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The effect of expansions and substitutions on group decision-making
:~
mathematical
social sciences
EI.SEVIER
Mathematical Social Sciences 30 (1995) 263-271
The effect of expansions and substitutions on group decision-making D r o r a Karotkin, Shmuel Nitzan* Department of Economics, Bar-llan University. 52900 Ramat-Gan, Israel Received July 1994; revised January 1995
Abstract In the current paper we analyze the effect of expansions and substitutions on the (optimal) performance of a group that makes decisions in pairwise choice situations. Within our framework expansions cannot be detrimental to group performance. Proposition 1 provides a necessary and sufficient condition for the invariance of group performance to expansions of order m. This condition implies that group performance is never invariant to such expansions if the original group consists of an even number of homogeneous members. In such a case any expansion always favorably affects group performance. Substituting an individual with at least one more skillful individual obviously results in increased group performance. Our search for conditions ensuring the invariance or the inferiority of group performance to substitutions is therefore naturally confined to individually quality-reducing substitutions, i.e. some group member is replaced by individuals whose decisional competencies are inferior to his. Proposition 2 provides a sufficient condition for the inferiority of individually quality-reducing substitutions of order m, rn -> 2. Proposition 3 establishes that in a homogeneous group with an odd number of members, individually quality-reducing substitutions of order 2 always adversely affect group performance. Keywords: Group decision-making; Performance
Dichotomous choice; Expansions; Substitutions;
1. Introduction When a production function is monotonically increasing in inputs, there is a natural trade-off between any two inputs, the marginal rate of substitution * Corresponding author. 0165-4896/95/$09.50 ~ 1995 - Elsevier Science B.V. All rights reserved SSDI 0165-4896(95)00790-3
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between the inputs being negative. In the context of group or organizational decision-making, the individual decisional quality and the number of decisionmakers can be viewed as inputs to the collective decisional quality-the performance of the group. Employing the uncertain dichotomous choice framework, Nitzan and Paroush (1980) have shown that in a group of homogeneous individuals who make collective decisions by resorting to the simple majority rule, the marginal product of individual quality (the common individual decisional skill) and the marginal productivity of quantity (the number of decision-makers) are positive. There exists therefore a natural trade-off between quality and quantity of decision-makers. This natural trade-off is carried over to the asymmetric dichotomous choice setting where qualified majority rules are the decision rules applied by the group (Sah and Stiglitz, 1988). Nitzan and Paroush's (1984) study of this asymmetric setting contains an analysis of the trade-off between the common individual decisional skill and the number of decision-makers along iso-optimal qualified majority rule curves (the quality-quantity combinations generating a particular optimal qualified majority rule). Along such curves quantity and quality are always inversely related. More recently, Gradstein et al. (1990) studied a different trade-off, namely the sensitivity of the optimal quantity (the number of individuals that ought to participate in the collective decision-making process) to the common individual decisional quality. Their comparative statics analysis established that the optimal quantity is not necessarily inversely related to individual quality. The current study is also concerned with the quality vs. quantity issue within the symmetric uncertain pairwise choice setting (Nitzan and Paroush, 1982; Sah and Stiglitz, 1985, 1988; Shapley and Grofman, 1984), but not necessarily under the assumption of homogeneous individuals. Specifically, we study the effect on group performance of two types of changes in the distribution of decisional inputs in which the number of decision-makers is increased. In the first case the group of decision-makers is expanded. When the decisional skills of the rn individuals added to the group are lower than those of the existing group members, we refer to the expansion as an individually quality-reducing expansion of order rn. In the second case some group member (for notational convenience we choose the least competent member) is replaced by other individuals. Substituting a group member with at least one more skillful individual obviously results in increased group performance. Our search for conditions ensuring the invariance or the inferiority of group performance to substitutions is therefore confined to individually quality-reducing substitutions of order rn (rn t> 2), i.e. some group member is replaced by rn individuals whose decisional competencies are inferior to his. There are many instances in which voting or decision rights are opened up to include additional decision-makers. In an economic organizational context,
D. Karotkin, S. Nitzan / Mathematical Social Sciences 30 (1995) 263-271
265
augmenting the size of an existing group of decision-makers seems a natural possibility that ought to be considered if the organization is committed to efficiency. The need to substitute a decision-maker also arises quite frequently. For example, when a group member resigns, is fired or, simply, is unable to fulfill his decision-making task (e.g. absence due to health problems). Notice that decision-making groups are often formed on the basis of quality or expertise. That is, the candidates of higher individual decisional quality are selected to the group. When the selection is confined to a fixed set of candidates (such as members of the board of directors), this implies that added members will be of a lower quality. This further justifies our interest in individually quality-reducing substitutions. An expansion cannot adversely affect the performance of the group, regardless of whether it is individually quality-reducing or not. The possibility of enhancing group performance certainly exists when the group is augmented by more skillful individuals. However, it may also exist even when less skillful individuals are added to the group. