I illustrate by example a way in which equilibria under probabilistic voting are fragile with respect to assumptions about the non-policy components of voter preferences. I also offer intuition for the fragility using the social welfare functions which also describe the equilibria

Dolmas, Jim

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MPRA
Munich Personal RePEc Archive
Almost orthogonal outcomes under
probabilistic voting: A cautionary
example
Jim Dolmas
Federal Reserve Bank of Dallas
11 February 2014
Online at https://mpra.ub.uni-muenchen.de/53628/
MPRA Paper No. 53628, posted 12 February 2014 15:29 UTC
Almost orthogonal outcomes under
probabilistic voting: A cautionary
example
Jim Dolmas∗
February 11, 2014
Abstract
I illustrate by example a way in which equilibria under proba-
bilistic voting are fragile with respect to assumptions about the non-
policy components of voter preferences. I also offer intuition for the
fragility using the social welfare functions which also describe the
equilibria.
JEL CLASSIFICATION CODES: D72; D78
KEYWORDS: Probabilistic voting; political economy
1 Introduction
In probabilistic voting, the non-policy terms in voters’ preferences—the
random disturbances from the viewpoint of the candidates—can be mod-
∗Federal Reserve Bank of Dallas, 2200 North Pearl Street, Dallas, TX 75201. Tel.: 1-214-
922-5161. E-mail: jim@jimdolmas.net. URL: http://www.jimdolmas.net/economics.
This is a substantially revised version of a paper that previously circulated with the title,
“Probabilistic voting: A cautionary note”. I wish to thank an anonymous referee for
comments on an earlier version of this note, and participants at the 2011 Guanajuato
Workshop for Young Economists for comments on a lecture incorporating some of this
material. Disclaimer: The views expressed herein are those of the author and do not
necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve
System.
1
eled as additive or multiplicative.1 At least in ‘macro’ political economy
models, one sees little justification offered for a particular choice between
these alternatives.2 How different can equilibrium voting outcomes be,
depending on this choice? This note illustrates, by way of example, that
the choice of additive or multiplicative disturbances can lead to outcomes
that are not just a little different—we may, in fact, obtain equilibrium pol-
icy vectors that are nearly orthogonal to one another.
The example is a simple calibrated model of redistributive transfers fi-
nanced with taxes on consumption, capital income, and labor income. In
that sense it is a descendant of one the earliest applications of probabilistic
voting, Lindbeck and Weibull (1987). The model also bears some similari-
ties to Profeta (2007), a more recent application of probabilistic voting.3 To
put the difference in outcomes into some empirical context, the resulting
“welfare states” under the alternative assumptions—transfers as a share
of national income—are further apart than those of, for example, the US
and Denmark.
Since certain fragilities of probabilistic voting have been pointed out
before (Ball, 1999; Slutsky, 1986), it is worth emphasizing what this note
is not about. It is not about non-existence of equilibria (equilibria exist),
nor is it about non-uniqueness (equilibria are unique, given the structure
of voter preferences). The caution it suggests is not for the theorist, but
rather for the practitioner who, seeking to model voting outcomes over a
multidimensional issue space, is considering an “off the shelf” version of
probabilistic voting as a solution.4
In the next section, I give a thumbnail sketch of the model and present
1In the language of Banks and Duggan (2005), these are the ‘utility difference’ and
‘utility ratio’ models.
2See, for example, Yang (1995), Hassler et al. (2005), Profeta (2007), or Alesina et al.
(2013).
3In the model here and in Profeta (2007), individuals value consumption and leisure,
and are taxed to finance redistributive transfers. In Profeta, a multi-dimensional issue
space arises because income tax rates are individual-type-specific; here, taxes are paid
on consumption and two types of factor income, and agents differ in their factor endow-
ments.
4Probabilistic voting has long been used as a solution concept in models where the
dimensionality of the issue space prevents application of the median voter theorem. The
conditions for existence of a probabilistic voting equilibrium are typically much less strin-
gent than the conditions sufficient to guarantee a Condorcet winner when the issue space
is multi-dimensional (Enelow and Hinich, 1989; Coughlin, 1992).
2
the almost orthogonal outcomes we obtain under additive and multiplica-
tive disturbances. Section 3 offers some intuition for the disparate equilib-
ria, based on the relationship between probabilistic voting outcomes and
the maximization of social welfare functions. Additive or multiplicative
disturbances correspond to different social welfare functions (Banks and
Duggan, 2005); in models of redistribution, the different objectives tilt op-
tima either toward or away from the economy’s less affluent members. In
section 4, I give more details of the model, the calibration, and the calcula-
tions.
2 Almost orthogonal outcomes
Imagine a one-period economy with taxes on labor income, capital income
and consumption, which generate revenue that is used to finance some
exogenous government consumption and a lump-sum redistribution to
agents in the economy. Agents differ in the productivity of their labor
effort and in their shares in the economy’s capital income.
