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Phil. Trans. R. Soc. B
doi:10.1098/rstb.2008.0259
Behavioural social choice: a status report
Michel Regenwetter1,*, Bernard Grofman2, Anna Popova1, William Messner1,
Clintin P. Davis-Stober1 and Daniel R. Cavagnaro3
Q2
1
University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA
2
University of California at Irvine, Irvine, CA 92697, USA
3
Ohio State University, Columbus, OH 43210, USA
Behavioural social choice has been proposed as a social choice parallel to seminal developments in
other decision sciences, such as behavioural decision theory, behavioural economics, behavioural finance
and behavioural game theory. Behavioural paradigms compare how rational actors should make certain
types of decisions with how real decision makers behave empirically. We highlight that important
theoretical prediction in social choice theory change dramatically under even minute violations of
standard assumptions. Empirical data violate those critical assumptions. We argue that the nature of
preference distributions in electorates is ultimately an empirical question, which social choice theory
has often neglected. We also emphasize important insights for research on decision making by
individuals. When researchers aggregate individual choice behaviour in laboratory experiments to
report summary statistics, they are implicitly applying social choice rules. Thus, they should be aware
of the potential for aggregation paradoxes. We hypothesize that such problems may substantially mar
the conclusions of a number of (sometimes seminal) papers in behavioural decision research.
Keywords: behavioral social choice; decision making; voting paradoxes
1. RECONCILING THE SEGREGATED DECISION
SCIENCES
The decision sciences are currently segregated into
nearly disparate research areas. On the one hand,
researchers study individual choice; that is, decision
making at the level of the individual decision maker. On
the other hand, another research community studies
social choice; that is, aggregate decision making at the
level of groups or societies, especially in the form of
voting. These two research communities, individual
and social choice researchers, by and large associate
with different scientific societies and publish in
different journals. Another important distinction is
that between normative, i.e. rational theories of choice,
which satisfy certain theoretically motivated optimality
criteria, and behavioural, i.e. descriptive theories that
describe or explain empirically observed choice
behaviour. Figure 1 shows, these conceptual distinctions along two major axes. Different research paradigms fall into different sections of the implied 2!2
table. While many paradigms in the decision sciences
do not fit squarely in a single spot, we have nonetheless
attempted to place several important paradigms in their
most pertinent locations in the table. For instance,
utility theory, such as expected utility theory (von
Neumann & Morgenstern 1947; Savage 1954), is the
normative theory of rational individual decision making
under uncertainty or risk.
There has been limited cross-fertilization between
individual and social choice research areas. The most
important exception is the fact that social choice
theory has systematically incorporated rational utility
theory as a theoretical primitive. Another noteworthy
exception is the literature on justice and fair division
(Balinski & Young 1982; Schokkaert & Lagrou 1983;
Kahneman et al. 1986; Brams & Taylor 1996;
Schokkaert & Devooght 2003; Konow 2008). As
Regenwetter et al. (2007a) have emphasized, major
progress in the decision sciences may hinge on the
ability of the various disparate communities to integrate
their collective wisdoms and develop new synergies.
The need for a unified framework to the decision
sciences is indicated by the large dotted box in figure 1.
Several major movements have arisen, which
respond to the largely normative tone of prior theory
in the decision sciences. Behavioural counterparts of
normative theories (e.g. behavioural decision research,
experimental economics, behavioural finance, behavioural game theory and behavioural social choice, see
also the glossary at the end of the paper) have a decadelong tradition of contrasting normative proposals, such
as expected utility theory, against empirical human
choice data.
We provide a status report on behavioural social
choice research, and discuss the facilitating role this
paradigm can play in establishing a broader and more
unified decision sciences research programme. The
paper is organized as follows. In §2, we review past
work showing that the famous Condorcet paradox of
majority cycles may have limited behavioural support.
In §3, we discuss the fact that behavioural decision
research in the individual choice domain routinely uses
aggregation and therefore must become attuned to
social choice paradoxes in order to avoid artefacts
caused by unsound data aggregation. Section 4
* Author for correspondence ([email protected]).
One contribution of 11 to a Theme Issue ‘Group decision making in
humans and animals’.
1
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This journal is q 2008 The Royal Society
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M. Regenwetter et al.
Behavioural social choice
normative and
prescriptive
theory
individual
choice
utility theory and
decision analysis
descriptive
theory and
empirical data
Heuristics and biases
behavioural decision research
behavioural economics
behavioural social choice
collective
and/or
interactive
choice
finance, game theory,
mechanism design and fads, stock bubbles
social choice theory
experimental economics
behavioural game theory
Figure 1. A unified framework to the decision sciences.
provides a new result on the sampling properties of
social choice rules when dealing with the preference
distributions other than the symmetric distributions we
have coined ‘cultures of indifference’ (see the glossary
at the end of the paper for a definition). Section 5
illustrates a behavioural social choice analysis. In §6, we
propose a research paradigm that expands the notions
of ‘Condorcet efficiency’ and ‘Borda efficiency’ (see
glossary for definitions).
2. THE EMPIRICAL RARITY OF THE CONDORCET
PARADOX
Social choice theory (Arrow 1951; Black 1958; Sen
1970; Gehrlein & Fishburn 1976; Riker 1982;
Tangiane 1991; Saari 1995; Mueller 2003) has had as
its principal concerns the axiomatic structure (i.e. the
abstract mathematical properties) of voting rules and
social welfare functions, including impossibility results
(see glossary for definitions). A major concern in social
choice theory has been the problem of intransitivity, i.e.
a cyclical situation where there exist three choice
alternatives such that the first is socially preferred to
the second, the second is socially preferred to the third,
yet, the third is socially preferred to the first.
