Comparing Micro and Macro Rationality
Robert J. MacCoun*
*Professor of Public Policy, Richard & Rhoda Goldman School of Public Policy, University of
California at Berkeley, 2607 Hearst Ave., Berkeley, CA 94720-7320; voice 510-642-7518; fax
510-643-9657; email [email protected].
The final version of this essay was published as: MacCoun, R. (2002). Comparing micro and
macro rationality. In M. V. Rajeev Gowda and Jeffrey Fox (Eds.), Judgments, decisions, and
public policy (pp. 116-137). Cambridge University Press.
12/29/98 2:04 PM DRAFT
MacCoun (6/3/2004) - 2
Psychologists have used laboratory experiments to demonstrate that individual judgment
and choice behavior systematically violates the assumptions underlying rational choice theories.
In principle, these cognitive and motivational biases have important implications for economic
theory and policy. But critics contend that various aggregate-level processes (group discussion,
market transactions) attenuate or compensate for individual biases (e.g., Kagel & Roth, 1995;
Page & Shapiro, 1992. Thus, recent experiments by social psychologists and experimental
economists have tested whether interaction with others attenuates individual biases, sustains
them, or amplifies them. Are groups more or less biased than individuals? No simple conclusion
has emerged from this research (see reviews by Kerr, MacCoun, and Kramer, 1996a; Tindale,
1993). Some studies find a clear reduction of bias at the aggregate level, while others find that
groups are significantly more biased than individuals acting alone.
To identify some of the conditions that determine relative bias (group bias - individual
bias), and to help clarify discrepant research findings, my colleagues Norb Kerr and Geoff
Kramer and I (Kerr, MacCoun, & Kramer, 1996a, 1996b) have conducted theoretical "thought"
experiments. Our work suggests that relative bias is a function of the type of bias, the applicable
social combination process, group size, and the distribution of individual opinions prior to
interaction or exposure to others. At the core of our work is a stochastic modeling approach
called social decision scheme analysis (Davis, 1973; Stasser, Kerr, & Davis, 1989).
SOCIAL DECISION SCHEME (SDS) MODELING
Social decision scheme modeling (Davis, 1973; Stasser, Kerr, & Davis, 1989) suggests
that group decisions are a product of two different processes, a sampling process and a social
combination process. Following Davis (1973), let:
•
n ≡ number of decision alternatives/response options (a1, a2, ... aj, ... an);
•
r ≡ number of group members (i.e., rj is the number of members supporting alternative j); and
MacCoun (6/3/2004) - 3
•
p ≡ (p1, p2, ...pn) = distribution of individual decisions across n alternatives.
The number of possible distributions of the r group members across the n alternatives is then:
 n + r − 1 (n + r − 1) !
m=
=
r 
r !(n − 1)!

(1)
If the groups are composed randomly (an assumption I relax later in this essay), then it follows
from the multinomial distribution that the probability that the group will begin deliberation with
the ith possible distribution, (ri1, ri2, ...rin), is:
r

 ri 1 ri 2

 ri 1 ri 2
r!
r
rin
 p1 p2 ... pnin = 
 p1 p2 ... pn
(
!
!...
!)
r
r
r
 ri1ri 2 ... rin 
in
 i1 i 2

πi = 
(2)
The key theoretical component of the SDS approach is the social decision scheme, D, an
m x n transition matrix, where element dij specifies the probability that a group beginning
deliberation with the ith possible distribution of member preference will ultimately choose the jth
decision alternative. These D's are frequently misunderstood to represent formal or explicit
voting or decision rules, perhaps because many have labels like "simple majority scheme." In
actuality, these matrices simply summarize the net effect of all the many cognitive, sociopolitical, procedural, and coordinational processes (see Kerr et al., 1996a) that combine to
integrate the judgments of individual members into a group decision--processes that may not even
be recognized by the group members. SDS modeling is a useful theoretical framework--an
"environment" for thinking through the consequences of alternative theoretical processes--rather
than a specific theory per se (see Stasser et al., 1989).
