- Email: [email protected]
Group decision support systems implement Bayesian rationality
European Journal of Operational Research 47 (1990) 29-34 North-Holland
29
Theory and Methodology
Group decision support systems implement Bayesian rationality Bernard SINCLAIR-DESGAGNE
INSEAD, Boulevard de Constance, 77305 Fontainebleau, France
Abstract: When saying that group decisions are good, one often means that they are 'externally Bayesian'. Proposed by Madansky (1964), the externally Bayesian criterion requires that the decisions of the group look as if they were chosen by a Bayesian individual. A process due to D e G r o o t (1974) satisfies this criterion. We argue that this process corresponds to a group decision support system. This reinforces the claim that group decision support systems can improve group decision making.
Keywords: Bayesian decision rule, group decisions, decision support systems, Markov chains
1. Introduction G r o u p decision support systems " a r e technologies designed to help people make faster, more satisfying, and ultimately 'better' decisions than they may be making without them" (Jelassi and Beauclair, 1987). G r o u p decision support systems (GDSS) are often said to help groups of people take better decisions. They constitute, however, an expensive technology. Therefore, the exact nature and scope of their contribution to group decision making need to be carefully assessed. Consider for instance the situation where the members of a group agree on a utility function, but have different opinions concerning the prior distribution of a relevant unknown state of nature. In this context, Madansky (1964) stipulates that the group should be externally Bayesian: Any sequence of group decisions, he says, can be come to by some individual who is a Bayesian, who starts with a particular prior density, and who
December 1988; revised May 1989
observes the sequence of x ' s observed by the group. Can a group decision support system help fulfill this requirement? An affirmative answer to this question is developed below. We argue that a GDSS is indeed sufficient for implementing Bayesian rationality. The externally Bayesian criterion is formally stated and discussed in the next section. In the third section, we present a procedure, due to DeGroot (1974), which makes a group externally Bayesian. This procedure builds up a group consensus through anonymous communication among group members. We point out that it corresponds in practice to a group decision support system. Further research directions are sketched in our fourth and last section.
2. The externally Bayesian criterion Consider a group with members labeled i--1 . . . . . n. All group members agree that the common utility u(a, O)results from choosing action a
0377-2217/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationality
30
t=O i Pio(O),
t=l I
, x is observed f(x[O)
first group decision
revised priors Pil(O) second group decision
Figure 1. Timing of group decisions
from the set A, when 0 is the realised, a priori unknown, state of nature. Let 0 belong to a given possibility set O. In period 0, each expert of the group has a personal opinion about 0 that can be represented by a distribution function F,0, of density pio(O), on O. Now, Figure 1 illustrates the timing of group decisions. An action is selected from A in period 0; a new action is chosen in period 1, after a signal x about 0 is observed. All experts agree on the likelihood of x given 0, denoted f ( x 10). Suppose also that each one is a Bayesian. Hence, at time t = 0, 1, member i will recommend an action that maximises her expected utility
P,, u(a, o ) = f u ( a , O)pi,(O ) din(0),
with ,q(0) being a Bayesian update of *r0(0), that is
~(o) =
f(xlO)%(O)
ff(xlO)~o(O)
din(O)
The externally Bayesian criterion specifies a structure on the sequence of group decisions that rules out many intuitive ways to aggregate the experts' advice. One such intuitive but unsatisfactory procedure is the 'opinion pool' proposed by Stone (1961). This procedure builds up group prior densities by taking a given linear weighted average of the specialists' subjective densities; i.e.
~,(o) = wlpl,(o) + ... +w.p,,(o)
(1)
for t = 0, 1, and wI wn nonnegative summing up to 1. Madansky (1964) shows that a group using this method is not externally Bayesian, unless one of the wi's is unity so that the group is practically lead by a dictator. The reason is that the weights wi's are fixed over time. As it will now be proved, a better procedure is obtained when the weights are adjusted according to the experts' relative reputations. Interpret the weight wit as the probability that expert i's forecast at time t is the most accurate. In period 0, individual i's prediction about signal x is summarized by the density .....
where pa(O) is the new prior probability of 0, obtained from pio(O) via Bayes's formula
f ( x ] O)pio(O )
e,~(o) =
ff(x
I O)pio(O) dm(O)
(To make sure that this optimal action exists, one may assume that A is a compact set and P,u(., O) is continuous on A for each 0 and t = 1, 2.) Assume now that, based on its members' opinions, the group has chosen action a o in period 0, and switched to a~ in period 1 after x was observed. The group is called externally Bayesian if there exists a pair of group prior densities 7r0(0), ~r~(0) on 0 for which ao maximises the group expected utility
f,(x)
=ff(xlO)p,o(O) dm(0).