Intransitive cycles are often labelled a ‘social choice
paradox’. Social choice theory has generated many
estimates of the degree of intransitivity created by/
inherent in the aggregation of individual preferences
into collective decisions. At the heart of much of this
work is the Condorcet paradox of cyclical majorities
(see glossary for a simple example). In a cyclical
majority, no matter which candidate is elected, a
majority of voters will be disappointed because they
would prefer someone else to be the chosen candidate.
Perhaps even more importantly, cyclical majorities
seem to cast into doubt the very notion of meaningful
majority decision making (Riker 1982).
There are two important literatures dealing directly
with the Condorcet paradox. The first is based on
analytic or simulation results that look at theoretical
distributions. The most common assumption is the
impartial culture (see glossary), a distribution in which
all (weakly or) linearly ordered actor preferences
among some set of objects are taken to be equally
likely. This literature asks how often we must expect to
find intransitive social preferences and concludes that
the paradox should be ubiquitous (DeMeyer & Plott
1970; Gehrlein & Fishburn 1976; Riker 1982; Gehrlein
1983; Lepelley 1993; Jones et al. 1995; Van Deemen
1999; Mueller 2003).
The second major literature on the Condorcet
paradox provides theorems that show sufficient conditions to avoid cycles and reveals that these conditions
appear to be highly restrictive (Sen 1966, 1970). Two
major themes of both these literatures are (i) the
theoretical prediction that majority elections should be
plagued with cycles and (ii) the broadly advertised
policy recommendation (Shepsle & Bonchek 1997) to
beware the use of majority rule in real elections because
a majority winner is unlikely to even exist.
Regenwetter & Grofman (1998a) and Regenwetter
et al. (2002b) found virtually no empirical evidence for
the Condorcet paradox in survey or ballot data.
Therefore, in a series of publications (Regenwetter &
Grofman 1998b; Regenwetter et al. 2002a,c, 2003;
Tsetlin & Regenwetter 2003; Tsetlin et al. 2003) that
culminated in a Cambridge University Press book
‘Behavioral social choice’ (Regenwetter et al. 2006),
members of the present team of authors and others,
re-examined the arguments leading to the belief that
the Condorcet paradox should be an inevitable
concomitant of any majority rule voting process.
They showed the existing results, while mathematically
correct, were nonetheless misleading. For example,
they found that simulation results were based on ‘knifeedge’ theoretical assumptions, where even minuscule
deviations from the theoretical assumptions lead to
dramatic changes in predictions. Similarly, they found
that the theoretical sufficiency conditions for avoiding
the Condorcet paradox primarily tell us what ‘cannot
be ruled out under all possible circumstances’ rather
than providing realistic evaluations of the threat posed
by the Condorcet paradox. Regenwetter et al. (2003)
stated abstract and yet empirically plausible sufficient
conditions to avoid the paradox, and found empirical
evidence in survey and ballot data that the conditions
they had identified as sufficient to avoid the Condorcet
paradox were satisfied (or sufficiently nearly satisfied).
Recent experimental work on deliberative polls
suggests potential explanations of how deliberative
democracy may avoid the Condorcet paradox (List
et al. 2007).
3. THE IMPORTANCE OF BEHAVIOURAL AND
NORMATIVE SOCIAL CHOICE THEORY FOR
BEHAVIOURAL INDIVIDUAL DECISION
RESEARCH
We now turn to the role that social choice theory should
(but currently does not) play in individual behavioural
decision research. Consider the following three binary
non-negative gambles:
(
gamble A :
Phil. Trans. R. Soc. B
RSTB 20080259—4/11/2008—11:41—JMANOHAR—313796—XML RSB – pp. 1–12
90% chance to win $73;
10% chance to win $90:50;
ð3:1Þ
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M. Regenwetter et al.
3
Table 1. Three examples of CPT predictions for gambles A, B, C.
CPT(0.32, 0.98)
CPT(0.56, 0.97)
CPT(0.67, 0.50)
(
gamble B :
V(A)
V(B)
V(C)
preference order
gZ0.32, aZ0.98
gZ0.56, aZ0.97
gZ0.67, aZ0.50
69.37
67.06
8.71
69.00
87.90
7.80
68.83
69.95
8.80
ABC
BCA
CAB
85% chance to win $20;
ð3:2Þ
15% chance to win $385;
(
gamble C :
parameter values
85% chance to win $65;
ð3:3Þ
15% chance to win $130:
Imagine that individual decision makers in a laboratory
experiment make pairwise choices among these
gambles. Suppose that most participants choose A
over B, most choose B over C and yet, most also choose
C over A. This kind of cyclical pattern of choices poses
a challenge to standard decision theories and it has
motivated recent prominent developments of heuristic
decision theories, i.e. theories of decision making by
‘computationally simple rules of thumb’.
For illustrative purposes consider cumulative prospect theory (CPT, see glossary; Tversky & Kahneman
1992; Wakker & Tversky 1993). For non-negative
gambles such as A, B, C in (3.1)–(3.3), CPT transforms each probability p of an outcome via a probability
weighting function w, say,
wðpÞ Z
pg
; for some 0! g% 1;
ðpg C ð1K pÞg Þ1=g
ð3:4Þ
and each gamble outcome x via a utility function v, say,
of the form
vðxÞ Z x a ;
for some 0! a% 1:
ð3:5Þ
The probability weighting function w, depending on
the value of g, inflates low probabilities (i.e. predicts
risk-seeking behaviour when probabilities are low,
such as in a lottery) and deflates high probabilities
(i.e. predicts risk avoidance when probabilities are
high). The utility function v, depending on a, inflates
relatively small gains and deflates relatively large gains.
