Candidate quality in a Downsian Model with a continuous policy space
Abstract
This paper characterizes a mixed strategy Nash equilibrium in a one-dimensional Downsian model of two-candidate elections with a continuous policy space, where candidates are office motivated and one candidate enjoys a non-policy advantage over the other candidate. We assume that voters have quadratic preferences over policies and that their ideal points are drawn from a uniform distribution over the unit interval. In our equilibrium the advantaged candidate chooses the expected median voter with probability one and the disadvantaged candidate uses a mixed strategy that is symmetric around it. We show that this equilibrium exists if the number of voters is large enough relative to the size of the advantage.
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Candidate quality in a Downsian Model with a Continuous Policy Space.1 Enriqueta Aragonès2 Institut d’Anàlisi Econòmica, CSIC Dimitrios Xefteris3 University of Cyprus January 10, 2011 1Aragones acknowledges financial support by the Generalitat de Catalunya Grant number 2009 SGR 1126, the Spanish Ministry of Education grant number ECO2009-08820 and the Barcelona Graduate School of Economics. 2Institut d’Anàlisi Econòmica, CSIC, Campus UAB, 08193 Bellaterra (Spain). Email: [email protected] 3Department of Economics, Faculty of Economics and Management, University of Cyprus. P.O. Box 20537, CY-1678 Nicosia (Cyprus). Email: [email protected]
Abstract This paper characterizes a mixed strategy Nash equilibrium in a one-dimensional Downsian model of two-candidate elections with a continuous policy space, where candidates are office motivated and one candidate enjoys a nonpolicy advantage over the other candidate. We assume that voters have quadratic preferences over policies and that their ideal points are drawn from a uniform distribution over the unit interval. In our equilibrium the advantaged candidate chooses the expected median voter with probability one and the disadvantaged candidate uses a mixed strategy that is symmetric around it. We show that this equilibrium exists if the number of voters is large enough relative to the size of the advantage. Key words: spatial competition; mixed strategies; candidate quality
1Introduction Candidate quality is considered to be a critical variable in electoral competition. It affects the decisions of politicians regarding whether to run for office, campaign fund-raising, voter behavior, election outcomes, and, ultimately, policy outcomes. Quality differences between two candidates can arise for many reasons, including charisma, office-holding experience, incumbency, advertising, scandal, and any other non-policy dimension that may affect the relative attractiveness of two candidates. In Political Science candidate quality is also denoted by “valence dimension” and its importance has been widely demonstrated over several decades of careful empirical research.1 All else constant, high quality candidates will fare better than low quality candidates. Furthermore, quality differences produce significant changes in the nature of political competition. The equilibrium properties of spatial competition between two candidates who differ in quality have been analyzed theoretically. Recent papers by Ansolabehere and Snyder (2000), Aragones and Palfrey (2002), Groseclose (2001) and Hummel (2010) report a number of theoretical results about the equilibrium properties of spatial competition between two candidates who differ in quality.2 These papers use a framework for studying the effect of candidate quality on political competition, based on the standard Downsian model competition between two candidates with an important twist: any voter will strictly prefer the “higher quality” candidate to the “lower quality” candidate if the candidates locate so that the voter