Strategic Ambiguity, Campaign Contributions and
Candidate Location
Alberto Alesina and Richard Holden
September 2007
Abstract
We analyze a model in which voters are uncertain about the policy preferences of
candidates.
Two forces a¤ect the probability of electoral success: proximity to the
median voter and campaign contributions. We show that campaign contributions may
have no equilibrium e¤ect on electoral success, despite having large out of equilibrium
e¤ects.
Moreover, candidates may wish to announce a range of policy preferences,
rather than a single point. This strategic ambiguity balances voter beliefs about the
appeal of candidates both to the median voter and campaign contributors.
Preliminary and incomplete.
Alesina: Harvard University, Department of Economics.
email:
alesina@fas.harvard.edu.
Holden: Massachusetts Institute of Technology, Sloan School of Management.
E52-410, 50 Memorial Drive Cambridge MA 02142. email: rholden@mit.edu.
1
1
Introduction
The standard two candidate model of electoral competition has two key implications: convergence to the ideal policy of the median voter, and unambiguous (i.e clear and precise)
policy platforms.
Convergence to the median is complete in a Downsian (Downs (1957))
model in which the two candidates care only about winning elections. Convergence is partial
in a “partisan” model in which the candidates care about policy per se (Wittman (1983),
Calvert (1985)), and can make a commitment to their electoral platforms1 . Partial convergence means that the two candidates propose policies (much) closer to each other than their
ideal ones. As for ambiguity, Shepsle (1972) shows that with risk averse voters the candidates
have an interest in being as clear as possible in announcing their welfare maximizing policy.
Any uncertainty about their policy platform would a¤ect the parties’negatively in the eyes
of risk averse voters.
The observed patterns of real world elections in two party systems seem rather far from
the implications of these basic models. For instance no commentators of recent American
presidential elections would argue that the two parties have moved closer to each other; on
the contrary casual observation suggests that extreme groups are quite in‡uential in both
parties and presidential candidates have taken increasingly polarized positions. McCarty,
Poole and Rosenthal (2006) present a swathe of evidence on the evolution of polarization in
the United States and its large recent increase. Speaking of convergence in American politics
today seems completely out of touch with reality. In the recent (2007) French presidential
election the two candidates took positions very far from each other and showed no interest
in converging, not even in words. Even less plausible is the implication that candidates are
unambiguous in their pre-electoral policy statements. Just the opposite: most candidates
are very careful in not taking clear positions on key issues or even changing their position
when useful. Often, speaking in front of di¤erent audiences the same candidates make rather
di¤erent statements on the same policy issue. One Republican presidential nominee in the
1
See Alesina and Rosenthal (1995) and the references cited therein for a review of these models.
2
2008 campaign (John McCain) accused another one (Mitt Romney) of changing his views
on abortion every even numbered year. One Democratic nominee (Hillary Clinton) has been
as ambiguous as one can possibly be on her votes and views about the Iraq war.
If the
candidates wanted to be unambiguous they could easily do a much better job at it! In a
sense, extremism and ambiguity seems somewhat contradictory: and in fact the dance over
this trade-o¤ between trying to appeal to many voters and taking extreme positions (close
to the views of extreme groups) does appear as a feature of recent presidential elections in
the United States.
We propose a model in which candidate may adopt policy platforms even more extreme
than their ideal policies and in addition would choose to be ambiguous. More precisely, in
equilibrium, the candidates may o¤er voters an ideological interval space within which they
position their platforms, but they purposely do not reveal which policy they stand for (and
will implement if elected) within that interval. In the model we combine four elements: …rst,
partisan preferences of candidates, namely candidates have ideal policies that they would
like to implement; second, uncertainty about the true ideal policies of the candidates; third,
uncertainty about the distribution of voters preferences, and speci…cally about the position
of the median voter; fourth, campaign contributions that, holding everything else constant
a¤ect the probability of victory by in‡uencing the views of the median voter. Ambiguity
of platforms emerge in equilibrium as a result of the candidates balancing two forces: the
need to converge towards the median and the need to raise contributions that may ‡ow from
groups to the extreme of the ideological spectrum (and “feel strongly”about certain issues).
For the sake of argument we can think of contributions as “money” but they an also take
the form of unpaid time of activists, union strikes (mainly in a non US context) and the like.
Extremism arises when the e¤ect of campaign contributions from extreme groups dominate
the gain in probability of winning from convergence to the median voter.
In an attempt to
gain contributions without loosing too many voters in the middle of the political spectrum the
candidates may choose to be ambiguous regarding their policy stand even though everybody,
3
contributors and voters are aware of this incentives. Obviously, ex ante uncertainty about
the true ideal policy of the candidates is crucial for this result on ambiguity of platforms.
