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Foundations for Group Decision Analysis
Article in Decision Analysis · June 2013
DOI: 10.1287/deca.2013.0265
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Decision Analysis
Articles in Advance, pp. 1–18
ISSN 1545-8490 (print) — ISSN 1545-8504 (online)
http://dx.doi.org/10.1287/deca.2013.0265
© 2013 INFORMS
Foundations for Group Decision Analysis
Ralph L. Keeney
Fuqua School of Business, Duke University, Durham, North Carolina 27708, keeney@duke.edu
T
his paper derives a general prescriptive model for group decision analysis based on a set of logical and
operational assumptions analogous to those for individual decision analysis. The approach accounts for each
group member’s potentially different frames of their common decision, including different events and different
consequences of concern. Assuming that each group member accepts the decision analysis assumptions to
evaluate his or her analysis of what the group should do and that the group accepts an analogous set of decision
analysis assumptions for the group’s decision, it is proven that the group expected utility for an alternative
should be a weighted sum of the individual member’s expected utilities for the alternatives. After each group
member does his or her decision analysis of the group’s alternatives, the essence of the group decision analysis
is to specify the weights based on the interpersonal comparison of utilities and on the relative importance or
power of each individual in the group.
Key words: group decision analysis; expected utility; group decisions; decision analysis
History: Received on December 20, 2012. Accepted by former Editor-in-Chief L. Robin Keller on January 6,
2013. Published online in Articles in Advance.
1.
Introduction
rule, such as the candidate with the most votes wins.
In this case, there is a collection of individual decisions by the voters that leads, with no specific group
action, to a selected alternative. The third class of
decisions is social planning or social welfare decisions where an individual planner or organization,
after taking judgments and preferences of individuals
affected by the decision into account, makes the decision. The model developed here may have relevance
to each of these classes of decisions, but such uses are
not the topic of this paper.
To provide prescriptive guidance for group decisions, the inherent reasons for their complexity must
explicitly be addressed in the group decision analysis.
These reasons include the following:
1. The members of the decision-making group may
view their common decision differently. They may be
concerned about different consequences of the alternatives and feel that different uncertain events matter
to the choice.
2. Even when group members agree on the relevant
consequences and events, they may assign different
probabilities to those events and different values to
those consequences.
3. Each group member’s perspectives of the decision and his or her evaluations of the alternatives
Decision analysis is a methodology and set of procedures for building models to guide decision making.
These models are prescriptive in that they provide
insights about what alternative to choose to best
achieve stated objectives. The foundations for decision analysis, which are a set of logical and operational assumptions (Pratt et al. 1964), are well
developed for individual decisions. However, several
attempts to extend the appealing logic of decision
analysis to group decisions have not succeeded except
in special cases. This paper develops a general group
decision analysis model.
Group decisions, as defined in this paper, are decisions where a group of two or more individuals must
collectively select an alternative from a set of two
or more alternatives that best satisfies the group’s
objectives, and no individual has veto power. This
definition rules out three classes of decisions where
individuals make decisions that affect groups. One
class is negotiations, because the individual negotiators are trying to best satisfy their own objectives
rather than the group’s objectives, and each individual has veto power. A second class of decisions ruled
out is voting situations where all votes are tabulated
to select an alternative according to a prespecified
1
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must both be incorporated in any group decision
analysis. This implies that the interpersonal comparison of preferences for the alternatives among members of the group and the relative importance or
power of each individual in the group be recognized.
Our group decision analysis model is structured from
the beliefs and preferences of each of the individual members of the decision-making group. A set of
group decision analysis assumptions, analogous to
those for individual decisions, are developed for this
particular group model. Given the group model and
these assumptions, a group can analyze any decision
consistent with the principles of decision analysis.
2.
Background Literature
Decision analysis was initially developed for individual decisions. Its logical foundations involve
constructing a frame to model that decision for subsequent analysis based on expected utility theory,
which integrates the related concepts of probability
and utility. The development of each of these concepts has occurred over time, and each concept has
multiple contributors. Ramsey (1926) provided the
initial theory for decision making based upon these
two concepts. Von Neumann and Morgenstern (1947)
developed the formal theory of expected utility, which
was based on so-called objective probabilities. Savage
(1954) extended expected utility theory to incorporate
probabilities based on a willingness to act, which are
often referred to as judgmental or personal probabilities. A concise history of these developments is found
in the work of Raiffa (1968).
The foundational basis for an operational theory
of decision analysis for an individual decision maker
was fully developed by Pratt et al. (1964). Because
my motivation for this paper is to develop a general
decision analysis model to logically analyze group
decisions, my analysis uses the individual decision
analysis framework of Pratt et al. (1964) as a basis.
Pratt et al. (1964) frames an individual’s decision for
analysis in the following summary:
Frame for an Individual’s Decision Problem. An individual
decision maker must choose among a set of alternatives An , n = 11 0 0 0 1 N , and wants to choose the best
of this set. The decision maker has specified a set of
mutually exclusive and collectively exhaustive events
Ej , j = 11 0 0 0 1 J , one of which will occur, and a set of
Table 1
Decision Analysis Assumptions for Individual Decisions
(from Pratt et al. 1964)
Principles of consistent behavior
IT: Transitivity. As regards any set of lotteries among which the decision
maker has evaluated his or her feelings of preference or indifference,
these relations should be transitive.
IS: Substitutability. If some of the prizes in a lottery are replaced by other
prizes such that the decision maker is indifferent between each new prize
and the corresponding original prize, then the decision maker should be
indifferent between the original and the modified lotteries.
Principles for scaling preferences for consequences and judgments
concerning events
IP: Preferences. The decision maker can scale his or her preference for
any consequence c by specifying a number 4c5 such that he or she
would be indifferent between (1) c for certain and (2) a lottery giving a
probability 4c5 chance at c ∗ and a complementary chance at c o .
IJ: Judgments. The decision maker can scale his judgment concerning
any possible event Ej by specifying a number p4Ej 5 such that he or she
would be indifferent between (1) a lottery with consequence c ∗ if Ej
occurs, c o if it does not, and (2) and a lottery giving a probability p4Ej 5
chance at c ∗ and a complementary chance at c o .
consequences cnj , n = 11 0 0 0 1 N and j = 11 0 0 0 1 J , that
will result if alternative An is chosen and event Ej then
occurs.
Each consequence includes all things that matter to
the decision maker from the choice of an alternative.
The assumptions that Pratt et al. (1964) developed to analyze an individual’s decision are given in
Table 1. The principles of consistent behavior provide
the logical foundation for decision analysis. For interpreting the substitutability principle, it is important
to understand that Pratt et al. (1964, p. 356) defined
the word prize to mean “either a consequence or the
right to participate in another lottery whose payout
will be a consequence.” Given these two principles
and assumption IP, Pratt et al. (1964) prove that the
decision maker’s preferences over consequences c can
be represented by a utility function u that is a positive
linear transformation of  and scaled by
u4c ž 5 = 0
and u4c ∗ 5 = 11
(1)
where c ž and c ∗ provide lower and upper bounds on
the decision maker’s preferences for all consequences.
From assumption IJ, the p4Ej 5 terms are the probabilities that represent the decision maker’s judgments
about the likelihoods of the various events.
Individual Decision Analysis Theorem. Given
the assumptions for an individual’s decision analysis in
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Table 1, it follows that the decision maker should evaluate
alternative An using its expected utility
U 4An 5 = èj p4Ej 5u4cnj 51
n = 11 0 0 0 1 N 1
(2)
where alternatives with higher expected utilities are preferred and èj means to sum over all j.