is indifferent between the two candidates on the policy dimension. Groseclose (2001) shows that, for office motivated candidates, existence of pure strategy equilibrium is especially problematic for small-to-intermediate values of the quality advantage. When the policy space is an interval of the real line and voters’ preferences are Euclidean the payoff functions of the candidates are discontinuous. This discontinuity implies that the best response of the disadvantaged candidate is not well defined. In this set up Aragones and Palfrey (2002) described the equilibrium strategies for a particular case: when the policy space is a finite grid of 1See, e.g., Stokes (1963), Kiewiet (1983), and Kiewiet and Zheng (1993). 2There are also some earlier theoretical papers that studied related kinds of asymmetry, such as incumbency or partisanship, e.g., Adams (1998), Bernhardt and Ingberman (1985), and Londregan and Romer (1993). 1
points on the [01] interval and when the advantage is relatively small. Hummel (2010) studies the same environment with the only difference that he considers that the advantage might be higher than Aragones and Palfrey (2002). In this case the median voter might prefer to vote for the candidate with superior valence even if he prefers the policy of the other candidate. Hummel (2010) characterizes the optimal actions of both candidates in a particular equilibrium of the game but, unlike Aragones and Palfrey (2002), he does not fully characterize any equilibrium of the game. The characterization of the equilibrium strategies for a continuous policy space has not been studied so far and it is the main objective of this paper. The model we analyze follows the standard Downsian model with a continuous policy space and the following modifications: we assume that voters have quadratic preferences instead of Euclidean and we assume that the beliefs of candidates’ on the distribution of the median voter’s ideal point are unimodal (not necessarily uniform). By changing the voters’ preferences from linear to quadratic the payofffunctions of the candidates become continuous and the best response of the disadvantaged candidate is well defined. Within this set up we are able to find conditions for existence of mixed strategy equilibrium and we are also able to describe the equilibrium strategies. We find a family of unimodal distributions for the median voter’s ideal point that guarantees existence of an equilibrium in which the advantage candidate chooses a pure strategy that concentrates all the probability in the expected location of the median voter, while the disadvantaged candidate chooses a mixed strategy that allocates probability symmetrically around the expected location of the median voter. We find necessary and sufficient conditions for this equilibrium to exist. These conditions impose restrictions only on the candidates’ beliefs on the voters’ distribution of preferences. We find that this equilibrium exists only if the level of uncertainty about the location of the median voter is low enough relative to the size of the advantage, that is, when candidates believe that the median voter’s ideal point is around 1/2 with high enough probability. Theconditionswefind for existence do not restrict at all the size of the quality advantage, that is, the difference in quality between the two candidates, as it was the case in Groseclose (2001). 2
