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)<g,(x) for (V)xct-4,0).
The mathematical apparatus is implemented in Figure 1. Here you can easily

observe that the voters placed between [-5, -4) obtain a higher utility value when they
choose a more extreme candidate and the voters placed between (-4, 0) obtain a higher
utility value from the election of a candidate who lies within the region of responsibility as
he is not penalized for radicalism. In the case when a voter is placed exactly on the border
of the region of responsibility. under the assumptions of this form of the directional model
the voter has a relation of indifference between alternatives.



/0 $Eg!}4'!haDoJVoti%

Figure 1 . Utility of the voters placed on the same direction of the ideological dimension
as the candidates

Direction of the ideological dimension

--+-- g€x):5x-4 -- a-- g(x):-4x

Source: Author

Thus, as I prove in the previous paragraphs by appealing to the linearity and mono-
tony of the utility function as well as the values taken by the function in the minimum and
maximum points of the interval each voter placed within the region of responsibility will
vote for the party which is placed on the border of the region of responsibility and each
voter placed outside the region will vote for the most extreme candidate of the ideological
continuum, even if they apply the penalty function for radicalismsz.

As there are two positions which have the potential to become dominante8 on each di-
rection of the ideological dimension the problem which arises is to identify the conditions
in which each position actually becomes dominant. In this sense, knowing the configurati-
on of the distribution of voters is crucial because if the density of voters is larger between
the boundary of the region of responsibility and the ideological extremity, the latter be-
comes the dominant position instead of the boundary of the region of responsibility. An
example of such a situation, i.e. a predominantly extremist electorate is depicted in Figure
26g, which is designed specifically to be a visual aid to the reader in understanding the
conditions in which Rabinowitz and Macdonald's original claim that the boundary of the
region is always a dominant position are false.



tunic 2012 (Pefspecti9e poGitice

Figure 2. Density of voters in an extremist electoral pool

Source: Author

Because the distribution of voters is encompassed between the graph of the
distribution function and the x-axis, therefore it is geometrically in'egular. it can only be
detemiined by calculating the area between the endpoints of the interval, that is the ex-
tremity of the ideological dimension and the boundary of the region of responsibility and
the boundary of the region and the neutral point.

Thus we have three possible cases:

a) (r.a) > (x,r) 'w J .f(x) -J /(x) > 0

b) (I,a)<(x.r) oj /(x)
0

f/(x)<o

c) (x,a)
0

(i.r) Qf /(i) -J/(1) = o
Where:./(x) = voter distribution function, a = the extremity of the ideological di-

mension and /" = the boundary of the region of responsibility.
In common language, the boundary of the region of responsibility is a dominant

position only as a special case of the directional model. in the hypothesis in which tho
difference between the definite integral from the ideological extremity to the boundary of
the region and the definite integral from the boundary of the region to the neutral point i$
negative. By contrast, the extremity of the dimension represents the ideologically doml-



]2 Span.[q.b.q dl/gun%

nant position if the above mentioned difference is positive. If the di#erence is 0 than both
positions are dominant.

K Concluding remarks

Concluding, I believe that in this paper I have realized the theoretical enterprise formu-
lated in the introduction, proving that:

1 . Rabinowitz and Macdonald's claim that "any candidate whose intensity 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"" is false as there are n cases where the ideological extremity is
dominant and the boundary of the region of responsibility is not.

2. Macdonald, Listhaug and Rabinowitz's assertions that the directional model does
not require information regarding the distribution of voters in order to formulate theore-
tical predictions with regard to the optimal party placement'' is false as electoral results
depend, inter alia, on the distribution of voters with respect to the boundary of the region
of responsibility.

3. Also, the paper contributes to the improvement of the full proportionality theorem
formulated by Macdonald, Listhaug and Rabinowitz': by specifying the conditions in whi-
ch the two instances of the theorem occur, i.e. the optimal ideological placement on the
boundary of the region of responsibility and the optimal ideological placement outside of
the region of responsibility.

References

I The "distribution of voters" refers to the configuration in which each member of an
electoral pool is placed on an ideological dimension. It is representable as the graph of
a function which reflects on the x-axis the ideological placements and on the y-axis the
number of voters which correspond to each position.