The biasing of probabilities and utilities in CPT is
based on a large empirical literature that has reported
cognitive biases and limitations in humans when
dealing with choice under conditions of risk
or uncertainty.
For a binary non-negative gamble f, writing p1 for
the probability of winning the smaller amount x 1, and
p2 for the probability of winning the larger amount x 2,
let p2 Z wð p2 Þ and p1 Z 1Kwð p2 Þ. In CPT, for binary
non-negative gambles g, h,
g is preferred to h5 V ð gÞO V ðhÞ;
2
X
pi vðxi Þ; f 2 f g; hg:
where V ð f Þ Z
ð3:6Þ
i Z1
The two functions w and v depend on two parameters,
g and a. Table 1 gives examples of parameter values,
implied values V( f ) for the above three prospects A, B
and C, and the implied preference order among
gambles, from the best to the worst. Regardless of the
choice of parameters g and a, we will not be able to
accommodate the empirical cycle using equation (3.6).
This is because (3.6) implies transitive preferences.
Thus, it seems as if CPT could not explain the
hypothetical cycle in our example.
A prominent recently proposed decision heuristic
(i.e. ‘simple rule of thumb’) tackles this problem. The
priority heuristic (PH, see glossary) of Brandstätter
et al. (2006) theorizes that decision makers compare
gambles via a process that induces a lexicographic
interval order (see glossary for a definition). In a
nutshell, decision makers sequentially (‘lexicographically’) consider three attributes (the so-called
‘reasons’) and an aspiration level for each reason.
They visit the reasons in a specific order and stop their
decision process whenever an aspiration level is met
for the given reason currently under consideration
(Brandstätter et al. 2006). For the three gambles above,
the PH predicts that the decision maker chooses A over
B (by reason 1). The PH also predicts that the decision
maker will choose B over C (by reason 3). However, the
PH predicts that the decision maker chooses C over A
(by reason 3). Clearly, the PH accounts for 100 per
cent (all three) of the pairwise majority choices.
In the terminology of Brandstätter et al. (2006), the
PH is able to capture perfectly ‘the process’ by which
the decision makers arrived at their final choices. Using
a similar approach, Brandstätter et al. (2006) argued
that the PH is superior to several leading decision
theories because it models the cognitive process of
decision making and, compared with these competing
theories, it correctly predicts the largest number of
modal pairwise choices in several datasets from the
literature (Kahneman & Tversky 1979; Tversky &
Kahneman 1992; Lopes & Oden 1999; I. Erev et al.
2002, unpublished manuscript). Because the modal
choice among a pair of gambles is also the majority
choice, these conclusions are, in fact, based on
descriptive analyses within a pairwise majority aggregation approach.
Now, let us return to the imaginary decision
experiment. Before, we reported the data in the usual
aggregated fashion that one often finds in the
behavioural decision literature that studies individual
choice. However, if we consider the data in more detail,
an interesting new picture emerges. Suppose the data
came from three decision makers (DM 1, DM 2 and
DM 3) who made the combinations of choices shown
in table 2. Note that each decision maker acted
in accordance with CPT using (3.6) and using one of
the parameter choices in table 1. Most importantly, not
a single decision maker chose in accordance with the
PH. But majority aggregation, a popular method for
summarizing choice data ( Tversky 1969; Kahneman &
Tversky 1979; Tversky & Kahneman 1981, 1986;
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M. Regenwetter et al.
Behavioural social choice
Table 2. Three decision makers acting by CPT, with a two-third majority supporting PH.
gamble pair
DM 1
DM 2
DM 3
two-third majority
choice
A versus B
B versus C
C versus A
compatible theory
A
B
A
CPT(0.32, 0.98)
B
B
C
CPT(0.56, 0.97)
A
C
C
CPT(0.67, 0.50)
A
B
C
priority heuristic
Birnbaum 2004; Brandstätter et al. 2006), obscures
this fact. By majority, the PH alone is able to
accommodate 100 per cent of the (majority choice)
‘data’, while CPT, the theory according to which we
computed each choice, accommodates at best twothirds of majority choices. We are facing a Condorcet
paradox, where the pattern of majority choices does not
match the choice pattern of even one individual
decision maker.
Any aggregation of choice data could create artefacts
in the analysis of decision-making behaviour. Aggregation, especially by majority, is common in individual
behavioural decision-making research, including in
seminal papers (Tversky 1969; Kahneman & Tversky
1979; Tversky & Kahneman 1981, 1986; Brandstätter
et al. 2006). This means that much past research in
behavioural decision research is susceptible to aggregation paradoxes such as the Condorcet paradox. This is
a reason why behavioural decision researchers should
systematically incorporate social choice theoretical
considerations into their work.
4. SAMPLE SOCIAL CHOICE OUTCOMES AS
ESTIMATORS OF POPULATION SOCIAL
CHOICE OUTCOMES
We now proceed to a new, yet simple, result in social
choice theory. We explain the behaviour of scoring rules
in samples from nearly any conceivable kind of culture
(population distribution) with the only caveat that it
deviates in one crucial way from the cultures that have
dominated the discussion historically.
While statistics has played a major role in social
choice theory in the guise of sampling distributions
derived from various theoretical distributions, such as
the impartial culture (see glossary for the definition),
inferential statistics does not seem to be used systematically. Traditionally, social choice theorists have rarely
considered the need to draw statistical inferences from
empirical data about underlying population properties
of social choice functions in a given electorate. Yet, ever
since the close call in the 2000 US presidential election,
it has become clear that published ballot counts are
not a deterministic function of the distribution of
preferences in a population. Rather there are many
probabilistic components that affect turnout, ballot
casting and ballot counting. Two of the present authors
have studied this notion for more than a decade.