The past attempts to extend the decision analysis
logic to group decisions implicitly assume that each
of the members of the decision-making group accepts
a commonly held frame of their decision problem.
Notably, and subtly, it assumes that each member
of the group (i) is concerned with the same consequences of the alternatives and (ii) considers the same
events to be those relevant to the decision. Thus,
the standard approach has assumed that the group
accepts the assumptions in Table 1 and searched for
a group utility function and a group set of probabilities that can be used in a result analogous to (2)
to calculate a group expected utility for each of the
alternatives (e.g., Raiffa 1968). The next three paragraphs, following the literature review in Keeney and
Nau (2011), summarize the work most relevant to this
paper.
Hylland and Zeckhauser (1979) investigate the
potential usefulness of separately combining the
individual’s probability assessments into a group
probability assessment and the individual’s utility
functions into a group utility function. They also
assume that the resulting model should be consistent
with weak Pareto optimality, meaning that if all members of a decision-making group prefer one alternative over another, the resulting group decision model
must yield a higher expected group utility for the
Pareto dominating alternative. In addition, they also
assume that aggregations should not be dictatorial,
so group probabilities and the group utility function
should not be identical to those of any individual in
the group. They prove that no group decision model
is consistent with these requirements.
Seidenfeld et al. (1989, p. 225) succinctly summarized the problem: “An outstanding challenge for
Bayesian decision theory is to extend its norms
of rationality from individuals to groups.” Their
approach is to broaden the group compromises that
they consider to include any set of group probabilities and group utilities, which need not be separately constructed from the individual’s probabilities
and individual’s utilities, respectively. Their interest
is whether there is any set of such group probabilities and group utilities that would yield a group
evaluation of alternatives that is consistent with individual’s common preferences over those alternatives.
Seidenfeld et al. (1989) also invoke the weak Pareto
condition. They then prove that there is, in general, no compromise group utility function and group
probabilities for events that can replicate the common preferences of the individual members of the
group.
Mongin (1995) is interested in the implications for
group decisions of a group of individuals who each,
and collectively as a group, accept the decision analysis (i.e., Bayesian) assumptions. In addition to combining individual group members’ probabilities over
events into group probabilities and group members’
utilities over consequences into group utilities over
consequences, he combines group members’ expected
utilities for alternatives into group expected utilities
for those alternatives. With an additional assumption
of strong Pareto optimality, Mongin (1995) proves that
there are no possible combinations consistent with the
assumptions.
Positive results have been developed for special
cases. Notably, Harsanyi (1955) proved that when the
expected utility assumptions hold for both individuals and the group, Pareto optimality holds, and all
members of the decision-making group have common
probability distributions to describe possible consequences of the alternatives, then the group’s expected
utility for an alternative must be a weighted average of the individual’s expected utilities for that
alternative.
Harsanyi’s (1955) result is extended to situations
where both probabilities of events and the utilities for consequences can differ among members of
the decision-making group used by Mongin (1998),
Chambers and Hayashi (2006), and Keeney and Nau
(2011). Mongin (1998) and Chambers and Hayashi
(2006) assumed that the group can specify expected
utilities for (a) the same alternatives as the individual’s consider and (b) a Pareto optimality condition. Keeney and Nau (2011) assumed that the
group has expected utility preferences over hypothetical acts described by lotteries resulting in vector
consequences, where the components are the utilities
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perceived by each member of the group, and that the
group’s preferences are consistent with an assumption
analogous to Fishburn’s (1965) concept of mutual utility independence over these consequences. Because
the result in Keeney and Nau (2011) is the same as
that for the group decision analysis theorem following
in §5, it is useful to summarize the important distinctions between the theories. These distinctions concern
the problem formulation, the decision frame, and the
assumptions used.
The origin for the distinction between the result in
Keeney and Nau (2011) and the result here is the problem formulation. The problem formulation addressed
by Keeney and Nau (2011) is the standard approach
where each group member is concerned with the
same set of events and perceives the same consequences for each alternative–event pair. The members may differ in their assignment of probabilities for
events and their utilities for the various consequences.
The problem formulation of this paper is broader. In
addition to assigning different probabilities to specific
events and different utilities to specific consequences,
it allows different group members to be concerned
with different events and to perceive different consequences following from any alternative–event pair.
Hence, the problem formulation in Keeney and Nau
(2011) is a special case of that addressed here.
Corresponding to the earlier distinction, the group
decision frame is broader in the current paper.
It incorporates each of the member’s decision frames
of their group decision by essentially using a union of
those decision frames. In Keeney and Nau (2011), the
decision frame for the group decision is exactly the
same as that of each member for their joint decision.
Concerning the assumptions used to calculate an
expected utility for the group, Keeney and Nau (2011)
first calculated the expected utilities perceived by
each of the individual members for the group’s alternatives. To combine these, Keeney and Nau (2011)
used a decision frame of a hypothetical group decision problem, where alternatives are defined by
objectively provided probabilities over consequences
described in terms of a vector of the expected utilities that each member perceives for this hypothetical group decision. Then an assumption analogous to
Fishburn’s (1965), and a weak Pareto type of assumption provide the result.
Decision Analysis, Articles in Advance, pp. 1–18, © 2013 INFORMS
The group alternatives in this paper are the real
alternatives available in the group decision, and the
probabilities are descriptions of the group’s judgments over the perceived real consequences that will
occur to the group. To calculate an expected utility for the group, assumptions analogous to the
decision analysis assumptions that each member
uses to individually analyze the group problem are
used. To relate the group preferences to the member’s preferences, an indifference assumption is used.
No Pareto assumption is needed, although the result
is clearly consistent with Pareto optimality.
The current paper has two results, stated in the
corollary in §5, that are not relevant to the group
decision frame in the Keeney and Nau (2011) paper.
With the broader formulation, it is shown that a group
probability distribution over events acceptable to all
group members is a product of each of the member’s probability distributions over events in their
individual frame of the group’s decision and that
a group utility function over consequences must be
a weighted average of each of the member’s utility
functions over the consequences that each member
feels are relevant to the group.
The main intent of the two papers is also different. The Keeney and Nau (2011) article is a potential
contribution to the Bayesian expected utility literature
mainly produced by and of interest to economists.
The current article is a contribution to the decision
analysis literature. The intent is to provide a logical
and practical foundation for spreading the application and use of decision analysis to group decisions.
To implement a theory and to have it be useful, it
is important that the assumptions are stated in terms
that pertain to the real decision being faced, the information needed to implement the result addresses the
complexities of the decision, and practical procedures
exist to assess the information. I believe the foundations for group decision analysis presented here make
such a contribution.
3.
Frame of the Group Decision
Problem
Results described in §2 are important and insightful, but they have not included a comprehensive
breakthrough that provides the foundations to extend
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and implement the decision analysis logic to group
decisions. Thus, if decision analysis is going to be
extended to group decisions, it cannot follow the
approach of framing the group decision as the individual decision is framed and then trying to construct
both a group utility function over consequences and
group probabilities for events. A different approach
must be followed.
This section describes a different and broader
approach to extend decision analysis to group decisions. It incorporates the different frames that group
members may have for their common decision, utilizes the logic embedded in the decision analysis principles for individual decisions to analyze the group’s
decision, and produces a result that can be applied
to group decisions using decision analysis techniques
and procedures. To illustrate different frames, consider a relatively simple decision concerning two people planning to have dinner together at a restaurant.