Table 1 presents five D's which have been shown to have either broad empirical support
in particular task domains (e.g., Simple Majority, Truth Supported Wins, Reasonable Doubt) or
utility as theoretical baselines (Proportionality, Truth Wins). The table assumes a particular
example, a 6-person jury trial with two verdict options, Guilty (G) and Not Guilty (NG).
MacCoun (6/3/2004) - 4
Proportionality Scheme. The probability of a particular faction prevailing is equal to the
relative frequency of that faction; i.e., dij = rjj/ri. It reproduces exactly at the group level what
was observed at the individual level, providing a useful "asocial" theoretical baseline.
Simple Majority Scheme. This is one representative of a family of D's in which there is
disproportionate "strength in numbers." For these schemes, there is disproportionate "strength in
numbers." Formally, if MC = a majority criterion, then for rij > MC, dij > rij/r. In the Simple
Majority D, MC = .5, and the alternative favored by more than half of the members will be
selected as the group decision; i.e., if rij/ri > .50, then dij = 1.0; if rij/ri < .50, then dij = 0.0.
Majority D's require a subscheme to handle ties where r is an even number; in this essay, I
assume equiprobability in the case of a tie. The simply majority scheme or slight variants have
been shown to do a good job of summarizing group judgments under a very broad array of
decision tasks, settings, and populations, particularly in judgmental situations where there is no
normative algorithm for defining or deriving a correct answer.
Table 1. Selected Social Decision Schemes
Initial Split
Proportion
-ality
Simple
Majority
Truth Wins1
Truth
Supported
Wins
Reasonable
Doubt2
G
NG
G
NG
G
NG
G
NG
G
NG
G
NG
6
0
1.00
0.00
1.00
0.00
1.00
0.00
1.00
0.00
1.00
0.00
5
1
0.83
0.17
1.00
0.00
0.00
1.00
1.00
0.00
1.00
0.00
4
2
0.66
0.33
1.00
0.00
0.00
1.00
0.00
1.00
0.67
0.33
3
3
0.50
0.50
0.50
0.50
0.00
1.00
0.00
1.00
0.19
0.81
2
4
0.33
0.66
0.00
1.00
0.00
1.00
0.00
1.00
0.06
0.94
1
5
0.17
0.83
0.00
1.00
0.00
1.00
0.00
1.00
0.00
1.00
0
6
0.00
1.00
0.00
1.00
0.00
1.00
0.00
1.00
0.00
1.00
1In this instance, alternative NG is the "true" response.
2The 4:2, 3:3, and 2:4 entries are taken from meta-analysis in MacCoun & Kerr (1988); the 5:1 and 1:5 entries are taken from Kerr
& MacCoun (1985).
Truth Wins Scheme. This is one of a large class of schemes in which factions favoring
one particular alternative have greater drawing power. In this particular scheme, if riT > 1, then
MacCoun (6/3/2004) - 5
diT = 1.0, else dij = rij/rj. (In Table 1, I assume that for the task in question, NG is the correct
verdict. Note, however, that the same logic might apply to a "Bias Wins" scheme, in which NG is
the incorrect verdict.) The truth wins scheme is of special interest because it depicts the optimal
case in which there is a normatively correct decision and the group selects it so long as at least
one member finds it.
Truth Supported Wins Scheme. Empirically, the truth wins model does a poor job of
describing actual group behavior, even in tasks with a demonstrably correct answer according to a
broadly shared normative framework (e.g., deductive logic). At best, "truth-supported wins"--i.e.,
the member finding the solution needs at least some initial social support or the group will often
fail to adopt the correct solution (see Laughlin, 1996; Stasser et al., 1989). Thus, truth finding is
a social process. Specifically, if riT > 2, then diT = 1.0, else dij = rij/rj.
Reasonable Doubt Scheme. The final example in Table 1 is another asymmetrical
scheme. This one was not derived theoretically but rather estimated empirically from a metaanalysis of about a dozen experimental studies of criminal mock juries (MacCoun & Kerr, 1988).