Thus we show that this mixed strategy equilibrium may exist even for small values of the quality advantage, which is when pure strategy equilibrium fails to exist. As in similar models, we find that in equilibrium the advantaged candidate obtains a larger probability of winning than the disadvantaged candidate. We also find that as the value of the advantage becomes larger the probability of wining of the advantaged candidate increases, the equilibrium strategies of the two candidates are more differentiated, and the conditions of existence of equilibrium are relaxed, that is, the number of voters (or the probability with which the expected median voter is around 12) needed for an equilibrium to exist is smaller. Finally, as the value of the advantage becomes smaller, that is, as the difference between the two candidates vanishes, the optimal strategy of the disadvantaged candidate moves closer to the advantage candidate’s. That is both players’ equilibrium strategies converge to the expected median voter as candidate ’s advantage shrinks to zero. Therestofthepaperproceedsasfollows. Thenext section describes the formal model. Section 3 presents the derivation of the equilibrium strategies and analyzes its properties. Finally, section 4 contains some concluding remarks. 2TheModel The policy space is the [01] interval. There are voters, where is an odd and finite number. Each voter has a utility function, with two components, a policy component, and a candidate image component. The policy component is characterized by an ideal point in the policy space, with utility of alternatives in the policy space a quadratic function of the distance between the ideal point and the location of the policy. The image component is captured by an additive constant to the utility a voter gets if the higher quality candidate wins the election. There are two candidates, and , who are referred to as the advantaged candidate and the disadvantaged candidate, respectively. Each candidate’s objective is to maximize his probability of winning the election. We assume that candidates believe that the ideal point of each voter is 3
an i.i.d. draw from a uniform distribution in [01] Thegametakesplaceintwostages. Inthefirst stage, candidates simultaneously choose positions in [01]. In the second stage, each voter votes for the candidate whose election would give him the highest utility. In case of indifference, a voter is assumed to vote for each candidate with probability equal to 12. Let denote the policy position chosen by candidate ,andletdenote the policy position chosen by candidate Then, the utility that a voter with ideal point obtains if wins the election is given by ()=−(−)2and his utility if candidate wins is given by ()= −(−)2where 0denotes the size of candidate ’s advantage. Since the behavior of the voters is unambiguous in this model, we define an equilibrium of thegameonlyintermsofthelocationstrategiesofthetwocandidatesinthefirst round. A pure strategy equilibrium is a pair of candidate locations ( )such that both candidates are maximizing the probability of winning, given the choices of the other candidate. A mixed strategy equilibrium is a pair of probability distributions ()over [01] such that there is no mixed strategy for that guarantees higher probability of winning than ,givenand there is no mixed strategy for that guarantees higher probability of winning than ,given. Notice that in this set up, if then all voters with −(−)2−(−)prefer to vote for candidate . Therefore, we have that all voters with an ideal point + 2+ 2(−)=b( ) prefer to vote for candidate Since the ideal point of each voters is drawn from a uniform distribution, this implies that the probability that a voter votes for the advantaged candidate is given by ( )=min{b( )1}and the probability that a voter votes for the disadvantaged candidate is given by ( )=max{01−b( )} Similarly if we have that all voters with an ideal point + 2+ 2(−)=b( )prefer to vote for candidate This implies that the probability that a voter votes for the advantaged candidate is given by ( )=min{11−b( )}and the probability that a voter votes for the disadvantaged candidate is given by ( )=max{0b( )} 4
