Voting Decision as a Constrained Choice Problem
A generic random utility model with a varying probability of
inclusion in the choice set
In prevalent models of issue voting, each voter
compares cardinal utility that he/she will derive Chooser i’s problem is
1
if specific political parties are elected.
max Vij + ln[Pr (j ∈ Ci )] + ijn,
j∈J
α
I Choice sets are implicitly assumed to be the
where J is the set of all alternatives, Vij is the systematic component of
same for all voters and include all parties.
I We assume that:
chooser i’s utility from choosing j,and Ci is the set of alternatives that
I parties are not equally likely to be included in voters’
i considers in his/her choice.
Mert Moral
Ph.D. Candidate in Political Science
Binghamton University
[email protected]
[email protected]
Estimates: Parliamentary election in Norway, 1989
Proximity
Model 1
-0.152** (0.008)
-0.025** (0.004)
-0.016** (0.005)
-0.097** (0.007)
-0.031** (0.004)
-0.047** (0.004)
0.058** (0.010)
I
j∈Ci
e
P(i chooses j) = P
γ0+zT
T
ij γ )
dij β−ln(1+e
e dik
4.267** (0.239)
-2045.223
4104.447
4155.100
1466
-1395.896
2813.792
2893.390
1466
-2090.958
4195.916
4246.569
1466
-1399.655
2821.310
2900.908
1466
Models 1 and 3 are conditional logit estimates; models 3 and 4 are estimates of the conditional logit with a varying probability of inclusion in the
choice set. † Policy utility is measured as Euclidean distances; N Policy utility is measured as products. Standard errors in parentheses. Two-tailed
tests. * p<0.05, ** p<0.01
Predicted probability of inclusion in the choice set (Model 4)
k∈J
Then, the log-likelihood function is
0
1
2
3
4
5
6
Votes in 1983 / district Hare quota
0
1
2
3
4
5
6
Votes in 1983 / district Hare quota
1
.6
.4
.2
3
4
5
6
7
8
9
Party Policy Extremity (All Issues)
Strong partisan of another party
Conclusions
... following the directional model of voting:
Voter’s problem:
X
max
qjk sik bk ,
3
4
5
6
7
8
9
Party Policy Extremity (All Issues)
No or weak party ID
The predicted effect of changes in policies on changes in parties’ vote shares
Model.1
Model.2
Model.3
32.6
32.6
32.3
33.2
33.2
18.6
18.4
18.8
Labour
4.7
Liberal
6.3
Center
8.9
12.3
12.2
12.5
Christian People’s
6.3
8.8
9.4
9.4
9.1
9.4
8.8
8.9
9.6
10.3
10.2
8.9
12.2
Progress
10
23.4
12.2
11.4
11.5
11.5
12.2
30
Observed
40
0
23.4
25.3
25.5
21.6
11.9
11.9
11.9
12.2
12.4
12.4
12.4
23.9
23.9
23.8
11.9
11.8
12.5
18.4
3.0
4.3
20
8.9
8.6
8.5
8.2
12.9
15.3
10.8
29.2
28.0
9.8
9.8
9.8
6.3
6.6
6.5
6.3
11.8
12.4
11.2
23.4
22.9
11.9
8.9
10.3
6.8
4.7
4.7
4.6
4.6
8.7
8.9
8.6
23.4
Conservative
32.6
30.5
30.1
30.1
4.7
17.5
17.1
18.7
Socialist Left
32.6
6.3
17.9
17.5
18.3
Model.4
15.2
15.3
15.0
4.7
3.2
3.3
3.3
15.1
14.8
15.2
0
We would like to thank Michael D. McDonald, Olga V. Shvetsova,
Robin E. Best, David H. Clark and Ekrem Karakoc for helpful
suggestions. Any remaining errors are our own.
Vote share in 1983
=4.15 Hare quotas (90 p’tile)
.8
1
.8
1
.2
where fij (yi , D, Z; β, γ) =

"
#

P

γ0+zT
T
ik γ )
dT β − ln(1 + e γ0+zij γ ) − ln
dik β−ln(1+e
e
if i chose j,
ij
k∈J