The scheme shows that in criminal juries, there is an asymmetry such that, ceteris paribus, 50:50
splits generally result in an acquittal, and two-thirds majorities favoring acquittal tend to fare
better than two-thirds majorities favoring conviction. MacCoun and Kerr (1988) present
evidence that this asymmetry reflects the rhetorical advantages provided by the reasonable doubt
standard of proof in criminal trials.
How do these decision schemes translate the initial distribution of opinions into a final
group decision? The probability of group outcome j is:
Pj = Σπidij, or in matrix notation, P = πD
(3)
where π = (π1, π2, … πm) = the distribution of starting points for group deliberation, and P = (P1,
P2, … Pn) = the distribution of group decisions across the n alternatives. Figure 1 shows the
relationship between p (the probability that an individual will favor option G--in this case,
MacCoun (6/3/2004) - 6
wrongly convicting an innocent defendant) and P (the probability that the group will select option
G), for four of the six decision schemes. The Proportionality scheme produces a 45-degree
baseline against which to compare the disproportionate strength in numbers of Simple Majority,
and the asymmetrical drawing power of the NG option in the Truth Wins and Truth Supported
Wins schemes.
COMPARING JUDGMENTAL BIASES IN INDIVIDUALS VS. GROUPS
When will groups be more, less, or equally biased as individuals? Norb Kerr, Geoff
Kramer, and I used the basic SDS framework to analyze the implications of various theoretically
interesting or empirically well-validated D's for this question. Our approach is fairly complex
and can only be summarized here (for a detailed treatment, see Kerr et al., 1996a; for a somewhat
more accessible presentation using an electromagnetism metaphor, see Kerr et al., 1996b). For
simplification, we categorized the host of known judgmental biases into three basic categories:
sins of imprecision, sins of commision, and sins of omission. Here I focus exclusively on sins of
commission (what Hastie and Rasinski, 1988, call "using a bad cue"), in which a normative
MacCoun (6/3/2004) - 7
model (e.g., logic, probability theory, a legal rule of evidence) holds that a certain factor is
irrelevant to the required judgment, yet that factor tends to bias judgment. Examples include
effects of decision framing (e.g., in terms of relative gains vs. relative losses; Kahneman &
Tversky, 1984), preference reversals (e.g., choosing among options vs. ranking options; see
Tversky, Sattath, & Slovic, 1988), and the effects of extraevidentiary information that is clearly
irrelevant (e.g., an automobile-theft defendant's physical attractiveness; MacCoun, 1990). (Kerr
et al., 1996a, provide detailed analysis of the sin of omission and sin of imprecision cases.)
Note that we are discussing bias, not noise (random error). It is trivially true that, thanks
to the law of large numbers, statistical aggregation will tend cancel out random errors in
individual judgment. It should also be noted that the present analysis is restricted to
homogeneous biases--cases where all group members are exposed to or vulnerable to the biasing
stimulus, though they needn't manifest the bias equally. Arguably, this characterizes the vast
number of biases identified in the behavioral decision research tradition (see Kerr et al., 1996a).
(The effects of heterogeneous biases--e.g., personal traits-- on individual vs. group judgment are
analyzed by Davis, Spitzer, Nagao, and Stasser, 1978; Kerr & Huang, 1986).
To motivate the analysis, consider a mock jury experiment in which there are n = 2
decision alternatives; Guilty (G) vs. Not Guilty (NG). Let p = the probability that any given
individual will vote "guilty" either prior to, or in the absence of, group deliberation. Assume that
we are experimentally manipulating extraevidentiary bias (e.g., exposure to pretrial publicity that
misleadingly implicates the defendant) using two conditions. In the High Bias condition (H), the
biasing information (e.g., the publicity) is either present, or highly salient, or otherwise extreme.
In the Low Bias condition (L), the biasing information is either absent, or less salient, or at a low
level. (We'll ignore the specifics of the manipulation for the purpose of generality across types of
bias.) Jurors are unbiased to the extent that pH ≈ pL, and biased to the extent that pH > pL; the
magnitude of individual bias is then
MacCoun (6/3/2004) - 8
b = pH - pL, where |pH - pL| > 0
(4)
and group bias is defined as
B = PH - PL = (πHDH - πLDL), where |PH - PL| > 0
(5)
In the "sin of commission" case examined here, Relative bias (the degree to which groups are
more or less biased than individuals) is then defined as
RB = B - b
(6)
In the analyses that follow, group size held constant at 6.
