root, we must have that
. It remains to show that
@H(a; )
@
@H (a; )
= 9
@
2
@H(a; )
@a
< 0 at any root. We have
+ 46
84 a3 + (288
31
216 ) a2 + (162
= 1 satis…ed
171) ,
< 0 for all
Let us show that it is positive for all
is increasing in :
@ 2 H (a; )
@ 2
1
2; 1
2
= 9a3
h
, a 2 0; 4
p
10
3
i
. Indeed, for such values,
@H(a; )
@
108a2 + 23a3 + 81
9 3
a
108a2 + 23a3 + 81
2
27a3 108a2 + 81 = 27 (1
a2 > 0.
a) 3 + 3a
Consequently, it remains to prove that
@H (a; )
@
h
for a 2 0; 4
p
3
10
i
= 72a2
29a3
9<0
=1
, which is straightforward.
We can also prove that b is increasing in . Indeed, given a =
3b 2 b +1
,
4 3
we can rewrite
(A2) as
~ (b; ) =
H
3
+6
+ 12
3
2
43 + 48 b3 + 6
57
2
+ 63
b+ 8
3
3
57
2
2
+7
18
144 b2
+ 165
(A5)
= 0.
This function has no double root: indeed, otherwise there would be a point with
in which case b =
16 13 +2 2
2 ;
16 9
(16
2 )2
9
10
which, as we know, has no roots on
because this is true for
~ (b; )
@H
= 3
@
2
+ 12
= 0,
plugging this into (A5) yields
3 )2
2 (4
@ 2 H(a; )
@b2
4
261
3
2
1
2; 1
. Now, this means that
+ 1053
= 1. It remains to prove that
43 b3 + 18
2
2
1536 + 768 ,
~
@ H(b;
)
@
114 + 165 b2 + 36
~
@ H(b;
)
@b
< 0 for all
,
> 0. We have
2
114 + 63 b+ 24
2
36 + 7 .
One can show that this is positive at the root b.
Proof of Proposition 10. Without loss of generality, we can assume that e2
If so, renormalize the rectangle [e1 ; e2 ]
have a uniform distribution on [0; m]
x e1 y f1
f2 f1 ; f2 f1
[f1 ; f2 ] by mapping (x; y) 7!
[0; 1], were m is the ratio
e2 e1
f2 f1 .
If Incumbent chooses E (F ), he will talk about E (F ) with probability p =
and will thus be able to reveal his competence credibly with probability
2
3;
then the ratio r =
E= F
e1
E
=
F
3(2
f2
f1 .
; we then
p
)
4(2
= p+(1
(8 7 )
,
)
p)
>
= 1. Then, as in Lemma 5, we can prove that if both E and F
32
are picked by Incumbent with a positive probability, then the line separating them must be
upward-sloping, with slope 1, and the points on this line must satisfy
(x; y) =
E
x+
y
+ (1
2
E ) ZE
Suppose, to obtain a contradiction, that
F
y+
x
2
(1
F ) ZF
= 0.
(0; 0) > 0, so that the line
(x; y) = 0 lies
above the “diagonal” y = x. Then, evidently, ZF > ZE , but if so, we would have
(1
(0; 0) =
E ) (ZE
ZF ) < 0, a contradiction. Thus, (0; 0) < 0, and the intersection of the line
(x; y) = 0 with @ ~ i are some points (a; 0) and (m; m a), with 0 a < m. The equilibrium
condition is (where we set
=
E
=
and (x; y) = (a; 0)):
F
(a) + (1
a
+ (1
2
) ZE =
) ZF .
Then
1
m a
3
We have ZE = a +
This implies ZF = m
(ZF
ZE ) =
a
.
2
+ 23 (m
a) = m, and ZF may be found from
!
(m a)2
(m a)2
m+1
+ ZF 1
:
ZE
=
2m
2m
2
2m 2am+a2 +m2 1
,
2m 2am+a2 +m2
and thus ZF
m(1 m)
2m+2am a2 m2
ZE =
=
m(1 m)
.
