Psychological Review
2005, Vol. 112, No. 2, 494 –508

Copyright 2005 by the American Psychological Association
0033-295X/05/$12.00 DOI: 10.1037/0033-295X.112.2.494

The Robust Beauty of Majority Rules in Group Decisions
Reid Hastie

Tatsuya Kameda

University of Chicago

Hokkaido University

How should groups make decisions? The authors provide an original evaluation of 9 group decisionrules based on their adaptive success in a simulated test bed environment. When the adaptive successstandard is applied, the majority and plurality rules fare quite well, performing at levels comparable
to much more resource-demanding rules such as an individual judgment averaging rule. The plurality
rule matches the computationally demanding Condorcet majority winner that is standard in evaluations of preferential choice. The authors also test the results from their theoretical analysis in a
behavioral study of nominal human group decisions, and the essential findings are confirmed
empirically. The conclusions of the present analysis support the popularity of majority and plurality
rules in truth-seeking group decisions.

legislatures, and committees. The second most popular explicit
procedure in committees is an autocratic “leader decides”
rule (e.g., Smith & Bliege-Bird, 2000). However, when no
explicit decision rule is adopted by a group or when a designated leader appears to make the group’s decision, the implicit
decision rule is usually still essentially a majority rule (cf.
Davis, 1973, 1982; Kerr, Stasser, & Davis, 1979; Stasser, Kerr,& Davis, 1980). Even when a small group has an explicit ruleother than a majority rule, the largest initial faction, that is, theplurality, is usually the ultimate winner (cf. Hastie, Penrod, &Pennington, 1983; Kameda, Tindale, & Davis, 2003; Kerr &Tindale, 2004).
There are several versions of majoritarian rules used inhuman societies. The most popular version in America is amajority–plurality rule in which the candidate that receives
the most votes wins (plurality rule hereinafter). In America,
this is the rule that is applied to determine the presidential
candidate favored by the popular vote or, more important,the candidate selected to receive each state’s electoral college votes. The majoritarian rule, and the focus of muchtheoretical analysis, is called a Condorcet majority rule after Marie-Jean-Antoine-Nicholas de Caritat Condorcet, theFrench philosopher who cofounded modern social choice theory (Condorcet majority hereinafter). Under a Condorcet majority rule the candidate who receives more than one half of thevotes cast is selected (nb, if there are abstentions, the majority
winner can receive less than one half of all possible votes).
When there are more than two candidates, all possible twocandidate elections are held, and a Condorcet winner is declared
if any candidate wins all the pairwise elections by a simple
majority.
When there are only two candidates, majority and plurality
yield identical results. However, there are many examples,when there are more than two candidates, in which the majority
and the plurality rule do not select the same winners. For
example, in the 2000 United States presidential election popular
vote, there was no simple majority winner, though Gore was thewinner by a plurality rule (there were three candidates who
received more than 1% of votes cast: Gore, Bush, and Nader).

Human societies rely on groups to make many important
decisions. There is a deep-seated belief that groups are moreaccurate and more just than individuals. This belief is based on
the commonsense notion that a group has more problem-solving
resources than any individual member because “several heads
are better than one.” The belief also depends on the assumptionthat the group process is effective at eliciting and integrating
its members’ beliefs and preferences. The most popular decision rule in groups of all types is the majority rule. The majority
rule has many virtues: It is “transparent” and the easiest of allsocial decision rules to execute; it is based on a simple principleof equal participation and equal power; it encourages the expression of sincere personal beliefs, rather than conformity; andit yields more effective problem solutions than typical (andsometimes even most accurate) members could achieve. The
present article is an exploration of the capacity of the majority
rule and the closely related plurality rule to produce accurate
judgments.
The majority rule is popular across the full spectrum of
human groups from hunter– gatherer tribal societies (Boehm,
1996; Boyd & Richerson, 1985; Wilson, 1994) to modern
industrial democracies (Mueller, 1989). Certainly, in ad hoc
Western groups, it seems to be the decision rule most frequently
adopted to make formal social choices in popular elections,

The research was partly supported by Grant-in-Aid for Scientific Research 14310048 from the Ministry of Education, Culture,
Sports, Science, and Technology of Japan to Tatsuya Kameda. We
are grateful to David Budescu, Barbara Mellers, Michael Regenwetter, Robert Sorkin, Gary Stasser, Ryo Tamura, and Takafumi Tsukasaki for helpful advice on the research reported in this article and to
Joseph Whitmeyer for his theoretical insights into the conditions thatmoderate the relative performance of the majority rule and the best
member rule.
Correspondence concerning this article should be addressed to Reid
Hastie, Center for Decision Research, Graduate School of Business, University of Chicago, Chicago, IL 60637, or to Tatsuya Kameda, Department
of Behavioral Science, Hokkaido University, Sapporo, Japan 060-0810.
E-mail: reid.hastie@gsb.uchicago.edu or tkameda@let.hokudai.ac.jp494

MAJORITY RULES

It is likely that Gore would also have been the Condorcet
winner, but since pairwise elections were not held, there is no
way to know this for certain.1
One goal of the present analysis is to explore how the performance of the plurality rule and the Condorcet majority rule converge and diverge in ecologically plausible, truth-seeking, groupdecision tasks. In contrast to the case of political elections, ourevaluation assumes that there is a general objective utility to thegroup decision and that individuals all have the same ultimate
preferences. However, when there is preferential conflict, differences in individuals’ votes are produced by differences in individual utility functions (e.g., “I would be more satisfied if Gore werepresident, you would be more satisfied if Bush were president”)
and by differences in knowledge (e.g., “I do not realize it, but Bushwould actually provide me with a higher personal utility”). In thepresent analysis, we focus on the case in which there is no conflict
between members with respect to personal preferences, so an
outcome that is better for any one member is better for everyone.
For example, when a committee of investors decides whether topurchase shares of stock, all members can be assumed to seek thedecision that will maximize profits. However, disagreements
among members’ judgments about which investment will yield thehighest profit introduce differences in their choices and votes. Yet,ultimately, in these cases, the best group decision rule is the rulethat is most accurate in achieving generally valued outcomes.
It is likely that majority and plurality rules evolved in humansociety in group foraging situations in which a band of animal or
human foragers needed to choose a common direction to search.
There is evidence that groups of nonhuman animals make collective decisions in accord with a majority rule (Conradt & Roper,
2003; Seeley & Burhman, 1999) and, as we noted above, it is a
popular rule in primordial hunter– gather societies. In the present
research, we measure the success of several social decision ruleson the basis of the correspondence to reality or accuracy of
judgments and decisions rendered by each rule (see Hammond,
1996; Hastie & Rasinski, 1988, for discussions of the correspondence– coherence distinction).

Accuracy as a Criterion for Good Group Decision Rules
There is a general presumption that groups are more accurate
than individual decision makers. This presumption is part of therationale for the use of group decisions in many human enterprises.
One useful characteristic of group judgment and decision strategies is adaptive flexibility in choice of social problem-solving
strategies (Hinsz, Tindale, & Vollrath, 1997). For example, someintellectual task performances are best described by a “truth-wins”
or “truth supported wins” strategy; these are tasks in which themost competent group members can demonstrate the quality of
their superior solutions so that the group tends to choose the bestsolution generated by any of its members (Laughlin, 1980; Laughlin & Ellis, 1986). Another opportunity for superior group performance occurs when the task is one in which no single member has
sufficient information to solve the problem, but different members
have the necessary information. In such cases, if information
pooling is effective and the group follows a majority or plurality
decision rule, the group can solve a problem at which no individual
member could succeed (e.g., Stasser, 2003; Stasser & Titus, 1985).