Following Kerr et al. (1996a), I will use three-dimensional surface plots and their
corresponding two-dimensional contour plots to depict the magnitude of RB at any given
combination of pH (the response tendency in the High Bias condition) and pL (the response
tendency in the Low Bias condition). These plots get rather complicated; as a guide, Figure 2
shows the "floor" of these plots and the x and y axis labels that apply to all subsequent plots.
(Except where indicated, the vertical z axis in the surface plots depicts RB, where higher values
represent greater group bias relative to individual bias.) Note that half the parameter space is
MacCoun (6/3/2004) - 9
0
50
Region where
Alternative G
is usually chosen
anyway
% for Alternative G
in Control Condition
(baseline)
This region is undefined,
since pBias > pControl
100
undefined; by definition, pL can never exceed pH.
0
50
100
% for Alternative G
in Bias Condition
Region with maximum
individual bias
(relative to baseline)
Region where individual bias is fairly weak
MacCoun
Figure 2
6/3/2004
For clarity, I have demarked three regions of the remaining space--though their properties
actually vary gradually, not discretely. The lower left triangle (in white) is a region where the
individual bias (b = pH - pL) favoring G is fairly weak. The upper right triangle (also in white) is
a region where G would most likely be chosen anyway, even in the absense of bias--i.e., there's a
ceiling effect. The gray square is the region of greatest interest. This is the region of maximum
individual bias; G is favored in the High Bias condition but rarely so in the Low Bias condition.
(Note that this diagram provides a graphical illustration of Funder's 1987 argument that biases
needn't inevitably produce "mistakes.")
Figure 3 presents three pairs of surface plots and contour plots, corresponding to the
Simple Majority, Truth Wins, and Truth Supported Wins decision schemes. (See Kerr et al.,
1996a for a similar analysis of groups of size 11, and a somewhat different set of decision
schemes. In all the contour plots, positive numbers refer to locations where groups are more
biased than individuals, and negative numbers depict locations where groups are less biased than
MacCoun (6/3/2004) - 10
individuals. Not shown are the plots for the Proportionality scheme; recall that that scheme
simply reproduces individual judgment probabilities at the group level; hence the surface plot is a
flat plane at the RB = 0 level. It is helpful to imagine that plane as a baseline against which to
compare the surface plots--just as the Proportionality scheme provided a 45-degree baseline in the
two-dimensional plot of p-to-P in Figure 1 above.
MacCoun (6/3/2004) - 11
Recall that the Simple Majority scheme and its variants are the modal empirical pattern in
judgment tasks where there is no widely shared algorithm for determining correct answers-arguably, the category that encompasses most real-world judgments. The most striking feature of
the top panel of Figure 3 is that in the region of maximal individual bias, groups generally can be
expected to amplify rather than correcting individual bias. Recall from Figure 1 that a simple
majority scheme tends to amplify majority views near the .75 (pro-G majority) and .25 (pro-NG
majority) regions. In the middle region of the parameter space, High Bias groups have their proG majority view amplified, but Low Bias groups have their pro-NG majority view amplified.
The net result is a striking High-Low difference.
Only in those regions where the bias is either weak or affecting judgments near the
ceiling can groups be expected to be less biased than individuals, so long as some form of
disproportional strength in numbers process is occurring. Near the lower left corner, any pro-G
bias in High Bias groups is washed out by the pro-NG majority amplification; individuals in the
High Bias condition miss this correction and show the weak (around .10 to .30) bias relative to
their Low Bias counterparts. Near the upper right corner, High Bias groups are near the 1.0
ceiling and experience little amplification, yet Low Bias groups are near the .75 mark and their
pro-G tendency does get amplified. The net (High - Low) result is negligible group bias in a
MacCoun (6/3/2004) - 12
region where individuals show a small but non-zero bias. MacCoun (1990; Kerr et al. 1996a)
notes that unfortunately, it is in just these regions that earlier investigators had inadvertently
looked for evidence that "jury deliberation corrects juror biases"; later studies examining "close
cases" (where the base rate was near .50) reached just the opposite conclusion. And we should
take little comfort from the relative advantage of groups over individuals in these regions;
arguably these are the two regions where "biases" seem least likely to produce actual "mistakes"-i.e., wrong decisions where right decisions would otherwise have been made (cf. Funder, 1987).