2m (m a)2
Consequently, a is found from the equation
1
m (1 m)
a
2 = 2;
2m (m a)
since the left-hand side is decreasing in a, it has at most one solution. It has a solution a < m
if and only if
+ m > 1, i.e., m > 1
.
Consider the cases where all or almost all types choose the same issue. If all types choose
F , then ZF =
m+1
2 ,
and the condition that type (m; 0) does not deviate is
(m + zm ) + (1
m
+ (1
2
) ZE
1
2
This is satis…ed (for zm = ZE = 0) if m
1.
)
m+1
.
2
If all types choose E, then ZE =
m+1
2 ,
and the
condition that type (0; 1) does not deviate is
1
2
+ (1
)
m+1
2
This is satis…ed (for z1 = ZF = 0) if m
2
1
(1 + z1 ) + (1
1
> 1, since
) ZF .
> 23 . Therefore, there is only one
equilibrium with both E and F chosen with a positive probability if m >
1
<m
1
2
1,
1
2
1,
two equilibria if
and only one equilibrium where only F is chosen when m < 1
33
.
Proof of Proposition 11. The set of types that decide to wait should be symmetric around
the line ei = fi . In addition, if the incumbent decides to act …rst, he should choose E if and only
if ei > fi , and choose F otherwise. The remainder of the proof involves considering all possible
cases as in the proof of Proposition 1 and applying (9).
Proof of Proposition 12. Suppose not, i.e., that some types choose to act …rst. Notice
that it cannot be that all types act …rst: then waiting would make voters believe that the voter
is taken from the same uniform distribution on [0; 1]2 , which means that the types with low ei
and fi would wait.
Now suppose that the border between the regions where the agent acts …rst by choosing E
and the region where he waits is given by fi = g (ei ), and the border between F and waiting
is symmetric and given by ei = g (ei ). Consider an individual with (ei ; fi ) = (x; y) such that
y = g (x). By choosing to act …rst and announce E, he expects to get x +
g(x)
2 .
By waiting,
he has an equal chance of having to campaign on E and F ; in the …rst case he expects to get
x+ g
1 (x)+g(x)
2
1 (y)+g(y)
and in the second case y + g
2
2
= g (x) + x+g2 (x) (where g 2 (x)
g (g (x))).
Consequently, we must have
x+
g (x)
1
=
2
2
Simpli…cation yields x = g
x+
g
1 (x)
+ g (x)
2
1 (x) + g (x) + g 2 (x).
contradiction completes the proof.
34
+
1
2
g (x) +
x + g 2 (x)
2
.
But this is impossible, since x < g
1 (x).
This
References
Acemoglu, Daron, Georgy Egorov and Konstantin Sonin (2012) “A Political Theory of Populism,” Quarterly Journal of Economics, 128(2): 771-805.
Alesina, Alberto and Alex Cukierman (1990) “The Politics of Ambiguity,”Quarterly Journal of
Economics, 105 (4):829-850.
Amorós, Pablo, and M. Socorro Puy (2013) “Issue convergence or issue divergence in a political
campaign,” Public Choice, 155(3-4): 355-371 .
Aragonès, Enriqueta, Micael Castanheira, and Marco Giani (2012) “Electoral Competition
through Issue Selection,” mimeo.
Ashworth, Scott, and Ethan Bueno de Mesquita (2008) “Electoral Selection, Strategic Challenger
Entry, and the Incumbency Advantage,” Journal of Politics, 70(4), pp. 1006-1025.
Austen-Smith, David (1990), “Information Transmission in Debate,” American Journal of Political Science, 34 (1):124-152.
Banks, Je¤rey (1990) “A Model of Electoral Competition with Incomplete Information,”Journal
of Economic Theory, 50: 309-325.
