Projection Effects and Strategic Ambiguity in Electoral Competition

Projection Effects and Strategic Ambiguity in Electoral CompetitionAuthor(s): Thomas JensenSource: Public Choice, Vol. 141, No. 1/2 (Oct., 2009), pp. 213-232Published by: SpringerStable URL: http://www.jstor.org/stable/40270953 .

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Public Choice (2009) 141: 213-232 DOI 10.1007/sl 1127-009-9449-4

Projection effects and strategic ambiguity in electoral competition

Thomas Jensen

Received: 6 November 2008 / Accepted: 24 April 2009 / Published online: 5 May 2009 © Springer Science+Business Media, LLC 2009

Abstract Theories from psychology suggest that voters' perceptions of political positions depend on their non-policy related attitudes towards the candidates. A voter who likes (dislikes) a candidate will perceive the candidate's position as closer to (further from) his own than it really is. This is called projection. If voters' perceptions are not counterfactual and

voting is based on perceived policy positions then projection gives generally liked candidates an incentive to be ambiguous. In this paper we extend the standard Downsian model in order to investigate under what conditions this incentive survives in the strategic setting of electoral competition.

Keywords Electoral competition • Ambiguity • Voter perception • Projection

JEL Classification D72 D83

1 Introduction

According to theories from psychology (see, e.g., Granberg 1993; Krosnick 2002) people prefer to be in a state of cognitive consistency. Therefore a voter prefers to believe that he

agrees with political candidates he likes (for reasons not directly related to the candidates' current policy positions) and disagrees with candidates he dislikes. One way the voter can achieve this is by distorting his perceptions of the candidates' policy positions. He "pulls" the positions of liked candidates towards his own position and "pushes" the positions of disliked candidates away from it. This is called projection. More specifically, positive projection (pulling) is called assimilation and negative projection (pushing) is called contrast.

If we assume that voters' perceptions are not counterfactual then projection of a candidate's policy position can happen only when the candidate is ambiguous. So if projection effects exist and voters vote based on perceived policy positions then generally liked candidates have an incentive to be ambiguous because of assimilation. This paper investigates under what conditions this incentive survives in the strategic setting of electoral competition.

T. Jensen (El) Department of Economics, University of Copenhagen, Studiestraede 6, 1455 Copenhagen K, Denmark e-mail: [email protected]

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214 Public Choice (2009) 141 : 213-232

We formulate and analyze an extension of the standard Downsian model that allows candidates to take ambiguous policy positions and introduces projection effects in voters'

perceptions of such positions. Ambiguous positions are modelled by intervals of policies. Each voter has an exogenous positive, neutral, or negative attitude towards each candidate. Thus the attitude is not in any way related to the candidates' policy position. When we say that a voter likes (dislikes) a candidate it simply means that he has a positive (negative) attitude towards him. Voters' perceptions of ambiguous positions are represented by probability distributions. They use perceived expected utility to decide for whom to vote. By assuming that voters are risk averse we get that they dislike ambiguity per se, i.e., if there were no

projection effects. Thus any result predicting ambiguity is driven only by projection. Our first results answer the following question: Under what conditions can a candidate

defeat the median by being ambiguous, i.e., win the election by taking an ambiguous position when the other candidate's position is fixed at the median? Loosely speaking, we show that if a candidate is liked by some voters and not disliked by too many then he can defeat the median. For example, a candidate who is not disliked by any voters can defeat the median if he is liked by an arbitrarily small group of voters.

Secondly, we consider the question of existence or non-existence of winning strategies (which must be ambiguous) for a candidate with an advantage due to voter attitudes and projection. We assume that a candidate who is not disliked by any voters and liked by a majority is running against an opponent who is not liked by any voters. Furthermore, we make some

specific assumptions about voter utility functions and perceptions. Under these assumptions we show that the advantaged candidate has winning strategies if the assimilation effect is

sufficiently strong. When assimilation is not sufficiently strong then the advantaged candidate does not have a winning strategy and the model does not have an equilibrium, not even in mixed strategies.

There is a substantial empirical literature on projection, see, e.g., Merrill et al. (2001) and the survey by Krosnick (2002). A large number of studies use cross-sectional data to

produce evidence that is consistent with projection effects. However, as Krosnick points out, usually the evidence is also consistent with alternative hypotheses, most notably policy based evaluation and persuasion. Furthermore, panel data studies have not produced com-

pelling evidence of projection. Therefore Krosnick concludes that the existence of projection has not yet been demonstrated convincingly and that further empirical research is needed.

Recent survey experiments on candidate ambiguity (Tomz and Van Houweling 2009) show that voters' perceptions of ambiguous policy positions display selective optimism based on party affiliation. In our model, voters' selective optimism is not necessarily based on party affiliation. Still, the convincing demonstration by Tomz and Van Houweling that one type of personal similarity between a voter and a candidate can lead to optimistic perceptions of ambiguous positions does provide considerable support for the empirical relevance of our model.

A number of theoretical models of ambiguity in electoral competition exist in the literature. Shepsle (1972) shows that if a majority of voters is risk loving around the median then an incumbent positioned at the median can be defeated by a challenger announcing a lottery position with mean equal to the median. Page (1976) argues that candidates use ambiguity on issues of conflict to focus voters' attention on consensus issues. Later theoretical models have offered different explanations of ambiguity. For example, the unwillingness of an incumbent to reveal his true policy preferences (Alesina and Cukierman 1990), uncertainty about the median voter's position (Glazer 1990), candidates' preferences for flexibility in office (Aragonès and Neeman 2000), the unwillingness of candidates to commit to a certain

position during primaries because they may be better informed about the electorate later

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Public Choice (2009) 141: 213-232 215

(Meirowitz 2005), and context-dependent voting (Callander and Wilson 2008). None of the

explanations are similar to the one we propose here. Note, however, that Page (1976: 748) mentions the possible link between projection and ambiguity in a footnote.

