Candidate location and endogenous valence. Economics Education and Research Consortium Working Paper Series ISSN No 05/17.

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1 Economics Education Research Consortium Working Paper Series ISSN No 05/17 Cidate location endogenous valence Alexei Zakharov This project (No was supported by the Economics Education Research Consortium All opinions expressed here are those of the author not those of the Economics Education Research Consortium Research dissemination by the EERC may include views on policy, but the EERC itself takes no institutional policy positions Research area: Public Economics

2 JEL Classification: D72, D02 ZAKHAROV A.V. Cidate location endogenous valence. Moscow: EERC, This paper analyzes the effect that the ability of cidates to increase their valence has on cidate location in the Downsian model of elections. I show that if the cidates can increase valence at a cost after they have selected policy, then the cidates will select different policy platforms in order to avoid spending too much on valence. I then consider the factors that determine the degree of divergence, the amount spent on valence, the location of the indifferent voter. Keywords. Russia, valence, policy divergence, cidate equilibrium, median voter. Acknowledgements. I thank Roy Gardner Shlomo Weber for valuable comments recommendations. Alexei Zakharov Central Economics Mathematics Institute 47 Nakhimovsky pr/ Moscow, Russia Tel.: ( ax@mail.ru A.V. Zakharov 2005

3 CONTENTS NON-TECHNICAL SUMMARY 4 1. INTRODUCTION 5 2. MODEL SETTING AND SPECIAL-CASE RESULTS 7 3. EQUILIBRIUM 9 4. COMPARATIVE STATICS CIRCULAR PREFERENCES GROUP-SPECIFIC CANDIDATE VALENCE Symmetric valence advantage Incumbency advantage THREE OR MORE CANDIDATES Proportional representation Equilibrium with three parties DISCRETE VOTERS Arbitrary number of voters Two voters One voter CONCLUSION 30 APPENDIX 33 REFERENCES 46

4 Economics Education Research Consortium: Russia CIS 4 NON-TECHNICAL SUMMARY This research focuses on the analysis of two related elections phenomena. The first one is campaigning on nonpolicy issues (such as negative campaigning public relations. The purpose of nonpolicy campaigning is to convince the voter that the cidate is better fit to govern than her opponent. Unlike policy campaigning (promoting cidate s position on a certain divisive issue such as taxation or environment nonpolicy campaigning appeals to voters across the political spectrum. High prevalence of nonpolicy campaigning (especially negative campaigning has several adverse effects. First, not enough attention is paid to policy issues, the voters are less likely to make an informed choice. Second, negative campaigning may lead to voter apathy low voter turnout. This issue is especially relevant in countries that do not have democratic established traditions. The second phenomenon that this work investigates is the divergence of cidate positions on key policy issues. Well-known theory predicts that cidates (or political parties will be better off selecting policy platforms that appeal to the median voter which is not the case in most real-world elections. This research uses game-theoretic approach to investigate the effects of several factors on both the magnitude of nonpolicy campaigning the degree to which the policies of the cidates diverge. The first of these factors is the degree of population s heterogeneity. If all voters are similar with respect to the policies they prefer (or if there is a homogeneous majority, then the cidates cannot select different policy positions, PR negative campaigning becomes the only available campaigning tool. If the voters are differentiated (for example, rich poor voters are different with respect to the tax rate that they prefer, then the cidates may select different policy positions, negative campaigning becomes less important. The second factor is the proportionality country s electoral system. The degree of the electoral system s proportionality is determined by the laws that govern the election process as well as by the size shape of the electoral districts. If one cidate (or political party has a clear majority, then more negative campaigning will occur under a more proportional system. If the degree of support for the two leading cidates (or parties are approximately equal, then negative campaigning will be more effective under a winner-take-all type electoral system (since capturing a small number of swing voters will amount to winning the election.

5 Economics Education Research Consortium: Russia CIS 5 1. INTRODUCTION The modern tradition of the analysis of cidate behavior began with the well-known work of Downs (1957. Spatial models of electoral competition in this subsequent works are built on the assumption that cidates compete by choosing positions in the issue space. Each voter then supports the cidate whose position is closest to the voter s ideal. For two cidates, the well-known outcome of such models is that both cidates select positions identical to the ideal point of the median voter (if one exists 1. The median voter result is based on few intuitive assumptions, but in most real-life twocidate elections the policy positions of the cidates are different. There were several theoretical explanations to this puzzle. One explanation is that cidates may fear the entry by a third cidate once their positions are set (Eaton Lipsey, 1972; Palfrey, 1984; Weber, Another is that cidates may be motivated by policy as well as by office (Calvert, 1985; Wittman, 1977, If cidate is uncertain about the preferences of the electorate is sufficiently risk-averse, then they may select policies closer to their own ideals. Cidates may arise endogenously from the electorate (Osborne Slivinskii, 1996; Besley Coate, 1997, A voter will not decide to become a cidate if there already is a cidate with similar policy preferences especially if the cost of running is high. The policy implemented by the winning cidate may also be a product of compromise between a cidate a number of interest groups (Baron, 1984; Austen-Smith, 1987; Grossman Helpman, 1996; Coate, Some recent research focused on the role of cidate valence. A term coined by Stokes (1963, valence refers to (positive cidate characteristics that, unlike policy position, affect all voters equally. Cidate s valence reflect her track record, 2 reputation, name recognition as well as from personal qualities such as charisma perceived intelligence. Incumbency is a likely source of valence. An incumbent cidate enjoys an advantage over her opponent since she is thought to had acquired experience while in office, is well known, the risk-averse voter are more certain about her policy position. A number of recent works considered Downsian models with one of the two cidates having an exogenous valence advantage. If the cidates are motivated only by office, then there is no pure-strategy equilibrium in which both cidates have positive payoffs (Ansolabehere Snyder, If they are partially motivated by policy, then an equilibrium may exist; the advantaged cidate will locate closer to the center (Groseclose, A mixed-strategy equilibrium was analyzed in Aragones Palfrey (2002. The authors proved that as the magnitude of the advantage tends to zero, the policy positions over which the cidates romize converge to the median voter point. Local conditions for a pure-strategy equilibrium 1 It will generally not exist if the number of the policy space dimensions is greater than one. See, for instance, McKelvey (1976, 1979 Plott ( A review of literature on the sources of cidate valence is given in Groceslose (2001.