Happily, under a Truth Wins scheme (the middle panel of Figure 3), groups will be less
biased than individuals under a very large range of parameter space. Only at the most extreme
levels of High Bias will groups do as bad or worse than individuals--the region where relatively
few groups can be expected to have any advocates of the correct, unbiased option. Thus, when
"truth wins," collective judgments can indeed be expected to better approximate the "rational
actor" of normative models. Unhappily, as noted above, the evidence suggests that, at best, "truth
supported wins," and then only for tasks where arithmetic, basic logic, or some other normative
scheme is widely shared among group members (Laughlin, 1996). As seen in the lower panel of
Figure 3, under a Truth Supported Wins process, the region of amplified bias expands relative to
the Truth Wins scheme, for groups are less likely to begin with two correct members than one
correct member.
WHAT IF EXPOSURE TO BIAS CHANGES THE GROUP PROCESS?
The examples thus far all share a key psychological assumption: exposure to biasing
information doesn't alter the processes that influence the individual-to-group transition; i.e., the
same D matrix applies to both High Bias and Low Bias conditions. The inputs differ across
condition, but the processes that produce the output stay the same. Kerr et al. (1996a, 1996b) cite
various empirical examples where that assumption is violated. Here I extend that discussion by
explicitly modeling a situation where a different D is applicable to each condition. MacCoun and
MacCoun (6/3/2004) - 13
Kerr (1988) cite considerable evidence that in criminal juries, the reasonable doubt criterion
promotes an asymmetric D (see Table 1). MacCoun (1990) and Kerr et al. (1996a) present
evidence that some extraevidentiary biases (defendant unattractiveness, pretrial publicity that is
biased against the defendant) seem to eliminate that asymmetry. In essence, the jury no longer
gives the defendant "the benefit of the doubt."
Figure 4 depicts such a situation. The decision scheme in the Low Bias condition is the
asymmetric Reasonable Doubt D shown in Table 1--the typical pattern for criminal mock juries.
The decision scheme in the High Bias condition is the Simple Majority D. Figure 4 shows that
the net pattern of relative bias falls somewhere between the Simple Majority pattern (top panel of
Figure 3) and the Truth Supported Wins pattern (bottom panel of Figure 3). When both the
prosecution's case and the anti-defendant bias are weak, juries still provide fairer verdicts than
jurors. But when the anti-defendant bias is strong, juries are considerably more biased than
jurors.
Other situations where exposure to bias changes the group process seem plausible; e.g.,
one can imagine groups "anchoring and adjusting" (see Kahneman, Slovic, & Tversky, 1982) on
MacCoun (6/3/2004) - 14
a numerical value when one is provided, but constructing a quantitative estimate more
systematically when no anchor is available. Tindale (e.g., 1993) presents some evidence for the
operation of a "Bias Wins" decision scheme, but this seems plausible only in cases where a
heuristic or bias, once voiced, is so compelling to others that they are willing to change their
preferences, without any recognition that the stated argument reflects a bias. The notion that
biasing manipulations might change the group process is of considerable interest to those of us
interested in the interface between cognitive and social psychology. Nevertheless, I conjecture
that most real-life examples will occupy the region of parameter space bounded by "strength-innumber" majority schemes at one extreme and the asymmetrical truth-supported wins scheme at
another (i.e., Figure 3, top and bottom panels). If so, identifying process-altering biases and
explicating their effects is unlikely to fundamentally alter the basic patterns illustrated here and in
Kerr et al. (1996a, 1996b).