Our model is also somewhat related to the theoretical literature on candidate quality (see, e.g., Ansolabehere and Snyder 2000; Groseclose 2001; Aragonès and Palfrey 2002, 2005). In these models one candidate has an advantage which makes all voters prefer him over the other candidate if there is policy convergence. In our model a candidate can have an

advantage due to voters' attitudes. But he can make use of that advantage only by being ambiguous which makes voters who like him assimilate his position. Voters' attitudes do not directly influence their voting behavior.

The paper is organized as follows. In Sect. 2 we set up the model and present two exam-

ples of our general model of projection of ambiguous policy positions. Section 3 contains our results (all proofs are relegated to the Appendix). In Sect. 4 we discuss and conclude.

2 The model

Our starting point is a standard one-dimensional spatial model with two candidates. We will extend this model by allowing candidates to take ambiguous policy positions and by introducing projection effects in voters' perceptions of such positions. In the following we describe the model in detail.

2.1 The candidates

Before the election the two candidates announce policy positions. Each candidate can announce either a certain position or an ambiguous position. A certain position is represented by a point in the policy space R. An ambiguous position is represented by a compact interval of policies. Thus the strategy space for each candidate can be written as

S = {[A - a, A + a]\ A e R, a > 0}.

Announced positions are credible in the sense that the winning candidate must enact a policy in his announced interval. So certain positions are credible in the usual sense.

Each candidate's sole objective is to win the election; none of them care about policy. Formally the preference relation of each candidate over the outcome of the election is given by

win > tie >- lose.

Finally, we assume that the candidates are fully informed about the electorate and that this is common knowledge.

2.2 The electorate

There is a continuum of voters and each of them has a preferred point in the policy space R. The distribution of preferred points is given by a density function v. We assume that v is continuous and that the support of v is an interval (bounded or unbounded). Without loss of

generality we assume that the median voter is located at x = 0, i.e.,

/»0 poo i

I v(x)dx = I v(x)dx = -. J-OQ JO 2

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Each voter has a utility function on the policy space. Let the utility function of the median voter be uq. Then the utility function uXQ of a voter with preferred point at jco is defined by

uXQ (x) = uq(x - x0) for all x € R.

We assume that uq is symmetric around 0, continuous on R, and twice continuously differentiable on R \ {0} with

uf0{x)^0 forjcÔ

and

Uq(x)<0 for x ^0.

Thus all voters are strictly risk averse. We will now model how voters decide on which candidate to vote for. If each candidate

announces a certain position then each voter simply votes for the candidate announcing the

position with the highest utility. If at least one of the candidates announces an ambiguous position then it is less obvious how the voters should decide for whom to vote. We assume that they are expected utility maximizers. But it is not straightforward for them to use ex-

pected utility since an ambiguous position is represented by an interval of policies rather than a probability distribution over policies. For a voter to use expected utility to evaluate an

ambiguous position he has to somehow associate a probability distribution with the interval

representing the position. The distribution represents the voter's perception of the ambiguous position, or, to put it differently, the voter's belief about which policy the candidate will enact if elected. How voters perceive ambiguous positions is a crucial element of our model and we will use the rest of this section to describe it.

As mentioned in the introduction, the main idea is that a voter's perception of an am-

biguous position depends on whether he has a positive, negative or neutral attitude towards the candidate announcing it. Note that the attitude is exogenous and thus not at all related to the candidate's position. If the voter likes the candidate, i.e., has a positive attitude towards him, then he will put most of the probability mass on the points of the interval that are closest to his preferred policy {assimilation). If the voter dislikes the candidate then he will do the opposite {contrast). And if the voter neither likes nor dislikes the candidate then he will

spread the probability mass evenly across the interval. We will formalize this below. For all voters the neutral perception of an ambiguous position is given by the uniform

distribution on the interval. So the perceived expected utility of the ambiguous position [A - a, A + a] for a neutral voter with preferred point xo is

1 fA+a - / uxo{x)dx. &* JA-a

Since voters are strictly risk averse it follows that neutral voters dislike ambiguity, i.e., they always strictly prefer the midpoint of an interval to the interval itself.

To model assimilation perceptions, first consider a voter with preferred point x0 > 1 and a positive attitude towards a candidate announcing [-1, 1]. Thus we are modelling assimilation from the right of an ambiguous position centered at the median. The probability distribution that the voter associates with the ambiguous position is given by some cumulative distribution function F/ . So the voter's perceived expected utility of the candidate's

position is

f uX0{x)dFl{x). J-oo

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Public Choice (2009) 141 : 213-232 217

We assume that F/ satisfies the following two conditions. £/[-i,i] denotes the cumulative distribution function of the uniform distribution on [-1, 1],

- F,1 puts no probability mass outside [- 1, 1], i.e.,

F/(jc) = 0 foralljc<-l and F/(l) = l.

- Fl strictly first-order stochastically dominates the uniform distribution on [-1, 1], i.e.,

F\(x) < t/[_i,i](jc) for all x e R

(with < for some jc).

The first condition implies that there is no "counterfactual perception". Since the winning candidate must enact a policy in his announced interval the voter does not put any probability mass on policies outside the interval. The second condition is a convenient mathematical formulation of assimilation from the right. It means that, for any jc e [- 1, 1], F/ puts at least as much probability mass to the right of jc as the uniform distribution does (and strictly more for some jc). Thus we see that, relative to the neutral perception given by the uniform distribution, the voter pulls probability mass to the right, i.e., towards his own preferred position.