(2)
and a~ maximises
This density is sometimes called 'the predictive density of x '. At time 1, it now seems reasonable to update the initial weight, denoted wi°, so that more importance is given to expert i's opinion if her forecast about x was good. The new weight w) is then computed using Bayes's formula
P~ ,,(a, a) = f u ( a , o ) ~ ( o ) dm(0)
wi°f,( x ) wli = wOfl(x ) + ... +wOf~(x) .
Po u(a, O) = fu(a, 0)~o(0) d i n ( 0 ) ,
(3)
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationality This is indeed an appropriate method, as the following result shows.
A group is externally Bayesian when it takes as its prior distribution a linear weighted average of its members' subjective distributions, the weights being updated through formulae (2)-(3).
L e m m a 1 (Madansky, 1964).
Proof. The group prior density in period 1 is now
given by
rrl(O ) = w~p11(O ) + . . . +Wlpnl(O) ~ : l w i ° f i ( x ) Pil( O )
~.~_,w°ff(x [O)Pjo(O ) d m ( 0 ) n W 0 E,=I ,f(xlO)p,o(O)
f f ( x I O))~=lw°Pjo(8) d i n ( 0 ) f(xlO)%(O) /f(x[O)~ro(O) dm(O) " The last expression means that the group prior density is updated using Bayes's formula. Hence, the group is externally Bayesian. [] The above result is interesting because it suggests valuable alternatives to dictatorship in group decision making. Dictatorship cannot be rejected for ethical reasons here, since experts' judgments, not individual preferences, are being considered. The prevalence of one's prior probability distribution must be based on her knowledge of the state of nature, and matters of knowledge, unlike matters of taste, imply that some persons are more expert than others. It is, however, unusual that one's expertise is always overwhelming. There are many cases in group decision making where people have equivalent expertise but still distinct opinions. Then it seems better to enhance communication between members so that a consensus on the probability assessments is reached. Such a philosophy underlies the group decision support systems presented in the next section.
3. A procedure for reaching a consensus over group priors
The problem of how to get a consensus among the members of a group has generated much re-
31
search recently. The new interest arose primarily from Aumann's (1976) formalization of the notion of common knowledge. Intuitively, an event is common knowledge among the members of a group if each one knows it, each one knows that everybody knows it, each one knows that everybody knows that everybody knows it, and so o n . . . Aumann shows that if two forecasters have the same prior, if their respective cognitive schemes are represented by partitions of O, and if their posteriors (or updated priors) are common knowledge, then these posteriors must be equal, even when each individual gets different private information. Thereafter, assuming that each one of two experts knows the information the other might have obtained, i.e. each expert knows her fellow's partition of O, supposing also that the partitions are finite, Geanakoplos and Polemarchakis (1982) designed a process where the two experts finally agree on a posterior distribution after communicating to each other their successive revisions of the prior. The theory of common knowledge presupposes that the group members have an identical prior and receive different private information. We consider here the symmetric case where individuals have different prior beliefs while they get the same signal. In this context, DeGroot (1974) constructed a procedure that leads to a group prior distribution over which all members agree. This procedure is described and analysed below. Procedure. Initially, in period t = 0, each expert has an opinion about 0 that is represented by a density function p°o( O). First, each expert makes public her subjective density. Second, when all subjective densities are common knowledge, each expert revises her opinion. Supposedly, the new subjective densities are linear weighted averages of the revealed densities. To describe this process in formal terms, denote expert i's announced subjective density by P~o(O), and assume that she concedes a positive weight vii to each group member's opinion; once she knows her colleagues' densities, she adopts the new subjective density k+l+o) Pi0 ~ = viap~o(O ) + " " " +vinPnko(O).