6 Economics Education Research Consortium: Russia CIS 6 for the case when cidate valence has both fixed rom components were obtained by Schofield (2004. In this work, I assume that the cidates are able to increase their valence at a cost. Such expenditures include running TV radio advertisements promoting cidate s positive qualities (such as integrity or experience, billboards for better name recognition, public relations actions to improve the cidate s image, so on. Building a competent team of aides advisors is also a costly task, the higher is the quality of such team, the higher the cost (Carrillo Castanheira, The choice of one s policy position, on the contrary, is assumed to be costless. The voteshare effect of increasing valence is assumed to depend on whether the policy platforms of the cidates are similar or far apart. If the cidates are identical with respect to their policy platforms, then a cidate with the greater valence is a better choice for all voters; thus an increase in valence by one of the cidates may swing a large share of electorate in her favor. If the policy positions of the cidates are relatively far apart, then an increase in valence of one cidate has a much smaller effect on her voteshare. 3 In any equilibrium, the marginal cost of increasing one s valence must equal marginal benefits. Thus, if two cidates simultaneously decide on the amount by which to increase their valence, that amount ( the cost of increasing valence will be greater if the policy distance is larger. 4 Finally, I assume that the cidates decide how much resources to spend in order to increase their valence after they have chosen their policy positions. This assumption is reasonable. In any election campaign the cidates are not likely to change their positions on key policy issues near the end of the race (or even midway through the campaign. At the same time, the intensity of the campaigning usually reaches its peak just before the elections. It follows that a cidate is faced with a dilemma. On one h, shifting policy position away from one s opponent will decrease the cidate s expected voteshare. On the other h, greater policy distance means that the cidate will have to spend less resources in order to increase her valence. Thus the cidates may strategically choose diverging policy positions. 5 3 This is a direct consequence of the assumption that the voters have increasing marginal disutility from policy distance. Suppose that a cidate s position moves away from the voter s preferred policy. The voter s utility will decrease by a greater amount if the distance between the cidate the voter is large. 4 Carrillo Castanheira (2002 suggested a different mechanism by which costly valence can lead to a differentiation of cidate platforms. As in this work, it was assumed that the cidates are office-motivated, that selecting the policy platform is costless. It is also assumed that there is a single (median voter who cannot perfectly observe the valence of the cidates. A cidate can exert effort in order to affect the probability that the votes believes that that her valence is high. If one of the cidates is located at the median the other deviates from the median, then the deviating cidate needs higher valence in order to win, her marginal benefit of valence effort will be higher. Thus a cidate deviating from the median signals to the voters that her valence will be higher than that of her opponent. If the valence is sufficiently unobservable, then the deviating cidate will have a greater expected utility. 5 Similar reasoning was used to explain product differentiation in the product location problem (see, for instance, Hay, 1976; d Aspermont, Gabszewicz, Thisse, 1979; Prescott Visscher, The utility from the consumption of a product is assumed to depend on the taste (or location of the consumer, on the

7 Economics Education Research Consortium: Russia CIS 7 This work differs from its predecessors in several important respects. First, the cidates are assumed to be purely office motivated. If the cidates are even partially policy-motivated, then one has to assume that there exists some mechanism that forces the winning cidate to implement the policy she promised (instead of her own preferred policy. 6 Second, the cidates are assumed to be identical. No cidate enjoys an exogenous advantage (as in Groseclose, Besides voters cidates, there are no economic agents (such as interest groups with specific policy interests. Finally, the assumptions regarding the cost function of the cidates the disutility function of the voters are stard (increasing marginal cost marginal disutility, all of the results are analytical, general form functions are used everywhere with minor exceptions. This work will be structured as follows. In Section 2 we will formulate the model. In In Section 3 we provide the results for a general continuous election rule (the special case of which is proportional representation. In Section 4 we consider the comparative statics of the proportional representation model. Section 5 investigates the possibility of a circular policy space. In Section 6 we analyze the possibility of one or both cidates enjoying a valence advantage from some part of the population. Section 7 investigates the possibility of a three-cidate election. In Section 8 we analyze the model under the assumption of a discontinuous winner-take-all election rule, discrete voters, voter uncertainty regarding cidate s valence. Section 9 concludes. 2. MODEL SETTING AND SPECIAL-CASE RESULTS Two cidates, 1 2, compete in an election by choosing policy platforms y 1 y 2 in a one-dimensional policy space X R (we assume that X is an interval. The cidates also compete by making investments into their valence characteristics, ǫ 1 ǫ 2. The valence investments cost c(ǫ, where c( is a thrice differentiable function with c(0 = 0, c (0 = 0, c > 0 c > 0 are made after the political platforms are chosen observed. Following the stard Hotelling Downs framework it is assumed that policy announcements are credible commitments that cidates cannot renege on promised policies once elected. There exists a continuum of voters with preferences over policy valence characteristics of the cidates. The utility of voter s from the election of cidate i is ǫ i φ( v s y i where v s is the ideal point of voter s φ(x is voter s disutility from a policy x units away from her ideal point. 7 We assume that φ (x > 0, φ (x > 0. product s price. Each of the two or more firms decides which product (with what location or taste to produce. After making that choice, the firms are free to participate in Cournot price competition. If two firms produce products that are perfect substitutes, then the loss of profit due to price competition is large. Thus a firm might be compelled to produce a differentiated product at the cost of losing some of the consumers. 6 Put forward by Alesina (1988, this was the main criticism of the Calvert Wittman model. 7 This utility function was formulated by Enelow Hinich (1982 is a stard utility function for modeling voter utility from spatial nonspatial characteristics of a political cidate. A different way of introducing valence into model was proposed in Sahaguet Persico (2006. In a two-cidate election

