Referenda as Runing Mates: Modeling The ...



Stuck in Space:

The Neglected Importance of Issue Salience for Political Competition

Scott L. Feld

Department of Sociology

Louisiana State University,

Baton Rouge, Louisiana

Bernard Grofman

Department of Political Science

and

Institute for Mathematical Behavioral Science

University of California, Irvine

Prepared for presentation at the European Public Choice Society annual meeting, April 26-28, 2003, Aarhus, Denmark. This research was partially supported by National Science Foundation grant SBR 97-30578 (to Grofman and Anthony Marley), Program in Methodology, Measurement and Statistics. We are indebted to Clover Behrend for library assistance.

ABSTRACT

Virtually all work on electoral competition assumes that candidates compete in terms of the issue platforms they offer, and that they sequentially search for winning positions by changing the nature of their platforms. This standard Downsian approach neglects the fact that candidates are often largely trapped in terms of what policy platforms they can credibly espouse by their past positions and by the positions voters attribute to the parties they represent. If candidate locations are largely or entirely fixed, it might seem inevitable which one will win and which one will lose! However, candidates can also compete by persuading voters to change the salience of the different issue dimensions. We argue that competition over issue salience has been wrongly ignored. We show that, taken two issue dimensions at a time, this competition can be viewed in two mathematically equivalent ways: as a fight over what will be the new single axis of cleavage on which voter choices will be arrayed, or as a fight over the relative weights (w/1-w)of the two existing issue dimensions. Even though candidate locations are fixed, we show that, unless the voting game has a core and one of the candidates is located at that core, there will always be one or more axes of cleavage on which each of the candidates will be victorious. Thus, even though "stuck in space," no candidate is ever doomed to defeat. Candidates can seek to move the electorate toward a cleavage axis on which they are winning by persuading the electorate to change the relative salience voters attaches to each of the two existing issue dimensions. We provide a geometric framework to understand how issue dimension weightings affect candidate choice

We show that, even though there is always at least one axis of cleavage on which each candidate will win, there is also a paradox of non-monotonicity in that a candidate who seeking to change the relative salience of the two existing issue dimensions so as to move voter choices to an axis of cleavage on which s/he is winning must risk the near certainty that the "route" to that winning outcome does not involve a monotonic increase in votes for that candidate. In particular, we show that, contrary to intuition, weighting a particular dimension more heavily need not have a uniformly favorable impact on the election chances of the candidate who is on the more popular side of that issue dimension. Indeed, ceteris paribus, as we increase the relative weight of an issue dimension over the range from zero weight (in which case, say, candidate A wins) to 100% weight (in which case candidate B wins), we can get a non-monotonic pattern of electoral consequences in which first A, then B, then A, then B, etc., is the majority winner. . Thus, movement in the desired winning direction may actually generate losses before it generates gains; or a complex pattern of losses, then gains, then losses, then gains, etc. Nonetheless, if one candidate is extreme and one is moderate with respect to the set of voter ideal points, we also show that the moderate candidate is relatively well buffered against attempts by his opponent to change voter perceptions of relative issue salience, since only dramatic changes in salience weights will lead to that candidate's defeat. On the other hand, when candidates are roughly equally centrally located wrt voter ideal points, then issue salience matters a great deal. Even small changes in relative issue dimensions weights may change the outcome. In sum, while platform locations matter, so does salience. Both must be taken into account in understanding political competition.

As the Downsian model has been operationalized by most authors, it implies that voters have fixed issue locations and that candidates compete for votes by where they locate in the issue space, with the voter assumed to vote for the party/candidate who is located closest to the voter’s own most preferred location. Thus, the driving force of political competition is each candidate's choice of an issue platform.