We model assimilation of [-1, 1] from the left by symmetry. Therefore the assimilation

perception for a voter with preferred point jc0 < - 1 is given by the distribution function Ff1 defined by

F"1(jc) = 1- lim Fhy) foralljceR. y^(-x)-

Because with this definition we have that, for any jc e [-1, 1], Ff1 puts exactly as much

probability mass on [- 1, jc] as F/ puts on [- jc, 1]. Then consider assimilation of [- 1 , 1] by some "interior voter", i.e., a voter with preferred

point jc0 € (- 1, 1). The distribution function representing the perception of such a voter is denoted F*°. Again we assume that

F*°(jc) = 0 foralljc<-l and Fjr°(l) = l

to rule out counterfactual perception. Formalizing the pulling of probability mass towards jco is a bit more tricky in this case than it was for "exterior voters" but the idea is the same. We want the distribution function to put more probability mass on points close to jco than the uniform distribution does. More precisely we assume that, for all jc > 0, F*° puts at least as much probability mass on the interval (jcq - jc, jco + jc) as the uniform distribution does (and strictly more for some jc). Formally, for all jc > 0,

lim Ffo(jco + y) - F^(x0 - x) > U{-iA](xo + jc) - t/Mf nfo - jc) y^x

(with > for some jc).

To have symmetry of perceptions we assume that, for any jco € (- 1, 1),

F~*°(jc) = 1- lim Ff°(v) foralljceM.

Thus we are done modelling how voters assimilate the ambiguous position [-1, 1], Now we will extend the model to cover assimilation of all ambiguous strategies.

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218 Public Choice (2009) 141: 213-232

First consider assimilation of [-a, a] for some a > 0. In this case we use a simple scaling of the distribution functions defining assimilation of [-1, 1]. More specifically, the assimilation perceptions of voters with xq > 0 are defined as follows (assuming symmetry this is all we need).

- For a voter with xo>a the distribution function is denoted F£. It is defined by

F%(x) = F/ ( -

) for all x e JR.

- For a voter with 0 < xo < a the distribution function is denoted F%° . It is defined by

f;° (x) = f/(-J

for all x g R.

It is easily seen that these distribution functions satisfy conditions that are analogous to the ones we imposed on the Fj*°s. Counterf actual perception is ruled out because we have, for any 0 < Jto < a,

F;°(jc)=0 for all jc < -a and F?(a) = l.

And the F^°s are assimilation perceptions because

Faa{x)<U{.a,a]{x) forallxeR

(with < for some jc)

and, for any 0 < jco < a and all x > 0,

lim F^(x0 + y) - F^(x0 -x)> U{-aja](x0 + x)- £/[_«,«] (x0 - x) y-*x~

(with > for some jc).

Finally, we will define assimilation of intervals of the type [A - a, A + a] where AÔ. In that case we simply translate the distribution functions defining assimilation of [-a, a] by the constant A. For example, the assimilation perception of [A - a, A + a] by a voter with xq > A + a is given by the density function Fâ defined by

Fff(x) = F;(x-A).

Obviously the translated distribution functions satisfy the "translated" versions of the conditions satisfied by the F^°s.

Contrast perceptions are defined analogously to the way we have just defined assimilation perceptions. So, for example, the distribution function representing the contrast perception of voters to the right of some interval is strictly first-order stochastically dominated by the uniform distribution on the interval. We use the same notation for contrast perceptions as for assimilation perceptions except that we replace the F s by G s. So the distribution functions representing the contrast perception of [- 1 , 1] by voters to the right of the interval is denoted

G\ and so on. For simplicity we assume that voters with the same preferred policy have the same at-

titude towards each candidate. Therefore we can define the attitude functions L, , / = 1 , 2,

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Public Choice (2009) 141: 213-232 219

by

if voters at x have a positive attitude towards Candidate i 1 0 if voters at x have a neutral attitude towards Candidate / > .

- 1 if voters at jc have a negative attitude towards Candidate / |

Any combination of attitudes towards the two candidates is allowed. So it is possible for a voter to like both candidates, dislike both candidates, like one candidate and dislike the other, etc. We make the technical assumption that, for each / = 1, 2, the sets Lr1({l}), L^l({0}), and L^d-l}) are Lebesgue measurable (such that we can integrate over them). Then the fraction of voters that have a positive/neutral/negative towards Candidate / is

/ v(x)dx I f v(x)dx I f v(x)dx. Jl~\{\}) Jl~1({0}) Jl7\{-1})

We end this section with two examples of our model of assimilation. We will get back to these examples later on.

2.2.7 Example 1

In our first example, the assimilation of [- 1 , 1] by voters to the right is given by the distribution function F/ defined by

*H¥*.+¥ E.rU)}- where 0 < 8 < 1 is a parameter. This corresponds to the belief that with probability 8 the

policy will be x = 1 (the policy in the interval that is closest to the voter's preferred point) and with probability 1 - 8 the policy will be drawn from the uniform distribution on [- 1 , 1].

We want a voter with preferred point xq g (- 1, 1) to have the same type of perception, i.e., to believe that with probability 8 the policy will be x = xq and with probability 1 - 8 the

policy will be drawn from the uniform distribution. The distribution function corresponding to this perception is

*' (x)-\l^x + lM lfxe[Xo,l] j

It is easily seen that these distribution functions satisfy the required conditions. The ex-

ample is extended to cover assimilation of all ambiguous positions as described above (by symmetry, scaling, and translation).

