An Expository Development of a Mathematical Model of the Electoral Process
Author(s): Otto A. Davis, Melvin J. Hinich and Peter C. Ordeshook
Reviewed work(s):
Source:
The American Political Science Review,
Vol. 64, No. 2 (Jun., 1970), pp. 426-448
Published by: American Political Science Association
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AN
EXPOSITORY
DEVELOPMENT
OF
A
MATHEMATICAL MODEL
OF
THE
ELECTORAL PROCESS*
OTTO
A.
DAVIS,
MELVIN J. HINICH
AND PETER C.
ORDESHOOK**
Carnegie-Mellon
University
I. INTRODUCTION
The fundamental
process
of
politics is
the
aggregation
of
citizens'
preferences
into a
col-
lective-a
social-choice.
We
develop, inter-
pret,
and explain
non-technically
in this
ex-
pository
essay
the definitions,
assumptions,
and theorems
of a mathematical
model of
one
aggregative
mechanism-the
electoral
process.'
This
mechanism
is conceptualized
here
as a
multidimensional
model
of spatial
competition
*
This
research
was supported
by
a
grant
from
Resources for
the Future,
Inc.,
to
Carnegie-
Mellon University,
and a National
Science
Foundation Grant
to the University
of Rochester.
The authors
are indebted to
many persons for
comments and
criticism and
wish especially
to
thank Professors
Peter H. Aranson
and William
H. Riker, University
of Rochester,
Howard
Rosenthal, Carnegie-Mellon
University,
and
Michael
J.
Shapiro,
University
of
California,
Berkeley.
**
Visiting
at
the
University
of
Rochester,
1969-70.
1
See the
following:
Otto
A.
Davis,
and
Melvin J.
Hinich, "A Mathematical
Model
of
Policy Formation
in a Democratic
Society,"
Mathematical Applications
in
Political
Science
II,
J. L. Bernd,
ed. (Dallas: Arnold
Foundation,
SMU Press,
1966); "Some
Results
Related
to
a
Mathematical
Model of
Policy
Formation
in
a
Democratic
Society," Mathematical
Applications
in
Political
Science III, J.
L.
Bernd,
ed.
(Char-
lottesville: University
of
Virginia
Press, 1967);
"On the Power and
Importance
of
the Mean
Preference
in
a Mathematical
Model of Demo-
cratic
Choice,"
Public
Choice,
5
(Fall, 1968),
59-72;
"Some Extensions to a
Mathematical
Model
of
Democratic Choice,"
forthcoming
in
Social
Choice,
B.
Lieberman,
ed.
(New
York:
Gordon
and
Breach);
Melvin
J.
Hinich and
Peter C. Ordeshook,
"Abstentions
and Equi-
librium
in
the Electoral Process,"
Public Choice,
7 (Fall, 1969);
Social Welfare
and Electoral
Choice
in
Democratic Societies,"
(unpublished,
Carnegie-Mellon
University,
1969);
Peter
C.
Ordeshook,
"Some
Extensions to
a
Mathemat-
ical Model of
Electoral Competition,
and Impli-
cations
for
the
Theory
of
Responsible
Parties,"
Midwest Journal of Political Science,
(February
1970); Theory
of the Electoral
Process (unpub-
lished Ph.D. dissertation,
University
of Roches-
ter, 1969).
in which
competition
consists
of
candidates
affecting
turnout
and
the electorate's
percep-
tion of
each
candidate's
positions,
and
in
which the
social
choice
is
a
policy
package
which the
victorious
candidate
advocates.
This approach,
inaugurated
by
Downs's
An
Economic
Theory
of
Democracy,
and falling
under the
general
rubric
"spatial
models
of
party
competition,"
has been
scrutinized,
crit-
icized,
and
reformulated.2
To clarify
the
ac-
complishments
of this formulation
we identify
and
discuss
in section
2
the
general
democratic
problem
of ascertaining
a social
preference.
We
review
critically
in section
3
the definitions
and
assumptions
of our model.
We
consider
in
sections
4
and
5
the
logic
of a
competitive
electoral
equilibrium.
We assume
in section
4
that the electorate's
preferences
can
be
sum-
marized
and
represented
by
a
single
function;
the
analysis
in section
5
pertains
to
competi-
tion between
two
organizational
structures
or
two
opposed
ideologies
(i.e.,
when
two func-
tions
are
required
to summarize
and
represent
the electorate's
preference).