Skaperdas and Grofman (1995: 57) argue that this portrait of political competition omits at least four key factors: “(1) the information conveyed by candidates is not only about putative issue positions but also about candidate attributes such as competence or trustworthiness; (2) almost invariably, candidates not only characterize themselves and their own policy positions but also see to (mis)characterize their opponent and their opponent’s policy positions as well; (3) in a world of multidimensional issue competition, candidates not only seek to convey the positions they wish to be attributed to themselves and to their opponent but also often seek to persuade voters that some dimensions (some issues) are more important than others; and (4) voters do not believe all they are told.” Also missing from the simple Downsian model are features of choice such as the role of partisan identification (Campbell, Converse, Miller and Stokes, 1960), retrospective voting (Fiorina, 1981) and incumbency advantage (King and Gelman, 1991; Feld and Grofman, 1991). Here, however, our focus will be limited to extending the basic Downsian model by looking at one important aspect of political persuasion, attempts at manipulation of the salience of the various issue dimensions.

Downs himself, in perhaps the least cited aspect of his classic work, emphasizes the importance of political persuasion (1957, 83-84).[1] The importance of political persuasion has also recently been stressed by authors such Zaller (1992) and Lupia and McCubbins (1998), with a number of recent studies of the effects of negative campaigning (Ansolabehere and Iyengar, 1995; Skaperdas and Grofman, 1995; Wattenberg and Brians, 1999; Lau et al., 1999).

Political realignment is often modeled in terms of the introduction of new issue dimensions or the changing salience of the previously existing issue dimensions that structure electoral conflict (see esp. Schattschneider, 1960; Riker, 1982). More generally, Riker (1986) has argued that the heresthetic "framing" of issues helps decide who wins. Also relevant to the model we offer in this paper are the work of authors in the spatial modeling literature who study the jurisdictional structure in a legislature in a setting where each committee determines the outcomes on a single issue dimension (see esp. Feld and Grofman, 1988; Humes, 1993). These authors look at the geometric structure of the set of generalized medians, i.e., the platforms consisting of median locations on each issue dimension, as we rotate the underlying axes that define the issue space.

Perhaps most directly relevant to the work we present here is the model in Hammond and Humes (1993). They consider two-candidate competition in two-dimensional space in a game where each candidate is associated with a single dimension and "controls" the angle of rotation of one of the two axes of cleavages that define this two-dimensional space. In this game voters are posited to choose whichever candidate is closest to them on one of the dimensions. For fixed voter and candidate locations, they show examples in which outcomes will depend upon which (not necessarily orthogonal) axes of cleavage are chosen by each of the candidates. In closely related work, authors such as Jones (1994) and Lawson (2000) have looked at the impact of changing issue salience on voter choice.

Our general concern in this paper will be with incorporating issue salience into models of political campaigning in two-candidate competition over two (or more dimensions). We assume that either candidate positions are fixed and thus we know voter preferences so that we can decide which candidate has the winning position with respect to that issue dimension, or that, even if the candidate positions with respect to that issue dimension are not fixed, some issues are known in advance to tend to favor one candidate or party. For example, a presidential candidate with military experience might be favored if defense issues were especially salient. Or, as John Petrocik (1996) has suggested in his work on "issue ownership," one party might be widely perceived as historically far more devoted to, say, fostering economic equality than the other, so that claims made by the latter party that they were even more interested in achieving economic equality would be dismissed by the voters as non-credible. Thus, the party associated with reductions in economic equality might be favored if the issue of economic inequality were especially salient in a campaign.

In light of such considerations we would anticipate that candidates would compete in seeking to “activate” and make more salient those issues/issue dimensions which they believe will favor their cause and on which their position is more popular than that of their opponent. But, as we shall see, finding an optimal campaign strategy vis-a-vis issue salience can be considerably more complex.

II. Issue Salience in Two or More Dimensions

For simplicity we will present our results for two issue dimensions in which the positioning of voter ideal points is not collinear. However, the basic results can be generalized in a straightforward fashion beyond two dimensions.

Consider two-candidate competition over two issue dimensions. To avoid technical complexities caused by knife-edge results such as the possibility of ties, we assume that there are an odd number of voters and that no two voters are located at the same point, i.e., have identical preferences. Let us imagine that, on issue 1 (the x dimension), a majority of voters are closer to candidate B, while on issue 2 (the y dimension), a majority of voters are closer to candidate A. The simplest such case, with three voters, is shown in Figure 1 below, with the voters defining the vertices of the triangle that comprises the elements of the Pareto set.