8 Economics Education Research Consortium: Russia CIS 8 Ideal points of the voters are distributed with continuous twice differentiable density f( over support S X. The voter s votes for cidate 1 if ǫ 1 φ( v s y 1 ǫ 2 φ( v s y 2 for cidate 2 otherwise. Voter distribution f( disutility φ( are known to both cidates. The value of office (for now is assumed to be equal for both cidates normalized to one. The probability of winning office (or, equally, the share of the office rent depends on the share of votes received by the cidate. We denote by g(f the share of office rent received by a cidate who has the support of a fraction F of voters. We assume that g is increasing in F, differentiable the necessary number of times, that g(1 F = 1 g(f. g is the function that describes the electoral system. For instance, g(f = F corresponds to the proportional representation, while 0, F < 1, 2 1 g(f = F = 1, 2 2 1, F > 1 2 corresponds to the winner-take-all political system. The political process can then be formalized as the following three-stage game. Stage 1. Each cidate i selects his policy platform y i X. The policy platforms are selected simultaneously. Stage 2. Policy platforms are observed by both cidates. Each cidate i selects ǫ i 0. Valence investments are made simultaneously. Stage 3. Voters observe (y 1, y 2, ǫ 1, ǫ 2. Each voter votes for one of the cidates. We assume subgame perfection at stage 3. Thus the payoff to Cidate 1 is U 1 = g(f(v s c(ǫ 1, (1 where The payoff to Cidate 2 is ǫ 1 φ( v s y 1 = ǫ 2 φ( v s y 2. (2 U 2 = 1 g(f(v s c(ǫ 2. (3 The first two results consider the proportional-representation case when the valences of the cidates are exogenously given. The first result is that of Calvert (1985 slightly adapted for this model. It states that if the valences of both cidates are equal, then both cidates choose policy platforms equal to the ideal point of the median voter. game that they considered, a cidate s policy proposal amounted to a redistribution of a fixed amount of resource (budget among a continuum of voters. The cidate s valence was the size of the budget that the voters believe would be available for redistribution once once that cidate is elected. Thus, cidate valence affected cidate s budget constraint, not directly the utility of voters.

9 Economics Education Research Consortium: Russia CIS 9 Proposition 1 (Calvert. Assume that g(f = F c(x = for x > 0. Then the unique equilibrium exists with y 1 = y 2 = s where s is the median of distribution F(. The result is completely different if one of the cidates has a valence advantage over another. Groseclose (2001 have shown that no equilibrium exists in this case. The following proposition is an adaptation of his result. 8 Proposition 2 (Groseclose. Assume that g(f = F, c( = ǫ 1 < ǫ 2. If for every s X there exists r X such that ǫ 2 φ( v s v r < ǫ 1, no pure-strategy equilibrium exists. The logic of the proof is intuitive: a cidate with a valence advantage can always win all the votes by choosing his opponent s position. As long as the opponent can gain votes by deviating from his original policy, he will do so. The analysis of the more general model presented here immediately yields an important result. Proposition 3. For any policy platforms y 1, y 2, in a LNE cidates spend an equal amount of resources on valence: ǫ 1 = ǫ 2. This result follows from the fact that the voters care only about the difference in valence between the cidates, not about the absolute levels. If the office rents obtained by the cidates are identical, then the cidates should also have identical valences. 3. EQUILIBRIUM The next step in the analysis would be to consider the general versions of g(, f(, φ( c(. We assume that there is an atomless continuum of voters that there is no uncertainty regarding the valence of the cidates. Indeed, a small increase in valence spending by one cidate should not result in a discontinuous increase in the cidate s utility if the cidate s voteshare increases continuously with valence, office rents increase continuously with voteshare. Most of the results obtained in this section are concerned with local Nash equilibria (LNE. A local Nash equilibrium is such a pair of strategies that any cidate cannot improve her utility by making infinitecimally small changes to her strategy. Any Nash equilibrium in its proper sense is also a local Nash equilibrium, but not vice versa. However, it is possible to derive necessary conditions for the existence of a proper (global Nash equilibrium. Proposition 4. For every c(, φ( there exists d such that the proper Nash equilibrium does not exist if the voter s policy preferences are distributed on an interval of length less than d. 8 A similar result appears in a number works by other authors.

10 Economics Education Research Consortium: Russia CIS 10 The argument behind this proposition is simple. If the voters are concentrated in a narrow interval, then (no matter what policies are chosen, a small increase in valence spending brings the cidate a large increase in votes; thus, the equilibrium cost of campaigning is large may exceed the expected gains from office. It may be one of the reasons why representative democracies fail in countries with very homogeneous populations (such as in traditional societies. We then formulate the necessary sufficient conditions for a local Nash subgame-perfect equilibrium. It is first necessary to give a definition. We will call a policy extreme if it is the greatest or the smallest number in X. We shall call an equilibrium interior if neither policy platform is extreme. In the first two propositions we derive the necessary sufficient conditions for an internal local equilibrium. Proposition 5. In any equilibrium (interior or not, the valence investments are equal: where ỹ = (y 1 + y 2 /2 d = (y 2 y 1 /2. ( ǫ 1 = ǫ 2 = c 1 f(ỹgf (F(ỹ, (4 2φ (d This result is intuitive. A marginal increase in valence investment of one cidate will give that cidate the same additional number of votes that he would lose from a similar increase in his opponent s valence. As long as cidates value office equally have identical means of increasing their valence (identical cost functions in our model, their investments should be identical as well. The second result defines the set of local interior equilibria: Proposition 6. The vector of strategies (y 1, y 2, ǫ 1 (y 1, y 2, ǫ 2 (y 1, y 2 is a local interior equilibrium if only if f (ỹg F (F(ỹ + f(ỹ 2 g FF (F(ỹ = 0, (5 φ (d 3 ( φ (d f(ỹgf (F(ỹ f(ỹg c 1 2φ F (F(ỹ = 0, (d (6 f (ỹg F (F(ỹ + 3f(ỹf (ỹg FF (F(ỹ + f(ỹ 2 f (ỹg FFF (F(ỹ > 0. (7 This result is surprisingly powerful. It states that in any local interior equilibrium ( hence in any interior equilibrium, the position of the indifferent voter is robust with respect to the cost of valence investments c( disutility from policy s distance from the voter s optimal point φ(. The indifferent voter s equilibrium position depends only on the distribution of voter preferences F( on the electoral system g( (see Fig. 1.

11 Economics Education Research Consortium: Russia CIS 11 f(y y 1 y 2 y 3 Fig. 1. y 1, y 2, y 3 Interior local equilibria. It is now useful to consider a very important special case of proportional representation. Proportional representation is both prevalent in the real world politics is especially important for multiple-player extensions of this model. 9 system that is linear in F in the vicinity of the equilibrium. Moreover, this result holds for any electoral Corollary 1. Suppose that, in some neighborhood of an equilibrium, g F (F(ỹ = α, where α > 0 is some constant. Then the vector of strategies (y 1, y 2, ǫ 1(y 1, y 2, ǫ 2(y 1, y 2 is a local interior equilibrium if only if f (ỹ = 0, (8 f (ỹ > 0, (9 φ (d 3 ( f(ỹα φ (d f(ỹα = 0. c 1 2φ (d (10 In this case, the voter who is indifferent between the policy platforms of both cidates is located in a local minimum of the density function. In real-world terms it predicts that if the voters (with regard to policy preferences are clustered around several values, then members of one cluster will all vote for the same cidate. One can provide an explanation for this phenomenon. Suppose that both cidates select some policy platforms, that a cidate (say, the one on the left deviates from his chosen policy toward the center. This move has three effects on that cidate s utility: 1. The voteshare effect the utility of that cidate increases because he obtains additional votes. We know that both cidates always select equal levels of valence spending. Thus the indifferent voter is always located in the middle of the two cidates. If one cidate s position shifts, then the position of the indifferent voter also shifts in the same direction. 9 For more than two cidates, proportional representation is the only electoral system to satisfy a number of reasonable regularity conditions, see below.

12 Economics Education Research Consortium: Russia CIS The policy distance effect an increase in the equilibrium level of valence spending because of smaller policy distance between the cidates. 3. The voter density effect a change in the equilibrium level of valence because of a different voter density in the neighborhood of the indifferent voter. Suppose that both cidates consider moving their positions a small distance to the center. The voteshare effects will be identical, so will be the policy distance effects. The voter density effects, however, will be different unless the density of the voters in the neighborhood of the indifferent voter is constant. Since in an interior equilibrium the sum of all three effects for either voter must be zero, the density of voters in the neighborhood of the indifferent voter is zero. Since the voter density effect must be negative for larger deviations, the indifferent voter must be located in the local minimum of the voter density function. The existence of an interior equilibrium requires the existence of at least two distinguishable groups of voters (that is, f must have at least two local extrema. If this condition does not hold, then either the equilibrium does not exist, or in an equilibrium one of the cidates selects an extreme policy which does not satisfy her first-order utility maximization condition. Primaries elections is one example when cidates are limited in the choice of policy platforms that they can propose, since they must also meet their general party guidelines. We have the following result. Proposition 7. Suppose that X = [y, ȳ] f (y > 0 y S. Then, in any equilibrium, the following is true. 1. y 1 = y; 2. Cidate 2 captures the majority of votes. Similar symmetric results hold if f (y < 0. This result can be interpreted as follows. If the density of voters is increasing in one direction (say, right, then the cidate that is located further to the opposite direction (left in our case will choose the most extreme policy available (see Fig. 2. Suppose that the leftist cidate decides to shift her position closer to the center. Then, the rightist cidate will also find it worthwhile to do so. However, the reverse is not the case. Suppose that for the leftist cidate, the voteshare effect the sum of voter density policy distance effects of a move to the center are equal. Then, for the rightist cidate the voteshare effect form a similar move to the center is greater then the sum of the other two effects. This is because the voter density around the indifferent voter will be smaller if the rightist cidate moves to the center hence the equilibrium expenditure on valence will be lower. If the voteshare effect the sum of policy distance the voter density effects