Banks, Je¤rey S., and John Duggan (2005) “Probabilistic Voting in the Spatial Model of Elections: The Theory of O¢ ce-motivated Candidates,” in Social Choice and Strategic Decisions
(Austen-Smith and Duggan, editors), 15-56.
Baron, David P. (1994) “Electoral Competition with Informed and Uninformed Voters,”American Political Science Review, 88 33-47
Besley, Timothy (2005) “Political Selection” Journal of Economic Perspectives, 19: 43-60.
Bhattacharya, Sourav (2011) “Campaign Rhetoric and the Hide-and-Seek Game,”, mimeo.
Callander, Steven and Simon Wilkie (2007) “Lies, Damned Lies, and Political Campaigns,”
Games and Economic Behavior 60: 262-286.
Canes-Wrone, Brandice, Michael Herron, and Kenneth Shotts (2001) “Leadership and Pandering: A Theory of Executive Policymaking.” American Journal of Political Science 45:532-550.
Caselli, Francesco and Massimo Morelli (2004) “Bad Politicians” Journal of Public Economics,
88(3-4): 759-782
Colomer, Josep M., and Humberto Llavador (2011), “An Agenda-Setting Model of Electoral
Competition,” Journal of Spanish Economic Association, 2011.
Downs, A. “An Economic Theory of Democracy,” Harper, New York.
Dragu, Tiberiu, and Xiaochen Fan (2013) “An Agenda-Setting Theory of Electoral Competition,” mimeo.
Duggan, John, and Cesar Martinelli (2011) “A Spatial Theory of Media Slant and Voter Choice,”
35
Review of Economic Studies, 78(2), pp.640-668.
Glazer, Amihai, and Susanne Lohmann (1999) “Setting the Agenda: Electoral Competition,
Commitment of Policy, Issue Salience,” Public Choice, 99, 377-394.
Harrington, Joseph (1993) “The Impact of Re-election Pressures on the Ful…llment of Campaign
Promises,” Games and Economic Behavior 5, 71-97.
Heidhues, Paul and Johan Lagerlof (2003) “Hiding information in electoral competition,”Games
and Economic Behavior, Elsevier, vol. 42(1), 48-74.
Kartik, Navin and Preston McAfee (2007) “Signaling Character in Electoral Competition”,
American Economic Review 97, 852-870.
Krasa, Stefan, and Mattias Polborn (2010) “Competition Between Specialized Candidates,”
American Political Science Review, 104(4), pp. 745-765.
Lizzeri, Alessandro and Nicola Persico (2005) “A Drawback of Electoral Competition” Journal
of the European Economic Association, 3(6): 1318-1348
Mattozzi, Andrea and Antonio Merlo (2007) “Mediocracy,” CEPR Discussion Papers 6163.
Morelli, Massimo, and Richard van Weelden (2011) “Re-election through Division,”, mimeo.
Osborne, Martin J. and Al Slivinski (1996) “A Model of Political Competition with Citizen
Candidates” Quarterly Journal of Economics, 111:65-96.
Petrocik, John R. (1996) “Issue Ownership in Presidential Elections, with a 1980 Case Study”
American Journal of Political Science, 40:825-850.
Petrocik, John R., William L. Benoit and Glenn J. Hansen (2003) “Issue Ownership and Presidential Campaigning, 1952-2000,” Political Science Quarterly, 118(4):599-626.
Polborn, Mattias, and David T. Yi (2006) “Positive and Negative Campaigning,” Quarterly
Journal of Political Science, 1, pp. 351-371.
Prat, Andrea (2005) “The Wrong Kind of Transparency,” American Economic Review, 95(3),
862-877
Riker, William H. (1982) “The Two-Party System and Duverger’s Law: An Essay on the History
of Political Science,” American Political Science Review, 76(4), pp. 753-766.
Riker, William H. (editor) (1993) “Agenda Formation,”University of Michigan Press:Ann Arbor.