2.2.2 Example 2

In our second example, the assimilation of [-1, 1] by voters to the right is given by the

density function // defined by

where 0 < y < 1 is a parameter. So, when assimilating [-1, 1], voters with jco > 1 put a constant probability density of ^- on the interval [0, 1] and a constant probability density

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220 Public Choice (2009) 141: 213-232

of ^y~ on [- 1 , 0). It is easily seen that the distribution function corresponding to the density function // is

And then it is straightforward to show that the distribution strictly first-order stochastically dominates the uniform distribution on [-1, 1].

The assimilation of [-1, 1] by voters with x0 € [0, 1) is given by the density functions

/*°,jcoe [0,1), denned by

/*<> = // if*oe|j,l) and

[^ if |jc0 - Jc| < 5 J L 2/

Given our definition of assimilation for exterior voters this is a natural way of defining it for interior voters. In each case a voter puts a constant probability density of ^ on the half of

the interval that is closest to his preferred point and a constant probability density of ^ on the rest. It is straightforward to check that the distribution functions given by the ff°s satisfy the required conditions. As with our first example it is extended to cover assimilation of all ambiguous positions as described above.

3 Results

In this section we will address the questions in the list below. Our answers will help us understand how and why the introduction of ambiguous positions and projection effects

changes the predictions of the standard model.

1 . Under what conditions does the median voter theorem break down because candidates can take advantage of the assimilation effect by being ambiguous? More specifically, when can a candidate defeat the median by being ambiguous, i.e., win the election by taking an ambiguous position when the other candidate's position is fixed at the median?

2. Under what conditions does a candidate with an advantage due to voters' attitudes have a winning strategy? That is, when does such a candidate have an ambiguous position that wins the election for him no matter what position the other candidate takes?

3. What can we say about existence and properties of Nash equilibria in the game where the two candidates simultaneously announce positions?

All proofs are relegated to the Appendix.

3.1 Defeating the median

Our first result shows that a candidate who is liked by a strict majority of voters can defeat the median, i.e., win the election when the other candidate's position is fixed at the median. Furthermore, it shows that he can do so by being ambiguous around the median. Note that no additional assumptions on voter utility functions or voter perceptions are needed. The result holds even if voters are very risk averse and the assimilation effect is very small.

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Public Choice (2009) 141: 213-232 221

Theorem 1 Suppose Candidate i is liked by a strict majority of voters, i.e.,

I v(x)dx > -.

Then there exists some a! > 0 such that, for any 0 < a < a! , Candidate i defeats the median

by announcing the ambiguous position [-a, a].

Being liked by a strict majority of voters is a necessary condition for a candidate to be able to defeat the median by an ambiguous position of the type [-a, a]. That follows imme-

diately from the assumption that all voters are strictly risk averse. However, the following result shows that a candidate who is liked by less than a majority may be able to defeat the median by adopting an ambiguous position that is not centered at the median. The sets Xf and Xr, / = 1 , 2, are defined by

Xt = {x> 0\Li(x) = 0} U (jc|L,(jc) = 1}

and

XT = {x< Q\Li{x) = 0} U [x\Li(x) = 1}.

We let E\ denote the expected value of F I .

Theorem 2 Suppose X* contains the preferred points of a strict majority of voters, i.e.,

I v(x)dx>-. Jxf l

Let A e (0, £"}). Then there exists some a' > 0 such that, for any 0 < a < a! , Candidate i

defeats the median by announcing the ambiguous position [a A- a,aA + a\.

By symmetry it follows that if a strict majority of voters has preferred points in XJ and we let A e (- E\, 0) then the same conclusion holds. Thus we see that a candidate who is liked by just a few voters may be able to defeat the median. For example, this is the case if he is not disliked by any voters.

The last result in this section shows that if neither Xf nor Xr contains the preferred points of a strict majority of voters then there exist voter perceptions (given by Fj*° and G*°, xo e [- 1 , 1]) such that Candidate i cannot defeat the median. Thus we have a necessary and sufficient condition for Candidate / to be able to defeat the median for any type of voter

perceptions.

Theorem 3 Suppose that neither X* nor XJ contains the preferred points of a strict ma-

jority of voters, i.e.,

f 1 f 1 / v(x)dx < - and / v(x)dx < -.

Jx+ 2 Jx- 2

Then there exist F*° and G\° , xq e [- 1, 1], (satisfying the assumptions on assimilation and contrast perceptions) such that Candidate i cannot defeat the median.

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222 Public Choice (2009) 141: 213-232

3.2 Winning strategies

Our first observation is that Candidate i does not have a winning strategy if the set of voters who like him, LJX ({ 1 }), is not a strict majority. Because in that case Candidate j can always get at least a tie by announcing the midpoint of Candidate / 's position. So winning strategies can exist only if at least one candidate is liked by a strict majority of voters. Here we will consider only the case where one candidate (Candidate 1) is not disliked by any voters and liked by a strict majority and the other candidate (Candidate 2) is not liked by any voter. Thus we have

L\(x)>0 and L2(x)<0 foralljceR

and

/ v(x)dx>-.

Assuming that Candidate 2 is not liked by any voter means that checking if some position of Candidate 1 is a winning strategy becomes a lot simpler. Because then we only need to check if Candidate 2 can get at least a tie against it by taking a certain position.

We will also make some specific assumptions about voter utility functions and assimilation perceptions. With respect to utility functions we assume that

uo(x) = - |jc|a forsomea?>l.

With respect to assimilation perceptions we assume that they are defined as in one of the two examples presented earlier. Remember that in the first example the strength of the assimilation effect is measured by the parameter <5, in the second example it is measured by the parameter y .