Regenwetter et al. (2006) and its component papers
that were published earlier, as well as Regenwetter &
Tsetlin (2004) and Regenwetter & Rykhlevskaia
(2007) have promoted the need to consider social
choice data from an inferential statistical point of view.
In this section, we show that the sampling
approaches to studying the behaviour of scoring rules
in large electorates are extremely dependent on
underlying theoretical assumptions. This provides
additional motivation as to why behavioural analyses
of social choice procedures are critical for our understanding of their real-world performance. Recall that
the most famous distributional assumption in social
choice theory is the impartial culture assumption,
according to which an electorate can be considered to
be a random sample with replacement from a uniform
distribution over linear (or weak) orders on the set of
candidates. According to this assumption, if one
randomly samples a voter from the population and if
there are n candidates, then all of the n! possible orders
of candidates are equally likely to match the preference
of that voter.
Consider the impartial culture from a statistical view
point. When computing Condorcet’s majority rule, or
any scoring rule (see glossary for definitions), such as
plurality or Borda, at the distribution level of the
impartial culture, we obtain a perfect tie among all
candidates. Yet, random samples of any size, if they
contain an odd number of voters, will reproduce that
majority tie among any two candidates with probability
zero, when voters are assumed to have linear-order
preferences. Condorcet’s majority relationship is not a
consistent estimator of the population majority
relationship when samples may originate from knifeedge distributions (see glossary for a definition) similar
in nature to the impartial culture.
Consider any social choice method, such as
Condorcet’s majority rule, or a scoring rule, aka
positional voting method. A culture of indifference (with
regard to that procedure) is a probability distribution over
preference relationships (linear orders, weak orders
(WOs) and partial orders) with the property that one or
more pairs of candidates are tied at the distribution
level, according to that social choice method. When
considering majority rule, for example, a culture of
indifference is any probability distribution over binary
relationships of any kind, such that for at least some
distinct pair of candidates, A, B, the total probability of
those preference relationships in which A is preferred to
B equals the total probability of those preference
relationships in which B is preferred to A. Regenwetter
et al. (2006) have shown that the majority rule
outcomes of large electorates (drawn from an underlying theoretical culture) will converge (with increasing
size of the electorate) to the majority preferences in the
underlying culture as long as that culture is not a culture
of indifference. The purpose of this section is to show
the analogous result for scoring rules.
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Cultures of indifference are knife-edge assumptions
that lead to chronically paradoxical behaviour (in the
statistical and social choice sense) of social choice rules
in random samples. Unfortunately, much work on the
likelihood of voting paradoxes has hinged on cultures of
indifference. This has created a common perception
that voting paradoxes are extremely likely. We consider
this a profound and far-reaching artefact of unrealistic
theoretical modelling assumptions.
Let S be a finite set of candidates. We define a
probability space (B, F , P), where B is a family of order
relationships on S (e.g. linear orders, WOs or, say,
asymmetric and acyclic binary relationships), F Z 2S
the power set of S and P : F 1 ½0; 1 is a culture,
and, in particular, a population probability distribution
over B. A scoring rule takes any preference relation
from B and gives a numerical score to each candidate in
S. Regenwetter & Rykhlevskaia (2007) have developed
general scoring rules for a general class of binary
relationships, using the notion of a generalized rank of
Regenwetter & Rykhlevskaia (2004).
More formally, a scoring rule is a set of functions
f fA : A 2 Sg, where fA : B/ R and fA(R) is the score
given to candidate A for the preference relationship
R. For a given scoring rule, we define the random
variable X A
i to be the score of candidate A for the ith
draw in a sequence of independent and identically
distributed (i.i.d.) draws from a population with
distribution P. That is, for R2B,
XA
i ðRÞ Z fA ðRÞ:
The sample score for candidate A is the average score of
A in a sample of size n. It is denoted by
n
X
XA
i
:
SA
Z
n
n
i Z1
B
For a sample of size n, whenever S A
n O S n , we say that A
is socially ordered ahead of B in the sample, by the
given scoring rule. The resulting order by any given
scoring rule for a given sample is called the sample social
order according to that scoring rule. The population score
of a candidate A is the expectation of X A
i . We write
jSj
X
SA Z E X A
fA ðRi ÞPðRi Þ:
i Z
i Z1
Whenever SAOSB, we say that A is socially ordered
ahead of B in the population, by the given scoring rule.
The order prescribed by a scoring rule for the entire
population is called the population social order of that
scoring rule.
The following result shows that if the population
social order has no ties, then S A
n converges in
probability to SA as n grows arbitrarily large. Hence,
SA
n is a consistent estimator of SA. We conclude, as
a consequence, that the sample social order converges
to (is a consistent estimator of ) the population
social order whenever the population is not a culture
of indifference.
Theorem 4.1. If the population social order of a given
scoring rule has no ties, then the sample social order of that
scoring rule converges to the population social order as the
sample size increases.
M. Regenwetter et al.
5
Proof. Since the random variables ðXiA ÞnZ1;.;N are
i.i.d. with finite mean and variance, the weak law of
large numbers implies that S A
n converges in probability
to SA. Hence, for any e, dO0 there exists Nd;e 2 N
such that
for nO Nd;e ;
Pr jS A
n K SA jO e ! d;
for any candidate A 2 S:
Pick any pair of candidates A, B2S. Since we assume
that there are no ties in the population social order,
we assume without loss of generality that SAOSB. Pick
dO0 and eO0, such that e! jSA K SB j. Then
B
Pr SnB R SnA ! Pr jS A
n KSA j C jS n KSB jO jSA KSB j ;
! Pr jS A
n KSA jO 3 AND
jS Bn KSB jO ðjSA KSB jK3Þ ;
% Pr jS A
n KSA jO 3 ! d for nO Nd;3 :
Thus, the sample social order converges to the
population social order.