495

A useful property of the majority rule in truth-seeking tasks isthat it encourages the sincere expression of personal beliefs. Imposition of a majority rule can shatter error-prone informationcascades by encouraging group members to rely on their personal
beliefs rather than conforming to others’ opinions. For example,
Hung and Plott (2001; see also Anderson & Holt, 1997) created
information cascades in a Bayesian judgment task. Each participant in the experiment observed a private information sample fromone of two urns. Their task was to identify the source urn fromwhich the sample had been drawn. When early respondents in asequential judgment task favored one urn, subsequent respondents
were likely to ignore the information in their private samples andconform to the prior choices. Ironically, a Bayesian analysis showsthat this is what people should do, ignore their own private
information and conform to the prior members’ opinions. Thus,
this task has the insidious property of producing rational but
erroneous information cascades when the first members havedrawn misleading samples, a condition that occurs on approximately one quarter of the trials under the conditions in Hung andPlott’s experiments. However, most important, Hung and Plottdemonstrated that when incentives were shifted to reward groupaccuracy and a majority rule was imposed, erroneous cascades
were almost completely eliminated.
However, groups rarely outperform their best members (Gigone& Hastie, 1997; Hastie, 1986; Hill, 1982; cf. Steiner, 1972). In themost common group “judgmental tasks,” involving uncertainty
and inconclusive or delayed feedback about the quality of solutions, groups usually perform at about the level of a typical median
member. The modest performance of groups in judgmental tasks issurprising on both intuitive and theoretical grounds. For example,
Condorcet’s (1785/1994) jury theorem (distinct from his voting
paradox) implies that if members of a group are individually able
to predict an uncertain outcome at a level better than chance, thena group of sincere voters, relying on a majority decision rule, willapproach perfect accuracy, as the number of members increases(cf. Austin-Smith & Banks, 1996; Bottom, Ladha, & Miller, 2002;
Grofman & Feld, 1988; Nitzan & Paroush, 1985; Young, 1988).
Thus, in theory, the majority rule is a powerful device for amplifying the accuracy of modestly accurate individual judgments.
However, decades of research by social psychologists have provided explanations for the less-than-ideal performance of mostsocial groups. “Process loss” saps the resources provided by individual members through coordination failures (Steiner, 1972),
social loafing (Latané, Williams, & Harkins, 1979), groupthink(Janis, 1972; Turner & Pratkanis, 1998), and interpersonal competition (McGrath, 1984).
There have been only a few behavioral tests of the accuracy or
adaptive success of group decision rules. The best example of an
accuracy test is provided by research conducted by Sorkin and his
1
A more dramatic example is provided by the French presidential
election in 2002, with nine major candidates, in which Chirac (who won
approximately 20% of the popular vote), Le Pen (17%), and Jospin (16%)
were the top three candidates. Chirac, the plurality winner, won over Le
Pen in a two-candidate runoff election. Though it is probable that Jospin
would have easily won over Le Pen in a pairwise election and would havebeen a strong contender in a two-candidate runoff against Chirac.

HASTIE AND KAMEDA

496

colleagues, who studied performance in a visual signal detection
task by individuals and small groups with assigned social decisionrules (Sorkin, Hays, & West, 2001; Sorkin, West, & Robinson,
1998). Others have manipulated group decision rules in legal
decision-making tasks, but these tasks do not permit an analysis of
accuracy, as there is no firm criterion for correctness of verdicts(e.g., Davis, Kerr, Atkin, Holt, & Meek, 1975; Hastie et al., 1983;
Saks, 1977). In Sorkin’s experiments, five- and seven-member
groups performed a difficult two-alternative, visual detection task
under a simple majority rule, as well as under supermajority andunanimity rules. Groups reliably outperformed individuals, andsimple majority rule groups were most accurate on a d⬘ metric for
detection sensitivity. However, groups did not perform at the level
predicted by an ideal observer model (based on the individual
members’ levels of accuracy), and Sorkin speculated that some“social loafing” occurred in the groups. In recent theoretical work,
Sorkin has argued that the unanimity rule may have advantages
when extensive information pooling and deliberation are part of
the group decision process (Sorkin, Shenghua, & Itzkowitz, 2004).
Notice that the majority rules introduce an asymmetry into
two-alternative decisions when there is a default option (“select‘no signal,’ unless a majority votes ‘signal’”). Such a default
option effectively moves the group decision criterion to more strictlevels as the majority quorum increases from simple (one-vote)majority toward unanimity. This means that the types of errors
made under the different decision rules are distributed differently
between Type I (signal present/no signal present, false alarms) andType II (no signal/signal present, misses) as the rules change. Thed⬘ measure is a useful global accuracy score under these conditions, but it would be a misstatement to say that the simple

one-vote majority is most accurate for all possible accuracy scores.
Guarnaschelli, McKelvey, and Palfrey (2000) also tested theaccuracy of majority and unanimity decision rules in the context ofa simulated jury decision task. Their Bayesian judgment task wasan abstract version of the acquit/convict decision addressed bycriminal trial juries. Three-person and six-person groups were
instructed to infer which of two urns was the source of a sample ofcolored balls, but the analogy to a jury decision was never madeexplicit to the subjects. As in Sorkin’s signal detection task, thejury decision task creates an asymmetry between the two possible
decision options: Innocent (acquit, “presume innocent”) is thedefault, and a quorum (majority or unanimity) is required to defeat
the default option and convict the defendant. Thus, accuracy must
be assessed conditional on the true state of the world (truly
innocent, truly guilty). As might be expected when the true state
was innocent (the blue urn had been selected), the unanimity rule
had an advantage; when it was guilty, the majority rule had theadvantage. Thus, the empirical result is equivocal on the questionof global accuracy. Perhaps the most important contribution of the
Guarnaschelli et al. study is the result that allowing discussion
consistently increased accuracy (straw polls before a final bindingvote). Furthermore, their empirical study is an antidote to a previous, controversial article that argued, on the basis of a theoreticalmodel (not behavioral data), that the unanimity rule without discussion was universally inferior to the majority rule (Feddersen &Pesendorfer, 1998).
The present research develops a correspondence-based evaluation of performance accuracy under nine group decision rules(summarized in Table 1). We construct a test bed environment inwhich simulated groups attempted to choose the best alternative

Table 1
Nine Group Decision Rules

Group decision rule
Average winnerMedian winner
Davis’s (1996) SJS
weighted average
winnerBorda rank winnerCondorcet majority
rule
Majority/plurality
rule
Best member rule
Random member
ruleGroup satisficing
rule

Algorithm summary
Each member estimates the value of each alternative, and the group computes eachalternative’s mean estimated value and chooses the alternative with the highest mean.
Each member estimates the value of each alternative, and the group computes eachalternative’s median estimated value and chooses the alternative with the highest
median.
Each member estimates the value of each alternative, and the group assigns a weighted
average value to each alternative (exponential weighting function assigns higher
weights to central, shared estimates) and chooses the alternative with the highest
weighted average value.
Each member ranks all alternatives by estimated value, and the group assigns a Borda
rank score to each location (the sum of the individual ranks for each alternative) andchooses the alternative with the lowest (most favorable) score.
All pairwise elections are held (e.g., 45 for 10 candidates), and the alternative that wins
all elections is the Condorcet winner (it is possible for there to be no unique overall
winner).
Each member assigns one vote to the alternative with the highest estimated value, and thealternative receiving the most votes is chosen.
Member who has achieved the highest individual accuracy in estimating alternative values
is selected, and this member’s first choices become the group’s choices.
On each trial, one member is selected at random, and this member’s first choices become
the group’s choices.
On each trial, alternatives are considered one at a time in a random order; the firstalternative for which all members’ value estimates exceed aspiration thresholds ischosen by the group.