Since we assume that there are voters, the probability with which the advantaged candidate wins the election is given by the probability that the advantaged candidate obtains the votes of at least a majority of the voters. Because each voter will vote for candidate with probability ( ) the probability with which candidate is elected may be computed by the sum of the Bernoulli distributions corresponding to at least a majority of successes over trials, that is, ( )= P =+1 2¡ ¢( )(1 −( ))− Similarly we could also show that the probability with which the disadvantage candidate wins the election is given by ( )= P =+1 2¡ ¢( )(1−( ))−= P =+1 2¡ ¢(1−( ))( )−= 1−( ) Observe that ( )and ( )are continuous functions of ∈[01] and ∈[01]and therefore ( )and ( )are continuous functions of ∈[01] and ∈[01] as well. Finally, if = we have that ( )=1and ( )=0, that is, if both candidates choose the same location then the advantaged candidate wins with probability one, because in this case all voters would prefer to vote for him. Notice that the payofffunctions of the candidates in our set up coincide with the c.d.f. of a Beta distribution with parameters ==+1 2Such a distribution is unimodal and symmetric around 1 2This observation allows us to offer an alternative interpretation of our model. Suppose that we have any number of voters, even a continuum, and a unique median voter. Suppose that the candidates’ beliefs about the distribution of the median voter’s ideal point are represented by this Beta distribution. In this case the candidates’ payofffunctions would be represented by the same family of c.d.f. parametrized by the parameter of the Beta distribution instead of the number of voters of the original set up. In this case ( )would represent the ideal point of the voter that is indifferent between the two candidates. Thus in both cases we have that the candidates’ beliefs about the location of the median voter’s ideal point are more concentrated around 1 2whenever the number of voters increases or, what is the same, when the parameter of the Beta distribution increases. 5
The payofffunctions of the candidates in our set up also coincide with those of the Condorcet jury members (see, for example, Kirstein and Wangenheim, 2010). This coincidence will prove to be helpful for our analysis. 3 Equilibrium Strategies When =0, neither candidate has an advantage, and we are in the standard Downsian world, where in equilibrium the two candidates locate at 1 2and each wins with probability 1 2. In general, when 0, there does not exist a pure strategy equilibrium. Different versions of this result have been stated and proven in Groseclose (1999) and Berger, Munger, and Potthoff(1999). The intuition is simple. If the disadvantaged candidate’s location is perfectly predictable, the advantaged candidate can copy that strategy and win for sure. Therefore, at least the disadvantaged candidate must be mixing. The result is true unless is sufficiently large that canlocateatthemedian and guarantee a payoffof 1. Proposition 1 If ≥1 4there is a pure strategy equilibrium in which wins with probability one. (All proofs may be found in the appendix.) In our case, if 1 4, then there will be no pure strategy equilibrium. The aim of this paper is to show that if ∈¡01 4¢there exists a mixed strategy Nash equilibrium3in which the advantaged candidate chooses a pure strategy and the disadvantaged candidate chooses a mixed strategy. In particular we show that in this equilibrium the advantaged candidate chooses a pure strategy corresponding to the ideal point of the expected median voter, =1 2while the disadvantaged candidate mixes between the two policy locations =1 2−√and =1 2+√each with equal probability. We find that this equilibrium exists as long as the number of voters is large enough relative to the size of the advantage .Wealsofind the minimal number of voters that guarantees 3If candidates have private information with continuous types, then this mixed equilibrium can be “purified.” That is, there will exist an equilibrium in pure strategies, where the equilibrium locations of candidates will vary with this private information (Aragones and Palfrey 2005). 6