Note that, like in an arms race or in advertising, in equilibrium campaign contributions may
not a¤ect the probability of victory of the two candidates, even though they would a¤ect
their choice of policy, making them more extreme We can parametrize the extent (of lack
thereof) of ambiguity (i.e. the size of the ideological interval o¤ered by candidates) and the
degree of convergence as a function of parameters that can be easily interpreted such as the
distribution of voter preferences, the amount of uncertainty about the position of the median
voter, the marginal e¤ect of campaign contribution on electoral outcomes, the distance of
parties’ideal policies, and their symmetry or asymmetry relative to the median voter’s ideal
policy, and the importance of o¢ ce holding rents.
Therefore the paper can also explain
how the evolution over time of certain parameters, like for instance the importance and role
of contributions may a¤ect the equilibrium even holding constant the ex ante (i.e. before
contributions) distribution of voters’preferences.
Many before us have noted the inability of simple “traditional” models to explain the
richness of diverse observations o¤ered by electoral contests. The issue of (lack of convergence) has been attributed to the inability of making commitments to moderate pre electoral
platforms (Alesina (1988)) but this model would imply convergence in a repeated game and
is not compatible with parties taking positions even more extreme than their ideal policies.
Alesina and Rosenthal (2000) discuss extremism of presidential candidates when facing an
adverse Congress, an issue which we do not address here. Glaeser, Ponzetto and Shapiro
(2005) derive the adoption of extreme policies from the incentive that parties have to get
people to vote. Extremists vote only if the policies propose to them are not too middle of
the road.
Campante (2007) presents a model in which campaign contributions in‡uence
how much parties choose to redistribute and presents evidence consistent with the fact that
more polarization in the population lead to more contributions which polarize parties’policies. McCarty, Poole and Rosenthal (2006) discuss the importance of contributors to the
4
increased polarization of American politics and relate both of them to the increased inequality. They show that both contributions from Political Action Committees (PACs) and soft
money from individuals are highly ideological. In particular, soft money comes from extremists and its importance has risen recently.2 On ambiguity, Alesina and Cukierman (1990)
present a model in which an incumbent in o¢ ce has an interest in introducing noise so as
not to allow the voters to learn his ideal policy by observing his policies perfectly while in
o¢ ce. In that model, however, only the incumbent can be ambiguous: there is no strategic
game in platforms. Several papers discuss the choice of policy (especially monetary policy)
in situation where the public does not know the type of policymakers, but these are not
models of electoral competition.3
The paper is organized as follows. Section 2 presents the basic model without uncertainty
about candidates’ ideal policies but with contribution. This part of the model delivers
contributions driven extremism. Section 3 derives ambiguity of candidates’platforms, that
is we illustrate the possibility that candidates choose a (possibly large) interval in the issue
line rather than a single point. Section 4 discusses extensions, applications and limitations
of the model and the last section concludes. All proofs are in the appendix.
2
The Model
2.1
The Basic Setup
Consider two candidates who have to choose a binding electoral platform on the ideological
(policy) space de…ned by the unit interval [0; 1] : The “left-wing”candidate (player X) has
a policy bliss point xb 2 [0; 1=2] and the “right-wing”candidate (player Y ) has a policy bliss
point yb 2 [1=2; 1] : Simultaneously and non-cooperatively player X chooses location x and
player Y chooses location y: For now we will take x and y to be real numbers (i.e. points)
2
3
See these authors for a more complete review of the political science literature on this issue.
See Persson and Tabellini (2002) for a survey.
5
on the unit interval, but later we will allow them to be sub-intervals.
Denoting p as the
probability that candidate x wins the election we assume that the preferences of player X
can be represented by the utility function
UX =
p (x
xb )2
(1
p) (y
xb )2 :
p) (y
y)2
p (x
yb )2 :
Similarly for player Y we have
UY =
(1
Electoral outcomes–encapsulated by the probability p–are determined by two forces. One
is the distribution of voter preferences and the position of the median voter.