Note. SJS ⫽ social judgment scheme.

Individual
cognitive
effortSocialeffort

High

High

High

High

High

High

High

High

Low

High

Low

Low

High

MediumLow

Low

MediumMediumMAJORITY RULES

out of a choice set of 10 alternatives. Members of the simulatedgroups had judgment policies that enabled each member to estimate a value for each of the 10 alternatives. These individual
estimates were then combined with a group decision rule to makea group choice from the set of 10 alternatives. The use of 10
alternatives, rather than a default versus nondefault dichotomouschoice, made all of the options equivalent and changed the task
from the Sorkin and Guarnaschelli et al. studies of majority ruleaccuracy. The larger choice set also allowed us to compare theperformance of choice rules that would be indistinguishable withsmaller (for example, two-candidate) elections.

Social Choice Theory Analyses of the Goodness of Group
Decision Rules
Historically, most theoretical evaluations of group decision rules
have focused on preferential conflict resolution and relied on
analytical methods to evaluate idealized performance of alternative
rules. The focal decision involves differences in individual groupmembers’ personal utility functions that lead to conflicting preferences—political elections are the classic example of this situation. Condorcet, Jean Charles de Borda (the other cofounder ofmodern social choice theory), and other philosophers writing at thetime of the French Revolution were concerned with finding a goodsocial choice rule to use in the new French republic. Condorcet
favored the simple majority rule, but he discovered that undersome conditions, in multicandidate elections, a unique Condorcet
majority winner does not exist. Indeed, he showed the majority
rule could even produce an intransitive ordering of candidates.
Condorcet’s “Voter’s Paradox” is a demonstration that groups withmore than two members who have different but individually rational, stable, transitive preferences could exhibit voting cycles
under a majority decision rule. A voting cycle is an intransitivity
such that, for example, the group prefers Candidate A over Candidate B, B over C, but C over A; even though any individual
member’s preferences are completely transitive. Condorcet wasdisappointed, as he had hoped that a democratic voting system
could be proven superior to a monarchy on logical grounds.
Instead, he discovered a fundamental flaw in the majority rule.
Voting cycles have been identified in empirical settings, including actual elections and legislatures (e.g., Chamberlin, Cohen, &Coombs, 1984). The paradox was later analyzed extensively by
Arrow (1951), who showed that four simple and desirable properties of any democratic system could not be satisfied by any social
choice procedure (MacKay [1980] and Saari [2001] provide accessible expositions of Arrow’s analysis and its implications;
McLean and Hewitt [1994] provide a review of relevant analyticaland empirical results).
Although no known voting rule survives the social choice analysis unscathed, there have been many efforts to show that somerules are better than others. Several proofs exist showing theefficacy of the Condorcet majority rule, if voters’ preferences fit
certain constraints. Black (1958) showed that if candidates can beordered consistently, with different voters’ ideal points located onthat dimension (each voter’s preferences are “single-peaked” on acommon dimension, e.g., conservative—liberal for American political candidates), majority cycles will never occur. Sen (1966)
proved that more general constraint, “value restriction,” on indi-497

vidual preferences would also guarantee no cycles (for every threecandidates there is one candidate that no voter ranks in the middle).
There are also several empirical studies of actual elections thatshow potential cycles are rare, given realistic preference orderingsof candidates by voters (e.g., Feld & Grofman, 1992; Regenwetter,
Adams, & Grofman, 2002; Tsetlin & Regenwetter, 2003).
Another argument in support of the Condorcet majority rulederives conditions under which the rule is likely to produce maxima or equilibriums in a general social welfare function, calculated
from individual voters’ utility functions. For example, Orbell andWilson (1978) showed that under certain conditions (e.g., when theopportunity cost of cooperation is low) the majority rule is effective at finding the social welfare-maximizing equilibrium solution.
However, these conditions appear to be rare in actual social conflict situations.
In the present analysis, we treat the Condorcet majority rule andthe plurality rule as closely related. However, if all possible
candidates are considered simultaneously, the plurality rule is ascoring rule in that every candidate is assigned a fixed score, basedon the number of voters who ranked it first. This means theplurality rule is not subject to intransitive cycles, as is the Condorcet majority rule. However, the plurality rule is sensitive tochanges in the composition of the choice set. For example, adding
a third, low-popularity candidate can have dramatic effects on theoutcome of an election between two much stronger candidates, as
witnessed by the Gore–Bush–Nader result in the 2000 presidential
election.
Our research demonstrates that in truth-seeking decisionmaking groups—in which members have generally adaptive,
but not perfect judgment policies—the plurality rule is very
likely to select the Condorcet winner if one exists; and one
almost always does exist. As we see it, the overriding motivation for voters to select the objectively best outcome acts in asimilar manner to constraints on the individual preference functions (proposed by Black, Sen, and others; see previous discussion). Constraints such as single-peakedness and value restriction reduce the rate of pathological outcomes for a rule like theCondorcet majority. Similarly, constraints on preferences introduced by adaptive judgment “functions” for members of agroup (that is, all the members perceive some glimmer of thetrue state of the world) make certain forms of preference
conflict very unlikely to occur in truth-seeking groups.

Group Decision Rules
Four of the group decision rules under examination involve the
combination of all individual estimates for each alternative into ascore, then the decision rule selects the highest rated alternative.
We call these “choice with group estimation” algorithms: (a)average winner, in which individual estimates are averaged foreach alternative and the alternative with the highest average rating
is chosen; (b) median winner, in which the individual estimates are
combined with a median estimator for each alternative; (c) social
judgment scheme winner, in which a weighted average value iscalculated for each alternative according to an exponential weighting scheme proposed by Davis (Davis, 1996; Davis et al., 1997);
and (d) Borda count winner, in which each member’s rank order ofHASTIE AND KAMEDA

498

alternatives is input into summation calculation and the value
assigned to each alternative is the sum of the individual rankings.2
Five of the group decision rules use individual judgments directly, without first computing a group estimate of the value ofeach alternative. Because it is not necessary for the group to
integrate all individual judgments of every alternative before it
renders a decision, these rules are computationally more efficientthan choice with group estimation rules. We call these “choicewithout group estimation” algorithms: (e) Condorcet majority rule,
all pairwise elections are held, and the alternative that wins allelections is selected, if such a winner exists (otherwise one of thealternatives, that received at least one first-choice vote is selected
at random); (f) plurality rule, each member votes for their firstranked alternative, and the alternative with the most votes ischosen (that is, each member is allotted one equal-weight vote, andthe decision rule is to select the choice option that receives themost votes); (g) best member rule, the member with the highest
individual accuracy over several previous trials is selected as thegroup leader, and that member’s highest rated alternative is selected on each trial; (h) random member rule, a member is chosen
at random from the group, and the group choice is governed solely
by that member’s judgments; and (i) group satisficing rule, a levelof aspiration threshold is set based on prior experience in foraging,then alternatives are considered sequentially, one at a time, and thefirst alternative that all members estimate to be above threshold isselected (several threshold-setting rules were tested, and the mostsuccessful, used here, selected the first alternative that was judged
to be one standard deviation above the mean payoff value, for all
alternatives).
Performance of the nine group decision rules is evaluated byconducting a simulation in which groups make several thousand
choices in the simulated environment, and our evaluation is basedon the overall success of each rule in selecting the most valuable
alternative on each trial. These methods originated in mathematics
and engineering (e.g., Fishman, 1996) and were pioneered in the
behavioral sciences by Axelrod (Axelrod & Hamilton, 1981);
Payne, Bettman, and Johnson (1993); Fiedler (1996); and Gigerenzer (Gigerenzer, Todd, & the ABC Research Group, 1999) inapplications to test the adaptive success of individual decision
strategies.