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Definition: voter preference over two or more unweighted dimensions - A voter prefers candidate A to candidate B if that voter's total distance to candidate A is less than the voter's total distance to candidate B on the given dimensions.

In general, if there is more than a single dimension of political competition, we know that there will not be a stable majority equilibrium for two-candidate competition (McKelvey, 1976, 1979). In Figure 1, the “swing” voter (in the bottom right) is closest to candidate B, so candidate B will win. In the usual Downsian story we would now have candidate A moving about the issue space looking for a location that will beat candidate B. Necessarily, such a location will exist (since candidate B’s win set is non-empty). But that new winning location for candidate A will in turn be vulnerable to a switch in location by candidate B. Etc.

However, the notion of candidates “hopping” about the space in search of a winning position, is very far from a realistic portrait of politics. Candidates are strongly constrained by their past positions and history and those of their political party.[2] This means that candidates may not always be able to make a credible shift to a winning position.

But even if candidate A in Figure 1 does not move location (perhaps because she cannot move because her credible options are so limited that there is no location in candidate B’s win set that is available to her), all is not lost. What candidate A can do is to seek to manipulate the importance that voters attach to the two issue dimensions. If she succeeds either in sufficiently inflating the importance of dimension 2 (the y dimension) or in sufficiently diminishing the importance of dimension 1 (the x dimension), or in doing enough of each, then candidate A, not candidate B, will have the winning position.

We can picture such changes in salience in a very straightforward geometric way as “stretchings” or “shrinkings” of one or both issue dimensions. In Figure 2 we show three such “salience manipulations” that are sufficient to convert candidate A’s position into that of a winner. The first (2a) shrinks dimension 1; the second (2b) expands dimension 2; the third (2c) does some shrinking of dimension 1 and some expansion of dimension 2.

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Shrinking one dimension while holding the other constant is equivalent to differentially weighting the dimensions.

Definition: voter preference over two weighted dimensions - A voter prefers candidate A to candidate B when the horizontal distance is weighted by h and the vertical distance by v if and only if the weighted total distance of the voter to candidate A (i.e., the square root of the sum of squared x distance weighted by h and the squared y distance weighted by v) is less than the weighted total distance to candidate B.[3]

While it is well known that candidates tend to emphasize the issue dimension that favors themselves, the strategic issues involved in two-candidate competition are much more complex than they first appear. We find that when one candidate moves the weighting of the dimensions in the direction of placing greater (relative) weight on the dimension in which she would receive the greatest number of votes if that were the only dimension of choice, but does not succeed in convincing voters that this is the only relevant dimension, increasing the weight on this dimension might actually lose votes or even lose the election.. That is, in general, the effects of differential weighting are not necessarily monotonic; i.e., increasing the relative weights on a particular dimension does not necessarily consistently favor the same candidate. Indeed, we can show that under certain conditions, increasing the weights of one dimension might increase, then decrease, then increase, then decrease, etc. the votes given to one candidate. That makes it very difficult in practice for candidates to campaign by attempting to increase the weight on a dimension ) on which (if this dimension were the sole dimension of choice) the candidate would be guaranteed a victory.

Suppose that we have two candidates, A and B, located at fixed positions in the space. We will show that, in these circumstances, weighting of the two dimensions in the space is equivalent to specifying the angle of a single dimension along choice is to be made. More formally:

Theorem 1: For two candidates, A and B, located at fixed points in the space, which we may without loss of generality label (-x0,-y0) and (x0,y0), candidate A is majority preferred to candidate B subject to dimensional weights of ratio h to v if and only if candidate A is majority preferred to B along the single issue dimension defined by the line at an angle -(h/v)(x0/y0), i.e., if and only if a majority of voter ideal points are on the A side of the line, y = -(h/v)(x0/y0)x.

