perspective iunie2012.pdf (Pefspectinepotit e j Does Voter Distribution Matter in the Directional Model? A Refutation of the Original Claims Abstract: !n this paper Idemonstrate that the distribution qvoten in©uences the theoretical predictions formulated through the instrumentalityof the directional model ofspatiat analysis, thereby n:fuming the claims made by Macdonald, Listhaug and Rabinowitz(t99a) who argue that the distribution if voters is ontyretevantjor proximRy-based models. Fu:ther. lshow that the boundalyof the region afresponsibility is not always the ideoiogicalty dominant position. as Rabinowitz and Macdonald {ag8g} originally assert. this situation appearing only as a special case. determined by thejact that the area encompassed between the graph of the voter distributianjunction and the x -axis is numerically higher between the boundary of the region of responsibility and the point of origin then between the most extreme point of the ideological dimension and the boundary qthe region gresponsib+tity. Finatty, limprove thejUtt proportionatitytheorem by specjDing the necessaryconditionsjor each position described in the theorem to become ideologically ootimal Keywords: directional model, distribution of voters, dominant position. .butt proportionality theorem, ideologicaldynamics, region of responsibility. Introduction The purpose of this paper is to demonstrate that the distribution of voters' matters in any attempt to determine the optimal place- ment of political parties in the directional model of Rabinowitz and Macdonaldz. Although in the original description of the model Ra- binowitz and Macdonald assert that "any candidate whose inten- sity exceeds the radius of the region of acceptability will perform worse in any two-person election than a candidate who adopts the same directional stand and is located at the boundary of the region of acceptability": and Macdonald, Listhaug and Rabinowitz further assert that the directional model can provide theoretical predictions wi- thout "assumptions [...] that constrain the strategies of parties or the distribution of voters"' in the present paper I intend to present the conditions under which these assertions are erroneous and the distribution of voters proves to be decisive in determining the ideological dynamics of political parties. To achieve this objective I will start from a simple example in which I show that these assertions are in fact contradicted by the results of the electoral competition after which I provide a generalized de- monstration in support of this position. Alexandra Volacu SNSPA (volacujr.123@yahoo.com) 6 $2!!@'M«) #V.ti«& [l. The SpatiaiAnatysis ofElectora{ Competition The Spatial Analysis of Electoral Competition is a theoretical framework which uses methodological tools specific to Public Choice Theorys and mathematical models in order to analyze the ideological dynamics of political candidates and parties, the structure of the party system and electoral competition in all the aspects involved. Two of the most important methodological principles used in this framework aro: a) methodological indivi- dualism, a "principle, rule or program telling historians and social scientists how to define collective concepts, explain social phenomena and/or reduce macro to micro'o which postulates that 'only individuals have aims and interests'z and b) instrumental rationality, a "conception of rationality"e which identifies utility maximlzation as the underlying basis of individual action. Spatial Analysis traces its roots back to the works of Blacks and Downslo. the latter being responsible for the development of the first spatial model which analyzed electoral competition, starting from the economic models elaborated by Hotellingtt, Lerner and Sin- ger'z and Smithies's with the purpose of detemlining the optimal placement of economic agents in a competition involving duopoly. In his model Downs introduces a series of as- sumptions in order to infer universal and general statements regarding electoral behavior, at the cost of severely simplifying the real electoral process: a) voters and candidates are rational actors, with the term rational action meaning "that type of action which is effici- ently planned to achieve the political and economic objectives consciously selected by actors"14, b) the preferences of voters can be ordered from left to right in a similar manner for all voters's, c) the preferences of each voter is unimodal and slides monotonously both to the left and to the right of V(x,yye. d) parties do not have perfect mobility as they are unable to "ideologically Jump" over other competitorslr. e) parties compete on a sin- gle ideological dimension. i.e. economic interventionismle, f) the ideological dimension is representable through a linear scale which goes from 0 to 1 00 starting from the left extre- mity's. g) parties exhibit a vote-maximizing behavioreo, h) voters participate in the electi- ons only if it is rational from a utility standpoint, meaning that the benefits of their actions outweigh the costs2', i) the party positions are well defined, in the sense that each party can be identified through the set of policies which it advocates2z, j) the electoral system is plurality (first-past-the-post), the elections are held in a single circumscription and only one candidate is elected23 According to Downs the ideological dynamics of parties24 is inseparably linked to the distribution of voters: if the distribution is unimodal parties ideologically move toward the median voter, i.e. the voter which splits the distribution in two equal halves, if the distribu- tion is bimodal and the modes approach the extremities of the dimension the ideological competition is centrifugal and if the distribution