Monte Carlo Simulation Method
Basic Simulation
First, we describe the basic structure of our Monte Carlo simulation and then introduce systematic variations of its key parameters to provide specific tests of the performance of social decisionrules. The simulation world has two major components, environmental events, namely the amount of reward (gain or loss) available at various locations, and foragers, whose fitness depends on
accurate predictions of the environmental events (Toda [1962] was
the inspiration for the present application). The state of an environmental event (reward available) is known only probabilisticallyto perceivers through proximal, partially valid cues, so each individual faces the adaptive task of aggregating uncertain cue information to infer whether environmental locations are likely to be
rewarding or punishing.
To illustrate the basic features of our simulation, we rely on themetaphor of a primitive forager seeking sustenance at locations in

an uncertain physical environment. This task represents the essential features of the most general decision problem faced by anyorganism: which option among a set of candidates to choose, given
uncertain information about the payoff contingent on choosing
each option. There are many modern analogues of this resource
search problem, such as choosing among uncertain financial investments, searching for information or a product on the Internet,
or seeking a job or a deal in a market (e.g., Adam, 2001; Bateson& Kacelnik, 1998; Giraldeau & Caraco, 2000; Weitzman, 1979).
However, more fundamentally, most everyday decisions under
uncertainty can be mapped into this task framework. The front-end
individual resource judgment part of the model is a direct implementation of Brunswik’s general “Lens Model” framework forperception and judgment (Brunswik, 1955; Cooksey, 1996; Hammond & Stewart, 2001). When the number of options (locations) is
reduced to two, the task is identical to the multicue judgment
problems that have been central in Gigerenzer’s research on adaptive rationality (Gigerenzer & Goldstein, 1996; Gigerenzer et al.,1999).
Structure of the environment. Suppose there are 10 locations(for foraging) that differ in potential value. It is a fundamental
adaptive problem for a band of foragers to choose a profitablelocation in which to forage in the uncertain environment. As
displayed in Figure 1, in the simulation the profitability of alocation can be estimated only imperfectly through three proximal
numerical cues with different levels of predictive validity (e.g., aforager could make judgments under uncertainty of a location’svalue on the basis of information about past success, vegetation,
weather, predators, and so forth; or about an investment on thebasis of information about past prices, price/earnings, company
revenue, and so forth). In the basic simulation, we set up thestochastic features of environmental events as follows. We first
generated a random number from a normal distribution N(0, 30)
and then designated it as location j’s (j ⫽ 1 . . . 10) resource value,
Qj. However, this value cannot be known directly, revealing itself
only imperfectly by three proximal cues. These cues were generated by taking each location’s true resource value (Qj) and adding
normally distributed error to it, creating a cue value composed oftrue value ⫹ error. In the basic simulation, the normally distributed
error terms had standard deviations of 10, 30, and 50, and the cues
(C1, C2, C3) differed in validity as predictors of the true value ofeach patch ranging from .50 to .36. to .26 on a percent-varianceaccounted-for metric. As shown in the left portion of Figure 1, theoptimal linear combination of these cues for estimation (explaining
78% of the variance of Qj) was the following:
Qj ⫽ 0.40C 1 ⫹ 0.25C 2 ⫹ 0.15C 3 .

(1)Thus, Equation 1 is the best linear representation of the uncertain
environment implemented in the basic simulation, providing astatistical ceiling on accuracy in estimates of each location’s value.
2

The highest ranking or lowest sum wins; for example, suppose there
are three members who rank Alternative A, 2nd, 4th, and 5th in a set of 10
alternatives, Alternative A’s Borda count would be 2 ⫹ 4 ⫹ 5 ⫽ 11; (nb,the highest possible ranking count in this example would be 3, which
would occur if all members ranked the same alternative 1st). Saari (2001)
argues that the Borda count has advantages over other social choiceprocedures; Black, 1958, provides an historical introduction to the Borda
count rule.

MAJORITY RULES

Figure 1.

499

Structure of the basic simulation test bed.

Foragers. As evident from the discussion above, the adaptive
goal for each individual “forager” is to combine the cues, in thesame manner as the optimal linear combination rule, to yield an
estimate of each location’s profitability. Such an individual estimation process, called a judgment policy, can be represented by
how the person weights the three proximal cues to form an estimate. Our simulation implemented this feature by assigning judgment policies to foragers at random as follows (see the right
portion of Figure 1). Member i’s estimation of location j’s profitability is expressed (i ⫽ member, j ⫽ location, and k ⫽ cue)estimated Qij ⫽ wi.1 C ij1 ⫹ wi.2 C ij2 ⫹ wi.3 C ij3,(2)

where wi.1 is the weight forager i gives to Cue 1 in estimation; Cij1
is Cue 1’s value revealed to member i about location j. Each
perceived cue value, Cijk, has two components: a true value (Cjk),
which is common to all members, plus an environmentalperceptual error (eijk), associated uniquely with each member i’s
perception of the cue (Cijk ⫽ Cjk ⫹ eijk). The error component, eijk,
was generated randomly from N(0, 20) in the basic simulation.
Thus, the model allows for perceptual errors— cue values are notperceived veridically—and different judges will make different
errors. (One key question in the present research concerns theextent to which different group decision rules damp or correct forunsystematic errors in members’ perceptions and judgments.)
Expressing a forager’s estimation process as ⌺wi.kcijk, the mostimportant element is the weighting scheme, the wi.ks in the individual forager’s judgment policy used to aggregate the three proximal cues. We relied on Dawes’s (1979; see also Brehmer & Joyce,
1988) observation that, in judgment under uncertainty tasks suchas the one in our simulations, people appear to use simplified linear
aggregation rules. Instead of using optimal weights (for example,
Equation 1), people judge as though they rely on approximate
weights and often on equal weights, getting the predictive “direc-tion” right, but only approximating relative cue importance.
(Dawes [1979] also demonstrated that such “improper linear models” achieve levels of accuracy comparable to optimal linear aggregation rules in many situations; see also Gigerenzer et al., 1999,
for analyses of other simplified estimation rules.)Dawes’s conclusion implies that most people would weight thethree cues approximately equally in aggregation. On the basis of
this reasoning, our simulation used the following procedure in theimplementation of wi.ks. For each member of the foraging group,
we generated three random numbers and then standardized them sothat their sum equaled 1. The standardized fractions determinedthat member’s judgment policy. Thus, the modal judgment policy
under this procedure is equal cue weighting, (.33, .33, .33), butthere is considerable variation in individual cue weighting rules.
The important point is that a modal forager’s estimates in the basic
simulation are not statistically optimal but, on average, are basedon equal cue weights.
Evaluating group aggregation algorithms. Given a group of
foragers who are able to judge each location’s profitability, we are
now ready to combine these individual estimates into a social
choice by means of the nine group decision rules described earlier(see Table 1). To evaluate performance of the group decision rules,
we relied on two efficiency indices in the basic simulation. Our
first index is the difference in actual profitability between thechosen and the best location on each trial (best location’s Q –chosen location’s Q). This index represents opportunity loss
(missed profit) accruing from the choice. Our second index is thesquared value of the first index; the squared opportunity loss
penalizes larger losses more severely relative to small losses. The
second index reflects the adaptive principle that a small opportunity loss may be tolerable, whereas a large one is not, which isoften the case in animal and human foraging in which a large lossmeans ruin or even extinction (Bateson & Kacelnik, 1998; Yden-