Finally,
we
sug-
gest
in
section
6
a
conceptualization
of elec-
toral processes
which facilitates
extending
and
empirically
testing
our model.
II.
THE DEMOCRATIC
PROBLEM
OF
SOCIAL
CHOICE
The
early
literature
of
spatial
theory ex-
amines
a relatively
simple
problem,
a funda-
mental assumption
of
which
is that
a
single
dimension
describes sufficiently
the
preferences
of
citizens.
Assuming:
(1)
that candidates seek
to
win
elections,
(2)
that
all
participants
in
elections
(i.e.,
candidates
and citizens)
have
perfect
information,
and
(3)
that
the candidate
can
adopt
any position
on this
single
dimension;
this literature
seeks
to ascertain
the
positions
candidates
should
adopt.
Finding
such
a
po-
sition,
however,
requires,
first,
that
it
exists-
i.e.,
that some
position
be
dominant, by
which
2
Anthony
Downs,
An Economic
Theory
of
Democracy
(New
York:
Harper
and
Row,
1957).
For
additional
theoretical
developments
see:
Gerald
Garvey:
"The
Theory
of
Party
Equilib-
rium,"
this
REVIEW,
LX
(1966),
29-38; David
E.
Chapman,
"Models
of
the
Working
of a
Two-
Party
Electoral
System,"
Papers
on
Non-Market
Decision
Making
III
(Fall,
1967),
and
Public
Choice
(Fall,
1968)
426
1970
A
MATHEMATICAL
MODEL
OF
THE
ELECTORAL
PROCESS
427
we
mean
that
if
a
candidate
adopts
that
posi-
tion
then
he
is
guaranteed
at least
a tie
in
the
election
and
a positive
plurality
if
his
opposi-
tion
selects
some
position
other
than
the
dominant
one.
Unfortunately
the existence
of
such
positions
cannot
be
guaranteed
generally
and
additional
assumptions
are
required
for
it
to
exist.
Consider
the following
incomplete
argument
purporting
to
support
the
proposi-
tion
that,
if all
citizens
vote,
the median
prefer-
ence
of the
electorate
is the
dominant
position:
Let
0*
(see
Fig.
1)
represent
the
median
position
for
the
density
of
"preferences"
f(x);
thus
0*
divides
the
density
equally.
If
the
first
candidate
selects
the
position
01
=
0*,
the
second
candidate
selects
the position
02<0*,
and
everyone
votes,
the
first candidate
receives
a
positive
plurality;
he
is
preferred
by
all
citizens
to
the right
of
0*,
which
by
construction
is
one
half
of
f(x),
and
he
is
preferred
by
those
citizens
to
the
left
of
0* who
are
nearer to
0*
than
to
02
(and
who
provide
his margin
of victory).
Since
dominant
positions
exert
a
powerful
attraction
to
candidates,
the
argument
that
candidates
should
converge
to
the
median
might
appear
to
be trivial.
However,
such
an
argument
is
incomplete;
it requires
additional
assumptions,
one
of
the
most
important
being
that
the
form
of each
citizen's
preferences
is
"single
peaked".
Specifically,
the
argument
that
all
citizens
to the
right
of
9*
prefer
01
with 02
<01
=
0*,
implicitly
assumes
the
exis-
tence
of
a
specific
class of
orderings
of
the
alternatives
on
the
horizontal
axis.
If
preference
is
indicated
on
the
vertical
axis,
the
preference
orderings
in
this class
are
represented
by
func-
tions
which
change
direction
at
most
once
from
increasing
to decreasing
(i.e.,
are
single
peak-
ed).
If this
assumption
is not
satisfied
we
may
be
unable
to identify
a dominant
position
so
that
a
paradox
of voting
is said
to
exist.
The illustration
of
this
assertion,
which
is
a
special
case
of
Arrow's
General
Impossibility
Theorem,
is
so
simple
that
it
bears
repeating.3
3
Kenneth
J.
Arrow,
Social
Choice
and
Indi-
vidual
Values
(Now
York:
Cowles
Commission
Monograph
,a12,
Wiley,
1951).
See
also:
Duncan
Black,
The
Theory
of
Committees
and
Elections,
(Cambridge:
Cambridge
University
Press,
1968);
and
with
R.
A.
Newing,
Committee
Decisions
with
Complementary
Valuation
(London:
W.
Hodge,
1951).