0
otherwise
Vote share in 1983
=.15 of a Hare quota (10 p’tile)
.8
j
Policy Extremity as
that of Socialist Left Party
.6
i
fij (yi , D, Z; β, γ),
Policy Extremity as
that of Conservative Party
.4
lnPr (Y, D, Z|β, γ) + c = c +
XX
k∈K
Directional Theory of Issue Voting.” American Political Science Review
83(1): 93-121.
Merrill, Samuel, and Bernard Grofman. 1999. A Unified Theory of
Voting: Directional and Proximity Spatial Models. Cambridge; New
York, NY: Cambridge University Press.
5.222** (0.346)
γ0+zT γ
ik )
β−ln(1+e
1. Electoral viability, extremity of parties’ policy offerings and strong
partisan attachment affect voters’ choice sets.
2. Conditional logit with a varying probability of inclusion in the choice
j∈Ci
set provides a better fit than conditional logit, when assessing the
k∈K
proximity and directional theories of voting.
where qjk is the position of party j on issue k,
3. In contrast to the conditional logit, conditional logit with a varying
sik is the position of voter i on issue k, Ci is
probability of inclusion in the choice set provides a more realistic
the set of alternatives that voter i considers in
picture of issue voting.
his/her vote choice, and bk ≥ 0 is the weight
of issue k in voting decisions.
Acknowledgements
Rabinowitz, George, and Stuart Elaine Macdonald. 1989. “A
I
0.091* (0.041)
0
... following the proximity model of voting:
Voter’s problem:
X
2
max
(qjk − sik ) (−bk )
Setting β = αδ,
-0.880** (0.080)
0
I
Choice probability and log-likelihood
modelN
Model 4
0.276** (0.015)
0.046** (0.008)
0.063** (0.011)
0.108** (0.013)
0.053** (0.006)
0.077** (0.007)
-0.048** (0.016)
1.744** (0.334)
-0.569** (0.085)
.2
Choice among the alternatives in the
effective choice set
1. Vij = e
γ0+zTij γ −1
2. Pr (j ∈ Ci ) = [1 + e
]
3. ijn ∼ i.i.d .Gumbel (0, α)
Directional
Model 3
0.237** (0.012)
0.040** (0.004)
-0.007 (0.009)
0.096** (0.010)
0.065** (0.006)
0.061** (0.006)
-0.138** (0.012)
5.655** (0.331)
-0.740** (0.081)
0
The set of alternatives that voter i considers in
his/her vote choice is influenced by:
I Party’s viability in voter’s district
The logic of electoral coordination suggests
that voters tend to disregard parties that have
little chance to win seats.
I Extremity of party policies
In the directional theory of voting, voters tend
to disregard parties that are too extreme.
I Strong affinity to a political party
Voters who have a strong attachment to a
specific party tend to disregard other parties.
dTij β
1
Varying probability of inclusion in the
choice set
Conditional logit with a varying probability of inclusion in
the choice set: components
.8
1. the determinants of the composition of a voter’s choice
set, and
2. the effect of party policy positions on voters’ choices
under these assumptions.
.2
We derive and apply a conditional logistic
regression with a varying probability of
inclusion in the choice set to examine
0
I
Used in the form of the multinomial logit with implicit availability/perception of choice alternatives
(Cascetta, Ennio, and Andrea Papola. 2001. “Random utility models with implicit
availability/perception of choice alternatives for the simulation of travel demand.” Transportation
Research Part C 9(4): 249-263.) and the constrained multinomial logit (Martı́nez, Francisco, Felipe
Aguila, and Ricardo Hurtubia. 2009. “The constrained multinomial logit: A semi-compensatory choice
model.” Transportation Research Part B 43: 365-377.)
Pr(Included=1)
.4
.6
choice sets, and
I voters have different choice sets.
Left-right
Agriculture
Environment
Immigration
Health care
Alcohol
Crime
Penalty term
Constant
Party’s viability in
voter’s district
Extremity of
party policies
Strong affinity to
another party
Log lik.
AIC
BIC
N
model†
Model 2
-0.139** (0.008)
-0.021** (0.007)
-0.032** (0.006)
-0.049** (0.006)
-0.025** (0.003)
-0.037** (0.004)
0.006 (0.009)
Pr(Included=1)
.4
.6
Motivation
Andrei Zhirnov
Ph.D. Candidate in Political Science
Binghamton University
14.9
21.8
10
20
Baseline Prediction
30
40
0
10
20
Progress moves left
16.8
30
40
0
10
20
Progress moves right
30
40