Suppose
that voter i has a single-peaked symmetric concave utility function with bliss point ib . For
concreteness assume that voters have a quadratic utility function like the candidates so the
utility of voter i is
Ui =
E(t
ib )2
where t is the policy implemented and the expectation sign highlights the possibility
that the voter is not sure about which candidate will prevail. With this utility function
a dominant strategy for each voter is to vote for the candidate o¤ering the policy closer
to ib : The distribution of the bliss points determines the number of votes received by the
candidates for each pair of policy proposed.4
The median voter is expected to have a bliss point at 1=2; but there is an aggregate
shock to voter preferences which makes the location of the median voter stochastic. This
shock is uniformly distributed on the interval ["; "] ; and we parameterize the magnitude
4
Obviously with no uncertainty about voter preferences for each pair of platforms chosen one party would
win for sure, the one with a policy proposal closer to the median. In this case the only equilibrium is one in
which both candidates position themselves at the media and the probability of victory is 1=2:
6
of the shock by assuming that "
analysis we will assume that
" = 1=
with
2 (1; 1) : Throughout much of the
= 2; to ensure that certain regularity conditions are satis…ed.
This is largely a technical assumption, but we will discuss it in more detail below.
The
second force is the e¤ect of campaign contributions. We assume that there is a contribution
maximizing point for each candidate: 0 for candidate X and 1 for candidate Y: This is
an extreme assumption which is meant to capture the idea that those who contribute are
groups with strong ideological views who are trying to pull candidates away from the middle
of the political spectrum. We discuss this assumption in more detail below, but the fact
that the contribution maximizing point are the actual extreme of the ideological space is a
simpli…cation: it is enough that the contribution maximizing points are more extreme that
the parties’ideal points to obtain all of our results, as it will become clear below.
We model the e¤ect of contributions as a force that moves the median voter (in expectation) closer to the candidate that gets the larger relative contributions. The interpretation
is that contributions allow parties to invest in advertising and persuasion of voters, moving
undecided voters toward their side5 . More speci…cally, the distance from the candidate location relative to the contribution maximizing point a¤ects " and ": As candidate X moves
toward the contribution maximizing point relative to candidate Y; " moves down while "
"
remains constant. This is represented in the following formula for "
"=
1
2
1
2
+ 1
1
1
+
2 2
(1
y
x) :
thus parameterizes the importance of campaign contributions. When
= 0 there is
no e¤ect of campaign contributions, and the size of the e¤ect is increasing in : In words the
candidate that is closer to his contribution maximizing point manages to move to expected
median voter towards his side. Given our distributional assumptions (and assuming that x
5
See Alesina and Tabellini (2007) for a similar way of modelling the e¤ects of campaign contributions.
7
and y are known) the probability that candidate X wins the election is:
Pr (X wins) = Pr ("
=
2.2
x+y
2
x<y
1
2
")
1
2
1
2
+
+
1
2
(1
y
x)
(1)
:
1=
The Standard Case: No Contributions
In this section we review the (well known) model under the assumptions that
= 0: Under
these assumption the probability that candidate X wins the elections is
p=
x+y
2
1
2
1=
1
2
:
By taking the …rst order conditions6 for the two candidates one can show
Proposition 1 Assume
= 0: Then there exists a unique interior Nash equilibrium in
candidate locations. Also, in the unique interior Nash equilibrium: (a) if xb = 1
x =1
y (b) x > xb and y < yb ; (c) if xb < 1
then x > 1
yb then x < 1
yb then
y and (d) if xb > 1
yb
y
The proof is well known, see for instance Alesina and Rosenthal (1995). Point (a) shows
that if the candidates’ideal policies are symmetric around the expected median the chosen
policies are also symmetric. Point (b) implies partial convergence: both parties o¤er policies
that are closer to each other than their ideal points. Point (c) shows that if a party if farther
from the expected median than the other one in ideal policies it will be farther in policies
and it will have a lower probability of winning. As an illustration of this result, consider the
case where xb = 1=5; yb = 3=4: In this case x = 0:358 and y = 0:619:
The critical result here (in addition to existence of course) is the partial convergence
e¤ect. The parties move closer to each other than their ideal policies. It would make no
6
These are in the appendix.
8
sense in fact for a candidate to announce a policy more extreme than his ideal: by moving
closer to his ideal he would increase his probability of winning and would also propose a
policy closer to his ideal. Also note that in the model which we use we did not include a
preference for holding o¢ ce per se. The candidates care only about the policy outcome,
and they want to win so that they can implement the desired policy. If they also had an
incentive to win per se, the amount of convergence would increase since the candidates would
be more willing to trade o¤ the policy location for an increase in the probability of winning
by converging to the median.7
In other words, the ideological space that lies on the right (left) of the right (left) wing
party is completely irrelevant: no policies will be ever proposed in that space. If the two
candidates are relatively close in ideology, a large part of the ideologic spectrum is then never
travelled.
2.3
Extremism and Contributions
We now consider the impact of campaign contributions. The proofs of the propsitions are
all in Appendix.
In order to make comparisons between the equilibrium when there are no contributions
(as in the previous subsection) we will add the subscripts C and N C to the equilibrium
values of x and y (x and y ) to denote the regime with and without campaign contributions
respectively.
Proposition 2 Assume
= 2; and xb = 1
yb : Then there exists an
> 0 such that in
the unique interior equilibrium xC < xN C and yC > yN C but pC = pN C = 1=2:
Again, as an illustration of this suppose
= 1=2; xb = 1=4 and yb = 3=4: In this
case there is a unique interior equilibrium with xC = 0:3 and yC = 0:7 which are more
extreme policy positions than the no contribution (i.e.
7
See Alesina (1988) for further discussion.
9
= 0) positions: xN C = 0:375 and
yN C = 0:625: However, because of symmetry the probability that each candidate wins is
still pC = pN C = 1=2:
The intuition is simple. A new force is in place now: campaign contributions move the
median voter toward the candidate that gets more of them. This additional force reduces
the incentive to converge in policy is this is why then is less convergence with campaign
contribution than without. In the case of perfect symmetry of the two candidates campaign
contributions do not a¤ect the probability of winning of the two parties but they do a¤ect
their policy choices, making them more extreme.
In fact, if the campaign contributions e¤ect is large enough then the candidates may take
positions which are even more extreme than their preferred points. For instance, if
= 3=4;
xb = 1=4 and yb = 3=4 then xC = 0:214 and yC = 0:786:
This is not particular to the numerical example just presented but holds more generally
as the following result establishes.
Proposition 3 Assume
= 2; and xb = 1
yb : Then in the unique interior equilibrium
dxC =d < 0 and dyC =d > 0:
Thus if the campaign contribution e¤ect is large enough then candidates will adopt policies more extreme than even their ideal policies in an e¤ort to court campaign contributions.
Proposition 4 Assume
= 2; and xb < 1
any interior equilibrium for any
jxN C
xC j < jyN C
yb (respectively xb > 1
> 0 we have jxN C
xC j > jyN C
yb ).
Then in
yC j (respectively
yC j).
This result highlights a sort of centrifugal force.
The more extreme party cares more
about campaign contributions than the less extreme one and therefore moves more towards
contributors moving even more away from the median.
The intuition is that the most
extreme candidate knows that he has a low probability of winning and therefore has a
stronger incentive to increase it by getting more contributions to a¤ect the median voter
10
and move the expected median.
He has a relative disadvantage compared to the more
moderate opponent in converging and a relative advantage at getting contributions. Being
more extreme to begin with, he pays a lower cost in moving toward the extreme contributors.
As with related voting models in which candidates have preferences over policy outcomes
(e.g. Wittman 1983) there is generally the possibility of multiple equilibria, and existence
of equilibrium requires certain regularity conditions be met. We will return to these issues
below.
3
Ambiguity of platforms
Assume now that there is some uncertainty about the true beliefs (i.e. bliss points) of
the two candidates. For simplicity we assume that there are only two types of candidates;
extending this to a continuum of types is discussed informally below and left for future
research. Candidate X has bliss point xL with probability qx and bliss point xR such that
1=2
xR > xL
0 with probability 1
qx : Similarly, candidate Y has bliss point yR with
probability qy and bliss point yL such that 1=2
y L < yR
1 with probability 1
qy :
Extensions to a continuum of possible types are left for future research
We now consider the impact of both campaign contributions and multiple candidate
types. In doing so, we now allow candidates to choose not simply a point, but an interval.
Obviously in the previous model with no uncertainty about party ideal policies they had no
occasion to choose an interval in the ideological space. Let the choice of interval for candidate
xi of type i be xi ; xi ; and for candidate yi be yi ; yi :
We assume that candidates are
free to choose any policy within their announced interval, but may not choose a policy
outside that interval. This is the logical extension to the assumption that we used thus far
that candidates are committed to their policy platforms. In the present paper we do not
investigate the issue of how this commitment is enforced.8
The game which this setting gives rise to is a dynamic game of incomplete information
8
See Alesina (1988) for a discussion of this issue.
11
and our solution concept is Perfect Bayesian Equilibrium. We specify the following out of
equilibrium beliefs of the voters. If candidate X does not follow the equilibrium strategy
then she is believed to be candidate type xL if she deviates by expanding the range to the left,
and type xR if she expands the range to the right. Similarly, if candidate Y does not follow
the equilibrium strategy then she is believed to be of type yR if the expands the range to the
right and type yL if she expands the range to the left.9
If a candidate narrows the range
then she is believed to be the type of candidate whose preferred policy is within the range.
If both types expand beyond the range on both sides then voters glean no information (i.e.
their posterior is their prior).