Riker, William H. (1996) “The Strategy of Rhetoric: Campaigning for the American Constitution,” Yale University Press:New Haven.
Shepsle, Kenneth A. (1972) “The Strategy of Ambiguity: Uncertainty and Electoral Competition,” American Political Science Review, 66(2), pp. 555-568.
36
Appendix B — For Online Publication
The following Lemmas study the strategies of a player j in the following situation. Suppose he
knows that if he picks issue E, then his opponent will choose E with probability
picks F , then his opponent will choose F with probability
F
=
F
+ (1
F)
F.
Denote
E
=
E,
and if he
E + (1
and suppose that at least one of these values is non-zero:
E
E)
+
F
and
> 0.
In what follows, we let ZE and ZF be the beliefs that voters have of a player j if he
chooses E and F , respectively (in equilibrium, ZE = E (ej + fj j dj (ej ; fj ) = E) and ZF =
E (ej + fj j dj (ej ; fj ) = F ) whenever these expectations are well de…ned). Slightly abusing notation, let E be the set of (ej ; fj ) such that dj (ej ; fj ) = E and F be the set of (ej ; fj ) such
that dj (ej ; fj ) = F . In other words, (re)de…ne E; F
F =
j
n E.
j;
E = fx; y 2
j
: dj (x; y) = Eg, and
In the following Lemmas, suppose that both areas, E and F , have a positive mass, unlike
otherwise noted.
Lemma 2 Let G = E \ F = @E \ @F be the set of points that have points from both E and
F in any "-neighborhood. Then G is a simple curve which may be parametrized by a (closed)
interval T
[0; 2] such that t 2 T is mapped into (x; y) such that x + y = t.
Proof. Since area E has a positive mass, there is a point (x; y) 2 E with x < 1 and y > 0,
and thus point (1; 0) and all points (x; y) within some "-radius of (1; 0) belong to E; similarly,
all points within "-radius of (0; 1) belong to F . Since G is intersecion of two closed sets, it
is closed; de…ne t1 = min(x;y)2G (x + y) and t2 = max(x;y)2G (x + y) and let
be de…ned by
: G ! [t1 ; t2 ]
(x; y) = x + y. It is injective: indeed, if for two points (x1 ; y1 ) ; (x2 ; y2 ) 2 G,
we had x1 < x2 and y1 > y2 , then there would be a points (x3 ; y3 ) 2 E and (x4 ; y4 ) 2 F
with x3 < x4 and y3 > y4 (because we could take these points arbitrarily close to (x1 ; y1 ) and
(x2 ; y2 ), respectively). But this would contradict monotonicity: indeed, monotonicity would
imply that point (x4 ; y3 ) 2 E because x4 > x3 , but also that (x4 ; y3 ) 2 F because y3 > y4 ,
which is impossible. At the same time, it is also surjective. Indeed, take some t 2 (t1 ; t2 ) which
is not part of Im . Since Im
is a closed set (as
is a continuous function with a compact as
its range), take t3 = max [Im \ [t1; t]] and t4 = min [Im \ [t; t2 ]]. Let (x5 ; y5 ) =
(x6 ; y6 ) =
1
(t4 ) and let " = min (x6
pick (x7 ; y7 ) 2 E and within
x5 ; y6
"
2 -neighborhood
y5 ) > 0. Within
"
2 -neighborhood
1
(t3 ) and
of (x5 ; y5 ),
of (x6 ; y6 ), pick (x8 ; y8 ) 2 F . Then by monotonicity,
(x8 ; y7 ) 2 E, and (x7 ; y8 ) 2 F (if not, then (x8 ; y8 ) 2 E by monotonicity, a contradiction). Now,
B-1
observe that
"
+ y5
2
"
< x6 + + y5 +
2
x 8 + y7 > x 6
x 8 + y7
and thus
( x8 + (1
(x8 ; y7 ) 2 (t1 ; t2 ); similarly,
7 ) x7 ;
y7 + (1
7 ) y8 )
E \ F , a contradiction.