WHAT HAPPENS WHEN GROUPS FAIL TO FORM OR FAIL TO DECIDE?
Our earlier SDS analyses (here and in Kerr et al., 1996a) assumed randomly sampled
group composition for a given individual p(A) in the population. Similarly, empirical SDS
applications--and most individual vs. group bias studies--estimate the probability of each group
composition from observed groupings following random assignment to groups. (A few empirical
studies have composed groups non-randomly for experimental purposes; e.g., MacCoun & Kerr,
1988.) And for simplicity, our SDS "thought experiments" assume that all groups, once formed,
reach a judgment. Here I explore some consequences of relaxing these assumptions.
The random composition assumption is sensible for addressing the theoretical question:
How does a group's decision differ from that of its individual members? More specifically, does
the group decision process amplify or attenuate individual-level bias? It is less appropriate for
addressing the more applied question: Are group decisions in the world more or less biased than
those of individuals? A problem with the random composition assumption is that realistically,
MacCoun (6/3/2004) - 15
not all possible groups will have the opportunity to form; not all groups that have the opportunity
to form will form, and not all groups that form will reach a judgment. There is a continuum of
possibilities:
1) No opportunity: a given configuration will have a lower probability of encountering each
other than predicted by random sampling;
2) Failure to form: a given configuration will fail to group even when given the
opportunity;
3) Disintegration: a given configuration will fall apart before reaching a group decision;
4) Fragmentation: a given configuration will splinter into smaller, more homogeneous
groups;
5) Deadlock: a given configuration will decide not to decide, or fail to reach a decision
(e.g., hung juries);
6) Non-unanimity: a given configuration will form and reach decisions by overriding the
objections of minority faction members--or the latter will overtly consent but covertly
disagree with the group's decision; or
7) Unanimity: a given configuration will group and reach a unanimous decision, either by
preexisting agreement or through genuine conversion of minority faction members.
Scenarios 5, 6, and 7 are modeled by traditional SDS analyses (see Stasser, Kerr, &
Davis, 1989). Note that in the unanimity scenario, the operative social decision scheme (D)
needn't require unanimity; D is a representation of the group process (or rather, a summary of the
consequences of that process), not a formal decision rule. Thus, the empirical success of simple
majority D's--the strength in numbers effect--occurs because minority factions tend to either join
the majority, or are overridden in a non-unanimous group decision.
Scenario 4 is both plausible and interesting, but I will not explore it in any detail here. In
short, a given population capable of producing, say, k 6-person group decisions, will instead
produce greater than k group decisions, many from groups of fewer than 6 members. These
MacCoun (6/3/2004) - 16
groups will be smaller but more homogeneous. If a simple majority scheme is operative, the
proportionality near .50 in large heterogeneous groups will give way to majority amplification
near .25 and .75 in the smaller homogeneous groups.
Scenarios 1, 2, and 3 can be modeled separately, but since they have similar
consequences, I model them identically by introducing a simple weighting function, λ, which
equals the proportion of group members belonging to the largest group faction. The idea is that
if a minority faction is proportionately large enough, there is some probability that the groups
won't form, will form but then fall apart, or will remain intact but fail to reach a decision on
certain topics. In "forced composition" situations (e.g., military units), it is unlikely that
heterogeneous groups will fall apart; instead, it is more likely that small minorities are ostracized
(MacCoun, 1996). But in unforced situations, it is probable that heterogeneous groups are less
likely to form in the first place.
Several lines of research on group formation, composition, and cohesion suggest that this
is plausible (see Levine & Moreland, 1998; MacCoun, 1996). These studies are often limited by
convenience sampling, and for logistical and economic reasons, they rarely examine the full
range of possible groupings in a population. Perhaps most directly relevant for present purposes
are cellular automata models of social influence processes (e.g., Latané, 1996; Epstein & Axtell,
1996; Axelrod, 1997), which show that under a variety of plausible assumptions, social influence
processes will result in a "clustering" of opinion members across social space. If so, "interior"
members will have less opportunity to "group" with members of outgroups than predicted by
random sampling; only "border" members may end up in overlapping groups. (Alternatively,
people may group based on one issue then find less agreement on a second issue.)
The analysis presented below is fairly limited. I applied the lambda weighting function
to only one decision scheme (Simple Majority). And it is possible that some alternative
weighting function might better represent the processes by which groups fail to form, fall apart,
MacCoun (6/3/2004) - 17
or fail to reach a decision.1 Nevertheless, this seems like a useful starting point for an analysis of
relative bias when there's non-random grouping.
Note that since the adjusted Pi equals piλi di, one can conceptualize λ in two different
ways: As a weighting of the sampling process (i.e., πiλi as in Scenarios 1 and 2 above), or as a
weighting of the decision process (i.e., λidi as in Scenario 3). Table 2 illustrates this weighting
for the case of 6-person criminal juries where p = .66. Note that because of attrition, the
weighted probabilities of a group of a given initial split (πiλi ) no longer sum to 1.00 in Scenarios
1 and 2. Similarly, the weighted transition probabilities (λidi) no longer sum to 1.00 in Scenario
3.2
1
So far, I've tested the fit of λ to only one data set; Kerr and MacCoun's (1985) data on hung jury rates for
12, 6, and 3 person mock criminal jury deliberations (N's = 167, 158, and 158 deliberations, respectively).
For 12-person deliberations, the average difference between λ and the proportion reaching a verdict given
each distinct initial split (12:0, 11:1, etc.) was 0.05, with a correlation of .74 between λ and the proportion
reaching a verdict. For 6-person deliberations, the average difference was -0.03 with a correlation of .90.
For 3-person deliberations, the average difference was -0.09 with a correlation of .96. This is particularly
impressive because λ is symmetrical and cannot account for the fact that in criminal juries, factions
favoring acquittal have extra drawing power. Thus, for 12-person deliberations, λ underpredicted the
probability of reaching a verdict for groups with majorities favoring acquittal (3:9 and 2:10), and
overpredicted the probability of reaching a verdict for groups with minorities favoring acquittal (10:2, 9:3,
8:4, and 7:5).
2
Of course, one could normalize these product terms to make them sum to 1.00, but this would be beside
the point of the present inquiry, which is to examine the effect of this attrition on the individual-group
comparison. Normalizing the product terms produces a weighted π that correlates strongly (e.g., .96 in the
case shown in Table 2) with its unweighted counterpart, π. Similarly, the normalized weighted d correlates
.88 with the unweighted d. The resulting relative bias plots differ only subtly from their unweighted
counterparts.
MacCoun (6/3/2004) - 18
Table 2. Effect of Lambda Weighting for 6-Person Groups, Assuming Baseline p = .66
Initial Split
πi
λi
πIλI
Simple
Majority di
λidi
Pi (i.e.,
πiλidi)
6, 0
0.08
1.00
0.08
1.00
1.00
0.08
5, 1
0.26
0.83
0.21
1.00
0.83
0.21
4, 2
0.33
0.67
0.22
1.00
0.67
0.22
3, 3
0.23
0.50
0.11
0.50
0.25
0.06
2, 4
0.09
0.67
0.06
0.00
0.00
0.00
1, 5
0.02
0.83
0.01
0.00
0.00
0.00
0, 6
0.00
1.00
0.00
0.00
0.00
0.00
1.00
N/A
0.69
N/A
N/A
0.57
SUM:
Figure 5 shows the probabilities of reaching either of two alternative decisions (A and B)
for individuals (p) and groups (P) when the Simple Majority group process is vulnerable to
attrition as modeled by the lambda weights. It is instructive to compare this figure to Figure 1.
Relative to the unweighted Simple Majority process, the weighted process yields lower
probabilities of group majorities prevailing--for either decision. The hump-shaped curve shows
the attrition that results from the weighting process.