is multimodal parties move toward each of the modes of the distribution. Although Downs' model had a significant value mainly because of the originality of applying this kind of analytical tools to political theory. it has been subjected to substantial critiques which either refer to the oversimplification of rea- lity or to the omission of certain variables (e.g. Stokes' critique on the unidimensionality of the model2s, Wittman's critique regarding the different potential party goals26, Hinich and Ordeshook's critique regarding the failure to distinguish between vote-maximizing objec- tives and plurality-maximizing objectives:', Palfrey's critique regarding the failure to take nto account new entry potential20, Osborne's critique regarding the cognitive limitation of {u?tte 2012 q'erspecthe ?oRtice 7 voters29, etc.). In light of the considerable number of deficiencies present in the Downsian model, the field has experienced a series of alternative models which, similarly to Downs' position, start from the premises that the distance between the ideal position of the and the position of the candidate determine the utility function of voters30, e.g. Davis, Hinich and Ordeshook3' who consolidate and expand the results of the Downsian predictions for two-party systems, Hinich and Pollardaz. Endow and Hinichn, Greenberg and Weberw Poole and Rosenthalas, Grofmann, Erikson and Romero37, etc. 111. Tile DirectionatModet The directional model was proposed by Rabinowitz and Macdonald as a critique to classical spatial analysis, considered by the authors "seriously flawed"3a. The main premi- ses on which Rabinowitz and Macdonald build their enterprise is that the utility function of voters is not designed based on an estimation of the euclidean space distance, instead being the result of two factors: 1. direction, meaning that voters attach a positive utility only to candidates who are perceived to be placed on the same side of the ideological dimension and 2. intensity, meaning that voters like candidates which propose policies more intensely. Assuming that candidate intensity is relevant immediately derives from the way in which Rabinowitz and Macdonald conceptualize political space, as a "symbo- lic field"ao and from the assumption that voters perceive policies in a diffuse m8RR©r40 Rabinowitz and Macdonald therefore argue that voting is influenced by the affects of Individuals and in this sense an increased level of intensity in policy proposals ensures candidates a better electoral positioning. But the model is still incomplete. Because in this form the theoretical predictions generated by the model would favor radical extremism (as all parties would have incentives to propose aggressive policies) its authors incorpo- rate a construct which they term "region of responsibility"4t, asserting that any candidate who crosses the region of responsibility is penalized by the electorate for adopting an 'unreasonable" position'2. Although the authors do no specify a penalty function in the original model Macdonald, Listhaug and Rabinowitz define an example of such a function as: P = r(I p I -r) , where P = penalty function, r = radius of the region of responsibility and p : position of the candidate, claiming however that it does not have a universal nature43. Knowing its assumptions as well as the structure of the penalty function we can de- scribe the directional model from a mathematical standpoint in the following manner: b'(r,C) = }' ' (' - P = }.v, - c. - r(I c, I -r) where U r'KC) = the utility of voter V when electing C, r= the total length of the vector of the voters position in an ideological space with n dimensions, C = the total length of the vector of the candidate position in an ideological space with n dimensions, P = penalty function, r = the position of a voter on an i-dimension, C. = the position of a candidate on an i-dimension, /" = radius of the region of responsibility. The directional model was in its tum subjected to a series of criticisms, out of which Westholm's position is particular strong: the region of responsibility is a property of the entire electorate although individual preference and perception functions cannot be universalizedm: the lack of a universally specified penalty function45, the degree of falsifiability of the theory is severely reduced by comparison with previous modelsa, the empirical analysis of the model is based on interpersonal4r comparisons rather than intrapersonal4e compahsons4e. 8 Spatiail'TReaD o$'Votiltg In the contemporary literature the majority of papers which deal with the directional model are engaged in the proximity-directional debate, taking the form of comparative studies between sets of data analyzed through both theoretical perspectives, such as Platt, Poole and Rosenthal on the American Congresss', Gilljam on Sweden5', Pierce on Frances2, Kramer and Rattinger on Germanys3, Blahs et al on Canadas4, Karp and Banducci on New Zeelandss, Cho and Endersby on the UK56, Claasen in an experimental frameworks', Tiemann in the elections for the European Parliaments', etc. IV. Does voter distribution matter in the directional model? According to Listhaug, Rabinowitz and Macdonald one of the main strong points of the directional model is that it can generate predictions regarding the ideological dynamics of parties without appealing to the distribution of voters. This can be de- duced from the phrase "the theorem (n.a. regarding the optimal placement of parti- es in proportional systems) makes no assumptions about the nature of the penalty mposed on parties beyond the region of acceptability nor about the shape of the voter distribution"se and from the per a contrario interpretation of the statement that proximity-based theory makes no definitive