&
In summary, first, we have seen that, in cultures of
indifference, social choice rules display highly irregular
behaviour in the sense that they need not be consistent
estimators of culture social orders. Yet, second, in
cultures that are not cultures of indifference (i.e. nearly
any culture one could think of ), large electorates will
display exactly the same social choice behaviour as has
been theoretically assumed at the level of the culture to
begin with. This holds for majority rule (Regenwetter
et al. 2006) and, as we have proved here, for scoring
rules. Note also that we have allowed individual
preferences to be binary relationships of any kind, not
just weak or linear orders.
5. EIGHT BEHAVIOURAL SOCIAL CHOICE
ANALYSES
We now expand recent developments in behavioural
social choice along the lines of Regenwetter et al.
(2007b). Those authors analysed four large sets of
empirical ballot data from the 1998–2001 annual
presidential election ballots of the American Psychological Association (APA), that were collected under
the Hare system. Each election featured on the order of
20 000 voters and a rather politicized electorate.
Regenwetter et al. (2007b) analysed these ballots
using a series of different models and using bootstrap
methods for statistical inference.
Table 3 illustrates some key points of such a
behavioural social choice analysis, similar to that of
Regenwetter et al. (2007b). First, since social choice
outcomes can depend very heavily on theoretical
assumptions, we carry out empirical analysis using at
least two sets of fundamentally different assumptions
about the nature of preferences and about the vote
casting process. Second, we evaluate the statistical
replicability of our findings. Third, we contrast the
famed theoretical incompatibility of competing social
choice procedures with a high degree of agreement
among methods in empirical data.
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Behavioural social choice
Table 3. Behavioural social choice analysis of 1998–2005 APA presidential election data, each with five candidates and tens of
thousands of voters. (We report results from the WO model (WO) and the Zwicker model (ZW). The WO results for 1998–2001
have appeared in Regenwetter et al. (2007b), all others are new. Social orders with bootstrapped confidence R98% are italics.
The order 31 524 indicates that candidate 3 is the winner, followed by candidates 1, 5 and 2, whereas 4 is the loser.)
1998
Condorcet
WO
ZW
Borda
WO
ZW
plurality
WO
ZW
1999
2000
2001
2002
2003
2004
2005
32 145 (0.9) 43 215 (0.6) 52 134
32 415 (0.8) 43 215 (0.8) 52 134
53 124 (0.7) 54 213 (0.7) 42 531
51 324
54 123 (0.9) 42 531
41 532
41 532
21 453
21 453 (0.7)
32 145
43 125 (0.9) 52 134
32 415 (0.6) 43 215 (0.9) 52 134
53 124
51 324
42 531
42 531
41 532
41 532
21 453 (0.9)
21 453 (0.8)
54 213 (0.8) 42 351
54 213
42 351
41 352
41 352
12 543
12 543 (0.02)
35 124 (0.7) 43 152 (0.8) 53 214 (0.7) 53 124
31 524
43 125 (0.5) 52 314 (0.9) 53 124
Regenwetter et al. (2007b) included the WO analysis
of the 1998–2001 data. The WO model assumes that
all ranked candidates are preferred to all non-ranked
candidates on the ballot, and that the voter is
indifferent among all candidates s/he does not rank.
In the table, we have added four new datasets (2002–
2005) and a new model which we call the Zwicker
model (Dr W. Zwicker, personal communication).
The ballot data are partial ranking counts. Zwicker
suggested interpreting the data as follows: count A as
strictly preferred to B if and only if both options have
been ranked and A has been ranked as preferable to B.
The Zwicker model (ZW) does not assume any
preference among any pair of candidates of which one
or both were left unranked. Thus, ZW translates partial
rankings into strict partial orders. For scoring rules, the
general results of (Regenwetter & Rykhlevskaia 2004,
2007) allow us to assign appropriate scores to all
candidates from every ballot, even to those candidates
that have not been listed in the voter’s partial ranking.
We have drawn 10 000 random samples, with
replacement, of sample size equal to the original ballot
count, from a hypothetical population distribution
(culture) estimated via either the WO model or the
Zwicker model. In each sampled set of ballots, we have
computed the Condorcet, Borda and plurality outcomes. This is a non-parametric bootstrap of the
confidence we can have in the empirical social welfare
outcomes under the three rules. The bootstrap is a way
to simulate possible sources of uncertainty in election
outcomes, such as unreliabilities in turnout, ballot
casting and ballot counting. Intuitively speaking, it
shows how sensitive the final tally is to small
perturbations in the ballot distribution.
For each dataset, and for each model, we report
the modal social welfare order, as well as its approximate bootstrapped confidence. When the confidence
exceeds 98%, then we leave out the value, and simply
display that social order in italics. First, in all eight
elections, and independently of the model, we avoid the
Condorcet paradox with confidence near 100% (the
table omits some details). Second, we find some degree
of model dependence regarding the exact nature of the
social orders. For example, the Condorcet order by
WO and ZW differ in 1998, 2001 and 2002, due to
tight pairwise margins for plurality. Nonetheless, this
model dependence has no bearing on the empirical
54 213
54 123
absence of a cycle. Third, note that plurality, which
uses the least information in the ballots, comes with
sometimes extremely low statistical confidence, i.e.
even small changes in the ballot distribution can affect
the social order. Fourth, table 3 suggests that there is a
fair degree of agreement among the three voting rules.
In particular, in nearly every case where the three social
orders can be estimated with high confidence, they
yield identical winners and identical losers. This stands
in direct contrast with the literature that predicts very
substantial disagreements among the three rules.