One individual wants to have a substantial discussion at a convenient location (e.g., not far away and
with easy parking), and the other individual is interested in great food at an exciting location. Because
they have different objectives, they will be concerned
about different consequences of their joint decision.
This will also result in different events being relevant to each individual, as the first individual would
be concerned with how quiet the restaurants are and
about travel times to the restaurants, whereas the second individual may be concerned with whether the
restaurants have exotic fusion dishes and a trendy
clientele.
The decision process for a more complex decision
may typically involve several group meetings over
time with individual effort between meetings, as each
individual essentially must evaluate the desirabilities
of the alternatives for the group. Subsequently, the
group must collectively decide which alternative to
choose. Hence, it is useful to conceptualize this process as two distinct sequential stages. In stage 1, each
group member frames the group’s common decision
and evaluates alternatives using the individual decision model illustrated in §2. In stage 2, the group
uses the member’s evaluations to collectively evaluate
alternatives. This could be done using a discussion,
a voting mechanism, or the more systematic group
decision analysis discussed here.
Figure 1
Basic Individual Decision Problem
(a) Frame of individual
decision
E1
(b) Model of individual
decision with all elements
p (E1)
c11
A1
u(c11)
A1
E2
E1
p (E2)
c12
p (E1)
c21
A2
u (c12)
u (c21)
A2
E2
c22
p (E2)
u (c22)
To clearly describe the two stages of the group decision problem, it is first useful to illustrate the steps
in an individual decision analysis model. Figure 1(a)
illustrates the frame of an individual decision, for
the simplest case of two alternatives and two events,
in terms of a decision tree structured directly from
the statement of the individual’s decision problem.
It has three elements: alternatives, events, and consequences. Figure 1(b) presents a model of that decision that incorporates the decision analysis principles
for scaling preferences and judgments. These introduce two new elements into the model: probabilities
of events and utilities of consequences.
At the beginning of stage 1 for any group decision, the individual members of the group may not
agree on any of the elements of the group decision.
The members may view their joint decision as having different alternatives, events, consequences, probabilities, and utilities in terms of Figure 1(b). During
the meetings, discussions, and information gathering
of stage 1, each member develops his or her own
model of the group’s decision that may account for
the ideas of other members. Fortunately it should be
relatively easy for a group to agree on the set of alternatives, and, as will be shown, that is the only agreement necessary to conduct a group decision analysis.
If individuals initially identify different sets of alternatives for a joint decision, the union of the individual alternatives in each of the member’s sets is the
appropriate set of alternatives to use for the group
decision. If some members feel that specific alternatives suggested by others are inadequate, they have
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the possibility and responsibility to evaluate those
alternatives as poorly as they feel appropriate.
However, even after substantial discussions, the
group members may not agree on either events or
consequences that are relevant to the group decision.
Accounting for this reality is distinct from essentially
all of the previous work on group decision analysis,
which either explicitly or implicitly assumes that all
group members recognize the same events and perceive the same consequences. Furthermore, even if
group members happen to agree on the events and
consequences, they may not and need not agree on
their judgments of the probabilities of those events or
their utilities for the consequences.
By the end of stage 1, each group member has completed his or her analysis of the group’s alternatives,
and stage 2 begins. Because each individual’s analysis
has taken into account all of the information relevant
to the group decision (i.e., alternatives, events, consequences, probabilities, and utilities) from that individual’s perspective, at this stage no group member
should alter his or her analysis in any way because of
any other group member’s information or analysis.
To illustrate the structure of a group decision faced
in stage 2, consider the simplest case of a group decision that involves two individual group members I1
and I2 , two common alternatives A1 and A2 , two
events for each group member, and a consequence
for each alternative–event combination. To keep notation simple, suppose that the events of relevance to I1
and I2 are different and labeled E and F , respectively,
and the corresponding consequences are different and
labeled c and d. The two members’ separate frames
of their joint decision are illustrated in Figure 2.
The frame of the group decision should be constructed from the two group members’ frames of their
group decision. The logic for this is based on the following principles, which are part of the reason for
and consistent with the two-stage decision process
described previously.
Origin of Judgments Principle. All original judgments
must be developed in the minds of individuals.
This principle refers to all judgments such as recognizing or creating alternatives, describing consequences, identifying events, specifying probabilities
for events, and constructing utilities for consequences.
The logic for this principle is that judgments can only
Decision Analysis, Articles in Advance, pp. 1–18, © 2013 INFORMS
Figure 2
Individual Frames for Two-Member Group Decision
(a) Frame of individual l1
E1
(b) Frame of individual l2
F1
c11
A1
d11
A1
E2
E1
F2
c12
F1
c21
A2
d12
d21
A2
E2
c22
F2
d22
be developed in a mind, and only individuals have
minds. Groups do not have a mind, so all original
judgments must occur in the mind of an individual. However, a group of individuals can decide to
express group judgments, which leads us to a related
principle.
Group Judgments Principle. Any group judgments
must be constructed by the group based only on the
judgments of the individuals in the group.
The logic for this principle is as follows: If no
group member recognizes some aspect that might be
relevant to a group decision, then clearly the group
cannot recognize and consider it. If a potentially relevant aspect is recognized by one group member, but
neither that group member nor any other member
cares about it, the group should not care about it. Succinctly, this principle implies that the group should
not care about anything that no member in the group
cares about, and the group should care about anything that at least one group member does care about.
From these principles, the two group members’
frames of their joint decision in Figure 2 are combined
in Figure 3, which illustrates two equivalent frames
of the two-member group decision faced in stage 2.
In the group decision, there are two alternatives A1
and A2 , four possible joint events represented by
(E1 F 5, and eight possible group consequences, represented as vectors of the form (c1 d5 expressing the
various possible combinations of consequences in the
original decision as perceived by the respective members. From Figure 3(a), it is easy to note that if the F
events and d consequences, which are relevant only to
member I2 , are eliminated, the group frame reduces to
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Figure 3
Equivalent Frames of Two-Member Group Decision
(a) Frame with conditional events
(b) Model with joint events
Group
probabilities
F1
E1
F2
A1
F1
E2
F2
F1
E1
F2
A2
F1
E2
F2
pG(E1, F1)
(c11, d11)
pG(E1, F2)
(c11, d12)
A1
pG(E2, F1)
(c12, d11)
pG(E2, F2)
(c12, d12)
pG(E1, F1)
(c21, d21)
pG(E1, F2)
(c21, d22)
A2
pG(E2, F1)
(c22, d21)
pG(E2, F2)
(c22, d22)
that of member I1 shown in Figure 2(a). This indicates
that the group decision frame includes a complete
characterization of everything I1 cares about, namely,
only E events and c consequences. It follows that if
I1 placed any direct value on d consequences, this
would involve double counting, because everything
I1 cares about is already accounted for with the c consequences. In addition, I1 would not care about the
probabilities for F events, as these can have no implications for the judgments of I1 about the chances of
different c consequences occurring. The analogous situation applies for I2 .
To analyze the two-member group decision in Figure 3(b), the group needs joint probabilities for all
possible combinations of (E1 F 5 events, represented by
pG 4Er 1 Fs 5, and group utilities for each of the possible
(c1 d5, represented by uG 4c1 d5.
The decision frame for the M-member group decision is constructed from each member’s individual
frames for their group decision analogous to the simple two-member example in Figure 3. An individual’s
decision frame for their group decision is shown in
Figure 4 and is defined as follows.