(4)
Third, one goes back to the first step, with pk0+ 1(.) as expert i's subjective density. The com-
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationafity
32
munication/revision process is repeated again and again, until the personal densities do not sensibly vary any more. Convergence of the procedure to an agreedupon density function is ensured by the following result.
now be established using the theory of Markov chains. The matrix V is a stochastic matrix with positive entries. Thus, by Theorem 2.1 of ~inlar (1975), there must exist a unique vector of weights u - (u 1. . . . . u,) such that for all i, lim wi~k) = u/, k---~ oo
L e m m a 2 (DeGroot, 1974). Using the above proce-
or equivalently,
dure, group members end up giving implicitly the same weights u 1. . . . . u,, to each other's opinion. This leads to the agreed-upon density #o(0) given by
U- W = u .
The emergence of an agreed-upon group density is now at hand since for all i, n
... +u.p.o(O).
(5)
lim p~o(O)= tim E w,7)P,o(O) k--.~
Proof. Let us put the weights v~j together into a matrix V - ( v ~ j ) . At the second iteration of the procedure, formula (4) can be written as /,o(O) = v,,go(O)
+ . . . +VlnP°o(O)]
= [,ilOll "4- "'" "~OinOnl] "pOlo(O )
+
+ . - - + v , . v . . ] .p°o(O)
= E w7'pjo(O), j=l
where the w,~2) ' s,
+v,.v.+],
are just the terms of the matrix V 2 = V- V. Hence, by a trivial induction, one can say that, at the k-th iteration of the procedure, each group member i values his colleagues' opinions according to subjective weights that appear in row i of the matrix g k"
The first assertion of the l e m m a - - t h e appearance of common weights ul . . . . . u , - can experts with terminals
j=l
ujpjo(O)
= ~0(0).
+ . . . +Vin[VnlpO0(o) + . . . +VnnP°o(O)]
[v,lvlj + -"
=
+ . . . +v,.p.'0(0)
=Vn[vllP°o(O)
k ---.oo j = l
[]
Implementing the above procedure requires in general a group decision support system. To display all opinions to everybody, a large common viewing screen is necessary. To suppress pressure coming from, for example, shyness, low status, power struggle or controversial ideas, devices that allow anonymous inputs are essential. To make information circulate efficiently among group members, electronic messaging can help. Finally, to behave as a Bayesian, to compute densities, each group member may need a computer terminal with decision-theoretic software. Actually, at least two particular GDSS can be imagined. One is suggested by the procedure itself. It would elicit the subjective densities, make them public, let the individuals revise their opinion, elicit the new subjective densities, make them public, and so on, until a consensus is reached. This GDSS is illustrated in Figure 2. Another GDSS is suggested by the proof of Lemma 2 and is de-
electronic
messaging
large common screen
Q
Figure 2. A GDSS that iteratively elicits an agreed-upon prior density
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationality
scribed in Figure 3. It would elicit the subjective weights vii and the subjective densities p;0(O), calculate the vector u = (u 1. . . . . u,), compute the agreed-upon prior distribution through formula (5), and then announce it. The main benefit of using these group decision support systems is stated in the following result. Theorem 1. Suppose all group members are Bayesian. The group is externally Bayesian if it uses one of the above decision support systems before taking a decision at time O. Proof. By Lemma 2, if the group uses one of the above GDSS, it reaches a consensus over a prior density distribution. Let this agreed-upon density at time 0 be /%(0). Then each expert and the group itself have the same expected utility
e0 u ( a ,
8)=
fu(a, 0 ) ~ 0 ( o ) d i n ( 0 ) .