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500

berg, 1998). Table 2 also reports a third measure based on thepercentage of trials on which each group aggregation algorithm
correctly chose the best location across the 100 trials. Finally,
because of its significance in the traditional social choice analyses,
we report a measure of Condorcet efficiency, the percentage oftrials on which each rule selected the Condorcet winner (calculated
only on trials on which there was a unique winner).
In the basic simulation, each foraging team was given 100
foraging trials. Thus, these performance indices were averaged
over the 100 opportunities, yielding a mean per-trial opportunity
loss index for each team for the various group aggregation algorithms we tested. We repeated this procedure for 1,000 groups(each with 100 foraging trials) in the simulation and calculated
overall average performance for each of the nine group decisionrules.
The best member and satisficing rules required special treatment, because both depended on information from performance on
the trials to instantiate the rules. The 100-trial block was used to
calculate values to select the best member for the best member
rule. We simply identified the member with the overall best
performance on an individual opportunity loss score and let thatmember serve as the best member. We calculated individual opportunity loss scores on the basis of each member’s highest estimated location on each of the 100 trials; that is, each member’s

first choice location’s value (Q) was subtracted from the best
location’s Q to calculate the member’s opportunity loss index. The
satisficing rule-stopping criterion was selected to maximize groupperformance. Several criterion-setting rules were tried, and themost successful was based on the theoretical standard deviation forthe locations’ profit values (used to generate profit values by thesimulation). The first location encountered, in a random sample
sequence, that was estimated to have a value one standard deviation or more about the mean value by all members, was selected oneach trial. If no location was “satisfactory,” a location was chosen
at random on that trial (this occurred on less than 3% of the trials).

Results of the Basic Simulation
Recall that we have two types of group aggregation algorithms,
with and without group estimation, differing in the amount ofcomputation required to calculate the group choice. Some algorithms (e.g., averaging, median rule) require groups to aggregate
members’ individual estimates for each of 10 locations first, before
they can yield a final group choice. Other algorithms (e.g., majority rule, best member rule) do not have an aggregation phase andsimply operate only on members’ first choice preferences (or alimited number of preferences). In the basic simulation, we exam-

Table 2
Performance of Each Group Decision Rule for the Simulation Test Bed (N ⫽ 1,000 100-Hunt Trials)
Opportunity loss
Group decision ruleM

SD

Squared opportunity
loss
M

SD

Best alternative
picked %Condorcet efficiency(% picking theCondorcet winner)

5-member groups
With group estimation (continuous or rank)
Averaging
MedianDavis’s (1996) SJSa
Borda rule
Without group estimation
Condorcet majority (unique winner only)
Condorcet majority (all)Majority/plurality
Best member
Random memberGroup satisficing

⫺4.27
⫺4.73
⫺6.00
⫺4.51

1.03
1.08
1.28
1.03

87.59
102.22146.56
95.07

32.71
35.85
48.34
33.05

65.8
64.8
62.3
65.0

91.3
90.0
82.8
94.4

⫺4.43
⫺4.65
⫺4.80
⫺5.29
⫺7.00
⫺6.45

1.10
1.10
1.06
0.88
1.31
1.35

93.54
99.54
105.88120.87186.19179.52

34.69
35.94
35.95
30.15
52.99
66.19

65.7
64.8
64.2
62.7
57.8
59.5

93.7
75.3
73.9
81.5

12-member groups
With group estimation (continuous or rank)
Averaging
MedianDavis’s (1996) SJSa
Borda rule
Without group estimation
Condorcet majority (unique winner only)
Condorcet majority (all)Majority/plurality
Best member
Random memberGroup satisficing
a

⫺3.92
⫺4.13
⫺4.63
⫺3.99

0.90
0.94
1.01
0.89

75.79
82.23
98.66
77.96

26.19
28.34
33.27
26.46

67.4
67.0
65.9
67.0

92.0
91.4
87.5
92.9

⫺4.02
⫺4.03
⫺4.15
⫺4.76
⫺7.09
⫺4.75

0.90
0.90
0.92
0.76
1.22
1.00

79.33
79.77
83.36
102.71
189.44106.19

26.80
27.13
27.77
24.96
52.31
35.97

67.0
66.9
66.7
64.6
57.6
64.2

91.6
72.6
70.0
85.4

The social judgment scheme (SJS) estimate is an average of all members’ quantitative judgments, with each judgment weighted by its centrality in the
distribution of all judgments (see Davis, 1996, for the specific calculations).

MAJORITY RULES

ined performances of these decision algorithms for two different
group sizes, namely 5- or 12-person groups.
Table 2 displays mean performances of the nine group decisionrules. As can be seen from the table, aggregation algorithms withgroup estimation (top half of the table) tend to perform better thanalgorithms without group estimation (bottom half) on all measuresof performance. It is not surprising that the averaging rule isoverall winner on adaptive success measures, as the rule is essentially an additive linear algorithm operating in an environment inwhich cues to resources are linearly related to true resource values.
We report two Condorcet winner evaluations, one calculated on
only the trials (97%) in which there was a unique winner and theother calculated across all trials by means of picking a winner fromthe tied top candidates when there was no unique winner (thesecond calculation is used in all comparisons with other rules). In
general, the more computationally demanding algorithms—averaging, median, Borda, and Condorcet winner— outperform thesimpler algorithms.
The plurality rule stands out for its high level of achievement,
given its simplicity. This rule is fast and frugal in terms ofnecessary computations (operating only on members’ first preferences), and its performance is comparable to much more computationally demanding algorithms, such as the averaging, median,and Borda rules. It even outperforms the social judgment schemeaggregation rule, which is a continuous scale analog of majority
aggregation (Davis, 1996; Davis et al., 1997). These conclusions
hold for both 5-person and 12-person groups.
Another important result is that the plurality rule also beats thebest member rule, whose choices adhere most closely to those of
the optimal cue utilization equation (Equation 1). Recall that in our
simulation, the modal members’ judgment policy is a nonoptimal,
equal cue-weighting combination rule. The observed superiority ofplurality rule to the best member rule is reminiscent of Dawes’s(1979) conclusion concerning “the robust beauty of improper
linear models,” but at the group rather than the individual,
decision-making level.
The Condorcet winner has been central in analyses of preferential social choice, and it is interesting to observe its performance inan accuracy-based test. It performs quite well, placing third behind
the Borda and the ecologically rational averaging rule. It is also
instructive to see that the Borda and plurality rules achieve comparable and high levels of Condorcet efficiency, picking the uniqueCondorcet winner about 95% of the time when there is one.