Frame for a group member’s decision. Group member
Im , m = 11 0 0 0 1 M, must evaluate the set of alternatives
Group
utilities
uG(c11, d11)
uG(c11, d12)
uG(c12, d11)
uG(c12, d12)
uG(c21, d21)
uG(c21, d22)
uG(c22, d21)
uG(c22, d22)
An , n = 11 0 0 0 1 N , one of which the group will choose.
The group member has specified a set of mutually
exclusive and collectively exhaustive events Emj , j =
11 0 0 0 1 Jm , one of which will occur, and a set of consequences cnmj , n = 11 0 0 0 1 N and j = 11 0 0 0 1 Jm , that will
result if alternative An is chosen and event Emj then
occurs.1
Each of the group member’s decision frames represent their personal perspective of the group decision at the time a group decision will be made. Their
individual decision frames allow them to evaluate all
of the group’s alternatives for input to that group
decision.
Figure 5 illustrates the group decision frame.
The events of interest to the group are members
of a set E of mutually exclusive and collectively
exhaustive events Eg , g = 11 0 0 0 1 G of the form
4E1j 1 0 0 0 1 Emj 1 0 0 0 1 EMj 5, where Emj , j = 11 0 0 0 1 Jm are the
events specified by member Im . The consequences of
1
Because the events relevant to member Im are different in general
from those relevant to any other group member, it may be more
appropriate to put a subscript m on all subsequent j notations.
Because this would result in subscripts to subscripts, and because
an adjacent subscript m always appears with the subscript j, the
second subscript will not be used.
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Figure 4
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This group frame includes every group member’s
frame for the group decision. To see this, recall that
group member Im does not care about any events
other than Emj or about any aspects of the consequences other than those of the form cnmj . If the events
and consequences without the subscript m, which are
those not of concern to Im , are eliminated from Figure 5, the problem frame reduces to that for Im shown
in Figure 4.
Decision Frame of Individual Group Member Im for the
Group Decision
Em1
A1
c1m1
Emj
An
cnmj
4.
EmJ
AN
cNmJ
interest to the group are described by a vector c =
4c1 1 0 0 0 1 cm 1 0 0 0 1 cM 5, where cm is a consequence that
member Im , m = 11 0 0 0 1 M perceives for the group.
Frame of the group decision. A decision-making group
of M members, M ≥ 2, must choose from a set of alternatives An , n = 11 0 0 0 1 N , one of which will be chosen. One of the mutually exclusive and collectively
exhaustive events Eg , g = 11 0 0 0 1 G will occur, and a
consequence cng , n = 11 0 0 0 1 N , g = 11 0 0 0 1 G will result
if alternative An is chosen and event Eg then occurs.
Figure 5
The Group Decision Analysis
Assumptions
To analyze the group decision in Figure 5, a group
utility function over the consequences (c1 1 0 0 0 1 cM 5
and a joint probability distribution over the relevant events are needed. The group decision analysis
assumptions in Table 2, which are analogous to those
for individual decisions in Table 1, provide the logical
basis to derive a solution to the group decision.
Note that the assumptions of scaling are existence
assumptions rather than constructive assumptions as
in Table 1 for the individual’s decision analysis. The
group utility function is constructed from the individual group member’s utility functions, and the
group probability distribution is constructed from the
individual group member’s probability distributions.
Both of these are done as part of the proof of the
group decision analysis theorem in §5. The reasonableness of the assumptions prior and the resulting
Group Decision Frame for the Group Decision
A1
E11
An
E1j
Em1
Emj
EM1
(c111,…,c1m1,…,c1M1)
EMj
(cn1j,…,cnmj,…,cnMj)
AN
E1J
EmJ
Group probabilities
EMJ
(cN1J,…,cNmJ,…,cNMJ)
Group utilities
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Table 2
Decision Analysis Assumptions for Group Decisions
Principles of consistent behavior
GT: Group transitivity. As regards any set of lotteries among which the
decision-making group has evaluated its feelings of preference or
indifference, these relations should be transitive.
GS: Group substitutability. If some of the prizes in a lottery are replaced
by other prizes such that the decision-making group is indifferent
between each new prize and the corresponding original prize, then the
decision making group should be indifferent between the original and the
modified lotteries.
Principles for scaling preferences for consequences and judgments
concerning events
GP: Group preferences. The decision-making group can represent its
preferences over the consequences (c1 1 0 0 0 1 cM 5 in terms of a group
utility function.
GJ: Group judgments. The decision-making group agrees that there
exists a representation of its judgments about the possible occurrence of
any combination of the events (E1 1 0 0 0 1 EM 5 in terms of a joint probability
distribution function.
utility function and group probability distributions
are discussed in §6 after presenting the main result.
To relate the individual’s preferences to the group’s
preferences, we need one additional identity assumption defined as follows.
Group Identical Indifference. If some of the prizes
in a lottery are replaced by other prizes such that
the probabilities of all consequences relevant to each
group member in the original and modified lotteries are identical, then the decision-making group
should be indifferent between the original and modified lotteries.
Note that the group identical indifference (GII)
assumption is less restrictive than a Pareto assumption that would require that the group must be indifferent between lotteries when each of the individual
group members was indifferent.
5.
Group Decision Analysis Theorem
The main result of this paper is the following theorem.
Its prescriptive usefulness and procedures to use it
are discussed in the following sections.
Group Decision Analysis Theorem. Given a
decision-making group of M ≥ 2 members facing a decision with alternatives An , n = 11 0 0 0 1 N , and assuming that
(a) each member accepts the decision analysis assumptions for individual decision-making in Table 1 for his or
her analysis of the group decision in Figure 4,
(b) the decision-making group accepts the group decision analysis assumptions in Table 2 for their group decision in Figure 5, and
(c) the decision-making group accepts the group identical indifference assumption,
then the group expected utility of any alternative An ,
denoted UG 4An 5, is
UG 4An 5 = èm wm Um 4An 5
= èm wm 4èj pm 4Emj 5um 4cnmj 551
(3)
where
pm 4Emj 5 is member Im ’s1 m = 11 0 0 0 1 M1
probability for event Emj 1
um 4cnmj 5 is member Im ’s1 m = 11 0 0 0 1 M1
utility for consequence cnmj 1
Um 4An 5 is member Im ’s1 m = 11 0 0 0 1 M1
expected utility for alternative An 1
and the wm , m = 11 0 0 0 1 M are scaling factors that sum to
one, where 0 ≤ wm ≤ 1.
The proof involves several steps. From assumption (a), each Im can do his or her own analysis
of the group’s decision, which provides the individual’s probabilities and utilities for the group’s decision. Given assumption (b), the group can analyze its
decision, depending on the group’s probability distribution over all possible combinations of events and
the group’s utility function over consequences perceived by each of the group’s members. Then, using
assumption (c), it is shown that the group preferences for alternatives depend only on the marginal
probability distributions over the set of events, from
which it follows that the group utility function uG
must be of the additive form uG = èm wm um , where
wm are nonnegative weighting factors. Hence, the
expected utility of each real alternative can then
be calculated as a weighted sum of the expected
utilities of that alternative from each individual’s
perspective.
Proof. Given assumption (a), from (1), each group
member Im has a utility function that can be scaled by
um 4cmž 5 = 0
and um 4cm∗ 5 = 11
m = 11 0 0 0 1 M1
(4)
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where any two of the cmž consequences need not be,
but may be, the same, and the same circumstance
holds for the cm∗ . Also, each group member has a
set of probabilities pm 4Emj 5 for all possible events that
he or she considers can influence the eventual cm .