13 Economics Education Research Consortium: Russia CIS 13 f(y y 1 = y ỹ y 2 ȳ Fig. 2. Equilibrium when voter density is increasing the cidates are policy-constrained. are equal for the rightist cidate, then the leftist cidate will prefer to move away from the center. As a result, the leftist cidate will always be pushed to select the most extreme policy. The majority of voters are located to the right of the indifferent voter. Thus we may expect the cidate who chooses a more moderate position to obtain the majority of the votes. 4. COMPARATIVE STATICS We have determined that in an interior local equilibrium, the location of the indifferent voter is corresponds to some local minimum of the voter density function. The other two properties that characterize the equilibrium are the policy distance the equilibrium level of valence spending. These two variables depend both on the density of voters in the neighborhood of the indifferent voter, an on the properties of the electoral system (characterized by the function g( in the neighborhood of the voteshare to the right of the indifferent voter. The direction of these relationships depends on third-order properties of the disutility function φ( c(. The third-order condition on the disutility function can be interpreted as follows. Suppose that a voter with the ideal policy v is uncertain about a cidate s policy. He believes that the cidate will select policy y δ with probability 1/2 policy y + δ with probability 1/2. Since the voter is risk-averse (φ ( > 0, the cidate must have a positive valence advantage of ˆǫ(v, y, δ in order to deliver the same utility to the voter, as the cidate who selects the policy y with certainty. Suppose that the third derivative of the disutility function is negative: φ ( < 0. Then, a voter s utility is less affected by policy uncertainty if the difference between the expected policy the voter s ideal policy is large. This means that ˆǫ declines with y v.

14 Economics Education Research Consortium: Russia CIS 14 The third-order condition on the cost function determines whether an increase in the level of valence spending causes a less than or greater than proportional increase in the marginal cost. Consider the effect of an increase in the density of voters around the indifference voter given fixed cidate positions y 1, y 2. If the third derivative of the cost function is negative (c ( < 0, then the proportional increase in the equilibrium level of valence will be greater (see Fig. 3. c (ǫ 2f (ỹ 2φ (d f (ỹ 2φ (d ǫ ǫ Fig. 3. Increase in the equilibrium level of ǫ if d is fixed f(ỹ doubles. The effect of the voter density in the neighborhood of the indifferent voter on policy distance on the equilibrium level of valence is described as follows. Corollary 2. Suppose that the system is in an internal equilibrium that the population density in the neighborhood of the indifferent voter ỹ increases by a small equal amount. Then the following is true. 1. The policy distance d between the two cidates will increase if c ( < 0 φ ( < The level of valence investment ǫ will increase if c ( (5φ ( + φ ( φ ( > 0, will decrease if c ( (5φ ( + φ ( φ ( < 0. An increase in the voter density around the indifferent voter has two opposing effects. First, it results in a greater voteshare effect, since each cidate will derive greater benefit from moving her position closer to her opponent s. Second, the equilibrium level of valence will increase. Knowing this, the cidates will strategically choose more distant positions. The second effect dominates if two conditions are met. First, the increase of voter density given fixed policy platforms should result in a significant increase in the level of valence spending. Second, the marginal disutility of voter should should not increase with the policy distance too quickly. If these two conditions are not met, then the net effect of an increase in the voter density is unclear.

15 Economics Education Research Consortium: Russia CIS 15 An increase in the voter density also has several effects on cidate valence. Given fixed cidate platforms valence should increase. However, cidate platforms may diverge, with the opposite effect on cidate valence. The results suggest that the first effect is always more powerful than the second one. Under certain conditions an increase in the density of voters induces cidates to choose political platforms located further apart, as they try to avoid excessive spending on valence. The results show that this move will not fully compensate the cidates. We now turn to investigate how do changes in the electoral system affects the equilibrium. The electoral system is the relationship between the voteshare the expected office rents of the cidates. First, it is the rules by which the elections are played out (party list, open list or single ballot. Party list system will allow minority parties to win larger share of votes if the distribution of voter preferences is similar across regions. The number of seats in the parliament the size of a voting district under a single-ballot system. If the size of the average voting district is large the number of seats in the parliament is small, then a minority party is likely to win a smaller share of seats. Finally, the shape of the electoral districts can be manipulated in order to benefit one of the parties. If the supporters of a minority party are concentrated in a very small number of districts or diluted across a large number of districts, then the minority party sins a smaller share of the vote. The results of the analysis are summarized in the following corollary. Corollary 3. Suppose that the electoral system varies with some parameter t, that we are in an interior equilibrium. Then the following statements hold. 1. Suppose that in the vicinity of the indifferent voter g F (F(ỹ increases with t while g(f(ỹ is unchanged. Then, the policy difference d increases with t. 2. Suppose also that φ ( < 0 c ( < 0. Then ǫ increases with t. 3. Suppose that in the vicinity of the indifferent voter g FF (F(ỹ is negative increases with t, that f (ỹ 0. Then the location of the equilibrium voter ỹ shifts to the left with an increase in t. Both an increase in voter density around the indifferent voter an increase in the marginal utility of votes around the indifferent voter will increase the distance between the cidates policy platforms. The changes in the electoral rule has the same effect on the equilibrium as the change in the density of the indifferent voter. Whether such move will increase or decrease equilibrium valence investment is unclear since, on one h, greater distance from the indifferent voter entails less valence investment; on the other h, greater density or greater marginal utility of votes makes valence investment more attractive.