Proof of Theorem 1: Without any loss of generality, for any pair of orthogonal dimensions, and for any points A and B, axes can be located such that the origin is at the midpoint between A and B, and such that A is located at (-x0,-y0) and B is located at (x0,y0). (Note that if one candidate is located below and to the left of the other, then the first candidate is labeled A. If one candidate is below and to the right of the other, then the situation is abstractly identical to its mirror image that can be labeled in this way.)[4] Then, any given voter whose ideal point is (x, y) prefers A to B if and only if his/her weighted distance to A is less than his/her distance to B; i.e.

h(x+x0)2 + v(y+y0)2 < h(x-x0)2 + v(y-y0)2

i.e.,

.

hx 2+2h x x0 +hx0 2 + vy 2 +2vy y0 + vy0 2 <

hx 2 - 2hx x0 +hx0 2 + vy 2 - 2vy y0 + vy0 2

i.e.,

2h x x0 +2vy y0 < - 2h x x0 - 2vy y0

i.e.

4h x x0 < - 4v y y0

Further, if we assume that y0 > 0 and v > 0 (A and B and v can always be assigned to make this so), then the voter prefers A to B if and only if:

x - (h/v) x0/y0

or

x>0 and y/x < - (h/v) x0/y0 .

We show the geometry underlying the relevant constructions in Figure 3. The above result implies that there exist values of h/v such that any voter on the A side of line u would prefer A to B for those values.

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It should be apparent that being on the A side of the v line in Figure 3 is equivalent to preferring A to B on the single dimension perpendicular to u. Similarly, any voter on the A side of line v would prefer A for a different value of h/v and this is equivalent to preferring A to B on a different single dimension, perpendicular to v. Thus, it should be clear that, for two orthogonal issue dimensions, for fixed candidate locations, every weighting, h/v , is equivalent to deciding over some single issue dimension. q.e.d.

It is easy to see from the above result that, if the vertical dimension is given a zero weight, then the decision is determined by which candidate is closest to the median voter along the horizontal dimension --and similarly, if the horizontal dimension is given a zero weight, then the decision is determined by which candidate is closest to the median voter along the vertical dimension. If the two dimensions are equally weighted, then the candidate closest to the median voter on the dimension defined by the line passing through both A and B is preferred by the majority.

Definition: A median line is a line such that half of the voter ideal points are on or above the line and half are on or below the line.

Corollary to Theorem 1: For weights of h and v, if the line y= -(h/v)(x0/y0)x is a median line, then there is no majority preference between A and B for this particular value of h/v. However, one candidate is majority preferred for slightly larger values of h/v, while the other candidate is majority preferred for slightly smaller values of h/v

Proof : By definition, a median line is a line that passes through a voter ideal point and divides the other voter ideal points in half. When the line y= -(h/v)(x0/y0)x is a median line through the midpoint between A and B, the voter on that median line is indifferent between A and B with the relative weights of h/v, and there is a tie among the other voters. Increasing or decreasing the relative weights is equivalent to rotating the dividing line between the voters preferring A from those preferring B. Therefore, rotating from a median line one way puts the decisive voter on one side, and rotating in the opposite direction puts the decisive voter on the other side. q.e.d.

Before we prove the next theorem we prove a needed lemma.

Lemma 1: Every point in the space is on at least one median line, and is on an odd number of median lines

Proof:. Consider Figure 4, where p is the point of interest and the x’s represent voter ideal points. Start with a line 0 through p, as shown.