"
= x6 " + y5 x5 + y5 = t1 ,
2
"
= x6 + (y5 + ") < x6 + y6 = t2 ,
2
(x7 ; y8 ) 2 (t1 ; t2 ). Thus, for any
2 (t1 ; t2 ). However, there is
2 [0; 1], we have
for which this point is in
Consider the following expression:
(x; y) =
x+
E
Lemma 3 Let (x; y) 2 Gn@
j
y
+ (1
2
E ) ZE
F
Proof. Suppose, to obtain a contradiction, that
(the opposite case is considered similarly). Take " =
y0
2
F ) ZF .
(B1)
(x; y) = 0.
(x; y) 6= 0. Suppose that
(x;y)
3 ;
(x; y) > 0
then, since curve G is continuous,
2 (0; ") for which there exists a unique point (x0 ; y 0 ) 2 G with x0 = x
there exists
(x0 ; y);
(1
and suppose that (ej ; y) 2 G ) ej = x and (x; fj ) 2 G ) fj = y,
i.e., there are no other points in G with ej or fj . Then
y 0 2 (y
x
2
y+
"; y) (let "0 = y
y 0 < "), and also
we know, by de…nition of G, that
(x0 ; y) >
(x0 ; y)
(x; y)
and
". Now consider the point
2 F , and by continuity of
(x0 ; y) > 0.
,
By construction, we have E (ej j (ej ; fj ) 2 F; fj = y) = x2 , and E (fj j (ej ; fj ) 2 E; ej = x) =
. Consequently, if the player chooses E, he gets uE =
chooses F , he gets uF =
F
y+
x
2
F ) ZF .
+ (1
however, that
uE
uF
=
(x; y) +
=
(x; y)
>
(x; y)
>
(x; y)
E
E
x0 +
y0
2
+ (1
E ) ZE ,
But (x0 ; y) 2 F , thus uF
x0
x+
y0
and if he
uE . Note,
y
2
"0
+
E
2
"
"
"+
2
3" = 0,
We get a contradiction which completes the proof.
The next lemma considers the case where (x; y) lies on a horizontal or vertical segment of G.
Lemma 4 Suppose that G contains a vertical segment not entirely on the border: for some
a 2 (0; 1), fy : (a; y) 2 Gg = [b; c]. Then
y 2 [b; c],
(a; b)
0 and
(a; c)
0, in particular, for some
(a; y) = 0. Similarly, suppose that G contains a horizontal segment not entirely on
the border: for some a 2 (0; 1), fy : (x; a) 2 Gg = [b; c]. Then
particular, for some x 2 [b; c],
(x; a) = 0.
B-2
(b; a)
0 and
(c; a)
0, in
Proof. Suppose that G has a vertical segment with (x; y) where x = a, y 2 [b; c] and suppose
a 2 (0; 1). If there is y~ 2 (b; c) such that (a; y) 2 E for y < y~ and (a; y) 2 F for y > y~,
then it must be that
uE =
E
a+
y~
2
(a; y~) = 0. Indeed, a person (a; y) with y 2 (b; c) expects to get
+ (1
E ) ZE
choosing F . Now, we have uE
from choosing E and to get
F
a
2
y+
+ (1
uF for y close to y~ but less than y~ and uE
y~ but greater than y~. Taking limits, we get
small enough, then there is some
from
uF for y close to
(a; y~) = 0.
Consider the case b > 0. Take a small " and consider the point (x0 ; y 0 ) = (a
b0
F ) ZF
"; b0 )
such that (a
2 G; for almost all ",
"; b + "). If " is
b0
is unique (take
such ") and, moreover, 0 < b0 < b; if we take " small enough, then b0 will be arbitrarily close to b.