Hence, each group member should evaluate alternatives using his or her own expected utility
Um 4An 5 = èj pm 4Emj 5um 4cnmj 51
n = 11 0 0 0 1 N 1 m = 11 0 0 0 1 M0
uG 4c1 1 0 0 0 1 cM 5 = èm uG 4cm 5 = èm wm uGm 4cm 51
(5)
A similar result follows from assumption (b). The
group should evaluate alternatives using
UG 4An 5 = èE pG 4E1j 1 0 0 0 1 EMj 5uG 4cn1E 1 0 0 0 1 cnME 51
(6)
where E = 4E1j 1 0 0 0 1 Emj 1 0 0 0 1 EMj 5 is a member of the
set E of all possible combinations of events; cnmE is
the consequence relevant to Im given alternative An is
chosen and E occurs; pG is the group probability of
any group consequence (cn1E 1 0 0 0 1 cnME 5 given alternative An is chosen and E occurs; and uG is the group
utility for consequence (cn1E 1 0 0 0 1 cnME 5.
Because of how pG 4E1j 1 0 0 0 1 EMj 5 in (6) is constructed (see Figure 3), its marginal probability distribution over Emj must be pm 4Emj 5, m = 11 0 0 0 1 M,
which provides the marginal probabilities over the
(cn1E 1 0 0 0 1 cnME 5 for alternatives An , n = 11 0 0 0 1 N . Now
define qG 4E1j 1 0 0 0 1 EMj 5 = çm pm 4Emj 5, so qG is a joint
probability distribution assuming probabilistic independence of all Emj , m = 11 0 0 0 1 M.
Consider two alternatives Ap and Aq described by
different joint probability distribution functions pG
and qG . As these have the same marginal probabilities over cm , m = 11 0 0 0 1 M, from the group identical indifference assumption, the group members each
consider alternatives Ap and Aq to be identical, so
the group must be indifferent between alternatives Ap
and Aq . This has two important implications. First, a
group probability distribution over events E that all
members can agree on is
pG 4E1j 1 0 0 0 1 EMj 5 = qG 4E1j 1 0 0 0 1 EMj 5 = çm pm 4Emj 50
sense,” meaning that the group is indifferent between
lotteries over multiple attributes (i.e., referring to the
different individual’s perceived consequences as different attributes in this case) when the marginal probability distributions over each attribute are identical.
From Fishburn’s (1965) Theorem 2, it follows that the
group utility function uG must be additive, so
(7)
Second, the group’s preferences for alternatives must
only depend on the marginal probability distributions over those consequences. This is Fishburn’s
(1965, p. 38) condition of “independence in the utility
(8)
where we can scale uG by
ž
uG 4c1ž 1 0 0 0 1 cM
5=0
and
∗
uG 4c1∗ 1 0 0 0 1 cM
5 = 11
(9)
ž
ž
ž
uG 4cm 5 is defined as uG 4c1ž 1 0 0 0 1 cm−1
1 cm 1 cm+1
1 0 0 0 1 cM
),
m = 11 0 0 0 1 M, uGm 4cm 5 is defined by
uG 4cm 5 = wm uGm 4cm 51
m = 11 0 0 0 1 M1
(10)
uGm is scaled by
uGm 4cmž 5 = 0
and uGm 4cm∗ 5 = 11
m = 110001M1
(11)
and the wm are nonnegative scaling factors where
∗
0 ≤ wm ≤ 1, m = 11 0 0 0 1 M. Evaluating (c1∗ 1 0 0 0 1 cM
5 with
(8) and using (9) and (11) yields
w1 + w2 + · · · + wM = 10
(12)
Note from (4) that u1 is scaled from 0 to 1, so
u1 4c1 5 can be used as a probability. Now, suppose
the group faces a choice between (a) an alternative
described by a group probability distribution yieldž
ing (c1∗ 1 c2ž 1 0 0 0 1 cM
) with a probability of u1 4c1 5 or
ž
ž
ž
(c1 1 c2 1 0 0 0 1 cM ) with the probability of 61 − u1 4c1 5] and
ž
(b) a sure consequence of (c1 , c2ž 1 0 0 0 1 cM
). The expected
utility to I1 of both the lottery and the sure consequence c1 is u1 4c1 5, so I1 is indifferent between these
choices. Obviously, each other group member Im ,
m = 21 0 0 0 1 M, is also indifferent. Hence, by group substitutability, the group must be indifferent. Equating
the group utilities yields
ž
ž
u1 4c1 5uG 4c1∗ 1c2ž 10001cM
5+61−u1 4c1 57uG 4c1ž 1c2ž 10001cM
5
ž
= uG 4c1 1c2ž 10001cM
50
(13)
Substituting (9) and (10) into (13) yields
u1 4c1 5w1 uG1 4c1∗ 5 = w1 uG1 4c1 50
(14)
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Now, substituting (11) into (14) yields
u1 4c1 5 = uG1 4c1 50
(15)
An analogous logic to (15) yields
um 4cm 5 = uGm 4cm 51
m = 21 0 0 0 1 M0
(16)
Substituting (16) into (8) yields
uG 4c1 1 0 0 0 1 cM 5 = èm wm um 4cm 51
(17)
which is the form of the group’s utility function over
consequences specified by combining the member’s
utility functions over the group consequences.
Substituting (7) and (17) in (6), the expected utility to the group of an alternative An can be calculated from
UG 4An 5 = 6çm pm 4Emj 576èm wm um 4cnmj 57
= èm wm Um 4An 51
n = 11 0 0 0 1 N 0
(18)
Substituting into (18) from the individual decision
analysis result in (5) yields
UG 4An 5 = èm wm Um 4An 5 = èm wm 6èj pm 4Emj 5um 4cmnj 571
n = 11 0 0 0 1 N 1
(19)
which completes the proof and indicates that
only the marginal probability distributions over cm ,
m = 11 0 0 0 1 M, are required to compute the group
expected utility. ƒ
Corollary 1. A group probability distribution (6) that
all group members can agree upon is pG 4E1j 1 0 0 0 1 EMj 5 =
çm pm 4Emj 5 as specified in (7). A group utility function
consistent with the group value judgments must be of
the additive form uG 4c1 1 0 0 0 1 cM 5 = èm wm um 4cm 5 indicated
in (17).
As demonstrated in the proof of the theorem, the
group probability distribution pG incorporates probabilistic independence among the probabilistic judgments of the members Im for events Emj , m = 11 0 0 0 1 M.
It could be that there are probabilistic dependencies among these judgments, but no member of the
decision-making group cares about this, because the
chosen pG includes all the probabilistic judgments of
concern to Im , m = 11 0 0 0 1 M, and the resulting analysis would be identical for any probability distribution
acceptable to the group.
It is useful to recognize that one could replace the
GII assumption in the statement of the Group Decision Analysis Theorem with a weak Pareto indifference (WPI) assumption and the result (3) would still
hold. The logic is as follows: A weak Pareto indifference assumption would state that if all of the group
members were indifferent between two alternatives,
then the group should be indifferent between those
alternatives. If two alternatives had identical probabilities over the consequences of relevance to each
group member, then all of the group members would
be indifferent between the alternatives. Hence, WPI
would imply that the group would be indifferent
between those alternatives, so the GII assumption
would hold, from which the group decision analysis
theorem follows.
6.