16 Economics Education Research Consortium: Russia CIS CIRCULAR PREFERENCES We then consider an important extension to the model consider the population s preferences distributed on a circle (see Fig. 4. We recall that in the two-cidate Downsian model with a circular issue space uniform density, any position of cidates corresponds to a pure-strategy equilibrium, in which each cidate receives one half of the vote. The question is whether a similar result holds for this work s model. Formally, we assume that X = [0, 1 that the utility of voter s from the election of cidate i is as follows. ǫ i φ( v i v s v i v s < 1 U(s, i =, 2 ǫ i φ(1 v i v s v i v s 1. (11 2 In all other respects the model is identical to that of the previous section. We assume proportional representation. For simplicity s sake we take g(f(ỹ = F(ỹ. Without the loss of generality we assume that y 1 < y 2 that (y 1 +y 2 /2 < 1/2. Thus we have ỹ 1 = (y 1 +y 2 /2 < 1/2, ỹ 2 = 1/2+(y 1 +y 2 /2 < 1/2. Cidate utility is written down as follows. U 1 = 1 F(ỹ 2 + F(ỹ 1 c(ǫ 1, (12 U 2 = F(ỹ 2 F(ỹ 1 c(ǫ 2. (13 We solve the model by first calculating the equilibrium valence investments. Proposition 8. Suppose that voter cidate utility are given by (11, (12, (13. Then, in a local equilibrium valence investments are equal: ( ǫ 1 = ǫ 2 = c 1 f(ỹ1 2φ (d 1 + f(ỹ 2, (14 2φ (d 2 where d 1 = (y 2 y 1 /2 d 2 = 1 (y 2 y 1 /2. We then obtain equilibrium conditions for cidate position: Proposition 9. Suppose that voter cidate utility are given by (11, (12, (13. Then the vector of strategies (y 1, y 2, ǫ 1 (y 1, y 2, ǫ 2 (y 1, y 2 is a local interior equilibrium if only if f(ỹ 1 f(ỹ 2 2 f (ỹ 1 φ (d 1 + f (ỹ 2 = 0, (15 φ (d 2 ( f(ỹ1 + c 1 2φ (d 1 + f(ỹ ( 2 f(ỹ1 2φ (d 2 2φ (d 1 + f(ỹ 2 2φ (d 2 ( f(y1 φ (d 1 f(y 2φ (d 2 = 0. (16 φ (d 1 φ (d 2

17 Economics Education Research Consortium: Russia CIS 17 y 2 d 2 d 1 ỹ 1 d 1 ỹ 2 y 1 d Fig. 4. Circular issue space. The result is weaker than the result for an interval policy set, since the locations of the indifferent voters depend on the disutility function, even if the election structure g( is locally linear. However, the result has several implications. First, since cidate valence is equal, we should expect indifferent voters to have diametrically opposing preferences. Second, if the voter density is uniform around the circle (f (y = 0, then any point on the circle can be an indifferent voter in a local Nash equilibrium ( a global equilibrium if the voter density is not too large. Third, the positions of the cidates are generally not diametrically opposing. 6. GROUP-SPECIFIC CANDIDATE VALENCE We assume that there are several groups of voters that vary in their degree of support for each of the cidates. For example, a member of a national or a religious minority is more likely to support a cidate with the similar background. We are interested in knowing how does the existence of such groups affects the amount of valence spending the policy distance between the cidates. More particularly, we want to know the effects of the relative size of such groups of the magnitude of support that each group offers to its preferred cidate. We assume that there are two groups of voters of size λ 1 λ. Each group of voters considers one cidate to have a valence advantage over the other cidate. Thus the utility of a voter from Group i with policy preference y from the election of cidate from group j is u ij (y = ǫ i + ǫ i φ( y j y (17 if i = j u ij (y = ǫ i φ( y j y (18

18 Economics Education Research Consortium: Russia CIS 18 otherwise. Thus ǫ 1 ǫ 2 denote the valence advantages that the groups lend to their preferred cidates. Note that there values need not be equal. A one group may see its cidate to be only slightly more professional than the other cidate, while the other group may believe that its cidate s advantage in terms of credit experience is very large. Since the valences of the cidates are group-specific, each group has its own indifferent voter. Denote by ỹ i : u ii (ỹ i = u ij (ỹ i, j i (19 the indifferent voter of group i. We will assume that y 1 > y 2 that y 1 > ỹ 1,2 > y 2. As we assume proportional representation, the utility of the cidates is given by U 1 = 1 λf 1 (ỹ 1 (1 λf 2 (ỹ 2, (20 U 2 = λf 1 (ỹ 1 + (1 λf 2 (ỹ 2. (21 Here, F 1 ( F 2 ( are the distributions of voter preferences in Groups 1 2 (as before, we assume that the distributions are atomless, with densities f 1 ( f 2 (. In an equilibrium, cidates spend an equal amount of resources on valence. Proposition 10. In any equilibrium with cidate payoffs (20 (21 valence spending is given by ( ǫ 1 = ǫ 2 = c 1 λf 1 (ỹ 1 φ ( y 1 ỹ 1 + φ ( ỹ 1 y 2 + (1 λf 2 (ỹ 2, (22 φ ( y 1 ỹ 2 + φ ( ỹ 2 y 2 where ỹ 1 ỹ 2 are given by (19. Further equilibrium analysis with the general form of cost disutility functions is complicated for the reasons that we are about to see. In particular, the analysis is less transparent because the effect of a shift in the positions of a cidates on the indifferent voter s position now depends on the relative valence advantage of the cidate (see Fig. 5. The position of the cidate with a valence advantage has a greater effect on the indifferent voter s position, since the indifferent voter s utility will decline by the greater amount if the advantaged cidate s position is changed. Consider the general problem with quadratic cost disutility functions φ(x = x 2 /2 c(x = x 2 /2. We have the following result. Proposition 11. Suppose that c(x = x 2 /2 φ(x = x 2 /2. Then, y 1 y 2 are an equilibrium if the following conditions are satisfied. where d = y 1 y 2. (λ ǫ 1 f 1 (1 λ ǫ 2 f ((λf 1 + (1 λf 2 ((λf 1 + (1 λf 2 = 0, ( ( λ ǫ1 f 1 (1 λ ǫ 2f 2 λf 1 + (1 λf 2 = 0, (24 d 2 d 2 d