Now, E (fj j (ej ; fj ) 2 E; fj = x0 ) =
b0
2
and E (ej j (ej ; fj ) 2 F; fj = y 0 ) =
F in equilibrium. At the same time, he expects to get uE =
chooses E and to get
b+"+
F
a
2
+ (1
F ) ZF .
a
E
"+
a
2
b0
2
This player chooses
+ (1
E ) ZE
if he
(a; b)
0.
Taking the limit " ! 0, we get
Now consider the case b = 0. In this case, again take a small " and consider the point
(x0 ; y 0 )
= (a
it is some zx0
"; "). Now, E (fj j (ej ; fj ) 2 E; fj = x0 ) need not be well-de…ned, but in any case
the same time, he expects to get uE =
get
F
(1
"+
a
2
F ) ZF ;
a
2 This
z
+ 2x0 +
0 and E (ej j (ej ; fj ) 2 F; fj = y 0 ) =
+ (1
F ) ZF .
E
a
"
Consequently, we have
taking the limit " ! 0, we again get
We can similarly prove that
(a; c)
E
player chooses F in equilibrium. At
(1
E ) ZE
(a
(a; b)
if he chooses E and to
") + (1
E ) ZE
"+
F
a
2
+
0.
0 and
(a; c)
0
In the case of a horizontal segment y = a, x 2 [b; c], we can similarly prove that
(a; b)
0
and therefore there is (a; y) 2 G with
and
(a; c)
0. Thus, in any case,
(a; c) = 0.
0, and thus there is (x; a) with
In what follows, let r =
E= F
(x; a) = 0.
2 [0; +1] (since we assumed that
to zero at the same time). The next results shows that for
points with
(a; b)
1
2
E
and
F
are not equal
< r < 2, G is precisely the set of
(x; y) = 0.
Lemma 5 If r 2
1
2; 2
, then f(x; y) 2
j
n@
1
2 ; 2 . Then the
2r 1
2 r ; moreover,
set of points with
j
:
then G is either a horizontal line or a vertical line.
Proof. Suppose r 2
straight line with slope
(x; y) = 0g = G n @
j.
If r
1
2
or r
2,
(x; y) = 0 is an upward-sloping
(x; y) is strictly increasing in x and strictly decreasing
in y. Consequently, G may contain no horizontal or vertical segments, as this would contradict
Lemma 4. If so, Lemma 3 implies that all points (x; y) 2 G n @
If r = 2, then the set of points with
j
satisfy
(x; y) = 0.
(x; y) = 0 de…nes a vertical line. In this case, G
cannot have a non-vertical upper-sloping part (by Lemma 3), and every vertical segment must
lie on the set
(x; y) = 0. A horizontal segment could have one end on the line
B-3
(x; y) = 0. If
this is the left end, then Lemma 4 implies that
impossible, since
(x; y)
0 on the right end. However, this is
is strictly increasing in x in this case. We would get a similar contradiction
if the right end of the horizontal segment satis…ed
If r > 2, then the set of points with
(x; y) = 0. Thus, G is a vertical line.
(x; y) = 0 de…nes a downward sloping line. This means
that G may have only one point of intersection with this set, and then Lemma 3 implies that G
must be either a horizontal or a vertical line. If it is horizontal, then its left end should satisfy
(x; y)
0 and its right end should satisfy
the fact that
(x; y)
0 by Lemma 4. Again, this contradicts
is strictly increasing in x. Thus, G is a vertical line in this case, too.
The cases r =
1
2
and r <
1
2
are considered similarly.
The next Lemma characterizes the conditions under which the curve G separating regions
E and F may be a horizontal or a vertical line.
Lemma 6 There exists an equilibrium for which the line G separating regions E and F is
the vertical line connecting points [a; 0] and [a; 1], where 0 < a < 1 if and only if a 2
h
i
1
1
F
;2
. Similarly, there exists an equilibrium for which the line G sepa1+ F
E
E
E
E
rating regions E and F is the horizontal line connecting points [0; b] and [1; b], where 0 < b < 1
h
i
1
1
E
if and only if b 2 1 + E
;2
.