Comments on the Result
The purpose of this paper is to provide a logically
sound and practical result that can be used to guide
prescriptive decision making for groups. To examine logical soundness, it is necessary to appraise the
problem formulation, assumptions, and result of the
group decision analysis theorem. To examine practicality, it is necessary to appraise how the theory can
be implemented. Logical soundness is addressed in
this section and implementation is discussed in §8.
6.1. Comments on the Problem Formulation
The problem formulation is based on the two-stage
decision process described in §3. In the first stage,
each individual group member analyzes the group’s
decision using the decision analysis framework for
individual decision making. This framework has been
accepted for more than 40 years as the basis for prescriptive decision making under uncertainty for individuals (Savage 1954, Pratt et al. 1964, Howard 1966,
Raiffa 1968).
The consequences cm of concern to group member Im in his or her own decision frame of the group
decision are meant to include all implications that
will matter to that group member from the choice of
an alternative. The consequences clearly characterize
what utilities are needed from the individual. They
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also indicate the events that matter to that individual,
namely, those events that would affect the consequences that might occur. And these events characterize what probabilities are needed from the individual.
Each individual is the expert on recognizing the consequences of importance in the group decision from
his or her own perspective and for providing the corresponding individual probabilities and utilities.
During the first stage of any group decision, significant interaction would likely occur among group
members. As a result, each group member will be able
to account for other members’ perspectives on consequences, events, probabilities, and utilities. At the end
of this stage, each group member’s decision frame
will have a common set of alternatives, but may be
different in terms of all other elements (e.g., events,
consequences, probabilities of events, and utilities for
consequences). At this point, each group member, and
the group as a whole, are agreeing that the specified consequences, events, probabilities, and utilities
of group member Im , m = 11 0 0 0 1 M, are fixed and
appropriately represent his or her judgments and
preferences.
In the second stage, the decision frame for the
group decision is constructed from all of the group
members’ frames, and when viewed in terms of only
what is relevant to an individual group member,
reduces to that individual’s frame of the decision. This
group decision frame, expanded compared to those
used in previous research, accounts for the reality that
individual group members may perceive that different consequences and different events are relevant to
the group’s decision in addition to possible different
utilities for the consequences and possible different
probabilities for the events.
In the special case where all group members are
concerned with exactly the same consequences and
events, the subscript m for member on consequences
cnmj and events Emj could be dropped from the problem formulation. If this were done, the main result (3)
reduces to
UG 4An 5 = èm wm Um 4An 5 = èm 4wm èj pm 4Ej 5um 4cnj 551
(20)
and holds when members have the same or different
probabilities and utilities.
In any group decision, whether or not a model is
used to provide insight for making the decision, group
members may strategically misrepresent information
about their judgments or preferences. Obviously, the
quality and content of the insights potentially available from any analysis using the group decision analysis model depends on whether members strategically
misrepresent information. If they do not, the potential insights guide the choice for the group based on
balancing perspectives of the members. If a member
does misrepresent information, a potential insight is
recognition that this is the case. Often, members of
the group would know each other, so they can recognize when another member’s judgments are out
of line with available information or preferences are
not consistent with previously stated views. Based
on the circumstances, members can appraise whether
this is because of strategic misrepresentation or a misunderstanding of some of the complexities of the
decision. In applications, group choices about sharing the member’s judgments and preferences should
be made by considering the transparency of the
model, the influence of one member on others, and
strategic misrepresentation. As with any application
of a model, the group must subsequently decide
whether and how the results of the model should
be used in providing insight to help make the group
decision.
The relative desirability of alternatives from each
decision maker’s perspective is described by his
or her expected utilities of those alternatives. The
interpersonal comparison of utilities and the relative
importance of the different members of the decisionmaking group are both incorporated in the scaling
factors denoted by wm in (3). The decision-making
group must collectively decide on the values of those
scaling factors as described in §8.
6.2.
Comments on the Group Decision
Assumptions
Because each group member has transitive preferences, if the group has expressed its preferences for
certain lotteries, then it seems reasonable to assume
that the group should wish to make decisions consistent with the group transitivity assumption GT.
Note that this assumption does not address situations
where the group has not expressed preferences for
a lottery, nor does it restrict in any way how group
preferences are constructed.
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The reasonableness of the group substitutability
assumption GS follows the same logic. Because each
group member accepts substitutability, if a group
replaces prizes in a lottery with new prizes deemed to
be equal in value to the original prizes, this should not
change the group preferences from that for the original lottery. This assumption does not specify how the
group determines that prizes are indifferent, but concerns only group preferences when substitute prizes
are indifferent to original ones.
The group judgments assumption GJ is that there
exists a group judgment about the possible occurrences of all event combinations (E1 1 0 0 0 1 EM 5 in terms
of a joint probability distribution. The reasonableness
of this assumption is supported by the following.
First, as demonstrated in the proof of the group decision analysis theorem, only the marginal probability
distributions over each of the Em , m = 11 0 0 0 1 M are
needed in the proof. Second, from the structure of
the group decision problem, each group member Im
only cares about Em and does not at all care about
Er 1 r 6= m. Thus, any joint probability distribution over
(E1 1 0 0 0 1 EM 5 that has the marginal probability distributions provided by Im for each Em , m = 11 0 0 0 1 M
should be acceptable to all group members. Hence,
the group probability distribution in (7) constructed
from the group member’s probability distributions is
an appropriate group probability distribution for the
decision.
The group preferences assumption GP is a group
commitment to fulfill its collective responsibility to
make a decision in a reasonable manner and not
inadvertently select a significantly inferior alternative.
Because each member accepts the decision analysis
logic for their individual evaluation of alternatives,
whereas preserving the possible distinctions in events,
probabilities, consequences, and utilities, it is reasonable to assume a group utility function exists as long
as it is of a form that does not require agreements
on probabilities and utilities that do not reflect reality. Also, because only group member Im is concerned
about consequences cm , it makes sense to use the
utility function of Im over cm as the group’s utility
function over cm . Consistent with assumption GP, the
group utility function that represents the group’s preferences must be of the additive form (17) with scaling
factors specified (discussed in §8).
The logic for the group identical indifference GII
assumption is as follows. The original and modified lotteries referred to in the assumption determine possible consequences of two alternatives. From
any group member’s perspective, the probabilities of
the possible consequences of the two alternatives are
identical. Thus, consistent with the group judgment
principle, the group as a whole should consider them
to be identical. The group should evaluate two identical alternatives, which should be perceived as the
same alternative, as equivalent, so they should be
indifferent between the two alternatives.
6.3.
Comments on the Group Decision Analysis
Result
To contrast use of the group decision analysis theorem
with previous work, four cases will briefly be considered. Case 1, the most general case, is when the only
thing in common in the members’ frames of the group
decision is the set of alternatives. Members have different consequences, events, probabilities, and utilities. In this case, there is nothing but the expected
utilities of the alternatives that could be combined to
evaluate the alternatives from the group’s perspective, which is the result of the group decision analysis
theorem.
Case 2 is a special case of case 1, where the members agree on the consequences and events, but may
disagree on the probabilities for these events and the
utilities of the consequences. This frame is the one
that has been implicitly assumed in most of the previous investigations of foundations for group decisions.
Consistent with the group decision analysis theorem,
no agreement on probabilities of events or on utilities
of consequences is required. The member’s probabilities and utilities are used in the member’s decision
analyses, and their expected utilities of the alternatives are weighted in (3) to provide a group expected
utility.
Case 3 is a special situation of case 2, where the
members also agree on the probabilities of events,
but may disagree on the utilities of the consequences.