A general
exposition
of the paradox
and
its
implications
is given
by William
H. Riker,
"Voting
and
the Summation
of
Preferences:
An
Interpretive
Bibliographical
Review
of
Selected
Developments
During
the
Last Decade,"
this
REvIEw,
LV
(December,
1961),
900-911.
Consider
three
citizens
whose
preferences
do
not
satisfy
the
single
peakedness
assumption:
Citizen's
Preference
Citizen
Ordering
I
A--- )C-)B
2
C-OB-+A
3
BAC
Although
each
citizen
has
no
difficulty
defining
a
preference
ordering
among
the
alternatives-
A,
B,
and
C-no
alternative
is
dominant.
B
defeats
A,
C
defeats
B,
and
A defeats
C, so
that
the
social
preference
ordering-
A->C->B->A
-is intransitive.
Thus,
if A,
B,
and
C
are
the
alternative
positions
for
a
candidate,
he
cannot
find
a
strategy
which
guarantees
him
at least
a
tie.
The
possibility
that
such
a
paradox
exists
poses
a
problem
for
majority
decision-making.
Although
most
standard
procedures
for
ag-
gregating
individual
preferences
(e.g.,
voting)
yield
a
unique
social
choice,
if
preferences
are
not
single
peaked
such
choices
depend,
for
example,
on
the order
in which
the
alternatives
are
presented.4
Thus,
if
we
cannot
guarantee
the existence
of
dominant
positions
in
the
con-
text
of
electoral
campaigns,
the
outcome
of
an
election
may
depend
on
the
temporal
order
in
which
candidates
select their
strategies.
Downs,
who
was
perhaps
the
first
to
intro-
duce
into the
contemporary
political
literature
the
problems
which
the
paradox
poses,
con-
siders only
the
world
of
one dimension.
A
simple
example
demonstrates,
however,
that
the
prob-
lems
which
the
paradox
introduces
are com-
pounded
as
more
dimensions
are considered5.
Consider
Figure
2
in which
the vertical
and
horizontal
axes
index
two
relevant
dimensions.
Assume
that
the
electorate
consists
of
three
f
(x)
a*
1
FIG.
1
4Duncan
Black,
op.
cit., pp.
21-25.
a
Duncan
Black
and
R.
A.
Newing,
loG.
cit.
428
THE
AMERICAN
POLITICAL
SCIENCE
REVIEW
VOL.
64
Xi
VI.
FIG.
2
voters,
with their
preferred
positions
denoted
by
vi,
v2,
and
v3,
and
that there
are
two
candi-
dates (who
do
not vote).
Finally,
assume
that
the concentric
circles drawn
about
vi,
v2,
and v3
represent
the
indifference
contours
of
each
voter's
preference
function.
Thus,
a voter is
in-
different
between two alternatives
if
they
lie
on
the
same
contour,
and
he
prefers
one alterna-
tive
to
another
if it
lies on
a
contour
closer to his
preferred
position-i.e.,
a
contour
of
a
smaller
radius.
Now
let
the
first candidate
adopt
any
strategy,
say
01.
Observe
that the
position
62
de-
feats
01
in a
majority
vote
since
it
lies on indif-
ference contours
of smaller
radius than 06
for
the two voters preferring
vi
and
V2.
But
voters
2
and 3
prefer
6i*
to
02
for
similar
reasons while 06
defeats
1i*,
etc.
Obviously
this cycle
continues
indefinitely.
No
dominant
position
exists,
and
the
position
a
candidate
should
adopt
depends
on
the
position
selected
by
his
opponent.
Although
this
result
may
not be
inordinately
surprising,
it
demonstrates
an
important
dis-
tinction between
the
unidimensional
and multi-
dimensional cases.
Consider Figure
2
again
but
assume that
citizens
cannot
vote
on
x1
(i.e.,
the
value
of
xi
is
fixed).
This is
equivalent
to
assuming
that
only
motions
on a line
parallel
to the
x2
axis
may
be
considered.
Preferences
on
this
line,
by
construction,
are
single peaked
so
that
the value
of
x2 preferred
by
V2
is the
dom-
inant
choice.
Alternatively,
if x2
is
fixed,
the
value
of
x2
preferred
by v3
is
the
social
choice.
Thus,
even
though
a
dominant position
exists
for
each of the
dimensions taken
individually,
the
composite
of these
dimensions
yields
an in-
transitive social
preference.
We
can
easily imagine
such
a
situation
when-
ever citizens
are
permitted
to vote both
on
the
amount
of some
service to
be
provided
pub-
lically
and
on
the
fiscal
institutions
for
funding
such
a service.