predictions about the location of parties unless assumptions are made that constrain the strategy of parties or the distribution of voters"eo. Further, in their original construction Rabinowitz and Macdonald explicitly claim that no party which is placed on the boundary of the region of responsibility can be defeated by a party which is placed outside this region61. the resulting conclusion being that this position is always the position toward parties converge. On the other hand, in the case study on Norway Macdonald, Listhaug and Rabinowitz assert that "in a fully proportional representation system'z all parties should be located at or beyond the region of acceptability"ea. Even so, the authors do not argue that in this case the ideological dynamic is influenced by voter distribution, the actual conditions in which a party should be placed outside the region of responsibility remaining unspecified. In the following pages I intend to demonstrate first of all that the optimal placement of parties can be successfully localized only if the distribution of voters is known. the- reby refuting the claim that voter distribution is irrelevant and secondly to break the full proportionality theorem in the two hypothetical cases suggested by the authors, formulating the conditions in which parties should be located on the boundary of the region of responsibility and the conditions in which parties should be located outside the region, depending on the distribution of voters. In order to easily understand the role played by the positioning of voters let us consider the following example in which for simplicity we have a single ideological dimension, two candidates which are competing: A and B and three voters: x, y and z. We consider that the ideological dimension is representable on a segment encompassed between {-5} and {5}, with 0 as a neutral point. A is placed on {-5}, B is placed on {-4}, x is placed on {-5} and y is placed on {-l}. With the above elements fixed, we will consider two different ca- ses. In the first one z is placed on {-3} and in the second it is placed on {-5}. Also, for operational reasons we consider that the region of responsibility has a radius of 4, in a similar fashion to Macdonald, Listhaug and Rabinowitz's examples. In the case of voter x we have: C,r(x,.4)=x.,4--/"(1 H l--/')=--5(--5)--4(j--SI --4) =21(1) and U(x,.B)=x. B =--5.(--4) = 20(2). From(1),(2):>(x,J)>(x,.B) . In the case of voter y we have : C/(p,H)=p-.4-/"(l-dl--f")=--1.(--5)-4(1--51--4)=1 (3) and U(.y,B) =y-.B = --1.(L-4) = 4(4). From (3),(4)a'(y,B) >(y,H) . In case lof voter z we iunte 2012 (Perspecthe poGttia have: U(z,.4)=z.,4--r(1.4l--r)=--3.(--5)--4(1-51-4)=11(5) and U(.y,,B)=y =-3.(-4) = 12 (6). from (5), (6) H; (z.B)>(z,H). In case 2 of.voter z we have: C/(z,.4)= z. .4 -- r(I H l-r) = --5.(--5) -- 4(1--5 1--4) =(5#) and C./O',a) =y-B = -5.(--4) =20(6#). From(5#),(6#) ='(z,H) >(z,B) According to the utility calculus of voters previously presented it is easy to observe that in both cases x prefers candidate A and y prefers candidate B. However, z's preferences are different between the two cases, depending on its positioning relative to the boundary of the region of responsibility. If he is placed within the region of responsibility z attaches to party A a utility of ll units and B a utility of 12 units, thereby preferring B. However, if z exits the region of responsibility. situation reflected by the positioning on {-5}. z attaches a utility of 21 to A (even if he applies the penalty for extremism) and a utility of 20 to B. preferring A to B. The result is clearly influenced by the distribution of voters in this case. the positioning of z within the region of responsibility or outside of the region determining the aggregation of preferences for A or B respectively. The result can be generalized by calculating the utility determined by the election of candidates for each voter on a direction of the ideological dimension. We consider, simi- larly to the previous example a single ideological dimension6s, representable on a seg- ment encompassed between {-5} and {5} with 0 as a neutral point. Let .f be the function of voter distribution / :l--5,0) -+ i , with the classical description J'(x) = ax ' +6x +c Without a loss of generality let us consider that the radius of the region of responsibi- lity has the same value as in the previous example. i.e. r = {-4} and the two parties are placed on the extremity of the ideological dimension and the boundary of the region of responsibility: ..4 = {--5}, B = {-4lw. Also, because unlike the previous example we do not have only 3 positions which correspond to the voters but n positions which correspond to n voters, we have: g:l-5,0)-+ i, g,(x)=x.J-r(IHl-r)=-5x-4, g,(x)=x.B=-4x. For all voters which are placed between the ideological extremity and the boundary of the region of responsibility we have g : [--5,--4) -+ i, g.,(--5) = --5 . (--5) -- 4 = 21, gp(-5 ) = --4 . (--5) = 20, g..(--4) = -5 . (---4) -- 4 = 1 6 si ga(-«+) = --4 . (-4) = 1 6 o =g,( S)>g,(-S) ald g,(-4)=g,(-4) (1). But g,(1..)-g.(X.)=-5X.. 4+5X. +4= = --5(x... -- x.) < 0 and g,(xn.) - g,(x.) = -4x.. 1. 4x; = -4(x.,. -x..) < 0 a CVbn. >.x. r.(x)] and g,(x)] (2) From (1)1 {2)0 fx(x)) g,(x} nr(v) x E [--5,:4). For all voters which are placed between the boundary of the region of responsi- '"'!.?:gIfS'H:21K':\lEE,T?g.IE,81-443!.,. 1gh-l8lJIJ:ie,g,(-5)= -4.(--+) = 16. g..(-4) = --5 .(0)=' 4= -:4 si ka(=4) =:-4 .(0) = 0 o o g.(4) > g,(-4) sig.(0) = rB(0)(3).But, $imiiariytotnenrstca$e, g.(1.*.)-g.(1.)= -5x... - 4+$x. + 4 = :5(x.,...-l.)<0 and g,(x..)- g,(4,)= -4lw. +4x. = £=4(x«..!-=x.)<0:>(V)x..i >x. :gJ(x)] ando,(xj](4). " From (3), (4)0 g,(x) (x,r) 'w J .f(x) -J /(x) > 0 b) (I,a)<(x.r) oj /(x) 0 f/(x)