In this case, the result is the same as that found by
Harsanyi (1955) using a Pareto optimality assumption
in addition to individual and group decision analysis
assumptions.
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Case 4 is the most restrictive situation of a group
decision, where the group members agree on everything: consequences, events, probabilities, and utilities. In this case, the group analyzes the decision
exactly as each of the group members analyze the
decision, and the expected utilities of alternatives for
the group are the same as those calculated in each
member’s analysis.
Although not assumed, the result of the group decision analysis theorem is consistent with weak Pareto
optimality, which is a basic assumption in much of
economic literature on the group decision making,
including the works of Harsanyi (1955), Hylland and
Zeckhauser (1979), Seidenfeld et al. (1989), and Mongin (1995). Clearly, when each group member prefers
an alternative Ai to an alternative Ak , then each of
their expected utilities Um 4Ai 5 must be greater than
Um 4Ak 5, so by (3), the group expected utility UG 4Ai 5
must be greater than UG 4Ak 5, therefore the group
must prefer Ai to Ak .
There are no restrictions in this group decision
framework about whether group members can have
common or overlapping events or consequences. For
a group consequence that is described by (c1 1 0 0 0 1
cm 1 0 0 0 1 cM 5, there may be common aspects in each
cm . On a corporate acquisition decision, each group
member may include the profit due to any acquisition as part of the consequences he or she considers relevant to the group decision. Hence, in each
cm , m = 11 0 0 0 1 M, that profit is included. Overall, the
acquisition profit is included M times in the group’s
consequence. However, because cm is a complete summary of what matters to Im , um only considers cm , so
there is no double counting of preferences. The point
is particularly clear in the special situation where each
group member is only concerned with profit and has
a utility function over profit. Because the sum of the
group member’s weights is one, the group’s weight
on profit is one, so there is no double counting.
The group decision analysis theorem allows for a
dictatorial group utility function, meaning one where
one scaling factor wm = 1, and all of the others are
zero. Because the group must collectively specify all
of the scaling factors, in practice, we would usually expect each of the wm to be positive. Any group
will essentially decide how to utilize the member’s
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evaluations of alternatives, and could choose to follow a single individual’s analysis if they collectively
decided to do so. Because of this, there seemed to
be no reason to exclude the dictatorial possibility by
introducing another assumption.
7.
Relationship to Arrow’s
Impossibility Theorem
Arrow (1951, 1963) investigated a specific group decision formulation to combine the preferences of individual members of a group for the alternatives to
obtain a group preference for those alternatives.
In general, he wanted to obtain group preferences P
for all An , n = 11 0 0 0 1 N given the individual member’s
preferences Pm , m = 11 0 0 0 1 M, so
PG 4An 5 = f 4P1 4An 51 0 0 0 1 PM 4An 551
(21)
where f is a function. Arrow’s interest was where
the preferences P were rankings of the alternatives.
He postulated five assumptions and proved that they
were inconsistent. Hence, an impossibility theorem
resulted, which has been very influential and continues to generate great interest.
Suppose one maintains the general formulation
(21), but changes the preferences of the individuals and the group to be ratings instead of rankings
of the alternatives. Then, if one adopts assumptions
analogous to Arrow’s using ratings, specifically the
expected utility of alternatives, it has been proven that
the group decision analysis result (3) follows (Keeney
1976). Thus, if one assumes that each member of the
group can specify the expected utilities of the group’s
alternatives, the result of the group decision analysis theorem can also be derived from a different
set of logical, but not decision analytic, assumptions.
This is understandable if one recognizes that the formulation (21) with expected utilities of alternatives
calculated by the individual group members explicitly assumes that group’s preferences for alternatives
depend only on the overall ratings (i.e., expected utilities) of the group members, and implicitly assumes
that the group’s preferences for alternatives depend
only on the individual member’s marginal probability
distributions over the consequences of alternatives.
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8.
Comments on Implementation
To implement the group decision analysis result (3)
requires information from each of the group members and from the group as a whole. Each member
must verify that the decision analysis assumptions
for an individual decision in Table 1 are appropriate for his or her own analysis of the group decision and then conduct that analysis. Because these
are each individual decision analyses, the procedures
to do this are well developed and understood (e.g.,
Raiffa 1968, Kirkwood 1997, Clemen and Reilly 2001).
The group decision model does not require or allow
individuals to assign probabilities to events or utilities to consequences that are not directly relevant to
that individual.
The group as a whole must verify the appropriateness of the group decision analysis assumptions in
Table 2 and the group identical indifference assumption. Verification could occur in a discussion that clarifies the meaning of each assumption and decides on
whether the group should make their group decision
consistent with that assumption.
The group must also collectively construct the scaling factors wm , m = 11 0 0 0 1 M to weight the individual
member’s expected utilities in the group evaluation.
Specifying these weights is perhaps the most difficult
aspect of implementing the group decision analysis
framework because (1) it requires value judgments
about which the individual members may not agree,
and (2) because the sum of the weights must equal
one, the larger the weight given to one individual’s
evaluations of the alternatives, the smaller the sum of
the weights for the other individuals’ evaluations.
A brief description of the conceptual foundation for
specifying weights will provide a basis for discussing
the practical aspects of specifying the weights and
using the group decision analysis model. Each individual’s weight depends on two separate factors, the
relative importance of that individual in the group
and the interpersonal comparisons of group member’s utilities. These two factors can be addressed separately, and their implications are easily combined, as
we will see.
In many situations, the group would either explicitly or implicitly assume that the importance of all
group members is equal; the basis for such a judgment is simply that being a member of the group
15
accords one an equal vote. In other situations, the
importance of different members may differ depending on characteristics such as ownership, position,
or seniority. For example, if three individuals jointly
owned an investment with 40%, 35%, and 25% shares,
the relative weights for joint decisions may be the
same as these percentages.
The interpersonal comparison of utilities is a more
complex issue that has long been studied and discussed (e.g., Harsanyi 1955, Luce and Raiffa 1957, Sen
1970, Dyer and Sarin 1979). The essence of the interpersonal comparison of utilities issue is to compare
the utility to individual I1 of going from a utility of
u1 = 0 to u1 = 1, where u1 = 0 corresponds to the worst
consequence for the group from the viewpoint of I1
and u1 = 1 corresponds to the best such consequence,
to the utility to individual I2 of going from u2 = 0 to
u2 = 1. The group may specify equivalent changes in
the utility for I1 and I2 to make this judgment. For
instance, the group may feel that a change from u1 = 0
to u1 = 005 is as significant to I1 as a change from
u2 = 0 to u2 = 1 is to I2 . It follows that the change from
u1 = 0 to u1 = 1 is twice as significant to I1 as a change
from u2 = 0 to u2 = 1 is to I2 . This would imply that
the ratio of w1 :w2 would be 2:1 if the only relevant
concern was the interpersonal comparison of utilities.
Suppose this utility comparison was for a decision
where I1 owned 40% of an investment and I2 owned
60%; the importance contributions to weights were set
at 0.4 and 0.6, respectively. The implication would be
that the relative weights should be 2 × 004 for I1 and
1 × 006 for I2 , so the scaling factors would be w1 = 4/7
and w2 = 3/7, respectively.
In practice, if there is a reason for having the relative importance of group members be different, this
issue will likely come up in group discussions and
be explicitly considered. Following the origin of judgments principle, it takes one member to think of
this concern and bring it to the group’s attention for
consideration. This may result in selecting specific
judgments about relative weights as illustrated earlier. If no member brings up the issue of the relative
importance of group members for the decision, then
it will likely be implicitly assumed that all members
should be treated as equal in this respect.