Since group members are Bayesian, everyone will recommend an action that maximises this expected utility, so the group will take such an action, say a 0. At time 1, each Bayesian group member will have the same updated prior distribution (o) =
f(xlO)go(O)
ff(x Io)t,0(o) dm(e)
33
A group behaving in such a fashion is externally Bayesian. [] Although they are theoretically equivalent, the two previous GDSS may perform quite differently in practice. Careful experimentation would help clarify this matter. It might happen, for instance, that several group members would be puzzled by a GDSS, like the one depicted in Figure 3, which elicits their personal opinion and comes out with some distribution function. These individuals would perhaps fear that the group decision process has got out of hand. They could then ask for the GDSS to be restarted after x is observed, so that they could input their revised beliefs. If this is done, however, it is interesting to note that the group can still be externally Bayesian. Theorem 2. Suppose all group members are Bayesian. The group can be externally Bayesian when its priors at time 0 and 1 are generated through a group decision support system like the one depicted in Figure 3. Proof. Let the vij°'s be the experts' subjective weights at time 0. Also let u ° . . . . . u,0 be the agreed-upon vector of weights in period 0. By definition of the GDSS and the theory of Markov chains (~inlar, 1975, Theorem 2.1, p. 152), the u i0, s are the unique solution to the equations
Hence the group will adopt this prior, and it will then again choose an action, say a 1, that maximises the expected utility
ujO
P~u( a, 0)
Each expert being a Bayesian, all subjective l weights at time 1, denoted vii, must be obtained
=
f u( a, 0)1~( 0) d i n ( P ) .
experts, with
possibly
terminals
messaging
n
E i=l
u,O0 vij,
E u°=l.
(6)
i=l
common v i e w i n g screen linked to a computer
Figure 3. A GDSS that computes an agreed-upon prior density after a one-shot clicitation round
34
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationafity
from the subjective weights at time 0 through Bayes's formula, i.e.
v~j = vO f l ( x ) + . . . +vOnf,(x ) • Let the GDSS at time 1 simply rescale these subjective weights so that it considers vii1 to be equal to v ° j ( f j ( x ) / f i ( x ) ) . Then the GDSS must compute a vector of agreed-upon weights u~. . . . . ul, that satisfy the equations 1 ~ 1 1 Uj = 2=~ UiVij j=l
=
We made, however, several simplistic assumptions. For instance, in the procedure yielding a consensus, we assumed away that a member could change her subjective weights once she hears her colleagues' opinion. But " a t different points in time, the group is likely to interpret comments differently, use decision rules differently, or rely on power in different ways" (DeSanctis and Gallupe, 1987, p. 592). To deal with such features requires further research.
Acknowledgement
u:oOj(/,(x)//,(x)) i=l
Thanks to formula (6), a solution to this system is precisely
u0f,(x).
I would like to thank Professor Bengt Holmstrom for helping me refine some of the ideas that generated this paper. I am also grateful to an anonymous referee for suggestions that helped me improve the exposition. This work was supported by INSEAD research grant no. 2146R.
A rescaling of these weights so that they sum up to 1 yields
u°f,(x)
1=
ui
uOf,(x) + ... +u°f,(x)
References "
By Lemma 1, the group is then externally Bayesian. []
4. Conclusion Several authors have recently pointed out that "there is little conclusive knowledge regarding how structure can be added to a group decision process to yield better decision making" (DeSanctis and Gallupe, 1987, p, 595). Our present paper provides a partial reply to this statement. We describe two group decision support systems and we show how they can make group decisions achieve Bayesian rationality in particular contexts.
Aumarm, R.J. (1976), "Agreeing to disagree", Annals of Statistics 4, 1236-1239. ~inlar, E. (1975), Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs. DeGroot, M.H. (1974), "Reaching a consensus", Journal of the American Statistical Association 69, 118-121. DeSanctis, G., and Gallupe, R.B. (1987), "A foundation for the study of group decision support systems", Management Science 33, 589-609. Geanakoplos, J.D., and Polemarchakis, H.M. (1982), "We can't disagree forever", Journal of Economic Theory 28, 192-200. Jelassi, M.T., and Beauclair, R.A. (1987), "An integrated framework for group decision support systems design", Information and Management 13, 143-153. Madansky, A. (1964), "Externally Bayesian groups", Memorandum RM-4141-PR, RAND Corporation. Stone, M. (1961), "The opinion pool", Annals of Mathematical Statistics 32, 1339-1342.