Our …rst result shows that in the setting we have considered, there exists no equilibrium:
either pooling or separating.
Proposition 5 Assume
= 2: Then for any yL ; yR ; xL ; xR there does not exist an
0
such that there exists a Perfect Bayesian Equilibrium in which either xL = xL = xR ; xL =
xR = xR and y L = yL = y R ; yL = yR = yR ; nor does there exist a separating equilibrium.
The intuition for this non-existence result is as follows. Fix player Y ’s strategy. Because
of the uniform distribution assumption there is always one type of player X which has a
higher probability of defeating player Y in the separating equilibrium. If the contributions
e¤ect is large ( is large) then this is player xL ; if it is small then it is player xR : Because
the players get to implement any policy in their range ex post the probabilities of victory are
the only relevant consideration. So a separating equilibrium cannot exist because one player
always wants to pool, and can mimic the other player10 . Similarly, a pooling equilibrium
cannot exists because one player always wants to separate.
Non existence is however purely duje to lienarity. The uniform distribution assumption
9
In the appendix we show that this result holds with an alternative speci…cation of out of equilibrium
beliefs where candidates who deviate are assumed to be the extreme type. Speci…cally, if candidate X does
not follow the equilibrium strategy then she is believed to be candidate type xL and if candidate Y does not
follow the equilibrium strategy then she is believed to be of type yR :
10
There is a …rst-order gain from doing so and a second-order loss from the change in interval and the
consequent e¤ect on policy choice.
12
means that there is a linear trade-o¤ between the “median voter e¤ect”and the “campaign
contributions e¤ect”.
This means that the probability of candidate X (say) winning is
monotonic in x: Whether it is increasing or decreasing depends on
; i.e. the relative
magnitude of the two e¤ects. Instead of the uniform distribution, then, suppose that the
aggregate shock to voter preferences is given by a variant of the beta distribution11 with
CDF 1=2
(1
q)2 : For "
" = 1=2 the probability that candidate X wins is then given by
Pr(X wins) =
1
2
1
2
x+y
2
1
4
2
(1
y
x)
:
We then have
Proposition 6 Assume that the shock follows the above beta distribution.
Y ’s strategy.
Then there exist yL ; yR ; xL ; xR and
Fix candidate
0 such that there exists a Perfect
Bayesian Equilibrium in which xL = xL = xR ; xL = xR = xR :
The intuition is as follows. Given the ability to choose any policy within the announced
range ex post, the only di¤erence in payo¤s between the pooling equilibrium and a deviation
is the probability of winning the election. Under the above distributional assumption the
probability of winning is non-monotonic in location–it has an inverted U-shape. Thus if xL
is su¢ ciently left of the peak of the hump and xR is su¢ ciently right of it then both types
bene…t from being believed to be an intermediate type. Translating this into fundamentals,
the non-mononiticity which gives rise to the pooling equilibrium arises when the median
voter e¤ect (which is largest at x = 1=2 and smallest at x = 0) decreases slowly from 1=2
and the more rapidly, and when symmetrically the campaign contributions e¤ect (which is
largest at x = 0 and smallest at x = 1=2) decreases slowly from 0 and the more rapidly.
We should also point out that the result is somewhat stronger than it appears as stated.
Although we say there exist yL ; yR ; xL ; xR ; we have chosen particular parameters of the
distribution12 .
11
12
For di¤erent values of yL ; yR ; xL ; xR one can choose di¤erent parameters
This is sometimes referred to as the generalized Kumuraswamy distribution.
The general CDF is 1=2 (1 q a )b :
13
and sustain the pooling equilibrium. Moreover, we have …xed candidate Y ’s strategy and
considered the deviations by xL and xR : It is also possible to construct examples in which
all types yL ; yR ; xL ; xR pool, using di¤erent parameters for the distribution of shocks to the
median voter.
The non-linearity of the median voter and campaign contributions e¤ect seems plausible
enough to us. But we now also point out that a somewhat less standard assumption also
delivers the pooling equilibrium, even with the uniform distribution. Suppose that voters
have exactly the same beliefs as before–they have a prior about which type each candidate
is.
But now suppose that campaign contributors believe that it is more likely that the
candidate is “extreme”(e.g. that candidate X is type xL ). If this di¤erence in priors is large
enough then the pooling equilibrium exists. This is because both types get large campaign
contributions in the pooling equilibrium, because they are believed to be the extreme type
with high probability. If the extreme type deviates they gain nothing. But if the moderate
type deviates she loses campaign contributions. Thus both types can be better o¤ in the
pooling equilibrium.