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A majority of voter ideal points must be on one side of that line; in this case, below the line. Call one half-space Q, and the other R. Now, rotate the line around p in a clockwise direction, as shown in Figure 4 – the rotation to line 1 is shown. As the line rotates, so do the half-spaces, Q and R, defined by the line. As the line passes over voter ideal points, they switch from being part of Q to R or vice versa– in this example, moving from line 0 to line 1 passes over one voter ideal point in the upper left, who moves then from being part of R to part of Q. As the rotation continues, a voter ideal point in the lower right would move from being part of Q to part of R; etc. Once one completes a full 180 degree rotation, all of the voter ideal points that were initially part of Q have become part of R, and vice versa. So, the number of voter ideal points in Q must go from being a majority to being a majority as it passes some voter ideal point in the process of rotation. The line going through that voter ideal point must be a median line, because it alone determines which side has the majority. This proves the part of the theorem stating that every point is on at least one median line. Now, if continuing the rotation passes over another voter ideal point that makes Q the majority again, then the line through that point must be another median line. However, since we know that Q must end up being the minority half-space once the 180 degree rotation is complete, we know that every other median line that makes it the majority again must be followed by still another median line that makes it the minority again before the rotation is complete. Thus, there must be an odd number of median lines passing though p. q.e.d.

Now we can proceed to the proof of Theorem 2.

Theorem 2: For every pair of candidates, A and B, at fixed locations, there is some weighting of the dimensions, h/v, such that A is majority preferred to B.

Proof: From Lemma 1, there must be at least one median line passing through the origin (0,0). That median line corresponds to some (h/v). Then, from the corollary to Theorem 1, A is majority preferred if the relative weights (h/v) are on one side (either higher or lower) of those corresponding to that median line, and B is majority preferred on the other side. Thus, there are relative weights that result in majority preference for A and others that result in majority preference for B. q.e.d.

Definition: A turnover is a change in which candidate is the majority winner accompanying a change in the relative weights attached to the two dimensions.

Theorem 3: For every pair of candidates, A and B, at fixed locations, changes in weightings give rise to an odd number of turnovers, with one turnover for each median line passing through the midpoint between A and B.

Proof : As shown in the proof of Theorem 2, as, say, the weighting increases on h relative to v, A gains some voters, then loses others. This pattern of gains and losses continues as we further increase h relative to v. The actual majority changes from one candidate to the other each time the rotating line crosses a median line. Thus, there will be as many switches as there are median lines passing through the midpoint between A and B. But there are an odd number of median lines through any point. q.e.d.

While we have shown that, for any set of voter ideal points and any two candidates, A and B, and any set of coordinates defining orthogonal dimensions, one can find a set of weights such that if each voter votes for the candidate whose weighted distance is closer to that voter, candidate A (or B) will get a majority of votes, Theorem 3 has direct implications for whether each candidate can pursue the simple strategy of persuading voter to increase the relative weights attached to the dimension or dimensions on which they would see themselves as favored. Such a simple strategy exists only when a candidate's vote share monotonically increases as the relative weight attached to a dimension on which his is the winning position increases. But often such monotonicity is absent. Indeed, there are circumstances in which increasing the relative weight attached to, say h (i.e., increasing the ratio h/v) can first shift a candidate from winning to losing, then from losing to winning, then from winning to losing, etc.. In other words, even if a particular dimension favors a particular candidate, attempts by that candidate to get voters to weight that dimension more heavily may backfire because, for some increases in the weighting of that dimension, the candidate will actually be disadvantaged rather than advantaged! Thus candidates may have trouble effectively pursuing a strategy about the appropriate dimensional weighting that will unequivocally improve his vote share if voters move only some of the way (but not all the way) toward that weighting (or overshoot it) is one of the key results in our paper.

However, there is one further complication we now need to discuss. While it is true that, as we have shown above, there are as many majority turnovers as there are median lines through the midpoint of A and B, and (since there is always at least one such median line) there is always at least one turnover, some of these turnovers can occur only if we assign a negative weight to one of the dimensions. But, when the axes are fixed, then negative weightings should be regarded as infeasible. Weighting a dimension negatively means that a voter prefers a candidate who is further from the voter on that dimension. That is not consistent with our common sense assumption that voters prefer candidates whose views are like their own.