If the
electorate
is
provided
with
the
opportunity
to
vote
only
for
the
amount
of
the
service
to
be
provided
(with
a
given
fiscal institution)
or
only
for
the
fiscal
institution
(with
predetermined
level
of
public
activity)
an
unambiguous
social choice
may
be
revealed.6
Such
choices,
as
the
previous
illustra-
tion
demonstrates,
are
not
guaranteed
with
the
composite
of
these
two
issues.
Since
such
a
simple
example
demonstrates
that
dominant
positions,
in
general,
do
not
exist
for
a
multi-dimensional
world,
one
won-
ders
whether
they
might
exist
for
some
reason-
able
set
of
conditions.
Tullock,
for
example,
suggests
that
the paradox
occurs
with
less
fre-
quency
than
we might
otherwise
anticipate
from
Arrow's
analysis.'
Socialization
and
agree-
ment
on
basic
normative
precepts
diminish
the
probablities
of
multi-peaked
preferences,
and
certain
symmetries
of
preference
reduce
the
probability
of
a paradox
occuring
in
a
multi-
dimensional
world.
Similarly,
the
molasses-
like
variability
of
political
parameters,
and the
uncertainty
and
imperfect
measuring
devices
of
both
practitioners
and academics,
bring
into
question
the
relevance
of such
precise
mathematical
analyses
as those
of Arrow
and
Black.
Stated
differently,
we
do not
know
the
frequency
with
which
the
paradox
occurs
in
reality.
That
the
paradox
can
occur,
neverthe-
less,
raises
an
ominous
note
for
democratic
theory.
Specifically,
it
decreases
the
probable
parsimony
of
acceptable
models.
If
social
choices
depend
on
the
order
in which
mo-
tions
are
brought
forward
for
a
vote,
or
on
the
number
of
motions,
or
on
the number
of
citizens
voting,
then
those
ceteris
paribus
con-
ditions
commonly
scattered
through
academic
tracts
(such
as
this
one)
can
be
of
considerable
importance.
We
contend,
therefore,
that
po-
litical
lore,
empirical
generalizations,
or
simple
graphic
arguments
are
not
satisfactory
for
understanding
the
political
process.
The
pri-
mitive
inquires
of Hotelling
and
Smithies,
and
the
verbal
unidimensional
elaborations
of
6
Such
situations
are
examined
closely
by
James
M. Buchanan,
Public
Finance
in
Demo-
cratic
Process
(Chapel
Hill:
University
of
North
Carolina
Press,
1967).
7
Gordon
Tullock,
"The
General
Irrelevance
of
the
General Impossibility
Theorem,"
Quarterly
Journal
of
Economics
(May,
1967).
Richard
G.
Niemi
presents
an
excellent
formal
treatment
and
interpretation
of
the probability
of
a
paradox
occurring
in
"Majority
Decision-Making
with
Partial
Unidimensionality,"
this
REVIEW,
LXIII
(June,
1969),
488-497.
1970
A
MATHEMATICAL
MODEL
OF
THE
ELECTORAL
PROCESS
429
Downs,
are
inadequate.
We
also reject
the
argument
that
no
generalization
is possible,
since
such
an
assertion
precludes
all
scientific
inquiry.
An adequate
comprehension
of political
processes
requires
rigorous
theory
which
spe-
cifies
unambigously
the
relationships
between
relevant
parameters.
We
seek,
therefore,
a
model
which
promises
to
satisfy
eventually
our
notions
of
an adequate
thory
(or
which
at
least
is conformable
to
such
a
theory).
Given
this
objective,
we
now
consider
more
rigorously
the
definitions
and
assumptions
which
consti-
tute
the
foundation
of
our
model.
III.
DEFINITIONS
AND
ASSUMPTIONS
A
theory
which
seeks
to explain
how
parties
and candidates
do act
or
ought
to act is pred-
icated
on
the
citizens'
responses
to
the
candi-
dates'
strategies.
And
our
preoccupation
with
the
paradox
of
voting
in
section
2
suggests
that
such
a
theory
is
central
to
a
spatial
analysis
of
the
electoral
process.
If we assume
that
parties
and candidates
waltz
annually
before
a
blind
audience-that
the
electorate
is
neither atten-
tive
nor
responsive
to the candidates'
maneu-
vers-then
spatial
analysis
is
not
a
requisite
for
understanding
this
waltz.