Proposition 7 Assume that the shock follows the uniform distribution and that
= 2:Then
for any yL ; yR ; xL ; xR there exists a di¤erence in priors between campaign contributors and
voters and
0 such that there exists a Perfect Bayesian Equilibrium in which xL = xL =
xR ; xL = xR = xR and y L = yL = y R ; yL = yR = yR :
One might alternatively think of this e¤ect as coming not from a di¤erence in priors, but
a di¤erence in information or information processing. For instance, if candidates are able
to send a di¤erent message to campaign contributors than voters (and these are messages
are kept separate) then campaign contributors and voters may have di¤erent beliefs about
the policy preference of the candidate. This e¤ect works in just the same way as di¤erent
priors–the key is that posterior beliefs of campaign contributors and voters are di¤erent.
14
4
Discussion of the Model
We have made a number of modelling choices in order to highlight our point in the simplest
possible way. The assumption that contributions are maximized when the parties move to
the extremes of the ideological spectrum is a simpli…cation meant to capture the fact that
groups distant from the median have a strong incentive to pull parties away from the middle.
This is related to intensity of preferences. If no contribution were made the parties would
converge so those at the extreme have to pull them away in some other way other than with
their votes.
The point is even clearer in a multidimensional setting. In fact in that case the groups
which are extreme and feel strongly about one particular issue will contribute precisely on
that one. So for instance the gun lobby which feels strongly about that issue will pull a
party toward extreme positions on gun control, gay right activist will do the same focusing
on theory preferred issue and not caring about other ones and so on. As we argued above
contributions can take the form of soft money or other forms, time, unpaid work etc.13 In
fact our intuition suggest that our result should generalize to a multidimensional settings,
precisely for the reasons above. However it is well known that multidimensional voting models present a signi…cant increase in analytical complexity relative to unidimensional ones and
we have chosen not to explore formally this avenue, at least for now but this is an excellent
area for future research. One result that would be quite di¤erent in multi dimensional voting
models is the one about “wasted”campaign contributions. In a unidimensional voting model
campaign contribution pull the parties in opposite directions and may counterbalance each
other. But in a multi dimensional voting model only the groups who feel very strongly about
one issue may contribute. That is the gun lobby may contribute to have no gun control but
those who prefer gun control may not be organized to counteract.
We have claimed and proven existence of equilibrium, we have not established unique13
See Campante (2007) and McCarthy Poole and Rosenthal (2007) for recent analysis on the polarizing
e¤ects of soft money.
15
ness. There may be other equilibria. Indeed even in the model without multiple types we
know that the equilibrium set can be complex. Normally the possibility of many di¤erent
equilibria is seen as a limitation of a model. Perhaps it is, however the complexity and the
variety of outcomes that one observes in real world electoral competition suggest that indeed
depending on the particular combination of factors and forces at play di¤erent equilibria
may indeed materialize in di¤erent elections. Perhaps this is why electoral campaigns are
entertaining, because they are di¢ cult to predict. Certainly models that predict that there
would always be full or almost full convergence with very precise policy platforms, may have
unique equilibria, but they do not seem to capture much of real life electoral dynamics.
We have also made two technical assumptions, that "
also stated several results of the form “there exists an
" = 1=2 (i.e.
= 2): We have
> 0”: that is, a su¢ ciently large
campaign contributions e¤ect. If the campaign contributions e¤ect is too large or too small,
or if
is too large then the probability that one candidate wins can fall outside the [0; 1]
interval. Therefore we see the restrictions we impose as simply regularity conditions.
Finally an interesting extension of the model would allow for entry. In a model in which
existing candidates can occupy an interval rather than a point on the ideological line may
make entry more problematic for third candidates.
5
Conclusion
According to traditional and accepted models, in two parties electoral contest the party
platforms should be clear and unambiguous and close to each other. In reality often: the
two parties do not converge and they make ambiguous policy promises, attempting to cover
a vast ideological space.
This paper provides a model which is consistent with both observations. What drives
extremism and ambiguity is the parties’ need to trade o¤ two forces: the gains in votes
obtained by converging to the middle and the bene…t of campaign contributions which in-
16
‡uence voters’behavior. Contributions often accrue to the parties if they move away from
the middle ground and towards extreme groups which feel especially strongly about certain
issues. Ambiguous polices allow the parties to attract contributions without committing to
extreme policies that would alienate middle of the road voters. Obviously, rational voters
would recognize this incentive but with some ex ante uncertainty about the true beliefs of
the candidates, the latter may maintain in equilibrium a certain amount of ambiguity in
their platforms.