29
Theory and Methodology
Group decision support systems implement Bayesian rationality Bernard SINCLAIR-DESGAGNE
INSEAD, Boulevard de Constance, 77305 Fontainebleau, France
Abstract: When saying that group decisions are good, one often means that they are 'externally Bayesian'. Proposed by Madansky (1964), the externally Bayesian criterion requires that the decisions of the group look as if they were chosen by a Bayesian individual. A process due to D e G r o o t (1974) satisfies this criterion. We argue that this process corresponds to a group decision support system. This reinforces the claim that group decision support systems can improve group decision making.
Keywords: Bayesian decision rule, group decisions, decision support systems, Markov chains
1. Introduction G r o u p decision support systems " a r e technologies designed to help people make faster, more satisfying, and ultimately 'better' decisions than they may be making without them" (Jelassi and Beauclair, 1987). G r o u p decision support systems (GDSS) are often said to help groups of people take better decisions. They constitute, however, an expensive technology. Therefore, the exact nature and scope of their contribution to group decision making need to be carefully assessed. Consider for instance the situation where the members of a group agree on a utility function, but have different opinions concerning the prior distribution of a relevant unknown state of nature. In this context, Madansky (1964) stipulates that the group should be externally Bayesian: Any sequence of group decisions, he says, can be come to by some individual who is a Bayesian, who starts with a particular prior density, and who
December 1988; revised May 1989
observes the sequence of x ' s observed by the group. Can a group decision support system help fulfill this requirement? An affirmative answer to this question is developed below. We argue that a GDSS is indeed sufficient for implementing Bayesian rationality. The externally Bayesian criterion is formally stated and discussed in the next section. In the third section, we present a procedure, due to DeGroot (1974), which makes a group externally Bayesian. This procedure builds up a group consensus through anonymous communication among group members. We point out that it corresponds in practice to a group decision support system. Further research directions are sketched in our fourth and last section.
2. The externally Bayesian criterion Consider a group with members labeled i--1 . . . . . n. All group members agree that the common utility u(a, O)results from choosing action a
0377-2217/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationality
30
t=O i Pio(O),
t=l I
, x is observed f(x[O)
first group decision
revised priors Pil(O) second group decision
Figure 1. Timing of group decisions
from the set A, when 0 is the realised, a priori unknown, state of nature. Let 0 belong to a given possibility set O. In period 0, each expert of the group has a personal opinion about 0 that can be represented by a distribution function F,0, of density pio(O), on O. Now, Figure 1 illustrates the timing of group decisions. An action is selected from A in period 0; a new action is chosen in period 1, after a signal x about 0 is observed. All experts agree on the likelihood of x given 0, denoted f ( x 10). Suppose also that each one is a Bayesian. Hence, at time t = 0, 1, member i will recommend an action that maximises her expected utility
P,, u(a, o ) = f u ( a , O)pi,(O ) din(0),
with ,q(0) being a Bayesian update of *r0(0), that is
~(o) =
f(xlO)%(O)
ff(xlO)~o(O)
din(O)
The externally Bayesian criterion specifies a structure on the sequence of group decisions that rules out many intuitive ways to aggregate the experts' advice. One such intuitive but unsatisfactory procedure is the 'opinion pool' proposed by Stone (1961). This procedure builds up group prior densities by taking a given linear weighted average of the specialists' subjective densities; i.e.
~,(o) = wlpl,(o) + ... +w.p,,(o)
(1)
for t = 0, 1, and wI wn nonnegative summing up to 1. Madansky (1964) shows that a group using this method is not externally Bayesian, unless one of the wi's is unity so that the group is practically lead by a dictator. The reason is that the weights wi's are fixed over time. As it will now be proved, a better procedure is obtained when the weights are adjusted according to the experts' relative reputations. Interpret the weight wit as the probability that expert i's forecast at time t is the most accurate. In period 0, individual i's prediction about signal x is summarized by the density .....
where pa(O) is the new prior probability of 0, obtained from pio(O) via Bayes's formula
f ( x ] O)pio(O )
e,~(o) =
ff(x
I O)pio(O) dm(O)
(To make sure that this optimal action exists, one may assume that A is a compact set and P,u(., O) is continuous on A for each 0 and t = 1, 2.) Assume now that, based on its members' opinions, the group has chosen action a o in period 0, and switched to a~ in period 1 after x was observed. The group is called externally Bayesian if there exists a pair of group prior densities 7r0(0), ~r~(0) on 0 for which ao maximises the group expected utility
f,(x)
=ff(xlO)p,o(O) dm(0).