With the assumptions above we have the following result. Again E\ denotes the expected value of F/ .

Theorem 4 If assimilation perceptions are as in the first {second) example then there exists a 8* e (0, 1) (y* G (0, 1)) such that the following statements hold.

1. Suppose Candidate 1 is liked by all voters in some neighborhood of the median. If 8 > £*

(X > /*) then there exists some a' > 0 such that, for any 0 < a < af , [-a, a] is a winning strategy for Candidate 1.

2. If8<8*(y<y*) then Candidate 1 does not have a winning strategy. More specifically, if Candidate 1 announces the ambiguous position [A - a, A + a] then Candidate 2 can

get at least a tie by announcing

A + aE\ ifA<0

and

A-aE\ ifA>0.

The cut-off value 8* (/*) from the theorem is given by the equation

j uo(x)dFf(x) = uo(E\)

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Public Choice (2009) 141: 213-232 223

(see the proof for details). In other words, 8 = 8* (y = y *) is equivalent to the median voter

being indifferent between [-1, 1] announced by Candidate 1 and the certain position E\. Furthermore, 8 ^ 8* (y ^ y*) is equivalent to

j uo(x)dF?(x)ûo(E\).

So it follows from the theorem that if Candidate 1 is liked by all voters in some neighborhood of the median then he has a winning strategy if and only if the median voter strictly prefers to vote for him if he announces [- 1 , 1] and Candidate 2 announces E\ .

It is important to note that even if Candidate 1 is liked by all voters he may not have a winning strategy. A winning strategy only exists if the assimilation effect is sufficiently strong.

3.3 Nash equilibria

Here we will make some observations about the existence and properties of Nash equilibria in the game where the two candidates simultaneously announce positions.

First, consider the case where neither candidate can defeat the median. Then (s*, s%) =

(0, 0) (convergence to the median) is an equilibrium because neither candidate can win the election by deviating to another position. Furthermore, if neither of the candidates is liked

by exactly 50% of the voters then (0, 0) is the unique equilibrium. Because in that case each candidate can defeat any position different from the median (an ambiguous position can be defeated by its midpoint). Thus the median voter theorem holds in this situation.

Then consider the case where one candidate (Candidate 1) can defeat the median but the other (Candidate 2) cannot. Then Candidate 1 can defeat any position of Candidate 2 (we disregard the case where Candidate 2 is liked by exactly 50% of the electorate). Therefore we have that in any equilibrium Candidate 1 must win the election. Thus (s*,^) *s an

equilibrium if and only if s * is a winning strategy for Candidate 1 .

Finally, consider the case where both candidates can defeat the median. Without loss of

generality assume that Candidate 1 is liked by at least as many voters as Candidate 2. If Can- didate 2 is liked by less than 50% of the voters then we have that (s*, s%) is an equilibrium if and only if s* is a winning strategy for Candidate 1 (Candidate 1 can defeat any position of Candidate 2). If Candidate 2 is liked by more than 50% of the voters then consider the numbers

Pi= I v(x)dx, i = l,2, y #/. J{x\Li(x)>Lj(x)}

Pi is the share of voters having more positive attitudes towards Candidate / than towards Candidate j (either they dislike j and are neutral or positive towards i or they are neutral towards j and like /). If Pt > Pj then Candidate i can defeat any position of Candidate j by imitation. Thus it follows that (s*, s%) is an equilibrium if and only if s* is a winning strategy for Candidate i.

From the arguments above we conclude that when the median voter theorem breaks down then (except perhaps for some very special cases) a Nash equilibrium exists only if one candidate has a winning strategy. Since we have only considered pure strategies it is relevant to ask if allowing for mixed strategies would ensure existence of equilibrium. This is not the case. For example, suppose that the assumptions from the analysis of winning strategies are satisfied and let 8* (y*) be the cut-off value for 8 (y) from Theorem 4. If 8 < <5* (y < y*) then a mixed strategy Nash equilibrium does not exist. This claim is proved in the Appendix.

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224 Public Choice (2009) 141: 213-232

4 Discussion

Our goal in this paper has been to investigate theoretically whether positive projection (assimilation) of policy positions can explain why some politicians are ambiguous with respect to their issue positions. To do that we have extended the standard Downsian model of electoral competition by allowing candidates to take ambiguous policy positions and by intro-

ducing projection effects in voters' perceptions of such positions. By assuming that voters dislike ambiguity per se (if attitudes are neutral then voters dislike ambiguity because of risk

aversion) we have made sure that projection is the driving force behind any result predicting ambiguity.

In the standard model the median defeats any other position. The first step therefore was to determine the conditions under which a candidate can defeat the median by being ambiguous. We presented necessary and sufficient conditions. We saw that it may suffice for a candidate to be liked only by a small group of voters as long as he is not disliked by too

many. For example, a candidate who is not disliked by anyone can defeat the median if he is liked by an arbitrarily small group of voters.

Having at least one candidate that is able to defeat the median by being ambiguous is

certainly a necessary condition for predicting ambiguity. However, it is far from sufficient. Electoral competition is a strategic situation and there is no reason to assume that one candidate will announce the median if he can do better by taking a different position. Therefore our next step was to consider existence and non-existence of winning strategies (which must be ambiguous). We restricted attention to the case where one candidate is not liked by any voters and made specific assumptions about voter utility functions and perceptions. We saw that if the assimilation effect is sufficiently strong then a candidate who is not disliked by any voters and liked by a strict majority has winning strategies (intervals centered at the

median). But even if he is liked by all voters he does not have a winning strategy if the assimilation effect is not strong enough. So (under some specific assumptions) our model does predict that a generally liked candidate will be ambiguous if he is running against a candidate who is generally not liked and the assimilation effect is sufficiently strong. But even a candidate with the largest possible advantage due to voters' attitudes is not able to transform this advantage into a certain victory if assimilation is not sufficiently strong. That is a rather striking result.

While the possible non-existence of winning strategies even for a candidate with the

largest possible advantage is an interesting feature of the model it is also problematic. Be- cause, as our observations on Nash equilibria revealed, when at least one candidate can defeat the median then non-existence of winning strategies implies non-existence of pure strategy equilibria (except perhaps in some very special cases). So when the median voter theorem breaks down, but neither candidate has a winning strategy, then our model does not

give us clear predictions in terms of pure strategy equilibria. And turning attention to mixed

strategy equilibria is not a solution to this problem. So while we did get several interesting results there is clearly room for further theoretical work on the link between projection and

strategic ambiguity. We believe that the framework developed here provides a solid back-

ground for such work.

Acknowledgements This is a revised version of the first chapter of my Ph.D. dissertation. I thank two anonymous referees, the editor in chief, Enriqueta Aragonès, David Dreyer Lassen, Rebecca B. Morton, Christian Schultz, Peter Norman S0rensen, Mich Tvede, seminar participants at the University of Copen- hagen, and participants and faculty at EITM V, University of Michigan, Ann Arbor for helpful comments and suggestions. Naturally, all remaining errors are mine.

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Public Choice (2009) 141: 213-232 225

Appendix

Proof of Theorem 1 For each n e N let

XÏ = lx\Li(x) = hl-<\x\<n\.

Since a strict majority of voters likes Candidate i there exists aniVeN such that X? contains the preferred points of a strict majority of voters. We will prove that there exists an a' > 0 such that, for any 0 < a < a! , all voters with x0 e X? strictly prefer [-a, a] announced by Candidate i over the median (jc = 0).

For each a > 0,

max \u"XQ{y)\ u -a<y<a u

is a continuous function of xo on (a, oo). So for a < jj it follows by compactness that the function is bounded on [^, N]. Thus we can define

Ca= max max \u"(y)\.

Now let ^ < jco < N, 0 < a < jj, and x e [- a, a]. Then, by Taylor's theorem, we have

uxo(x) = uX0(0) + u'Xo(O)x + U-^-x2.

for some Ç e [- a, a] (actually between 0 and jc). And thus it follows that

uxo(x)>uxo(O) + ufXQ(O)x-^x2.

Using this inequality we get (E{ denotes the expected value of F/)

f° uXQ(x)dFaa{x) >

j° (uXQ(0) + uXQ(0)x

- ^-x2\dFaa{x)

= uX0(0) + ii^(0) r xdFaa{x) - ^ ̂ f x2dFaa{x)

J -a ^ J-a

= «^(O) + u'XQ(0)aE\ -

y !" x2dFZ(x)

> uX0(0) +u'X0(0)aE\ -

^-a2 ^ f dFaa(x)

^ J-a

= uxo(O) + U'xo(,O)aEl-tfa2

>uxls(0) + u'Q(-âE\-^-a2.

Since «q(- jj) > 0, E\ > 0, and Ca is decreasing with « it follows that, for a sufficiently small,

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226 Public Choice (2009) 141 : 213-232

f uX0(x)dFZ(x)>uX0(0) forall-|-<*o<JV. ™ J -a

™

So all voters with j, <xo<N and L/(jc0) = 1 strictly prefer [-a, a] announced by Candi- date i over the median for a sufficiently small. By symmetry the same holds for voters with - N < jc0 < - jj and L,- (jc0) = 1 . Thus it holds for all voters with jc0 £ X? . D

Proof of Theorem 2 Let A g (0, £}). Pick aniVeN such that a majority of voters have preferred points in

X?+ = Ix\xeX+,^<\x\<n\.

For a > 0 with a A + a < jj we can define

CAtO= max max K'(y)|. jj<\xO\<NaA-a^yâA+a

Then, by Taylor's theorem, we get that for all jc0 with jj < \xo\ < N and all a > 0 with

aA+a<jj,

uxo(x) > uX0(0) + u'XQ(0)x -

^-x2 for all xc[aA- a, a A + a].

For all voters with xo e X+, jj <xo < N the perceived expected utility of [a A - af a A + a] is at least

I paA+a - / K,o (*)</*.

Using the inequality above we get that, for all a > 0 with aA + a < jj,

i paA+a paA+a / C \ - / uxo(x)dx >

/ uX0(0) + <0(0)^ -

-^*2 C

)Jx

= uxa (0) + «;0 (0)a A - -£2- ^ / x2^x ^ JaA-a

> uX0(0) + <(0)aA -

î(flA + a)2

> ^(0) + «o(-^)

Afl - ^(1

+ A) V.

The last expression is strictly greater than m^0(0) for sufficiently small a. So it follows that, for a sufficiently small, all voters with jcq g Xf, jj < xo < N strictly prefer [aA-a,aA + a] over the median.

For a voter with jc0 € X+, -N < x0 < -jj the perceived expected utility of [a A - a, a A + a] (with - jj < a A - a) is

r +a «,d^w.