If we fix the set of issue dimensions and rule out negative weightings, the strategic situation may be simplified in that the exclusion of certain potential turnovers as infeasible (those associated with negative weights) may mean that one of the candidates may never be able to get a majority in any feasible weighting scheme. Alternatively, if we rule out negative weighting, there may now only be a single feasible turnover. If there is only a single feasible turnover, then one candidate should have the simple optimal strategy of seeking to increase the relative weight given by voters to one of the dimensions, while the other candidate has the equally clear strategy of seeking to reduce the weight given by voters to that issue dimension.

Theorem 4: One candidate is majority preferred to the other for all feasible (non-negative) weights if and only if the only median lines that pass through the midpoint between A and B are contained entirely within the upper right and lower left hand quadrants.

Proof: The previous proofs have shown that thereis a turnover where and only where there is a median lines corresponding to the line,

y= -(h/v)(x0/y0)x.

The weights are not feasible when h/v < 0; that corresponds to a line through the origin with positive slope; i.e. going through the lower left and upper right quadrants. If all the median lines go through those quadrants, then there are no median lines based upon feasible weights, and there are no feasible turnovers. With no turnovers, the same candidate must get the majority for all feasible weights. q.e.d.

Definition: For a given set of voter ideal points, the yolk is the smallest circle intersecting all median lines (Mckelvey,1986; Miller, Grofman and Feld, 1990).

Corollary 1 to Theorem 4: (a) If the yolk is located entirely within the upper-right (lower-left) quadrant, then A (B) is majority preferred for all feasible weights, i.e., there are no feasible turnovers. (b) If the yolk is located entirely within the lower-right (upper-left) quadrant, then A is majority preferred on one dimension, and B is majority preferred on the other. In this case there is always at least one feasible turnover and the number of turnovers must be odd.

Proof: (a) If the yolk is located entirely within the upper-right (lower-left) quadrant, then since all median lines go through the yolk, the only median lines that can go through the midpoint between A and B must only go through the upper-right and lower-left quadrants. From Theorem 4, we must have the same candidate winning irrespective of the weights. If the yolk is in the upper-right (lower-left), then A (B) always wins (is on the midpoint side of the median line.

(b) Since all median lines pass through the yolk, then all median lines passing through the midpoint between A and B must also pass through the yolk and therefore go through the upper-left and lower-right quadrants. There is always at least one median line and an odd number of such median lines through this midpoint, and each median line switches the majority winner. q.e.d.

Definition: An issue salience monotonicity paradox arises when there exist feasible values of h/v, such that increases in the ratio h/v change the majority preference from A to B, and other feasible values of h/v such that increases in the ration h/v change the majority preference from B to A.

Corollary 2 to Theorem 4: There is an issue salience monotonicity paradox if and only if there are two or more median lines through the origin (the midpoint between A and B) that pass through the upper left and lower right quadrants.

Corollary 3 to Theorem 4: If one candidate is majority preferred on one issue dimension while the other candidate is preferred on the other, we avoid an issue salience monotonicity paradox if and only if there is only one median line through the origin that passes through the upper left and lower right quadrants.

Definition: A simple issue salience paradox occurs when one candidate wins on both of the two issue dimensions, but loses for some particular feasible weighting of those dimensions.

Corollary 4 to Theorem 4: A simple issue salience paradox occurs if and only if there are an even number of median lines passing through the origin that goes through the upper left and lower right quadrants.

Proof: An even number of median lines passing through the origin that goes through the upper left and lower right quadrants implies an even number of turnovers. Thus, the candidate who is majority preferred on one dimension is turned over and back (perhaps multiple times). q.e.d.

So far , we have considered what happens if we treat axes as fixed and also what happens if we rule out negative weights. But, if candidates can manipulate choice of the axes as well as the weights to be attached to each, then, in effect, all weights become feasible, and Theorem 2 applies.

Theorem 5: For every pair of candidates, A and B, at fixed locations, there exist some set of orthogonal dimensions and some set of non-negative weights on the distances in those dimensions such that A (B) is majority preferred to B (A) .

Proof: There is always at least one median line through the midpoint between A and B. If the axes are rotated so that median line passes through the “feasible” quadrants, then there are weights that switch the majority preferred candidate at that median line. On one side, A is preferred to B; on the other, B is preferred to A. q.e.d.