This observation is consistent with the view often mentioned by commentators that a
long and detailed voting record, say in the Senate, is “baggage” that a¤ect negatively the
electoral; chances of a Presidential candidate, while newcomers may have an advantage. Our
model is consistent with this observation: it is more di¢ cult for somebody with a long and
detailed voting record to be ambiguous in his Presidential campaign and appeal to a broad
range of contributors and voters. A newcomer instead has more room to be ambiguous in
his platform.
References
[1] Alesina, Alberto (1988). “Credibility and Policy Convergence in a Two Party System
with Rational Voters,”American Economic Review 78, 796-805.
[2] Alesina, Alberto and Alex Cukierman (1990). “The Politics of Ambiguity” Quarterly
Journal of Economics,
[3] Alesina, Alberto and Howard Rosenthal (1995). Partisan Politics Divided Government
and the Economy, Cambridge University Press, Cambridge United Kingdom.
[4] Alesina, Alberto and Howard Rosenthal (2000). “Polarized Platforms and Moderate
Policies with Checks and Balances,”Journal of Public Economics 75, 1-20.
17
[5] Calvert, Randall (1985) “Robustness of the Multidimensional Voting Model: Candidate
Moderation Uncertainty and Convergence,” American Journal of Political Science 29,
69-95.
[6] Campante, Filipe (2006). “Redistribution in a Model of Voting and Campaign Contributions”, mimeo, Harvard University.
[7] Downs, Anthony (1957). An Economic Theory of Democracy, Harper and Row, New
York, NY.
[8] Glaeser, Edward, Giacomo Ponzetto and Jesse Shapiro (2005). “Strategic Extremism:
Why Republicans and Democrats Divide on Religious Values,” Quarterly Journal of
Economics 120, 1283-1330.
[9] McCarty, Nolan, Keith Poole and Howard Rosenthal (2006). Polarized America, MIT
Press, Cambridge, MA.
[10] Persson, Torsten and Guido Tabellini (2002). Political Economics, MIT Press, Cambridge, MA.
[11] Shepsle, Kenneth (1972). “The Strategy of Ambiguity: Uncertainty and Electoral Competition,”American Political Science Review 66, 555-569.
[12] Wittman, Donald (1983) “Candidate Motivation: a Synthesis of Alternatives,”American Political Science Review 77 142-177.
18
6
6.1
Appendix
Proposition 1 …rst-order conditions
The …rst-order condition for candidate X is
(x
xb )2
(y
xb )2
2
+ 2(x
xb )
x+y
2
1
1
+
2 2
= 0:
+ 2(y
yb )
x+y
2
1
1
+
2 2
= 0:
Similarly for candidate Y we have
(x
yb )2
(y
yb )2
2
6.2
Proof of Propositions 2-7
Proof of Proposition 2. The …rst-order condition for candidate X is
(x
xb )2
+2(x
1
2
xb )
1
1
+
2 2
x+y 1
+ (1
2
2
(y
x
1
2
1
1
+
2 2
1
1
+
2 2
1
+
2
1
2
1
1
+
2 2
1
1
+
2 2
1
+
2
xb )2
y)
= 0:
Similarly for candidate Y we have
(x
+2(y
yb )2
yb )
1
2
1
1
+
2 2
x+y 1
+ (1
2
2
= 0:
19
(y
x
yb )2
y)
Using the fact that
= 2 and xb = 1
yb and then solving simultaneously yields a unique
interior pair of solutions
1
2
1
=
2
xC =
1+
yC
1+
1
1
2yb 1
3 + yb (6
1 2yb
3 + yb (4
The no contributions equilibrium is given by setting
1
2
1
=
2
xN C =
yN C
4)
6 )
;
:
= 0; and hence
2yb 1
1 4yb
1 2yb
1+
1 + 4yb
1+
;
:
Therefore
xC
xN C =
=
1
2
1+
2 (4yb
1
negative we have xC
1
2
4)
3 (1 2yb )2
1) (1 3 + yb (6
The numerator is clearly positive for
Thus the sign of xC
2yb 1
3 + yb (6
4))
2yb 1
1 4yb
1+
:
> 0; and since yb > 1=2 we know that 4yb
xN C depends on the term 1
3 + yb (6
xN C and hencexC < xN C : At
the term is continuous in : Therefore there exists
1 > 0:
4) : When this term is
> 0 the term is 1
4yb > 0 and
> 0 such that the term is negative.
Identical reasoning applies for candidate Y which establishes that yC > yN C : It is clear that
xC = 1
yC and thus from (1) we have
Pr (X wins) = 2
1
2
1
4
which completes the proof.