The number of types is small (four), and no type constitutes a majority.
Though static and simplistic, the model is calibrated in a realistic way.
A more precise description of the economic environment appears later in
Section 4 below.
The policy decision is a vector of taxes and an associated transfer pay-
ment. The issue space is three-dimensional, since a choice of the three
tax rates implies a transfer level, via the government’s budget constraint.
Consider the following two very different outcomes (where the τC, τN and
τK are tax rates on consumption, labor income and capital income, and T
is the associated transfer as a fraction of aggregate output):τCτN
τK
 =
0.000.25
0.00
 (A)
which yields a transfer of
T = 0
versus  τˆCτˆN
τˆK
 =
0.510.03
1.00
 (M)
3
which yields a transfer of
Tˆ = 0.34.
These orthogonal outcomes were obtained as probabilistic voting equi-
libria from a single model with the structure described above, the only
difference being whether we assume the non-policy elements in voters’
preferences enter additively (A) or multiplicatively (M).
Obviously, the outcomes are far apart. Using the Euclidean distance
between tax vectors as a metric, cases (A) and (M) are further apart than
those of the most and least generous welfare states among advanced economies.5
3 Intuition: Additive disturbances, multiplica-
tive disturbances and social welfare functions
The intuition for the disparate outcomes under additive or multiplicative
disturbances can be seen by considering the equivalence between equilib-
rium outcomes under probabilistic voting and the maximization of a social
welfare function. The additive or multiplicative cases correspond to social
welfare functions that differ in the weight they place on the utility of the
less affluent.
The political environment consists of candidates from two parties vy-
ing for election. Candidates espouse policy platforms—tax vectors, in our
case—and are assumed committed to enacting their platforms, if elected.
In the probabilistic voting framework, voters’ preferences over election
outcomes depend on more than just candidates’ policy platforms.6
In the additive case, the utility a voter of type i gets from candidate A
winning the election and enacting policy τA is
VAi
(
τA
)
= ξAi + vi
(
τA
)
,
where vi ≥ 0 is the voter’s indirect utility function over the choice of tax
vector. In a two-party election, a voter of type i votes for candidate A over
5The corresponding tax vectors for the U.S. and Denmark, for example, are roughly
τUS = [0.06, 0.23, 0.27] and τDK = [0.21, 0.40, 0.40] (Carey and Rabesona, 2002), for a Eu-
clidean distance of 0.26, or 26 percentage points. The distance between τ and τˆ above is
1.14, or 114 percentage points.
6For example, voting intentions may depend on voters’ perception of the leadership
qualities of the candidates, as in models of valence (Schofield, 2004).
4
candidate B if
vi
(
τA
)
− vi
(
τB
)
> ξBi − ξAi ≡ ψi.
The ψi’s—which represent aspects of voters’ party preferences apart
from the explicit policies—are taken as random from the two candidates’
standpoints. Given a distribution of the ψi’s in the population, assume
that each candidate chooses his policy platform to maximize his expected
plurality, given the policy choice of the other candidate. Assuming that
F (ψi), the CDF of ψi, has the logistic form (independent of i), candidate
a’s expected plurality is then
2∑
i
fi
(
exp
[
vi
(
τA
)]
exp [vi (τA)] + exp [vi (τB)]
)
− 1, (1)
where fi is the fraction of type i agents in the electorate.
In contrast to the additive case just described, suppose instead that the
utility a voter of type i gets from candidate A winning the election and
enacting policy τA is
VAi
(
τA
)
= exp
(
ξAi
)
vi
(
τA
)
,
with ψi ≡ ξBi − ξAi still distributed logistically. In this case, A’s expected
plurality takes the same form as (1) above, but with exp (vi ( · )) replaced
by vi ( · ) —i.e.,
2∑
i
fi
(
vi
(
τA
)
vi (τA) + vi (τB)
)
− 1. (2)
It’s easy to verify that the first-order conditions for maximizing (1),
evaluated at a symmetric equilibrium (τA = τB) are identical to the first-
order conditions for maximizing a utilitarian social welfare function:
S (τ) =∑
i
fivi (τ) . (3)
Likewise, the first-order conditions for maximizing expected plurality with
multiplicative disturbances (2) are (at τA = τB) identical to the first-order
conditions for maximizing a social welfare function of the form
Sˆ (τ) =∑
i
fi log (vi (τ)) . (4)
5
These results are essentially Corollary 3 and Corollary 3′ from Banks and
Duggan (2005).
Note that in comparison to (3)—in which voter utilities are perfect
substitutes—the curvature present in (4) offers greater gains from the trans-
fer of utils from the relatively well-off to the relatively worse-off. As a re-
sult, the preferences of poorer agents will receive effectively more weight
under (4) than under (3).