We
conceptualize
each
citizen's
choices
and
actions
as the
outcome
of
a
two-stage
sequen-
tial decision process.
First,
we assume
that
the
citizen
evaluates
both
candidates'
(or
parties')
positions
in
terms
of his
own
preferences;
sec-
ond,
that
he decides
whether
to
vote or to ab-
stain.
If
he
votes
he
supports
his
preferred
can-
didate.
The
sequential
decision
process
is
ordered
in this
fashion
because the
model
postulates
that the
decision
concerning
whether
to vote
or to abstain depends
upon
the citizen's
comparative
evaluation
of
the candidates.
Every
formalization,
however,
reveals
the
ambiguities
associated
with
one's
initial con-
ceptualization
of
a
problem.
Consider,
first,
the
central
problem
of
ascertaining
the
method cit-
izens use
to
compare
candidates.
Downs,
as we
note earlier,
assumes
that citizens
compare
the
candidates' ideological
closeness
to themselves.
The
inadequacy
of
this
conceptualization
is
that
responses
to
campaign
issues cannot
be
characterized
as
necessarily
ideological.
Al-
8
V.
0.
Key,
Public
Opinion
and
American
Democracy
(New
York:
Knopf,
1963),
Ch.
7;
Phillip
E.
Converse,
"The
Nature
of
Belief
Systems
in
Mass
Publics,"
in
David
E.
Apter
(ed.),
Ideology
and
Discontent
(New
York:
Free
Press,
1964),
pp.
206-261;
"The
Problem
of
Party
Distances
in
Models
of
Voting
Change,"
in
M.
Kent
Jennings,
and
L.
Harmon
Zeigler
though
some
elections
might
involve
a
single
issue,
citizens'
preferences
cannot
be
ordered
unambiguously
on
a
single
continuum.
Opinion
cleavages
demonstrate
that
if
spatial
models
are to
retain
descriptive
and
predictive
value,
they
must
allow for
more than
one dimension
of
conflict
and
taste.
This
requirement
first motivated
our
anal-
ysis.
Instead
of assuming
that
each
citizen
pre-
fers one position
on
a common dimension,
we
assume
that
a citizen prefers
a
position
on
each
of many
dimensions.
We
represent
a
preferred
position
by
a
number,
x,
on the scale identified
with
each
dimension.
Consequently,
for the
ith
citizen
and
the kth
dimension
the
symbol
x?,*
in-
dicates
the
position
that
a
citizen, i,
most
pre-
fers
with
respect
to the
dimension, k.
We
repre-
sent the
ith
citizen's
preferred
positions
for
all
n dimensions
by
the
vector,
Fxil
(1)
L::iFt2
_Xln_
This
approach
facilitates
an analysis
more
nearly
consonant
with
empirical
evidence.
For
example,
the
complexity
of modern society,
the
indeterminate
implications
of many
policies,
and the
vagueness
of
political
utterances
guar-
antee
the
inability
of
even the
most
educated
citizen
to
obtain
a
thorough
knowledge
and
understanding
of
governmental
policy
and of
the candidates' positions
on issues.
Thus,
cit-
izens employ
criteria
other than
issues
for eval-
uating
candidates.
The
established fact that
responses
not related
to issues
(e.g.,
partisan
identification,
and
candidate
image)
play
sig-
nificant,
if not
dominant,
roles
in
determining
electoral
outcomes,
however,
does not vitiate
the
rationalistic
perspective
of
voting
behavior.
Since
our model
is
multi-dimensional,
we can
incorporate
all
criteria
which
we
normally
as-
sociate
with
a
citizen's voting
decision
pro-
cess-issues,
style,
partisan
identification,
and
the like.9
The
assumption
that
candidates,
(eds.),
The Electoral Process (Englewood
Cliffs:
Prentice-Hall,
1966), 175-207;
Donald E.
Stokes,
"Spatial
Models of
Party
Competition,"
this
REVIEW,
LVII (June, 1963),
368-377.
9
The relative
importance
of
issues, compared
to
image
and partisan
bias,
as
causal
determinants
of voting
behavior remains
an open
question.
Aggregate
analyses
of
cross-sectional
survey
data
demonstrate
clearly
the
predictive
dominance
of
partisan
identification.
V. 0.
Key,
however, con-
cludes
in The
Responsible
Electorate
(Cambridge:
The Belknap
Press
of
Harvard
University
Press,