Testing the Generality of the Initial Findings
In the following, we vary several key parameters of the basic
simulation to systematically examine the robustness of the comparative success of the plurality rule. In this extended simulation,
we focus on five of the original nine algorithms (average winner,
Condorcet winner, best member rule, random member rule, andplurality rule) and compare their performance under a wide rangeof simulation parameters. As the two group sizes and the two
efficiency measures used in the basic simulation yielded qualitatively parallel results, the extended simulation examines the performance of 12-member groups only on the mean opportunity loss
index.
Cue validity. A key parameter of the stochastic environment isthe degree of predictive validity of three proximal cues for esti-

501

mating the criterion (profitability of a location). In the basic
simulation, we set up the validity of the three cues so that the
proportions of variance in the criterion explained by the best cue,the second best cue, and the worst cue were 50%, 36%, and 26%,
respectively (Equation 1).
What if we vary the relative validity of the three proximal cues
systematically? Are the group decision rules affected differentiallyby changes in relative cue validity? In the extended simulation, we
kept the predictive validity of the best cue constant (explaining
50% of the variance), while changing the validity of the other twocues systematically. Figure 2A displays mean opportunity losses ofthe four decision rules under different levels of cue validity. The
abscissa represents discrepancy in predictive validity between thebest and the worst cue in terms of the ratio of explained variance.
The 1.9 point on the abscissa corresponds to the original cue
structure used in the basic simulation (1.9 ⫽ 50%/26%).
Figure 2A shows that, compared with the other two algorithms,
the average winner and the plurality rule perform in a similarmanner. In a sense, the plurality rule may serve as a computationally economical substitute for the averaging rule in group decision
making.
Second, the superiority of the plurality rule over the best member rule, observed in the basic simulation, diminishes and is
eventually reversed with increasing differences in cue validity. For
example, when the best–worst cue ratio is 3.4 (that is, the worstcue explains only 15% of the variance in the criterion, whereas thebest cue explains 50%), the best member rule outperforms theplurality rule. Recall that the best member’s judgment policy will
be closer to the optimal equation than a modal member’s judgment
policy. The plurality rule performs at a level closer to the modal
member than the best member rule, and the difference between
best member and modal member increases as the differences
between optimal best cue and worst cue weights increase. Thus,
with the increase in discrepancy of predictive validity between thebest and worst cues, the best member rule has an increasing
advantage over the plurality rule (and average winner rule).3
Performance with incomplete information. The simulation so
far has assumed complete information for each member; each
forager had access to information about all three cues for all 10
locations. What if each forager has access to cue information about
only a subset of the 10 locations? These incomplete information
cases represent the classic rationale for group decisions: If eachmember has only part of the information necessary to solve a
problem and different members have nonshared information, thenseveral heads are almost certainly better than one. To examine thisissue, we introduced a new situation in which information on eachlocation was available to each forager with a probability less thanunity. Each forager produces estimates for a subset of 10 locations,and the group as a whole must operate on incomplete individual
inputs. Figure 2, panels B and C, shows results from two suchincomplete information situations, in which the probability of
individual accessibility to each location is set at 0.8 or 0.6, (in3
We experimented with other manipulations of the statistical structureof the simulated environment, specifically, manipulating average cue intercorrelations. However, these variations had no effects on the relative
performance of the group decision rules, so we do not report them in detail
here.

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HASTIE AND KAMEDA

other words, on average, each member has information only about
8 or 6 of the full choice set of 10 locations).
First, compared with the other two decision rules, the average
winner, plurality, and Condorcet winner rules behave very similarly under incomplete information conditions. Second, missing
information has a larger detrimental impact on the best member
rule than on the plurality rule. Compared with the complete information case (see Figure 2A), the best member rule degrades
rapidly with increases in missing information, whereas the plurality, Condorcet winner, and the average winner rules are affected
only slightly. This implies that the plurality rule performs theinformation-pooling function effectively at a collective level by
relying on foragers’ first preferences. Finally, however, when
information becomes very incomplete (0.6 levels), the voting rules(plurality, Condorcet winner) fall behind the ecologically rational
averaging rule. It is obvious how the averaging rule serves such an
information-pooling function statistically, but the fact that thesame result can be achieved by the far simpler majority algorithm
is important and nonobvious. Again, this illustrates that the plurality rule can serve as a computationally efficient substitute forthe averaging rule in group decision making under uncertainty.
Distributions of resources in the environment. In the original
simulation, we generated locations’ profitability values randomly
from a normal distribution. Normality is a reasonable first hypothesis about actual uncertain environments, but what if the profitability distribution is skewed? The most extreme (and adaptively
most challenging) case of a skewed environment would be a“barren” environment in which only one location is profitable,
whereas the other nine locations are not. We examined performance of the five aggregation algorithms under such a barren
environment, while manipulating the other parameters in the samemanner as described above.
The results from previous tests were replicated under the barren
environment. Briefly, the averaging rule and the plurality rulefunction in a similar manner. The plurality rule (and the averaging
rule as well) beats the best member rule, as long as discrepancy incue validity is not extreme. Furthermore, incompleteness of information enhances the superiority of the plurality rule over the best
member rule even more dramatically in the barren environment
than it does in the richer environment. In a barren environment,
incomplete information is often fatal to the functioning of the best
member rule, whereas it is less detrimental to the plurality rule.

A Behavioral Study of Nominal Groups of Human
Decision Makers

Figure 2. Performance of group decision rules as a function of
judgment cue validity (cue discrepancy; best/worst cue ratio interms of explained variance). A: Complete information, p ⫽ 1.0.
B: Incomplete information, p ⫽ .08. C: Incomplete information,p ⫽ .6.

The computer simulation studies supported our hypothesis thatthe plurality rule is both economical in computational terms andachieves high levels of accuracy in adaptive group decision making under uncertainty. This result is in line with the empiricalfinding that majority aggregation can often summarize actualgroup decision outcomes well, especially when some uncertainty is
associated with the decision and/or task environment (see Kameda,
Tindale, & Davis [2003] for a review of group decision-making
studies conducted in industrialized societies such as the UnitedStates and Japan; see Boehm [1996] for a review of ethnographic
data on group decision making in primordial societies).
Although these two lines of findings converge nicely under theadaptation theme, there still remain some gaps between our com-

MAJORITY RULES

puter simulation results and the empirical group decision-making
literature. One important limit may derive from the artificiality of
individual judgments used in our computer simulation. In thesimulation, we generated individual judgments using a Brunswikian framework to evaluate various group aggregation algorithms. Our method of generating individual judgment policies(wi.k’s in Equation 2) was inspired by an empirical finding inbehavioral decision making, yet artificiality in simulating the individual judgment process may have biased our conclusions. It
would be desirable if we could use actual individual judgments
under uncertainty as inputs to group decision making algorithms.
For this reason, we collected data from human research participants who actually made a series of judgments and choices under
uncertainty. We then formed nominal groups by resampling individuals from this data set and compared the performance of fourgroup decision rules in the same manner as in the Monte Carlosimulations. It is important to note that we did not study actualinteracting groups; rather we used individual human judgments inplace of our individual judgment simulation and relied oncomputer-created nominal groups to study the effects of alternate
group decision rules.

Collection of Behavioral Data
Participants. Participants were 129 (82 male and 47 female)undergraduate students enrolled in introductory psychology
classes at Hokkaido University, Hokkaido, Japan. Participants
were paid contingent on their performances in the experiment(M ⫽ 382 yen, SD ⫽ 200 yen; approximately 120 yen ⫽ $1 US).
Experimental task. The experimental task was based on theBrunswikian framework used in the computer simulations. Participants made a series of judgments and choices under uncertainty inwhich the state of environmental event (profitability of a company)
was known only stochastically by means of three statistical cues.
Two kinds of environmental structure were created: one with asmall cue-validity discrepancy and the other with a larger discrepancy. The former is the original environment implemented in thebasic Monte Carlo simulation in which the ratio of the best andworst cues in terms of explained variances was 1.9. The latter is
one of the environments examined in the extended simulation inwhich the best and worst cue ratio is 3.4. (The best linear representation of this second environment is given by Q ⫽ 0.50C1 ⫹0.20C2 ⫹ 0.10C3.) The two environmental structure conditions
were introduced to verify that cue discrepancy had the same
impact on performance of group aggregation algorithms based onhuman judgments as had been observed in the Monte Carlo simulations (cf. Figure 2A).
Procedure. The experiment was run individually on laboratory computers. On arrival, participants were randomly assigned to
one of the two cue conditions mentioned above (low cue discrepancy vs. high cue discrepancy). The experiment had three phases.
In the first phase, participants were asked to choose the “mostprofitable company” out of 10 on the basis of three statistical cues
provided for each company (analogous to forage “locations” in theoriginal simulation). As in the simulation studies, perceived cue
values for each company could be different for each participantbecause of perceptual errors, although true cue values were held
identical across participants. Participants were told that their experimental payment would be contingent on choice accuracy.