(2)
and a~ maximises
This density is sometimes called 'the predictive density of x '. At time 1, it now seems reasonable to update the initial weight, denoted wi°, so that more importance is given to expert i's opinion if her forecast about x was good. The new weight w) is then computed using Bayes's formula
P~ ,,(a, a) = f u ( a , o ) ~ ( o ) dm(0)
wi°f,( x ) wli = wOfl(x ) + ... +wOf~(x) .
Po u(a, O) = fu(a, 0)~o(0) d i n ( 0 ) ,
(3)
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationality This is indeed an appropriate method, as the following result shows.
A group is externally Bayesian when it takes as its prior distribution a linear weighted average of its members' subjective distributions, the weights being updated through formulae (2)-(3).
L e m m a 1 (Madansky, 1964).
Proof. The group prior density in period 1 is now
given by
rrl(O ) = w~p11(O ) + . . . +Wlpnl(O) ~ : l w i ° f i ( x ) Pil( O )
~.~_,w°ff(x [O)Pjo(O ) d m ( 0 ) n W 0 E,=I ,f(xlO)p,o(O)
f f ( x I O))~=lw°Pjo(8) d i n ( 0 ) f(xlO)%(O) /f(x[O)~ro(O) dm(O) " The last expression means that the group prior density is updated using Bayes's formula. Hence, the group is externally Bayesian. [] The above result is interesting because it suggests valuable alternatives to dictatorship in group decision making. Dictatorship cannot be rejected for ethical reasons here, since experts' judgments, not individual preferences, are being considered. The prevalence of one's prior probability distribution must be based on her knowledge of the state of nature, and matters of knowledge, unlike matters of taste, imply that some persons are more expert than others. It is, however, unusual that one's expertise is always overwhelming. There are many cases in group decision making where people have equivalent expertise but still distinct opinions. Then it seems better to enhance communication between members so that a consensus on the probability assessments is reached. Such a philosophy underlies the group decision support systems presented in the next section.
3. A procedure for reaching a consensus over group priors
The problem of how to get a consensus among the members of a group has generated much re-
31
search recently. The new interest arose primarily from Aumann's (1976) formalization of the notion of common knowledge. Intuitively, an event is common knowledge among the members of a group if each one knows it, each one knows that everybody knows it, each one knows that everybody knows that everybody knows it, and so o n . . . Aumann shows that if two forecasters have the same prior, if their respective cognitive schemes are represented by partitions of O, and if their posteriors (or updated priors) are common knowledge, then these posteriors must be equal, even when each individual gets different private information. Thereafter, assuming that each one of two experts knows the information the other might have obtained, i.e. each expert knows her fellow's partition of O, supposing also that the partitions are finite, Geanakoplos and Polemarchakis (1982) designed a process where the two experts finally agree on a posterior distribution after communicating to each other their successive revisions of the prior. The theory of common knowledge presupposes that the group members have an identical prior and receive different private information. We consider here the symmetric case where individuals have different prior beliefs while they get the same signal. In this context, DeGroot (1974) constructed a procedure that leads to a group prior distribution over which all members agree. This procedure is described and analysed below. Procedure. Initially, in period t = 0, each expert has an opinion about 0 that is represented by a density function p°o( O). First, each expert makes public her subjective density. Second, when all subjective densities are common knowledge, each expert revises her opinion. Supposedly, the new subjective densities are linear weighted averages of the revealed densities. To describe this process in formal terms, denote expert i's announced subjective density by P~o(O), and assume that she concedes a positive weight vii to each group member's opinion; once she knows her colleagues' densities, she adopts the new subjective density k+l+o) Pi0 ~ = viap~o(O ) + " " " +vinPnko(O).