MacCoun (6/3/2004) - 19
This weighting process seems both simple and innocuous, but it has some rather
counterintuitive consequences. The top panel of Figure 6 repeats the Simple Majority analysis of
the top panel of Figure 3, but with lambda weighting to represent group attrition. Americans tend
to believe that an important advantage of group decisionmaking is the diversity of viewpoints it
bring to bear on a decision (MacCoun & Tyler, 1988), and this belief is not unfounded (Nemeth,
1986). Nevertheless, the top panel of Figure 6 suggests that, if anything, the loss that occurs
when heterogeneous groups fail to form, or fail to reach a decision, makes those groups that do
reach a decision appear to behave surprisingly like groups operating under the Truth Wins
scheme, despite the fact that a basic majority-amplification process is otherwise operative.
MacCoun (6/3/2004) - 20
Does this mean that those groups that actually form in the world are frequently less
biased than individuals? Well, yes and no. No, if the question is: Is the group decisionmaking
process less biased than individual decisionmaking. The reason is that the apparent benefits
depicted at the top of Figure 5 result not from improved decisionmaking, but from attrition--in
particular, attrition in a region where majority amplification would otherwise make groups more
biased than individuals. This is apparent in the bottom panel of Figure 6, where I compare the
MacCoun (6/3/2004) - 21
decisions produced by this "weighted group process," not to those of individuals, but to those
produced by unweighted groups; i.e., I subtract the surface at the top of Figure 3 from that at the
top of Figure 5.
Table 3 illustrates the effect of attrition, created by the weighting function, at three
different regions of the parameter space. In the first region, High Bias = .60 and Low Bias is .10.
The lower panel of Figure 6 suggests that this is where the most profound effects of weighting
occurred. In this region, individuals are quite biased, but groups (in the unweighted case) are
much less so, mostly because the individuals who start out at .10 end up in groups shifting toward
.01. Weighting preserves the latter phenomenon, but the loss of many High Bias groups creates a
reduction in G decisions (48% instead of 67%), resulting relative bias near 0. Thus, weighting in
this region has significantly reduced total group bias. It isn't that attrition has encouraged more
groups to reach the unbiased NG decision than would otherwise be the case. Rather, attrition has
weeded out groups from many bias-enhancing regions. At the group level, this hardly paints a
glowing picture of real-world group decisionmaking.
In the second region, High Bias = .33 and Low Bias = .10. Unweighted groups
significantly reduce this bias, because of a shift in both conditions toward the NG majority. The
weighting process has no effect on this shift; the probability of a G verdict drops simply because
some groups fail to form or reach decisions.
In the final region (the upper right corner of Figure 6), High Bias = .90 and Low Bias =
.60. Here there is little relative bias in the unweighted case. But weighting eliminates many of
the Low Bias groups that would have shifted in the direction of G, making groups look less
biased (relative to individuals) than in the unweighted case.
MacCoun (6/3/2004) - 22
Table 3. Effects of Weighting Function in Selected Regions
Unweighted Case
Condition
p
b
P
B
RB
Weighted Case
P
B
RB
Rbdifference
(Wtd-Unwtd)
High Bias
Low Bias
High Bias
Low Bias
High Bias
Low Bias
.60
.50
.10
.33
.60
.67
.17
.01
.23
.10
.90
.68
.20
.99
.68
.48
-.02
-.19
.12
-.11
-.07
.42
.12
+.11
.00
.19
-.04
.01
.30
.48
.12
.00
.31
.01
.90
.48
What if the question is: How does the population of group decisions in the world differ
from the population of individual decisions? From that perspective, the analysis presented here
suggests the hypothesis that, in an indirect, almost perverse fashion, something approximating
"truth wins" might describe the distribution of group decisions that actually occur. Truth wins to
the extent that majority amplification processes that would have pushed groups into greater bias
get attenuated by attrition. The apparent benefits of attrition occur by eliminating many diverse
groups where minorities would have caved in to their biased majority counterparts. The
remaining population of groups is less heterogeneous; it isn't that they are less biased as groups,
just that as individuals they might have been highly biased (or almost completely unbiased)
anyway.
This may seem counterintuitive, but partly because we often overestimate the benefits of
group decisionmaking, and underestimate the power of majorities to swallow up minority
viewpoints. From that perspective, perhaps we should be grateful that many group decisions fail
to occur.
MacCoun (6/3/2004) - 23
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