503

There were 50 such choice trials in total. After each choice,participants were informed whether the choice was correct, alongwith their cumulative reward up to that trial.
In the second phase, we asked participants to report on their
judgment strategies used to integrate cue information during thefirst phase. Five strategy alternatives were provided: equal cue
weighting (Dawes, 1979); take-the-best strategy (Gigerenzer et al.,1999); weighted additive (regression) strategy; configural weighting (e.g., Mellers, 1980); or any other strategy described by theparticipant. In all experimental conditions, the most common strategy was described as equal cue weighting, followed by unequalweight regression-like additive rule, and then configural-weighting
rules. Among the explicit responses offered to participants, thetake-the-best strategy was selected least frequently.
In the third phase of the experiment, participants were provided
only one company’s cue information on each trial and asked toestimate its profitability numerically. There were 30 such estimation trials. Participants were again instructed that their experimental payment would be contingent on the accuracy of their estimates(no performance feedback, however, was provided during theestimation phase).

Nominal Group Analysis
With the individual human judgment data, it is now possible toevaluate performance of the group decision rules by forming
nominal groups. Here, we focus on individual choice data collected in the first phase of the behavioral experiment. For each of
the two cue conditions, we randomly composed 1,000 nominalgroups by a resampling procedure and evaluated performances ofthree aggregation algorithms: the plurality rule, the best member
rule, and the random member rule. To reduce participants’ cognitive load, we did not ask them to provide profitability estimates for
10 companies in the initial choice phase. Instead, we asked only
for their top preferences on each of 50 trials. (As individual,
per-company judgments are not available, the average winnercannot be implemented in this analysis.)
Figure 3, panels A and B, displays results of the nominal group
analysis (5-person groups and 12-person groups, respectively) inthe small cue-discrepancy environment. Recall that participants
made 50 choices over time. In Figure 3, we summarized these 50
choice opportunities into 10 blocks and plotted mean performanceof the group decision rules for each block.
It is obvious that participants improved in accuracy over time.
For example, in 5-person groups (see Figure 3A), mean opportunity loss by the random member rule (a benchmark strategy ofrandomly picking one member’s individual preference as thegroup’s choice) was reduced from ⫺10.6 in the first block to ⫺3.8
in the last block.
How did such individual learning in the uncertain environment
affect performances of the plurality rule versus the best member
rule? As can be seen from Figure 3A, the superiority of theplurality rule to the best member rule was firmly established in theearly trial blocks. After the second block, the plurality rule yieldeda smaller mean opportunity loss than the best member rule, except
for turbulence in Block 8. This tendency is accentuated in 12person groups, as shown in Figure 3B. These patterns replicate theresults of the Monte Carlo simulation (see Table 2).

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most participants reported relying on equal cue-weighting policies. Besides calculating subjective self-reports, we also calculated a quantitative judgment policy for each participant byregression analysis of the estimation data collected in the lastphase of the experiment. Average beta weights, summarizing
impact of the three cues, were .31, .25, and .26 in the small
cue-discrepancy environment and .22, .30, and .26 in the large
cue-discrepancy environment. These patterns were statisticallydifferent from the optimal linear regression weights but indistinguishable from an equal cue-weighting rule. Together withthe parallel performance patterns of group aggregation algorithms described earlier, these results provide more empiricalevidence for the plausibility of the assumptions underlying ourMonte Carlo simulation.

General Discussion
The Marquis de Condorcet initiated a long-running evaluationof the quality of majority rule decisions with two monumentally
important conclusions. The bad news was stated in his voting
paradox: The majority rule cannot satisfy absolutely elementary
criteria for coherent, decisive social choice procedures. The good

Figure 3. Performance of nominal decision making groups as a functionof experience making judgments in a small cue-discrepancy environment.
A: 5 members. B: 12 members.

We next turn to the large cue-discrepancy environment. Figure
4, panels A and B, displays results of a nominal group analysis. As
can be seen from the graphs, performance of the plurality rule andthe best member rule was much closer here, in contrast to the small
cue-discrepancy environment examined above. The plurality rule
outperformed the best member rule in only 6 of 10 blocks. Thispattern is again qualitatively consistent with the original simulation results in that the superiority of the plurality rule to the best
member rule was gradually lost and could be even reversed inlarge cue-discrepancy environments (cf. Figure 2A).

Individual Judgment Policies
The nominal group analysis in which we used actual human judgments yielded results comparable to those we obtained in the computer simulation studies. This suggests thatthe assumptions underlying our computer simulations approximate the patterns of actual human judgments under
uncertainty.
One of the central assumptions in the Monte Carlo simulation
was that the modal member judgment policy could be capturedby an equal cue-weighting integration process. As noted above,

Figure 4. Performance of nominal decision making groups as a functionof experience making judgments in a large cue-discrepancy environment.
A: 5 members. B: 12 members.

MAJORITY RULES

news was stated in his jury theorem: The majority rule can enhance
the accuracy of “above chance accurate” group members, andunder some conditions, it could approach perfect levels of accuracy. Condorcet and later analysts relied on mathematical derivations of the implications of the majority rule to evaluate its usefulness under relatively limited conditions. What is new in thepresent analysis is our use of a stochastic simulation method thatyields new results concerning the comparative accuracy of the
majority rule and its robustness across variations in environmental
conditions.
Both computer simulations and a nominal group analysis thatused actual human judgments revealed that, among frugal algorithms, the plurality rule is adaptively the most efficient across awide range of environmental variations. Two comparisons, inparticular, provide strong evidence for the “robust beauty of majority rule.”

Plurality Versus Averaging
First, the plurality rule and the average winner strategy consistently performed in a similar manner, even though the underlying
computations are very different for the two algorithms. Variationsof key parameters in the simulation, such as discrepancy in cue
validity, degree of information completeness, skewness in environmental events, and different group sizes had essentially thesame effects on the performance of both aggregation rules. Although at a computational level averaging and plurality voting are
obviously different, at an abstract level they share some performance characteristics. Both have an additive character in thatevery member’s contribution is summed up in the central aggregation process. The plurality rule precedes the summation with anindividual “winner take all” computation that puts all the weighton the favored alternative. This could be viewed as a weightedaverage-like computation, but with extreme weights (0s and 1s).
While the first-step computation in the average winner strategy isan equal-weight calculation, the similar performance of the tworules leads us to emphasize the analogical correspondence between
them.
One essential functional similarity that leads the two rules tomimic one another is that both effectively “cancel out” random
errors in the environment. To illustrate, suppose for the momentthat individual judgment policy is optimal as expressed in Equation 1. Then, as long as environmental and perceptual errors areunsystematic, averaging of individual estimates provides an unbiased estimate of the true target value (profitability of a location in
the foraging domain). The same error cancellation function is also
served by the plurality aggregation of individual first preferences,
as unsystematic errors that happen to bias one member’s preferences are unlikely to be shared with others members. Thus, unsystematic perceptual errors can be reduced by both rules, though
the error-reduction process occurs on different levels of measurement for the two procedures (at an interval scale level for the
average winner vs. at an ordinal scale level for the plurality rule).
Another, less obvious functional similarity between the tworules involves the information-pooling function. Both the average
winner and the plurality rule were only slightly affected by missing
information. Even if a substantial amount of information was
missing, both rules did not degrade much, compared with otherrules (see Figure 2, panels B and C). Under the average winner