existence of this Nash equilibrium as a function of the size of the advantage. We start by demonstrating that the strategy proposed for candidate ,e=(=1 2−√with probability 1 2and =1 2+√with probability 1 2) is an optimal response to candidate choosing e=1 2We prove that this holds true for all values of Proposition 2 For all 0and for all 01 4we have that e=(=1 2−√with probability 50% and =1 2+√with probability 50%) is a best response to e=1 2 Next we have to show that the strategy proposed for candidate ,e=1 2is a best response to candidate choosing e=(=1 2−√with probability 1 2and =1 2+√with probability 1 2)Notice that when candidate is choosing strategy e=( 1 2−√w.p. 1 2;1 2+√w.p. 1 2) candidate ’s probability of election is given by: ( e)=1 2( 12−√)+1 2( 12+√) where ( 12−√)= P =+1 2¡ ¢( 12−√)(1 −( 12−√))− and ( 12+√)= P =+1 2¡ ¢( 12+√)(1 −( 12+√))− Moreover, observe that the function ( )increases with for all 1−p−2+2+1 remains constant (( )=1)for∈[1 −p−2+2+1p2+]and decreases with for all p2+ [Figure 1] That is, ( 12−√)is increasing in ∈[01−q1 4+2+√)constant in ∈[1 − q1 4+2+√ q1 4+2−√]and decreasing in ∈(q1 4+2−√ 1] and ( 12+√)is increasing in ∈[01−q1 4+2−√)constant in ∈[1 −q1 4+2−√ q1 4+2+√]and 7
a pure strategy equilibrium in which the advantaged candidate obtains the votes from all voters. Therefore, wins with probability one. ¨ Proofofproposition2: Given that =1 2we search for a value of that maximizes the payofffunction of the disadvantaged candidate, that is, the following expression. (1 2)= P =+1 2¡ ¢(1 2)(1 −(1 2))− Kirstein and Wangenheim (2010) show that () =¡−1 −1 2¢[( )(1 −( ))]−1 20for ≥1 2.Thuswehavethat(1 2)is strictly increasing in (1 2). Therefore, in order to find a value of that maximizes (1 2)it is enough to find the values 12that maximize (1 2)=max©0b(1 2)ª=max{01 4+ 2− 1−2}; and the values 12 that maximize (1 2)=max{01−b( )}=maxn03 4− 2+ 1−2o Notice that 12−√=argmax1 4+ 2− 1−2,and12−√∈£01 2¤when ∈(01 4]. Similarly, when 12we find that 12+√=argmax3 4− 2+ 1−2and 12+√∈£1 21¤when ∈(01 4] Therefore, the proposed mixed strategy for candidate is a best response to =12for any 0¨ Proofofproposition3: If =1candidate ’s probability of winning is given by 1( e)=1 2( 12−√)+1 2( 12+√) We have seen that 1( e)is increasing in ∈[0q1 4+2−√)and that 1( e)is decreasing in ∈(1 −q1 4+2−√ 1] and, therefore, the best response of must belong in ∙q14+2−√ 1−q14+2−√¸ 14
Observe that for all ∈(01 4)we have that 12−√q14+2−√121− q14+2−√12+√and that if =1candidate ’s probability of election 1( e)= 1 2( 12−√)+1 2( 12+√)can also be written as 1( e)=12(1 −+12−√ 2+ 2(−12+√))+12(+12+√ 2− 2(−12−√)) for ∈[q14+2−√ 1−q14+2−√] Thus, 1() =− 4(−12+√)2+ 4(−12−√)20if and only if ³−12+√´2³−12−√´2 This implies that 1( e)is decreasing for ∈[q14+2−√ 12) and it is increasing for ∈(121−q14+2−√]. [Figure 2] Therefore 1( e)is increasing for ∈[0q14+2−√],decreasingfor∈[q14+2−√ 12 ) increasing for ∈(121−q14+2−√]and decreasing for ∈[1−q14+2−√ 1].This implies that when =1the optimal responses of candidate are either =q14+2−√or =1−q14+2−√but not =1 2¨ Proofofproposition4: Let’s show that () 0for ∈[q14+2−√ 1 2)A similar analysis would prove that () 0for ∈(1 21−q14+2−√] Since ( e)=1 2( 12−√)+1 2( 12+√)we have that () =1 2 (12−√) +1 2 (12+√) where ( 12−√)= P =+1 2¡ ¢( 12−√)(1 −( 12−√))− 15