Besides the absence of a cycle, this is another important
divergence from common wisdom in social choice
theory. We discuss this more directly next, with a
special focus on the agreement about the winner.
6. GENERALIZATIONS OF CONDORCET
EFFICIENCY AND BORDA EFFICIENCY
Building on research about the Condorcet paradox, a
highly sophisticated, and often quite technical, literature is concerned with the Borda and Condorcet
efficiencies of voting methods. For instance, the
Condorcet efficiency of a voting method is the
conditional probability that the election winner
matches the Condorcet winner in a random sample of
ballots (from some theoretical distribution), provided
that there exists a Condorcet winner. More generally,
this literature studies the interrelationship among social
choice rules (Chamberlin & Cohen 1978; Gehrlein &
Fishburn 1978; Riker 1982; Merrill 1984, 1985;
Bordley 1985; Gehrlein 1985, 1992; Merrill & Nagel
1987; Nurmi 1988, 1992; Adams 1997; Gehrlein &
Lepelley 2000; Merlin et al. 2000; Mueller 2003). A
large part of this literature concentrates on cultures of
indifference. This literature predicts that many standard and competing voting procedures disagree with
one another a substantial part of the time. A related
theoretical and empirical literature studying variants of
the Condorcet jury theorem (Grofman 1981; Miller
1986; List & Goodin 2001), however, avoids cultures
of indifference. The empirical literature that compares
social choice procedures against each other (Yaari &
Bar-Hillel 1984; Felsenthal et al. 1986, 1993; Rapoport
et al. 1988; Leining 1993; Felsenthal & Machover
1995; Hastie & Kameda 2005; Tideman 2006) is small,
it by and large avoids considerations of statistical
inference, and usually considers sparse datasets.
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M. Regenwetter et al.
7
Table 4. Probability of agreement on the winner by Condorcet, Borda and plurality. (As benchmarks, we report simulation based
probabilities for the impartial cultures over WOs on five candidates (WO5) and for a uniform distribution on partial rankings of
five candidates (PR5). The remaining quantities are the bootstrapped (simulated sampling) probabilities using the 1998–2005
APA presidential election data. Confidences above 98% are in italics.)
benchmark
model
agreement
WO
ZW
agreement
WO
ZW
agreement
WO
ZW
agreement
WO
ZW
WO5
empirical data based
PR5
1998
1999
among Condorcet and Borda winners
0.68
0.65
O0.99
0.99
0.54
O0.99
0.89
among Condorcet and plurality winners
0.51
0.43
O0.99
0.79
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O0.99
0.86
among Borda and plurality winners
0.59
0.51
O0.99
0.79
0.56
O0.99
0.93
among Condorcet, Borda and plurality winners
0.46
0.38
O0.99
0.79
0.34
O0.99
0.84
2000
2001
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2004
2005
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
0.96
0.89
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
0.02
0.7
O0.99
O.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
0.05
0.67
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
O0.99
0.02
0.63
Arrow’s (1951) famous impossibility theorem can
be interpreted to mean that any choice of a consensus
method comes at the cost of giving up principles that
underlie other, competing, and mathematically not
universally compatible, voting methods. Saari (1994,
1995, 1999) has shown that one can create distributions that yield virtually any combination of
differences in results across voting methods. He has
developed an algorithm to specify such distributions
precisely. We propose a straightforward extension of the
study of Condorcet efficiency: what is the probability in
random samples from known cultures, and what is the
inferred (e.g. bootstrapped) population confidence
based on empirical data, that any two or more social
choice procedures, e.g. Condorcet and some scoring
rules, agree on (i) the winner, (ii) the loser, (iii) the
entire social order? Which social choice rules appear to
be in heavy empirical disagreement, and which appear
to be highly consistent in most empirical settings? What
characteristics of the empirical distribution appear to
drive the agreement and/or disagreement among
competing social welfare functions?
While much more work is needed to support any
general claims, we have some early indication that this
approach will reveal more puzzles about social choice.
Table 4 shows, as benchmarks, our simulated agreement probabilities for the impartial culture on WOs
over five candidates (WO5) and for the uniform
distribution on partial rankings of five candidates
(PR5). These benchmarks suggest that one should
not have high hopes of two or even all three among
Condorcet, Borda and plurality to yield the same
winner for five candidates. The table compares these
benchmarks to agreement probabilities we derived by
bootstrapping from the empirical ballot data. Our
analysis in table 4 suggests that in six out of the eight
ballots-based distributions, the corresponding probabilities virtually equal 100 per cent.
A large proportion of the theoretical literature is
based on cultures of indifference, where sample social
choice functions are not consistent estimators of the
population social orders. Once we move away from
cultures of indifference, large samples will have high
agreement among social choice rules if and only if the
cultures (population distributions) themselves have
social orders that agree. This is a direct consequence of
the fact that the Condorcet procedure and all scoring
rules are consistent estimators of the corresponding
population social orders, whenever we are not drawing
from a culture of indifference. Clearly, we face another
situation where theoretical predictions are direct
functions of the underlying theoretical assumptions:
in cultures other than cultures of indifference, if the
culture features agreement among Condorcet and/or
scoring rules, then large samples will replicate that
agreement with probability converging to 1. Even
though margins may be small in some empirical
elections that involve heavy campaigning, we do not
consider cultures of indifference to be realistic representations of real-world electorates.