19 Economics Education Research Consortium: Russia CIS 19 u u y 2 y 2 y 1 y 2 y 1 y 1 (a Disadvantaged cidate little effect on the position of the indifferent voter. (b Advantaged cidate large effect on the position of the indifferent voter. Fig. 5. The effect of a change in a cidate s position on the position of the indifferent voter. The immediate observation is that we should not, in general, expect the equilibrium to exist. Equal centripetal changes in the positions of the cidates result in different changes in the locations of the indifferent voters in the voteshare of each cidate (see Fig. 5. However, the changes in the equilibrium level of valence expenditures will be identical unless the voter density is not constant. These differences will have to be compensated by different voter densities in the neighborhoods of the indifferent voters. Thus, if the voter density is uniform, then an equilibrium will exist only if an exact relationship between other parameters is satisfied. There are two possible ways to simplify the model in order to obtain relevant results. First, one can consider the symmetric case, with voters drawn from the same distribution, population groups of equal size, equal valence advantages of the cidates. Second, one can consider the case when all voters give an advantage to one cidate Symmetric valence advantage Here we consider the case when there are two equal groups of voters with identical distribution of policy preferences. Each group linked to a specific cidate. Voters from that group consider that cidate to have a valence advantage over her opponent. We will assume that the voter distribution is uniform. Because of this assumption, we will not be able to consider the effect of the valence advantage on the location of the two indifferent voters. However, we will be able to trace its effect on the policy distance. We have the following result. Proposition 12. Suppose that cidate utility is given by (20, (21, (22 that λ = 1 2, ǫ 1 = ǫ 2 = ǫ, f 1 ( = f 2 ( = f. Then, in equilibrium we have f φ d 1 + φ (d 2 2c 1 (φ (d 1 φ (d 2 + φ (d 2 φ (d 1 = 1 (25

20 Economics Education Research Consortium: Russia CIS 20 ( c 1 f = ǫ, (26 φ (d 1 + φ (d 2 where d 1 = y 1 ỹ 1 = ỹ 2 y 2, d 2 = ỹ 1 = y 2 = y 1 ỹ 2. Also, we have if c ( < 0 φ ( < 0. d 1 ǫ > 0 d 2 ǫ < 0, 6.2. Incumbency advantage There are several ways to model the effects of incumbency upon the equilibrium level of valence investment equilibrium political platforms. The first one is to assume that the cidates move at different times; the first mover then has an advantage over his rival. This is a traditional approach that will be discussed in greater details in the last section. Here we consider the second approach. We assumes that the cidates move simultaneously but one cidate has a valence advantage (Stokes, 1963; Enelow Hinich, 1982; Harrington Hess, The reasons why incumbency may confer valence advantage (see Groceslose, 2001 are better name recognition, greater voter certainty good track record. What is more important, the valence of cidates in regional elections is one of the factors that the central government can does control through explicit political support given to some of the cidates. We first consider the case when the policy platforms are exogenous. Proposition 13. Suppose that y 1 < y 2 are exogenous, λ = 1, f 1 ( = f 2 ( = f(, f ( 0, ǫ is given by (22. Then an increase in ǫ 1 will increase the equilibrium valence investment of both cidates as long as f (ỹ > 0. As a cidate s exogenous valence increases, the location of the indifferent voter is likely to shift away from that cidate. If the population density around the new location is higher, then the valence spending by the cidates will be greater. With endogenous political platforms the marginal population density around the indifferent voter is zero, so the equilibrium valence investment does not change. Proposition 14. Suppose that y 1 y 2 are endogenous, λ = 1, ǫ 1 = 0, f 1 ( = f 2 (, cidate utility is given by (20, (21, (22. Then, we have ǫ 1 = ǫ 1 < 0,

21 Economics Education Research Consortium: Russia CIS 21 ǫ ǫ 1 = 0, ǫ 1 > 0. Thus, if the magnitude of the incumbent s advantage is small relative to distribution of voter policy preferences is relatively smooth, then a small exogenous increase in valence by one of the cidates is not likely to affect the other s valence spending. The indifferent voter will shift away from the advantaged cidate (increasing her voteshare, while the positions of both cidates will shift in the opposite direction. 7. THREE OR MORE CANDIDATES We assume that one cidate s political platform is fixed. The two other cidates are outsiders choose their policy platforms (to the left to the right of the central cidate simultaneously after the choice of the first cidate has been made (but not before the first cidate chooses the level of valence investment. Such a setting repeated itself in the recent Russian presidential parliamentary elections, where there has been a clear centrist front-runner (the United Russia party or Vladimir Putin, a number of left- right-wing outsiders. The US presidential primaries where one cidate is the incumbent president is another example. We assume that cidates exert resources to decrease the valence of other cidates. In this setting, valence investment ( negative PR by an outside cidate creates a positive externality for the other outside cidate, since it lowers the valence of the central cidate, the two outside cidates do not compete for votes directly. We will try to determine how does the distribution of voter preferences affect the equilibrium policy platforms of the outside cidates, as well as the equilibrium valence investments Proportional representation Our first goal is to establish proportional representation as We first explain why the analysis of the multiparty equilibrium is limited to proportional representation. It turns out that proportional representation is the only electoral system that satisfies a number of reasonable regularity assumptions. Suppose that the number of cidates contesting the office is N. We will denote by the political system a function G : N 1 N 1 that, for each election outcome, assigns a division of one unit of utility among the N cidates (here N 1 is the N 1-dimensional simplex. We will require the electoral system to satisfy some basic regularity conditions. We will introduce several notations. First, we will require the electoral system to be symmetric, so the