For some decisions, the interpersonal comparison
of utilities factor may also be unconsciously handled
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16
by implicitly assuming that the changes of utilities
from um = 0 to um = 1 are equally significant for all Im .
If this issue is brought up explicitly, it may be possible
to specify weights using agreed-upon value trade-offs
as discussed in the two-person example earlier and
the fact that all weights must sum to one. Baucells
and Sarin (2003) investigated minimum agreements
among group members that are sufficient to specify
the weights on the member’s utilities when the group
utility function is additive.
If through open discussion the group does not
reach an agreement on specifying the set of member’s
weights, one of two procedures may be useful. The
first is to have each group member construct a set
of scaling factors that he or she feels is appropriate.
Analysis with each of these sets may provide insights
to reach an agreed-upon decision.
The second procedure, which may utilize information from the first procedure, is to jointly set
bounds on the relative importance of group members and on their interpersonal utility comparisons or
directly on the relative scaling factors, both of which
lead to bounds on the weights. For example, a group
may unanimously agree that the significance to member I1 of going from a utility of u1 = 0 to u1 = 1 is
greater than, but not twice as great as, the significance
to member I2 of going from u2 = 0 to u2 = 1. This
would indicate that the relative weight of w1 to w2
must range between one and two. For a two-member
group of equally important individuals, the weight of
w1 must be between 1/2 and 2/3, with w2 = 1 − w1 .
The group decision analysis using (3) can be done
over the collective ranges of the weights that are
acceptable to all group members of the decisionmaking group. Depending on how restrictive the
weights are and on the specifics of the individual’s
expected utilities for the alternatives, the acceptable
ranges of weights may be sufficient to provide a
unique ordering of the alternatives. In other cases, it
is likely that the least desirable alternatives would be
eliminated from consideration based on the analysis
using the acceptable ranges of weights. If the expected
utilities of the individuals have greater positive correlation, given bounds will tend to provide more definitive group evaluations of the alternatives.
Keeney: Foundations for Group Decision Analysis
Decision Analysis, Articles in Advance, pp. 1–18, © 2013 INFORMS
For any model to be useful, in addition to being
able to obtain the information necessary to implement the model, it must be relevant to some important problems. Bordley (2009) discusses classes of
important business decisions where the members of
a decision-making group would reasonably identify
different events as relevant to the decision. The same
logic for events and consequences is relevant to many
governmental and public decisions as indicated by
the following case.
As it happens, the group decision analysis
model (3) was essentially applied to a very significant
decision years ago. Dyer and Miles (1976) worked
with the Jet Propulsion Laboratory to provide an analysis of NASA’s Mariner Jupiter/Saturn 1977 project
alternatives. This was a 300 million (in 1977 dollars)
project that included two spacecraft trajectories to be
launched within a couple of weeks of each other and
fly by both Jupiter and Saturn to conduct numerous
scientific experiments.
There were 10 groups of scientific studies to be conducted on this mission. A different scientific team had
responsibility for each group of studies. A Scientific
Steering Committee, composed of the leaders of each
of the 10 scientific teams, had responsibility for creating and choosing trajectory pairs. Because the scientific experiments concerned distinct fields of study,
such as infrared radiation, imaging science, ultraviolet spectroscopy, and cosmic ray particles, the teams
naturally had different consequences of concern and
different events that could affect those consequences.
These characteristics are exactly those assumed in
the group decision analysis frame of this paper, and
the previous standard group decision analysis frame
would not be appropriate for a decision with these
characteristics.
To evaluate alternatives, Dyer and Miles (1976)
used a few different procedures including what they
referred to as an “additive collective choice rule” that
had intuitive appeal and is essentially the group decision analysis result (3). They had the scientific teams
each provide the expected utilities for their respective
experiments given each trajectory pair. In addition
to other sensitivity analyses, the Steering Committee
selected two sets of scaling factors for the additive
collective choice rule and used each to lend insight
about the relative desirability of the alternatives. Dyer
Keeney: Foundations for Group Decision Analysis
17
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and Miles (1976) provided details about how the evaluation process helped in both guiding the Steering
Committee to select an alternative trajectory pair and
in honing that pair of trajectories to improve its value.
Their application definitely indicates the potential relevance of the group decision analysis result to important decisions.
9.
Summary
There have been several attempts over the past halfcentury to extend decision analysis from individual
to group decisions. None have resulted in a decision
analysis framework to analyze the general case of a
group decision. These attempts, including those by
Raiffa (1968), Hylland and Zeckhauser (1981), Seidenfeld et al. (1989), and Mongin (1995), each resulted in
impossibility theorems. In retrospect, a key contributor to the difficulty in finding a positive solution to
the group decision was the implicit assumption in all
of these efforts that the group decision had the same
frame as an individual decision, which implies that
the group members are each concerned with the same
set of consequences and with the same set of possible events. As a result, the search for a solution proceeded to specify group probabilities for these events
and group utilities for these consequences to produce
a group decision analysis. As group members could
have different judgments about probabilities of events
and different preferences for the consequences, simultaneously combining both different probabilities and
different utilities turned out to be problematic.
The approach taken in this paper addresses the
more realistic general decision problem where group
members may have different perceptions, and therefore different decision frames, of their common group
decision. The group decision frame explicitly incorporates each member’s frame, so it is broader than any
member’s decision frame. This allows each member
to incorporate his or her potentially different consequences and events of concern, as well as different
probabilities of events and utilities of consequences,
into the group decision. Using group decision analysis assumptions, analogous to those for an individual decision and relevant only for the specific
group decision constructed from the individual member’s frames, provides a decision analysis solution
for group decisions. This solution is that the group
expected utility for an alternative is the weighted sum
of the individual member’s expected utilities for that
alternative. This result incorporates and maintains the
integrity of each member’s decision analysis of what
he or she feels is in the group’s interest and, in addition, explicitly addresses how the evaluations of the
group members should be combined. The result is a
logically sound operational framework to conduct a
decision analysis of any group decision.
Acknowledgments
The comments of David Bell of Harvard University and
Robert Nau of Duke University were very helpful and much
appreciated.
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Ralph L. Keeney is a research professor emeritus at the
Fuqua School of Business at Duke University. His education includes a B.S. in engineering from the University of California, Los Angeles, and a Ph.D. in operations
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Keeney: Foundations for Group Decision Analysis
Decision Analysis, Articles in Advance, pp. 1–18, © 2013 INFORMS
research from Massachusetts Institute of Technology. His
research interests are in the areas of decision making and
risk analysis. He has applied such work to important personal decisions and as a consultant for private and public
organizations addressing corporate management problems,
environmental and risk studies, and decisions involving
life-threatening risks. Prior to joining the Duke faculty, Professor Keeney was a faculty member in Management and
in Engineering at MIT and at the University of Southern
California, a research scholar at the International Institute
for Applied Systems Analysis in Austria, and the founder of
the decision and risk analysis group of a large geotechnical
and environmental consulting firm. Professor Keeney is the
author of many books and articles, including Value-Focused
Thinking, Decisions with Multiple Objectives, coauthored with
Howard Raiffa, and Smart Choices, coauthored with John S.
Hammond and Howard Raiffa, which has been translated
into 15 languages. Dr. Keeney was awarded the Ramsey
Medal for distinguished contributions in decision analysis
by the Decision Analysis Society and is a member of the
U.S. National Academy of Engineering.