Our findings show how crucial empirical work will
be in untangling the puzzle surrounding the question of
agreement or disagreement among social choice
procedures. Axiomatic theory is, by and large, mute
about population distributions (i.e. about what are or
are not suitable assumptions to make about underlying
cultures that generate ballot frequencies). The nature
of preference distributions in electorates is ultimately
an empirical question. We hope that this paper will
encourage the social choice community to augment
their traditionally normative theoretical work with a
behavioural analysis component. However, we would
like to emphasize that empirical work should be carried
out in a fashion that is statistically sound. Published
ballot counts are not deterministic functions of the
underlying preference distributions and must be
subjected to adequate statistical inferential methods.
7. CONCLUSION AND DISCUSSION
Behavioural social choice research has collected
evidence that Condorcet cycles are surprisingly rare
in empirical survey and ballot data. This work has
leveraged statistical inference as a major methodological tool. Behavioural social choice theory has also
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Behavioural social choice
highlighted the role of model dependence, i.e. the fact
that conclusions about social choice procedures can
hinge on the theoretical assumptions that enter
theorems, simulations or statistical analyses of empirical data. Nonetheless, while inferred distributions of
preferences in electorates often depend on theoretical
assumptions in the analysis, the empirical absence of
majority cycles has been extremely robust across a
range of modelling assumptions. The same holds for
the agreement among competing voting methods.
In this paper, we have reviewed how various
branches of the decision sciences are nearly completely
disparate. Furthermore, we firmly believe that major
advances could be possible if the different ‘constituencies’ of the decision sciences were consolidated. For
example, we have highlighted how the standard
research paradigm of individual behavioural decision
research routinely relies on social choice aggregation
of individual choice data, often without regard to
possible social choice paradoxes. While it is too early to
tell, this practice could permeate the empirical
literature with artefacts.
In the social choice domain, we have discussed a
new but straightforward result about the statistical
nature of scoring rules. When sampling from a culture
that is not a culture of indifference (sufficiently), large
electorates’ social order, by any scoring rule, will
match (with probability arbitrarily close to one) the
social order by the same scoring rule found in the
underlying culture. Our previous work has shown that
the same is true for the Condorcet criterion. This has
major implications for the famed disagreement among
social choice rules. Whether Condorcet and/or various
scoring rules, such as Borda, agree or disagree with
each other in large samples will completely depend on
the assumptions made about the underlying culture.
In cultures of indifference, which have received a
disproportionate amount of attention, all of these
scoring rules display pathological sampling behaviours
because, in this case, the sample social orders are not
consistent estimators of the population social orders.
For example, with just two candidates and an
impartial culture, the social order in the impartial
culture is a two-way tie by Condorcet and by every
scoring rule. Yet, for samples of any size and for
asymmetric and weakly complete pairwise individual
preferences (i.e. when each decision maker strictly
prefers one or the other among the two choice
alternatives), if the sample size is odd, the probability
that the sample social order matches the social order
of the underlying population, is zero, regardless of
sample size.
Behavioural social choice analyses have now
revealed in a number of datasets that competing social
orders appear to be in nearly perfect agreement with
each other, often with high statistical confidence. This
suggests that realistic cultures should not be cultures of
indifference, and that the theoretical literature may
promote overly pessimistic views about the likelihood
of consensus among consensus methods. Axiomatics
highlight that competing methods cannot universally
agree with each other. Simulation results, as we have
shown, will completely hinge on the assumptions made
about the generating distribution that underlies the
ballot counts. Ultimately, it falls upon empirical
researchers to discover the properties of real-world
distributions of preferences in real populations, and to
characterize the conditions under which competing
social choice rules agree or disagree with each other.
Here, we have highlighted a behavioural social
choice approach to understanding the empirical and
statistical properties of preference aggregation and
voting methods. We think that this paradigm can be
usefully extended to other domains. For example, there
is a growing literature on statistical properties of belief
aggregation methods that builds on the Condorcet jury
theorem in much the same way that early social choice
work built on the Condorcet criterion for aggregating
preferences (Black 1958; Grofman et al. 1983). Much
early work in this area uses very strong assumptions
about statistical independence, while some recent work
uses Nash–Bayesian assumptions that seem to us
behaviourally implausible ( List & Goodin 2001;
Dryzek & List 2003).
Science often proceeds in a two-stage process where
empirical work and theoretical work go hand-in-glove,
each inspiring the other in an upwardly spiralling
ladder of knowledge. This is our hope for behavioural
social choice.
This material is based upon work supported by the Air Force
Office of Scientific Research, Cognition and Decision Program,
under Award no. FA9550-05-1-0356 entitled ‘Testing
Transitivity and Related Axioms of Preference for Individuals
and Small Groups’ (to M.R., PI ) and by the National
Institutes of Mental Health under Training Grant Award no.
PHS 2 T32 MH014257 entitled ‘Quantitative methods for
behavioral research’ (to M.R., PI ). Grofman’s contributions
to this research were supported by the Institute for
Mathematical Behavioral Sciences and the Center for the
Study of Democracy at the University of California, Irvine. D.
Cavagnaro carried out this work while a NIH postdoctoral
trainee at the University of Illinois at Urbana-Champaign.
Any opinions, findings and conclusions or recommendations
expressed in this publication are those of the authors and do
not necessarily reflect the views of the Air Force Office of
Scientific Research, the National Institutes of Mental Health,
the Insitute for Mathematical Behavioral Sciences, or the
Center for the Study of Democracy.
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Glossary
Behavioural economics, behavioural finance and behavioural
game theory: these fields differ from their traditional
counterparts by integrating realistic psychological factors,
such as bounded rationality and cognitive biases, into their
models.
Borda efficiency: a voting method’s Borda efficiency is the
likelihood that the election winner matches the Borda
(scoring rule) winner, when one exists, in a random
sample of ballots (typically from an impartial culture).