The implication of Theorem 5 is that, even with fixed candidate positions, and even where there is an initial set of dimensions such that there are no feasible weights by which candidate A can be majority preferred to candidate B, there is still always some new set of axes and some set of relative weights for those new axes that result in candidate A receiving a majority of votes. Of course, we would expect that it would ordinarily be very difficult for a candidate to simultaneously persuade voters about both the need to redefine the dimensional axes and about the relative weights which should be given each dimension. Still, such two-pronged heresthetic communications have, at least in principle, the potential for converting a loser into a winner.

Discussion

The standard Downsian approach to candidate/party competition emphasizes party/candidate location as the driving force in voter choice, with voters choosing the issue platform closest to the voter's own ideal point. But, often the location of candidates will be, if not completely fixed, at least constrained, with only so much "wiggle room."[5] Here, paralleling ideas in Petrocik (1996) on "issue-ownership," we have looked at situations in which certain issues favor a particular party or candidate. We have offered a widened notion of strategic choice, in which candidate decisions go beyond the choice of a policy platforms to include decisions about persuasive aspects of the campaign -- here, which dimensions to emphasize. In such situations, we have looked at the consequences of attempts to persuade voters to change the relative importance that they attach to different issues, taking the candidate issue positions as (largely or entirely) given.

There are many ways in which candidates and their supporters can seek to change the salience of different issues and thus to affect voter choices. For example, Salvanto (2000) has looked at the effects of ballot referenda in California on voter perceptions of the most relevant campaign issues. His analyses look both at the anticipated consequences of decision to place particular referenda on the ballot and on the decisions of some candidates to identify their own campaigns with particular ballot issues, a practice he call “referenda as running mates.” In California in 1994, the incumbent Republican governor, Pete Wilson, ran on anti-illegal-immigration platform and strongly associated his campaign with the campaign to pass Prop. 187, a referenda proposing to eliminate state spending on public services for illegal immigrants. By associating himself with a YES vote on the referendum he increased the salience of the immigration issue for those voting on the governor's race.

While this strategy appeared to make sense for Wilson, in that a substantial majority of voters were in favor of Prop. 187, the results in this paper show that we must be cautious in assuming that being associated with the winning side of an issue necessarily benefits a candidate as the (relative) weight attached to that issue increases among voters. There is the potential for loss of support when voters increase the weight attached to a dimension on which a candidate is favored, even though the candidate would win if that were the only dimension of choice.

The knowledgeable reader may have noticed that our results about the nonmonoticity of changes in dimensional weightings seem to be similar in spirit to results about cycling in the absence of a core point. But, we must be careful. Even with a core when issues are weighted equally, differential weighting of issue dimensions can change the outcome. This paper is, we believe, the first to have shown specifically how, given the geometry of voter choice, perverse effects of dimensional weightings can arise.

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Figure 3

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[1] Downs’ views on political persuasion are discussed in Grofman (1987) and Weatherford (1993). (See also Grofman and Withers, 1993.)

[2] For spatial models which explicitly recognize this fact see e.g., Aranson and Ordeshook, 1972; Coleman, 1972; Wuffle et al., 1989; Owen and Grofman, 1996)

[3] While we could present results in terms of a single parameter w, such that w was the weight on one dimension and 1-w on the other, because we will sometimes be considering negative weights the discussion below is easier to follow if we use separate parameters for each of the dimensions.

[4] When the position of B (x0,y0) happens to be (1,1), then this line is entirely a function of h/v, but the proof is written for the general caYZ\@ A ٬O

R

7F¨-ª-À-Á- I!L!($;$È$É$~&?&õ)ö)p*s*x,y,.3.°1É1Ë1Õ1×1

2@6A6É:â:Y;c;e;’;þCCŽC?C’C“C›CœCžCŸCƒD„D†Dse of any A and B where the axes are drawn through the midpoint between them.

[5] Cf. Wuffle et al. (1989).

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