To see this, suppose that τ? maximizes the utilitarian social welfare
function (3). Then, for τ near τ?, (4) can, to a first-order approximation, be
written as7
Sˆ (τ) ∼= η +∑
i
(
fi
vi (τ?)
)
vi (τ) ,
which has the same form as the utilitarian social welfare function (3)—up
to the constant η—but gives relatively more weight to types with lower
values of vi (τ?).
If the preferred policy vectors of our model economy’s poor and rich
are far apart, the probabilistic voting outcomes will be far apart. This is
the intuition for the disparate outcomes shown in the introduction.
4 The model behind the example
The model is a simple static model of an aggregate economy, with pref-
erences and technology that are standard in much of macroeconomics.
Agents maximize a utility function
u(c, n) = c (1− n)φ
subject to
(1− τN)wen+ (1− τK) sΠ+ T = (1+ τC) c,
where c is consumption, n is labor effort, w is the wage rate,Π is aggregate
profits, and T is the lump-sum transfer. The agent’s type is a pair (e, s),
where e is the type’s labor productivity, and s is the type’s share of aggre-
gate profits. There is a distribution f (e, s) of types, with ∑(e,s) f (e, s) = 1.
Let n(e, s) and c(e, s) denote the consumption and work effort of agent
type (e, s).
7Using the fact that log (vi (τ)) ∼= log (vi (τ?)) + (1/vi (τ?)) (vi (τ)− vi (τ?)).
6
The technology for producing output is Cobb-Douglas, Y = Nα, where
Y is aggregate output and N ≡ ∑(e,s) f (e, s)n(e, s)e is the aggregate effec-
tive labor input. Aggregate profits are given by Π = (1− α)Y and the
wage w obeys w = αY/N.
Exogenous government consumption is specified as a fraction g of ag-
gregate output, and generates no utility for individuals. The aggregate
resource constraint is thus
C = Y− G = (1− g)Y,
where C = ∑(e,s) f (e, s)c(e, s).
The government runs a balanced budget—
T + gY = τNwN + τKΠ+ τCC
—so the issue space can be taken as the set of three-dimensional tax vectors
τ = (τC, τN, τK). The lump-sum transfer can be used only for redistribu-
tion, not as a lump-sum tax to finance government consumption (T ≥ 0).
While the model is too simple to be taken as a good description of a
real-world economy, I nevertheless try to calibrate it to be roughly consis-
tent with U.S. data. I assume that α, which corresponds to labor’s share of
national income, is 0.6. I choose φ to be consistent with the average agent
devoting 30% of his time to work when the tax vector is τ0 = (0.05, 0.25, 0.25)
(a very rough approximation to the U.S. tax system).
For the distribution of agent types, I use two e values and two s values,
for a total of four agent types. I calibrate the marginal distribution of e to
match the cross-sectional standard deviation of log real wages from Katz
and Autor (1999). I calibrate the marginal distribution of s to match the
U.S. distribution of wealth by quintiles reported in Budrı´a-Rodrı´guez et al.
(2002). I set the correlation between e and s based on the wealth–earnings
correlation reported in Budrı´a-Rodrı´guez et al. (2002). None of the four
types constitutes a majority, of course, though the low-e, low-s type comes
close at 47 percent of the population.
I solve the model numerically on a 101× 101× 101 grid of tax vectors in
[0, 1] × [0, 1] × [0, 1]—so each τ takes values in {0.00, 0.01, 0.02, . . . , 1.00}.
Calculating the competitive equilibrium at each tax vector τ yields indirect
utility functions v(e,s) (τ) for each voter type. I solve for voting equilibria
7
by solving the equivalent social welfare maximizations described in sec-
tion 3.8
Case A in section 2 is the symmetric equilibrium of the two-candidate,
normal form game with payoffs implied by the expected plurality func-
tion (1). Case M is the equivalent object for payoffs given by (2). For the
reasons described in 3, the voting equilibrium in case A (the additive dis-
turbance case) leans toward the favorite outcome of the wealthier agents,
who prefer zero redistributive transfers and a tax on labor income just
sufficient to pay for the exogenous government consumption. The voting
equilibrium in case M (the multiplicative disturbance case) tilts toward the
favorite outcome of the poorest agents, who prefer large transfers financed
by taxes on capital and consumption.
References
Alesina, A., G.-M. Angeletos, and G. Cozzi (2013). Fairness and redistri-
bution: Reply. American Economic Review 103(1), 554–61.
Ball, R. (1999). Discontinuity and non-existence of equilibrium in the prob-
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8The programs for these computations are available at http://www.jimdolmas.net/
economics/current-work.
8
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9


Almost orthogonal outcomes under probabilistic voting: A cautionary example

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