JaA- a

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Public Choice (2009) 141 : 213-232 227

Using the "Taylor inequality" from above we get that, for all a > 0 with aA + a < jj,

paA+a / C \ ^0(x)dF:â"(x) >

/ «,0(0) + <0(0)x -

-^x2 C

L )dF:£-°(x) ..A-a JaA-a \ L /

> uX0(0) + ufXQ(0)(aA - aE\) -^(aA

+ a)2

> uX0(0) + u'o(jj)(A

~ E\)a ~ %^d

+ A)2"2-

The last expression is strictly greater than Wjco(O) for sufficiently small a. So it follows that, for a sufficiently small, all voters with jc0 e X+, -N < x0 < -jj strictly prefer [a A - a, aA+a] over the median.

Thus we have seen that, for a > 0 sufficiently small, all voters with jc0 € Xf + (a strict

majority) prefer [aA - a,aA + a] over the median.

Proof of Theorem 3 Suppose that assimilation perceptions are as in the first example (for some 0 < 8 < 1). Let contrast perceptions be given by

JF/ if*0 €[-1,0)1 1

[F-1 ifjc0€[0,l] I*

Suppose Candidate i announces some interval. We will show that at least 50% of the voters strictly prefer the median over the interval.

The interval can be written as

[a A -a,aA + a] for some A e R, a > 0.

For A = 0 all voters with L/(x0) < 0 (at least 50%) strictly prefer the median (by risk aversion). For A > E\ it is easily seen that

E{Fxa\a) >aA- aE\ > 0 for all jc0 € [a A -a,aA + a].

And then it follows by risk aversion that all voters with jco < 0 strictly prefer the median over the interval. Analogously it follows that if A < -E\ then all voters with x0 > 0 prefer strictly the median over the interval.

Thus the only cases left are A € (0, E\) and A e (-£}, 0). Suppose A € (0, E\). Then it is straightforward to check that for each voter with

xoe{x> 0\Li(x) = -1} U {jc < 0\Li(x) < 0}

the mean of the voter's perception of the interval is further away from jco than the median (0). Thus all these voters strictly prefer the median over the interval (by risk aversion). Since the set above is the complement of Xf these voters constitute at least a weak majority. If A e (-£"1,0) then it follows analogously that all voters with xo £ XJ~ strictly prefer the median over the interval.

In the proof of Theorem 4 we use the following lemma.

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228 Public Choice (2009) 141: 213-232

Lemma 1 Suppose the assumptions from the section on winning strategies are satisfied. If for some C > 0, f_{ uo(x)dF®(x) > uq{C) then there exists ane > 0 such that

j uXQ(x)dFx°(x)>uXo(C) for all x0 e[-l,€].

Proof We will do the proof only for the case where assimilation perceptions are as in the first example. The other case is analogous.

First note that by a simple continuity argument it suffices to show that

j uX0(x)dFx°(x)>uXQ(C) forall*o€[-l,0).

For each jc0 e [-1,0] define Cx\ Cxv° > x0 by

f uxo(x)dF?(x) = uxo(Cxo)

and

1 fl - / uX0(x)dx = uX0(Cxu°).

That is, for a voter at xOy Cx° (C^°) is the certainty equivalent of Fj*° (U[-UU) to the right ofx0.

Since jx_x uo(x)dF?(x) > uo(C) it follows that C° < C. If we can show that, for each

x0 e [-1,0), Cx° < C° then we have

/ uX0(x)dFX0(x) = uX0(Cx°) > uX0(C°) > uX0(C)

and we are done. Let Jt0 € [- 1 , 0). If Cl? < Cl then we have

8uX0(0) + (1 - <5)MjC0(C£ ) < Suxo(xo) + (1 - <5K0(C*°) = uX0(Cx°).

So the certainty equivalent (the one to the right of the preferred point) of the lottery "0 with

probability 8, C^ with probability 1 - 5" for a voter at jco is greater than or equal to Cx°. The certainty equivalent of the same lottery for a voter at 0 is C°. And since voters at jco are less absolute risk averse on [0, C^] than voters at 0 (this follows from the assumption that woW = - I*T f°r some a > 1) the certainty equivalent for a voter at 0 is greater than that for a voter at jco- Thus we must have Cx° < C°. So we see that it suffices to show that

C^0<C^forallx0G[-l,0). Again let x0 e [-1, 0). Define C*°_, C^°+ > x0 and C£_, C°u+ > 0 by

1 fxo+l 1 Cx - - / uxo(x)dx = ux»{Clf_), / uxo(x)dx = uXQ(Cxv\)

and

- - / uo(x)dx = «o(C^_),

/ uo(x)dx = uo(C°u+). Z + XoJ-\ -Xo JXQ+i

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Public Choice (2009) 141 : 213-232 229

Since /*°+1 uXQ(x)dx = /*°+1 uo(x)dx and voters at jc0 are less absolute risk averse on [xo + 1, 1] than voters at 0 it follows that

<%_<<%_ and CS?+<C°+.

Thus we have

2-ûX0(C°v_) + ̂ «,0(C°+) < i±^BlB(cy_) + q^MO

= «„«#).

So the certainty equivalent (the one to the right of the preferred point) of the lottery "C^_ with probability - ^, Cy+ with probability z|a" for a voter at jcq is greater than or equal to

CXy . The certainty equivalent of the same lottery for a voter at 0 is C^. And since voters at jc0 are less absolute risk averse on [C^_, C^+] than voters at 0 the certainty equivalent for a voter at 0 is greater than that for a voter at *o- Therefore we must have CXJ <Cy.