F
F
F
F
Proof. It su¢ ces to prove the …rst part of the result, as the proof of the second part is completely
symmetric.
Under the conditions of the Lemma, we have ZE =
a+1
2
+
1
2
=
For (x; y) such that x > a, we have E (fj j (ej ; fj ) 2 E; ej = x) =
a+2
a
1
a+1
2 and ZF = 2 + 2 = 2 .
1
2 ; for x < a, we can set this
value arbitrarily; denote it by E (fj j (ej ; fj ) 2 E; ej = x) = zx . At the same time, for any (x; y),
E (ej j (ej ; fj ) 2 F; fj = y) = a2 . For any (x; y) with x < a, we must have
E
(x + zx ) + (1
a+2
2
E)
F
y+
a
+ (1
2
F)
a+1
;
2
in particular, this should hold for y = 0 and x arbitrarily close to a; since zx
E
(a) + (1
E)
a+2
2
F
a
+ (1
2
F)
(B2)
0, we have
a+1
.
2
(B3)
For any (x; y) with x > a, we must have
E
x+
1
2
+ (1
E)
a+2
2
F
y+
a
+ (1
2
F)
a+1
;
2
(B4)
in particular, this should be true for y = 1 and x arbitrarily close to a, therefore,
E
a+
1
2
+ (1
E)
a+2
2
B-4
F
1+
a
+ (1
2
F)
a+1
.
2
(B5)
Now, (B3) is equivalent to
This implies 2
E
1
F
not equal to 0 together,
E
Ea
E
1, and (B5) is equivalent to
F
1, so E 2
h
> 0. Thus, a 2 1 + F
E
+
2
F,
F
Ea
E
+
F
1.
and in particular, since E and F are
i
1
F
;2
. One can easily verify
1
E
E
E
E
that for any such a (provided that a 2 (0; 1)) there is an equilibrium, where we de…ne zx = 0
and for x = a, we let (x; y) 2 E if and only if y < c, where c satis…es
a+
E
c
+ (1
2
E)
a+2
=
2
c+
F
a
+ (1
2
F)
a+1
2
(B6)
(existence of such value follows from inequalities (B3) and (B5); it is unique if
E
>2
F.
The next Lemma characterizes conditions under which there is an equilibrium where (almost)
all types choose E.
Lemma 7 There exists an equilibrium where all types choose E if and only if
and there exists an equilibrium where all types choose F if and only if
Proof.
1
2
+
1
2
E
+
It su¢ ces to consider the case where everyone chooses E.
= 1.
For any (x; y), we have E (fj j (ej ; fj ) 2 E; ej = x) =
1
2 E
1
2 F
+
F
1,
1.
We have ZE =
1
2,
and we can pick
E (fj j (ej ; fj ) 2 E; ej = x) = zy arbitrarily. We can also pick ZF arbitrarily.
Player (x; y) can choose E in equilibrium if and only if
E
x+
1
2
+ (1
E)
y+
F
zy
2
+ (1
F ) ZF .
This must be satis…ed for x = 0, y = 1; we can …nd suitable zy and ZF (for example, zeroes) if
and only if
1
2
E
E)
+ (1
F.
1
2 E
Thus, a necessary condition for such equilibrium is 1
0. At the same time, since
F
the type (0; 1) was most prone to deviation, this condition is also su¢ cient.
The next Lemma restricts the possible separtion lines G if r 2
Lemma 8 Suppose 1
(x; y) 2 G n @
j
.
r < 2. Then there exists (x; y) 2 G such that y
satisfy y > x, then
(x; y) 2 G such that x
1
2; 2
E
< 23 . Similarly, suppose
y, and if all (x; y) 2 G n @
1
2
j
Proof. Consider the case 1
r < 2 (the case
all points (x; y) 2 G satisfy
(x; y) = 0. Solving the equation
y=
2r
2
(1
1
x+2
r
<r
satisfy x
B-5
<r
y, then
1. Then there exists
F
< 32 .