(4)
Third, one goes back to the first step, with pk0+ 1(.) as expert i's subjective density. The com-
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationafity
32
munication/revision process is repeated again and again, until the personal densities do not sensibly vary any more. Convergence of the procedure to an agreedupon density function is ensured by the following result.
now be established using the theory of Markov chains. The matrix V is a stochastic matrix with positive entries. Thus, by Theorem 2.1 of ~inlar (1975), there must exist a unique vector of weights u - (u 1. . . . . u,) such that for all i, lim wi~k) = u/, k---~ oo
L e m m a 2 (DeGroot, 1974). Using the above proce-
or equivalently,
dure, group members end up giving implicitly the same weights u 1. . . . . u,, to each other's opinion. This leads to the agreed-upon density #o(0) given by
U- W = u .
The emergence of an agreed-upon group density is now at hand since for all i, n
... +u.p.o(O).
(5)
lim p~o(O)= tim E w,7)P,o(O) k--.~
Proof. Let us put the weights v~j together into a matrix V - ( v ~ j ) . At the second iteration of the procedure, formula (4) can be written as /,o(O) = v,,go(O)
+ . . . +VlnP°o(O)]
= [,ilOll "4- "'" "~OinOnl] "pOlo(O )
+
+ . - - + v , . v . . ] .p°o(O)
= E w7'pjo(O), j=l
where the w,~2) ' s,
+v,.v.+],
are just the terms of the matrix V 2 = V- V. Hence, by a trivial induction, one can say that, at the k-th iteration of the procedure, each group member i values his colleagues' opinions according to subjective weights that appear in row i of the matrix g k"
The first assertion of the l e m m a - - t h e appearance of common weights ul . . . . . u , - can experts with terminals
j=l
ujpjo(O)
= ~0(0).
+ . . . +Vin[VnlpO0(o) + . . . +VnnP°o(O)]
[v,lvlj + -"
=
+ . . . +v,.p.'0(0)
=Vn[vllP°o(O)
k ---.oo j = l
[]
Implementing the above procedure requires in general a group decision support system. To display all opinions to everybody, a large common viewing screen is necessary. To suppress pressure coming from, for example, shyness, low status, power struggle or controversial ideas, devices that allow anonymous inputs are essential. To make information circulate efficiently among group members, electronic messaging can help. Finally, to behave as a Bayesian, to compute densities, each group member may need a computer terminal with decision-theoretic software. Actually, at least two particular GDSS can be imagined. One is suggested by the procedure itself. It would elicit the subjective densities, make them public, let the individuals revise their opinion, elicit the new subjective densities, make them public, and so on, until a consensus is reached. This GDSS is illustrated in Figure 2. Another GDSS is suggested by the proof of Lemma 2 and is de-
electronic
messaging
large common screen
Q
Figure 2. A GDSS that iteratively elicits an agreed-upon prior density
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationality
scribed in Figure 3. It would elicit the subjective weights vii and the subjective densities p;0(O), calculate the vector u = (u 1. . . . . u,), compute the agreed-upon prior distribution through formula (5), and then announce it. The main benefit of using these group decision support systems is stated in the following result. Theorem 1. Suppose all group members are Bayesian. The group is externally Bayesian if it uses one of the above decision support systems before taking a decision at time O. Proof. By Lemma 2, if the group uses one of the above GDSS, it reaches a consensus over a prior density distribution. Let this agreed-upon density at time 0 be /%(0). Then each expert and the group itself have the same expected utility
e0 u ( a ,
8)=
fu(a, 0 ) ~ 0 ( o ) d i n ( 0 ) .