20
=
1
= Pr (Y wins) ;
2
Proof of Proposition 3. >From the proof of Proposition 2 we know that
1
2
1
=
2
xC =
1+
yC
1+
1
1
2yb 1
3 + yb (6
1 2yb
3 + yb (4
;
4)
:
6 )
It follows that
dxC
=
d
2 (1
3 (1 2yb )2
3 + yb (6
4))2
< 0;
and
dyC
=
d
2 (1
3 (1 2yb )2
3 + yb (6
4))2
> 0:
Proof of Proposition 4. Follows directly from inspection of the reaction functions.
Proof of Proposition 5. Fix the strategy of other types and consider a deviation by type
xL : In the pooling equilibrium her payo¤ is
UXP =
pP (xL
pP
=
where
0
B
pP = 2 @
xL )2
1
1 (qy yL + (1
pP (qy yL + (1
qy )yR
3
4
(1
qy y L
(1
qy )yR
xL )2
xL )2 ;
qx xL +(1 qx )xR +qy yL +(1 qy )yR
2
+
qy )yR
qx x L
1
4
(1
qx )xR )
1
C
A:
If she deviates in a way which removes her bliss point from the interval then a fortiori
she is worse o¤ than deviating in a way that keeps her bliss point in the interval.
It is
therefore su¢ cient to consider only the latter deviation. Suppose that player xL deviates
by expanding her interval on the left. Then the probability that player xL wins under the
21
deviation is
0
B
pSxL = 2 @
xL +qy yL +(1 qy )yR
2
+
3
4
(1
qy y L
(1
1
1
4
qy )yR
C
A:
xL )
Supposing that player xR deviates by expanding her range to the right this probability is
0
B
pSxR = 2 @
xR +qy yL +(1 qy )yR
2
+
3
4
(1
qy y L
(1
1
1
4
qy )yR
C
A:
xR )
The di¤erences in payo¤s are then
Note that pP
then
xL
pSxL
1
(yL + yR )
2
xL
pSxR
1
(yL + yR )
2
xR
xL
=
p
xR
=
pP
pSxL + pP
2
P
pSxR = 0; and hence if
xL
;
2
:
> 0 then
xR
< 0 and if
xR
>0
< 0: Thus if one player prefers the separating equilibrium then the other does
not and hence the separating equilibrium does not exist. Similarly, the pooling equilibrium
cannot exist because one player always prefers to deviate. Now consider the deviation which
involves narrowing the range. Again, each type is still able to choose their preferred policy
ex post, and the change in beliefs is the same as above so that the bene…t from this deviation
is the same as above. Hence neither the separating nor the pooling equilibrium exists.
Proof of Proposition 6.
Note that we need only provide an example of values, so let
= 2: Consider candidate X: The probability of winning in the pooling equilibrium is
pP =
1
2
1
2
1
4
2 1
1
(yL + yE )
2
1
1
(xL + xR ) +
2
2
The payo¤ to player xL in the pooling equilibrium is
p
P
1
1
1
yL + yR
2
2
22
2
xL
;
1
(yL + yE )
2
1
(xL + xR )
2
2
:
and similarly for player xR : The probability that player xL wins if she deviates is
pSxL =
1
2
1
2
1
4
1
(yL + yE )
2
2 1
1
2
xL +
1
(yL + yE )
2
2
xL +
;
and her payo¤ is
pSxL
2
1
1
yL + yR
2
2
1
xL
:
Again, the analogous expression for player xR is apparent. Let xL = 0:2; xR = 0:4 and …x
y at 0:8: Calculating the di¤erence in payo¤s for the players in the pooling equilibrium
compared to deviating we …nd
xL = 0:09 and
xE = 0:04: Thus neither has a pro…table
deviation.
Proof of Proposition 7. Consider candidate X: The probability of winning in the pooling
equilibrium is
0
B
pP = 2 @
qx xL +(1 qx )xR +qy yL +(1 qy )yR
2
+
3
4
1
qy y L
(1
1
4
qxC xL
qy )yR
1
C
A;
where qxC is the prior probability that campaign contributors place on the candidate being
type xL : If player xL deviates then the probability of winning is
0
B
pSxL = 2 @
xL +qy yL +(1 qy )yR
2
+
3
4
(1
qy y L
(1
1
4
qy )yR
xL )
1
C
A:
Again, we need only provide an example of values which make the deviation unpro…table.
If qxC = 1; qx = 1=2; qy = 1=2 and xL = 0:2; xR = 0:4; yL = 0:6; yR = 0:8 then
and
xR = 0:018: Symmetric reasoning applies for candidate Y of both types.
23
xL = 0:025