505

rule, such robustness against missing information is readily understandable. Even when information about a candidate is missing forsome members, the mean of individual estimates still provides areliable, unbiased estimate of the target value (location profitability). The average winner rule effectively pools whatever information is available to members. Our finding that a similar
information-pooling function is achieved by the plurality rule,
which operates only on members’ first preferences, is not intuitively obvious and is theoretically and practically important.
Given these essential functional similarities, plurality aggregation serves as a computationally more frugal substitute for averaging in group decision making under uncertainty. Whenever
averaging yields an adaptive group choice, the plurality rule serves
as its fast-and-frugal counterpart.

Plurality Versus Best Member
In most actual decision-making situations, usually the groupaggregation algorithms with group estimation are impractical interms of procedural costs; choice strategies that are cognitively
less demanding are the only viable options in many situations. The
plurality rule and the best member rule are the most representativesocial decision rules in the world of practical decision making.
Indeed, in the social choice tradition (Arrow, 1951; Mueller,1989), the two rules are pitted against each other (as democratic vs.
autocratic regimes) in terms of logical coherence. However, how
do these two rules compare from an adaptive perspective? This
question could be answered analytically, if no environmental uncertainty were involved in the decision setting (some example
derivations are available from the authors). However, in an uncertain environment, as represented by the Brunswikian framework,
analytical solutions would be formidable.
Thus, we conducted Monte Carlo computer simulations to systematically vary key parameters. The results showed that, even if
modal members’ judgment policies were nonoptimal in the statistical sense, the plurality rule outperformed the best member rule
under a wide range of parameter values. The only exception to thisobservation is in a complete-information case in which the cue
discrepancy is quite high. Under these conditions, the best member
rule yielded substantially better performances than the plurality
rule (see the 3.4 point in Figure 2A). However, we believe that thisreversal may be of little significance in real decisions. First, as we
discussed earlier, the assumption of complete information is usually violated in the everyday world. As is evident in Figure 2,
panels B and C, the plurality rule performs well under incomplete
information conditions, consistently outperforming the best member rule.
Our simulation assumed that individual judgment policies were
unresponsive to the environmental cue structure, with the modal
policy being equal cue weighting. However, this assumption may
be unrealistic if the discrepancy in cue validities is extreme.
Individuals do recognize dramatic differences in cue validities andshould adjust their judgment policies to unequal weight algorithms. For example, what if modal individuals shift their policy
from equal cue weighting to take-the-best strategy? Under thelarge cue discrepancy conditions, the take-the-best strategy is close
to the statistically optimal linear combination (cf. Gigerenzer et al.,1999). Then, the plurality rule, which is good at handling unsystematic errors, should perform close to the best member rule.

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Indeed, nominal group analysis using actual human judgments
provided some support for this speculation. Even in the large-cue
discrepancy condition (corresponding to the 3.4 case in Figure2A), the plurality rule performed as well as the best member rule.
This suggests that the adaptive efficiency of the plurality rule may
be even more robust in actual decision making than in our
simulations.

The Condorcet Majority Winner
The Condorcet winner has been central in social choice analysesof the performance of group decision rules. It is not a practical
rule; we know of no actual decision-making groups that attempt toapply it, but it has served as the standard against which othervoting and scoring rules have been measured, in theory and inpractice of political elections (cf. Regenwetter, Adams, & Grofman, 2002; Regenwetter, Marley, & Grofman, 2002). Our simulation provides a first test of its performance against an accuracy
criterion, and it fares well, being beaten only slightly by the Borda
and averaging rules. It is also instructive that the Borda andplurality rules track the behavior of the much more computationally demanding Condorcet winner calculations, matching it on
approximately 95% of the trials in which there was a uniquewinner.

Conclusions
In this article, we have explored the performance of fast-andfrugal group decision rules with stochastic simulations and anominal group analysis under a Brunswikian model of theenvironment-judgment system. Although we believe that the Brunswikian framework is a useful research paradigm to represent key
features of decision making under uncertainty, the paradigm can beenriched further by incorporating other adaptively critical environmental elements. One such element may be environmental instability. For example, the archeologist Potts (1996) argued that
ecological instability, including climate changes, fluctuations infood sources, and other survival-relevant conditions, was a fundamental condition of early hominid evolution. Environmental instability is also an essential feature in the modern society, in which
technologies, economies, and cultures are changing rapidly.
The present study assumed an uncertain but stationary environment. However, if temporal instability of the environmental structure is indeed realistic and adaptively significant, we need tosystematically study these conditions. For example, if the environmental structure changes over time, an individual judgment policy
effective at time t may be maladaptive at time t ⫹ 1. Suchinstabilities will provide a challenging test for the adaptive valueof the majority rule. How various group decision rules will perform in unstable environments is an open question. Is a majority
rule still adaptive in an uncertain and nonstationary environment(cf. Boyd & Richerson, 1985; Henrich & Boyd, 1998; Kameda &Nakanishi, 2002, 2003)? How is individual learning reflected ingroup adaptation to environmental shifts (Busemeyer & Myung,
1992; Camerer, 1999; Massey & Wu, 2003)?
Another theoretical issue concerns motivational aspects in groupdecision making. In this article, we focused only on cognitive
aspects of decision making and assumed no special role for members’ motivation; we essentially assumed that members are equally

motivated to provide good individual estimates. Although thisassumption is reasonable in individual decision making in which
one’s fate is contingent solely on one’s own decisions, it may notalways be the case in group decision making. When individuals’
inputs are pooled collectively, “social loafing” in which somemembers free ride on others’ efforts often degrades group performance. In the context of group decision making, members’ involvement in exhaustive information search and demanding deliberations may not be guaranteed. What happens to groupperformance if we add individual information search and voting
costs to the social decision system (Kameda & Nakanishi, 2002,
2003; Kameda, Takezawa, & Hastie, 2003)?
One of the most fundamental problems in social decision making is the question of how to aggregate people’s judgments orpreferences into a collective choice. Social choice theorists (Arrow, 1951; Mueller, 1989) have approached this issue from thelogical perspective; a group aggregation rule that potentially yields
logical incoherence (e.g., intransitivity, deadlocking) in decisions
is regarded as unacceptable. In contrast, we took an adaptive
perspective in evaluating group decision rules and found that the
majority rule, although not perfect in terms of logical coherence(e.g., Condorcet voting paradox), nonetheless supports robust
adaptively viable decisions under uncertainty. Perhaps, the “robust
beauty of the majority rule” explains its wide popularity in modern
as well as primordial societies. When exploring the Northwest
Territory in 1805, Captain Clark used the majority rule to decide
where to set his winter camp (Ambrose, 1996; Moulton, 2003).
Everyone in the expedition, including servants and native guides,
had an equal vote in the majority rule decision. This social choiceprocedure may have been adaptive as well as fitting the democratic
ideals he cited in his personal diary of the expedition.

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Received August 28, 2003
Revision received October 7, 2004
Accepted October 12, 2004 䡲