1 is considered similarly). By Lemma 3,
E ) ZE
2
1
2
x, and if all
F
(1
E
(x; y) = 0 for y, we get
F ) ZF
.
(B7)
2r 1
2 r
This de…nes an upward-sloping line for
r
1, we have
2r 1
2 r
1.
Suppose that all (x; y) 2 G satisfy x
G with @
y. Then we can denote the points of intersection of
by (a; 0) and (1; b) (with a < 1, b > 0 because of the assumption of positive mass).
j
If so, we have b =
2r 1
2 r
(1
a). We have
1, ZE
and thus ZF
2
b 2r
3
2 a b
2 1
+ + = +
3 3 3
3
ZE =
Since r
0, i.e., for r < 2. Moreover, since we assumed
r
1
+
b
r 1
=1+
b.
3
2r 1
1. On the other hand, in this case, Pr (dj = E) =
0 We can rewrite
E
y
2
x+
b(1 a)
2
1
2
Pr (dj = F ),
(x; y) = 0 as
ZE
y+
F
x
2
ZF + ZE
ZF = 0,
(B8)
which must be true for any (x; y), for which the indi¤erence condition is satis…ed, in particular,
(a; 0). A su¢ cient condition for the left-hand side to be positive is
(a
E
Substituting a = 1
2 r
2r 1 b
ZE )
and ZE = 1 +
b
E
2r
r 1
2r 1 b,
1
2
+
1
a
2
F
1 + ZE
we have
1+
F
1>0
2
2r
r
r 1
b +
b.
1
2r 1
This is positive whenever
r+
F
Since b
1
2
1
b
+2
+r
r
1 > 0.
1, this is positive if
1
2
Since
2r
F
F
(r
1) + r
0, this is true, and therefore such equilibrium is impossible.
Now, suppose that all (x; y) 2 G n @
intersection of G with @
j
satisfy x < y. Then we can denote the points of
j
by (a; 1) and (0; b) (with a > 0, b < 1 because of the assumption of
positive mass). If so, we have 1
b=
2r 1
2 r a.
We also have
a 2 b
a 2 1
ZF = + + = + +
3 3 3
3 3
Since 1
1 > 0.
r < 2, it must be that ZF
Pr (dj = E), we must have ZE
1 and 1
2r 1
2 r a
3
=1
r
2
1
a.
r
1. As before, since Pr (dj = F ) =
ZF
B-6
ZE
1, so ZE
2
a(1 b)
2
1
2
ZF . In equilibrium, we
must have (B8) satis…ed for all (x; y) for which the indi¤erence condition holds, in particular,
(0; b). The left-hand side is negative whenever
b
2
E
Substituting b = 1
2r 1
2 r a
1
2
2 + ZF
and ZF = 1
E
4r
2
(b
F
3
a
r
ZF ) + 2 (1
r 1
2 r a,
we have
1 +
F
2
r
r
2
r
ZF ) < 0
a +2
r
2
1
a.
r
This is negative whenever
1
2
Fr
(4r
3)
a
+ 2 + 2 (r
Now consider two subcases. If r > 1, then, since a
2 r
2r 1
1) < 0.
(B9)
(because b = 1
2r 1
2 r a
0), (B8)
holds if
3r (r
which is negative, provided that
F
>
2
3r
1)
F
+ 2 (r
1) < 0,
(B10)
and r > 1. On the other hand, if r = 1, then b = 1
and x < y for all points on G implies a <
2 r
2r 1
a,
strictly. Hence (B8) holds if (B10) is satis…ed
weakly, which is true since r = 1 in this case. Therefore, if
in equilibrium.
B-7
E
=
Fr
2
3,
this case is impossible