and ( 12+√)= P =+1 2¡ ¢( 12+√)(1 −( 12+√))− We can compute the derivative of ( 12−√)with respect to ()using the results in Kirstein and Wangenheim (2010) and obtain (12−√) =¡−1 −1 2¢[( 12−√)(1−( 12−√))]−1 2 Thus we have that the total derivative of ( 12−√)with respect to by composing it with its partial derivative, that is, (12−√) =(12−√) (12−√) which can be written as (12−√) =¡−1 −1 2¢[( 12−√)(1 −( 12−√))]−1 2(12−√) and similarly we have that (12+√) =¡−1 −1 2¢[( 12+√)(1 −( 12+√))]−1 2(12+√) Therefore, () = 2¡−1 −1 2¢[[( 12−√)(1 −( 12−√))]−1 2(12−√) +[( 12+√)(1 − ( 12+√))]−1 2(12+√) ] and we need to prove that for large enough values of we have that [( 12−√)(1−( 12−√))]−1 2(12−√) +[( 12+√)(1−( 12+√))]−1 2(12+√) ] 0 whenever ∈[q14+2−√ 1 2) This holds if and only if ∙(12−√)(1−(12−√)) (12+√)(1−(12+√))¸−1 2(12−√) +(12+√) 0 First of all we will show that ∙(12−√)(1−(12−√)) (12+√)(1−(12+√))¸−1 2 decreases with and it tends to zero as tends to infinite. Notice that for ∈[q14+2−√ 1 2)we have that ( 12−√)( 12+√)1 2 16
which implies that ( 12−√)³1−( 12−√)´( 12+√)³1−( 12+√)´ always holds, since it does not depend on . Thus ∙(12−√)(1−(12−√)) (12+√)(1−(12+√))¸−1 2 1and lim→∞ ∙(12−√)(1−(12−√)) (12+√)(1−(12+√))¸−1 2 =0 Sincewehavethat(12+√) 0we also have that ∙(12−√)(1−(12−√)) (12+√)(1−(12+√))¸−1 2(12−√) + (12+√) 0will hold for large values of Similarly we could show that for ∈(121−q14+2−√]we have () 0which completes the proof of the proposition. ¨ Proofofproposition5: From the last proposition we know that () 0for ∈[q14+2−√ 1 2)if and only if ∙(12−√)(1−(12−√)) (12+√)(1−(12+√))¸−1 2(12−√) +(12+√) 0 which can also be written as 2 ln− (12+√) (12−√) ln(12−√)(1−(12−√)) (12+√)(1−(12+√))+1 becausewehavethat(12−√) 0andwealsohavethat(12−√)(1−(12−√)) (12+√)(1−(12+√))1implies ln ∙(12−√)(1−(12−√)) (12+√)(1−(12+√))¸0 For =12to be a local maximum we need to have () ≥0for =12−where 0 and →0 Thus we have to compute lim→122 ln− (12+√) (12−√) ln(12−√)(1−(12−√)) (12+√)(1−(12+√))+1 17
Notice that when approaches 1 2we have that − (12+√) (12−√) →1and (12−√)(1−(12−√)) (12+√)(1−(12+√))→ 1thus we may apply l’Hopital’s rule and we obtain that lim→12 ln− (12+√) (12−√) ln(12−√)(1−(12−√)) (12+√)(1−(12+√))= lim→12 − (12−√) (12+√) − (12+√) (12−√) (12+√)(1−(12+√)) (12−√)(1−(12−√)) (12−√)(1−(12−√)) (12+√)(1−(12+√)) Observe that since for ∈(12−√ 12+√)we have that ( 1 2−√)=1−+1 2−√ 2− 2(1 2−√−)and ( 1 2+√)=+1 2+√ 2+ 2(1 2+√−) then ( 1 2−√) =−1 2− 2(1 2−√−)2and (1 2+√) =1 2+ 2(1 2+√−)2 which implies that µ− (12−√) (12+√) ¶= 1+ (1 2−√−)2 1+ (1 2+√−)2→1 and −(12+√)(1−(12+√)) (12−√) = 2 (1 2+√−)3−2 (1 2−√−)3+22 (1 2+√−)2(1 2−√−)21 (1 2+√−)−1 (1 2−√−) 1+ (1 2−√−)2 2→2 √ Similarly we find that (12+√)(1−(12+√)) (12−√)(1−(12−√))→ 1 4− 1 4−=1 and (12−√)(1−(12−√)) (12+√)(1−(12+√)) = =(1−2(12−√))(12−√) (12+√)(1−(12+√))−(12+√)(1−2(12+√))(12+√) (12−√)(1−(12−√)) (12+√)2(1−(12+√))2→4√ 1 4− becausewehavethat 18
(12−√) →−1; (12+√) →+1 and 1−2( 12−√)→−2√;1−2( 12+√)→−2√ Therefore, we obtain that lim→122 ln− (12+√) (12−√) ln(12−√)(1−(12−√)) (12+√)(1−(12+√))+1=2¡1−4 8¢+1= 1 4 Furthermore, we compute the sign of ln − (12+√) (12−√) ln(12−√)(1−(12−√)) (12+√)(1−(12+√)) for all ∈(q1 4+2−√ 1 2) and for all ∈(01 4)using Mathematica and we get that it is positive (see figure 7). This implies that if ln− (12+√) (12−√) ln(12−√)(1−(12−√)) (12+√)(1−(12+√))+1for →1 2then the inequality should hold for all ∈(q1 4+2−√ 1 2)as well. Thus, =1 2is a global maximum if and only if ≥1 4¨ 19
Figure 1: The probability that a voter votes for the advantaged candidate, p(x,y), as a function of the advantaged candidate’s policy choice (x) given a policy choice of the disadvantaged candidate (y). x p(x,y) ½ ( y+ d/y ) ½ ( (1-y)+ d/(1-y) ) y
Figure 2: The probability that candidate A, P1(x,D), as a function of the advantaged candidate’s policy choice (x) given the best response of the disadvantaged candidate (D) when n=1 and d=0.05.
Figure 3: The probability that candidate A wins, Pn(x,D), as a function of the advantagedcandidate’s policy choice (x) given the best response of the disadvantaged candidate (D) when n=5 and d=0.1, thus n>1/4d
Figure 4: The probability that candidate A wins, Pn(x,D), as a function of the advantaged candidate’s policy choice (x) given the best response of the disadvantaged candidate (D) when n=5 and d=0.05 thus n=1/4d.