Borda winner: a candidate who receives the highest score total
in a contest where each voter ballot consists of a complete
ranking of the candidates, and where each candidate
scores as many points from a given ballot as there are
candidates with a worse rank position on that ballot. The
Borda score has also been extended to situations where the
ballots provide binary relationships (e.g. partial orders),
not necessarily full rankings of the candidates (Regenwetter & Rykhlevskaia 2007).
Cognitive bias: a pattern of deviation in judgement that occurs
in certain situations. This deviation is relative to people
outside of the particular situation. For example, a framing
bias occurs when a change in the wording of a survey
question leads to a systematic change in the responses to
that question.
Condorcet efficiency: a voting method’s Condorcet efficiency is
the likelihood that the election winner matches the
Condorcet (majority) winner, when one exists, in a
random sample of ballots (typically from an impartial
culture).
Condorcet paradox: also called the ‘paradox of cyclical
majorities’, the Condorcet paradox occurs when group
preferences are not transitive even though individual
preferences are transitive. The simplest example involves
three voters and three alternatives. Label the alternatives
A, B and C, and assume that voter 1 has preference order
ABC, voter 2 has preference order BCA and voter 3 has
preference order CAB. Then majorities prefer A to B, and
B to C and C to A.
Condorcet winner: a candidate who beat each other candidate
in pairwise contest. A Condorcet winner need not always
exist; see ‘Condorcet paradox.’
Culture of indifference: with regard to a given voting procedure,
a culture of indifference is a probability distribution over
preference relationships (linear orders, WOs, partial
orders or other binary relationships) with the property
that one or more pairs of candidates are tied, at the
distribution level, according to that voting method.
Cumulative prospect theory: prospect theory is a descriptive
model of decision making between risky prospects. It was
developed by Kahneman & Tversky (1979) as a realistic
alternative to expected utility theory. Cumulative prospect
theory (CPT) was put forth by Tversky & Kahneman
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(1992) as a theoretical improvement of the original
prospect theory. The main observations underlying CPT
are that people have different risk attitudes towards gains
and losses, caring more about potential losses than
potential gains, and that people tend to overweight
extreme but unlikely outcomes.
Descriptive theory: a theory that describes how people actually
behave in some situations.
Experimental economics: the use of laboratory experiments to
test the empirical validity of various economic theories.
Game theory: a branch of applied mathematics that aims to
capture decision-making behaviour in situations where an
individual’s success in making a choice depends on the
choices made by others.
Heuristics: simple, efficient rules that have been proposed to
explain how people make decisions, come to judgements,
or solve problems, typically when facing complex problems or incomplete information. Heuristics often involve
educated guesses, rules-of-thumb and common sense.
Impartial culture: a population consisting of equal numbers of
voters holding each possible linear or WO preference over
candidates. An impartial culture implies a complete
majority tie among all candidates at the population level,
i.e. it is an example of a culture of indifference.
Impossibility results: a class of theorems showing that no social
choice rule can simultaneously satisfy a given set of
reasonable criteria. See, Arrow’s (1951) famous ‘impossibility theorem’.
Knife-edge distribution: refers to the situation in which small
disturbances to an assumed probability distribution in a
model can yield dramatically different analytical and
simulation results.
Lexicographic interval order: a lexicographic order with the
additional property that the categories of each attribute is
interval ordered. That is to say, for each attribute, there is a
mapping from the set of categories of that attribute to a set
of closed intervals on the real line with the following
property: category A is preferred to category B whenever
the left-hand endpoint of A’s interval is strictly to the right
of the right-hand endpoint of B’s interval. This type of
order relationship is noteworthy because it can generate an
intransitive preference order from intuitively reasonable
transitive preferences within each separate attribute.
Lexicographic order: a preference ordering applied to a set of
elements differing on two or more ordered attributes. The
M. Regenwetter et al.
11
order of elements is determined by sequentially comparing
each attribute. The first attribute in which two elements
differ determines the order between those two elements.
The name comes from its generalization of the ordering
of words in a dictionary. A word (i.e. a sequence of letters)
a1, a2, ., ak comes before another word b1, b2, ., bk if
and only if ai comes before bi in the alphabet, where i is
the first index for which ai is different from bi.
Mechanism theory: the study of how the rules of a game
influence the behaviour of the game’s participants, under
certain assumptions of rationality. Mechanism design is
the application of mechanism theory for the purpose of
designing a set of rules to encourage a specified pattern of
behaviour, such as truth telling or participation.
Normative theory: a theory that describes how rational agents
should behave in some situations.
Positional voting method (aka, scoring rule): a voting method in
which candidates are awarded points based on their
position in the ranking on each ballot, and the candidate
with the most total points is the winner. The Borda count
is an example of a positional voting method. Positional
voting methods have been extended to situations where
the ballots provide binary relationships (e.g. partial
orders) not necessarily full rankings of the candidates
(Regenwetter & Rykhlevskaia 2007).
Priority heuristic: descriptive model of decision making put
forth by Brandstätter et al. (2006). The model posits a
decision-making heuristic in which decision makers who
face a choice between two gambles will sequentially
consider three attributes (minimum gain, probability of
minimum gain, maximum gain) on which they compare
the gambles. The choice among two choice gambles is
terminated as soon as an attribute yields an unequivocal
preference, beyond some pre-specified aspiration level,
among the two choice alternatives.
Social choice theory: the study of rules for aggregating
individual preferences to form a collective preference.
Transitivity: in mathematics, a binary relationship is said to be
transitive if whenever an element x is related to an element
y, and y is in turn related to an element z, then x is also
related to z. In social science, transitivity is often applied to
preferences. To have transitive preferences, a person or
group who prefers A over B and B over C, must also prefer
A over C.
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