Proof of Theorem 4 It is straightforward to check that

f uo(x)dF*(x) = uo(El) J - l

is equivalent to

The expression on the left-hand side of the latter equation is negative for S = 0 (y = 0), positive for 8 = 1 (y - 1), and differentiable w.r.t. 6 (y) on [0, 1] with positive derivative. Therefore there exists a «5* € (0, 1) (y* e (0, 1)) such that

S%8* & J uo(x)dF*(x)%uo(E\)

(y^Y* ^ / uo(x)dF«(x)%uo(E{)\

With this definition of 8* (y *) we are ready to prove the statements of the theorem. 1. Suppose that, for some p > 0, Candidate 1 is liked by all voters with jco G (- fi, fi). It

suffices to show that if /^ uo(x)dF^(x) > uQ{E\ ) then, for a > 0 sufficiently small, [-a, a] is a winning strategy for Candidate 1 .

If /_!, iiotodFftjc) > uo(E\) then we can pick a C € (0, £f) such that

J uo(x)dF?(x)>uo(C).

From Lemma 1 it then follows that there exists an e > 0 such that

j uX0(x)dFÏ°(x)>uX0(C) farall*o€[-l,*].

The statement now follows from two claims below.

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230 Public Choice (2009) 141: 213-232

Claim A For sufficiently small a > 0, [-a, a] announced by Candidate 1 defeats any y e (-aC, aC) announced by Candidate 2.

Claim B For sufficiently small a > 0, [- <2, a] announced by Candidate 1 defeats any y £ (-aC, aC) announced by Candidate 2.

Proof of Claim A Choose an AT e N such that a strict majority of voters have preferred points in the set

X» = lx\Ll(x) = h^<\x\<N\.

It suffices to show that, for a sufficiently close to zero, all voters with jco G Xf, xq > 0 will prefer [-a, a] announced by Candidate 1 over aC announced by Candidate 2. Since C € (0, E\) this follows by Taylor's theorem as in earlier proofs.

Proof of Claim B Let a < p. From the homogeneity (of degree a) of uq and what we have shown above it follows that, for any xo e[-a, as],

I" uX0(x)dF?(x)=aa J f u*(x)dF* (x) > aau^(C) = uXQ(aC).

J -a J - 1

Thus we see that all voters with xq e [-a, as] strictly prefer [-a, a] announced by Candi- date 1 over any certain position y >aC.

Voters with xq e (-0, -a) has the same perception of [-a, a] announced by Candidate 1 as voters with xo = -a. And their absolute risk aversion on [-a, a] is lower. Therefore these voters also prefer [-a, a] announced by Candidate 1 over any y > aC.

By Taylor's theorem it is easily seen that voters with x0 < -ft and a neutral attitude towards Candidate 1 strictly prefer [-a, a] announced by Candidate 1 over any y > aC when a > 0 is sufficiently small.

Thus all voters with xo < as strictly prefer [-a, a] announced by Candidate 1 to any y > aC for sufficiently small a > 0. By symmetry it follows that all voters with jc0 > -as

strictly prefer [-a, a] announced by Candidate 1 to any y < -aC. That ends the proof of the claim.

2. It suffices to show that if f _{ uo(x)dF®(x) < uq{E\) and Candidate 1 announces [A - a, A + a] for some A < 0, a > 0, then Candidate 2 can get at least a tie by announcing A + aE\ (the other case, A > 0, then follows by symmetry).

We first show that /|, uXQ{x)dF**{x) < uXQ(E\) for all x0 e (0, 1]. For x0 e [E\, 1] we use the fact that xo > E\ > E\Q and risk aversion to get

J ux,(x)dFxx\x)<ux«(E?)<ux,{E\).

For xo € (0, E\ ) we can use the fact that jco is closer to E\ than 0 to get

J uXQ{x)dFxx\x) < j uo(x)dFl°(x)<uo(El)<uXo(El)

(the first inequality is easy to show for both examples of assimilation perceptions).

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Public Choice (2009) 141: 213-232 231

Then, using the homogeneity of u0 as above, it is easy to see that all voters with jc0 e (A, A +a] strictly prefer A +aE\ announced by Candidate 2 over the interval announced by Candidate 1 . Because of risk aversion the same is obviously true for voters with jco > A + a. Thus at least 50% of the voters strictly prefer A + aE\ announced by Candidate 2 over the interval announced by Candidate 1 .

Proof of claim about mixed strategy Nash equilibria Suppose 8 < 8* (y < y*) and that (Ai , A2) is a mixed strategy Nash equilibrium. We will show that this leads to a contradiction.

Let £ > 0. There exists an a' > 0 such that A2 puts less than e probability mass on the

positions (certain or ambiguous) with midpoint in (- a\ a') \ {0}. Therefore, by announcing the ambiguous position [-a, a] for a sufficiently close to zero, Candidate 1 defeats A2 with probability greater than 1 - s (Candidate 1 defeats all positions with midpoint not in (- a! , a') \ {0}). Thus it follows that in the equilibrium (Ai , A2), Candidate 1 must win with

probability one. Pick À € M, à > 0 such that, for any neighborhood B of (Â, à), A\ puts some probability

mass on

{[A-a,A + a]\(A,a)eB}.

Suppose A < 0 (if A > 0 the argument is analogous). The certain position À + àE\ is preferred to [Â - à, À + à] (announced by Candidate 1) by a strict majority of voters (because 8 < 8* (y < /*), see Theorem 4). Therefore^ strict majority also prefers Â + aE\ to positions that are sufficiently close to [Â - à, Â + à]. Thus there exists a neighborhood B of (À, à) such that any position in B is defeated by Â + àE\ . So if the strategy of Candidate 1 is Ai then Candidate 2 can win with a strictly positive probability by announcing À+àE{. Thus, in the equilibrium (Ai, A2), Candidate 2 must win with a positive probability. This is a contradiction.

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Projection Effects and Strategic Ambiguity in Electoral Competition

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