2 Rabinowitz and Macdonald. 1 989.
3 Ibid, P.108.
4 Macdonald, Listhaug and Rabinowitz, 1991, p.1113.
5 Public Choice Theory is the economic study of decisional processes which are not

directly connected to the economic market. For a monograph see Mueller (2003).
6 Udehn, 2002, p.497.
7 Agassi, 1960, p.244.
8 Sugden, 1991 , p.753.
9 Black. 1958.
10 Downs. 1957.
11 Hotelling, 1929.
12 Lerner and Singer, 1937
1 3 Smithies. 1 941 .
14 Downs, 1957, p.55.
15 Ibid, P.164.
16 Ibid, PP.164-165.
17 Ibid, P.1 72



iunie 2012 (Perspecthe po6ttce

18 Ibid, P.165.
19 Ibid. P.64.
20 Ibid, p.67. This assumption has an ambiguous nature as Downs does not appear

to clearly distinguish between vote-maximizing and plurality-maximizing, which is later
made by Davis. Hinich and Ordeshook (1970).

21 Miroiu, 2006, p.248.
22 Ibid, P.247
23 Ibid, P.248.
24 But the party system also.
25 Stokes, 1963.
26 Wittman, 1973.
27 Hinich and Ordeshook, 1 970.
28 Palfrey, 1984.
29 osborne. 1 995
30 Premises due to which these models are generically termed 'proximity models' as

they are representable in the euclidean space (Lewis and King, 1999. p.22).
31 Davis. Hinich and Ordeshook. 1 970.
32 Hinich and Pollard, 1982.
33 Endow and Hinich. 1982
34 Greenberg and Weber. 1985.
35 Poole and Rosenthal. 1 985.
36 Grofman. 1 985.
37 Erikson and Romero. 1 990.
38 Rabinowitz and Macdonald, 1 989, p.93.
39 Ibid, P.94
40 Rabinowitz, 1978. p.793.
41 Macdonald, Rabinowitz and Listhaug, 2001 . p.484; This construct was initially termed

'region of acceptability" (Rabinowitz and Macdonald, 1 989, p.1 08). However, in this paper I
will use the temp 'region of responsibility" in general. the exception being represented by the
cases where I directly quote papers from the period which precedes the 2001 redefinition of
the term, as it adequately reflects the perceived irresponsibility of candidates.

42 Rabinowitz and Macdonald, 1989, p.108.
43 This penalty function will be also used in the present paper considering that the

assertions regarding the irrelevance of the voter distribution function come from the same
paper, i.e. Macdonald, Listhaug and Rabinowitz (1 991).

44 Westholm, 1997. p.872.
45 Ibid, P.867.
46 Ibid. P.867.
47 Interpersonal comparisons are comparisons between the utilities of voters in voting

a single party.
48 Intrapersonal comparisons are comparisons between the utility attached to every

party by the voter.
49 Westholm, 1997, p.868.
50 Platt, Poole and Rosenthal., 1992
51 Gilliam, 1992
52 Pierce. 1 997.



ipad"€gj$WQVV.}@.

53 Kramer and Rattinger, 1997.
54 Blais, Nadeau, Gidengil and Nevitte, 2001 .
55 Karp and Banducci, 2002
56 Cho and Endersby, 2003.
57 Claasen, 2007.
58 Tiemman. 2011 .
59 Macdonald, Listhaug and Rabinowitz, 1991 , p.1113.
60 Ibid, P.1113.
61 Rabinowitz and Macdonald, 1989, p.108.
62 By full proportional system the authors describe a system in which the electoral

threshold is 0 or very close to 0. (Macdonald. Listhaug and Rabinowitz, 1 991 , p.11 27)
63 Macdonald, Listhaug and Rabinowitz, 1 991 , p.111 31 The authors term this propositi-

on the "full proportionality theorem" (Macdonald, Listhaug and Rabinowitz, 1991 . p.11 27).
64 Macdonald, Listhaug and Rabinowitz, 1991 , p.1112
65 I consider only one ideological dimension to simplify the demonstration but the

results do not differ for a multidimensional political field, the only modification being that
vectors are used instead of points to describe the position of voters and candidates.

66 if the generalization is valid for these two points it is automatically valid for any other
subset on [-5,0] with the boundary of the region of responsibi]ity on {-4} as the function is

67 As Westholm observes (1 997, p.872), this position is difficult to defend as an extre-
mist voter would have no incentive to apply a penalty to an extremist candidate.

68 The dominant position is defined as the position in which "the candidate is guaran-
teed at least a tie in the election and a positive plurality if his opposition selects some
position other than the dominant one" (Davis, Hinich and Ordeshook, 1 970. pp.426-427)

69 in order to simplify Figure 1 1 have only introduced a single direction of the ideologi-
cal dimension. i.e. the left direction.

70 Rabinowitz and Macdonald, 1989, p.108.
71 Macdonald, Listhaug and Rabinowitz, 1991 , p.1113.
72 Ibid, P.1127.

linear

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