22 Economics Education Research Consortium: Russia CIS 22 outcome does not depend on the identity of the cidate. Formally, G(F(X = F(G(X for all X N 1 F(. Here F( is a mapping that permutates the components of the vector argument. Second, we will require monotonicity: a cidate s share of office should be increasing in his share of votes. Thirdly, we will require the electoral system to be additive: the share of office for any group of cidates should only depend on the total amount of votes received by that group. It is an obvious requirement otherwise in some cases two cidates will be able to increase their total utility by trading votes. Formally, we require that X, Y N 1, i, j X i + X j = Y i + Y j G(X i + G(X j = G(Y i + G(Y j. (27 If there are only two cidates, then any function that is increasing symmetric relative to the 45-degree line represents a symmetric, additive monotonic electoral system. For that reason, a general case was analyzed in the previous sections of this work. If the number of parties running for office is greater than two, the problem of selecting an efficient electoral system (even in the theoretical setting of this model may become very complicated, while the results of equilibrium analysis may become intractable. Fortunately, there exists a theoretical result that will significantly narrow the search down. Proposition 15. Suppose that N 3. Then the only electoral system that is monotone, symmetric additive is the proportional representation: G(X = X. Corollary 4. Suppose that the electoral system is symmetric, monotone that additive property (27 is not satisfied. Then for every distribution of votes X there exists a pair of cidates (i, j such that G(X i +G(X j < G(X i +G(X j, where X i = X i +δ, X j = X j δ, X k = X k, k i, j δ is some real number. This result axiomatizes proportional representation for elections with three or more cidates. The symmetric property is, in essence, the requirement that the process of vote count is not biased in favor of any one of the cidates. The number of seats in the parliament received by a party that obtained, say, 30% of the popular vote should not depend on which party it was that obtained those 30%. Needless to say that this property is often violated as the one of the cidates or parties (usually the incumbent party receives explicit support from the body that organizes controls the elections. In modern Russian practice the most common form of such a violation is filling some of the empty ballots in favor of the preferred cidate. The additive property implies that no two parties will find it optimal to form a coalition, provided that the coalition will receive all only the votes that the two original cidates would otherwise receive. Moreover, if the additive property is not satisfied, then for every distribution of votes there will be a pair of cidates that will benefit from some sort of a coalition. This leads to a surprising conclusion. If one observes an election with three or more cidates, the cidates in the election are not prevented from making coalitions with one another,

23 Economics Education Research Consortium: Russia CIS 23 a coalition between any two cidates receives the same total number of votes as the two cidates combined, then the election operates under a proportional representation scheme. This means that the gains that a cidate (party receives from competing in an election are actually proportional to the share of votes that that cidate receives. The gains include not only the expected gains from winning the office or from controlling a share of parliament, but also the political capital obtained from making a strong showing in an election, as well as the gains from the formation of possible parliamentary coalitions after the elections. This argument follows the line of the well-known prediction (Duverger, 1964 that proportional representation promotes the formation of the greater number of political parties than does the majority voting Equilibrium with three parties We analyze a model similar to the one given in the previous section. We assume that instead of raising their own valence, cidates exert resources to decrease valence of their opponents. Let ǫ ij be the amount by which the valence of cidate j decreases because of actions by cidate i. The total cost of valence spending by cidate i is therefore N k=1 c(ǫ ik. We first obtain the results on equilibrium levels of valence spending. These results are similar to Proposition. Proposition 16. In a local Nash equilibrium the following statements hold. 1. If j / {i 1, i + i}, then ǫ ij = If j = i + i, then ( ǫ ij = ǫ ji = c 1 f(ỹ i, (28 φ (ỹ i y i + φ (y j ỹ i where the voter ỹ i is indifferent between cidates i i + 1. One should not expect a pure-strategy LNE to exist in the general case. In order to make further analysis analysis tractable, we assume specific functional forms: φ(x = x 2 /2 c(x = ax 2 /2. We also assume f(y to be piecewise constant. We assume that the policy platform of the cidate 2 is normalized to zero fixed. Proposition 17. With φ(x = x 2 /2, c(x = ax 2 /2, a piecewise linear density f(ỹ equilibrium policy platforms by cidates 1 3 valence investments are defined as follows. y f 1f 1 2ay 1 f 2 ay 1 y 2 f 1 ay 2 1 y f 2f 2 2ay 2 f 1 ay 1 y 2 f 2 ay 2 2 f 1f 2 f 2 a 2 y 2 1y f 1f 2 f 1 a 2 y 2 1y f 1f 2 f 1 a 2 y 3 1y 2 = 0, (29 f 1f 2 f 2 a 2 y 1 y 3 2 = 0, (30