Since group members are Bayesian, everyone will recommend an action that maximises this expected utility, so the group will take such an action, say a 0. At time 1, each Bayesian group member will have the same updated prior distribution (o) =
f(xlO)go(O)
ff(x Io)t,0(o) dm(e)
33
A group behaving in such a fashion is externally Bayesian. [] Although they are theoretically equivalent, the two previous GDSS may perform quite differently in practice. Careful experimentation would help clarify this matter. It might happen, for instance, that several group members would be puzzled by a GDSS, like the one depicted in Figure 3, which elicits their personal opinion and comes out with some distribution function. These individuals would perhaps fear that the group decision process has got out of hand. They could then ask for the GDSS to be restarted after x is observed, so that they could input their revised beliefs. If this is done, however, it is interesting to note that the group can still be externally Bayesian. Theorem 2. Suppose all group members are Bayesian. The group can be externally Bayesian when its priors at time 0 and 1 are generated through a group decision support system like the one depicted in Figure 3. Proof. Let the vij°'s be the experts' subjective weights at time 0. Also let u ° . . . . . u,0 be the agreed-upon vector of weights in period 0. By definition of the GDSS and the theory of Markov chains (~inlar, 1975, Theorem 2.1, p. 152), the u i0, s are the unique solution to the equations
Hence the group will adopt this prior, and it will then again choose an action, say a 1, that maximises the expected utility
ujO
P~u( a, 0)
Each expert being a Bayesian, all subjective l weights at time 1, denoted vii, must be obtained
=
f u( a, 0)1~( 0) d i n ( P ) .
experts, with
possibly
terminals
messaging
n
E i=l
u,O0 vij,
E u°=l.
(6)
i=l
common v i e w i n g screen linked to a computer
Figure 3. A GDSS that computes an agreed-upon prior density after a one-shot clicitation round
34
B. Sinclair-Desgagne / Group decision support systems implement Bayesian rationafity
from the subjective weights at time 0 through Bayes's formula, i.e.
v~j = vO f l ( x ) + . . . +vOnf,(x ) • Let the GDSS at time 1 simply rescale these subjective weights so that it considers vii1 to be equal to v ° j ( f j ( x ) / f i ( x ) ) . Then the GDSS must compute a vector of agreed-upon weights u~. . . . . ul, that satisfy the equations 1 ~ 1 1 Uj = 2=~ UiVij j=l
=
We made, however, several simplistic assumptions. For instance, in the procedure yielding a consensus, we assumed away that a member could change her subjective weights once she hears her colleagues' opinion. But " a t different points in time, the group is likely to interpret comments differently, use decision rules differently, or rely on power in different ways" (DeSanctis and Gallupe, 1987, p. 592). To deal with such features requires further research.
Acknowledgement
u:oOj(/,(x)//,(x)) i=l
Thanks to formula (6), a solution to this system is precisely
u0f,(x).
I would like to thank Professor Bengt Holmstrom for helping me refine some of the ideas that generated this paper. I am also grateful to an anonymous referee for suggestions that helped me improve the exposition. This work was supported by INSEAD research grant no. 2146R.
A rescaling of these weights so that they sum up to 1 yields
u°f,(x)
1=
ui
uOf,(x) + ... +u°f,(x)
References "
By Lemma 1, the group is then externally Bayesian. []
4. Conclusion Several authors have recently pointed out that "there is little conclusive knowledge regarding how structure can be added to a group decision process to yield better decision making" (DeSanctis and Gallupe, 1987, p, 595). Our present paper provides a partial reply to this statement. We describe two group decision support systems and we show how they can make group decisions achieve Bayesian rationality in particular contexts.
Aumarm, R.J. (1976), "Agreeing to disagree", Annals of Statistics 4, 1236-1239. ~inlar, E. (1975), Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs. DeGroot, M.H. (1974), "Reaching a consensus", Journal of the American Statistical Association 69, 118-121. DeSanctis, G., and Gallupe, R.B. (1987), "A foundation for the study of group decision support systems", Management Science 33, 589-609. Geanakoplos, J.D., and Polemarchakis, H.M. (1982), "We can't disagree forever", Journal of Economic Theory 28, 192-200. Jelassi, M.T., and Beauclair, R.A. (1987), "An integrated framework for group decision support systems design", Information and Management 13, 143-153. Madansky, A. (1964), "Externally Bayesian groups", Memorandum RM-4141-PR, RAND Corporation. Stone, M. (1961), "The opinion pool", Annals of Mathematical Statistics 32, 1339-1342.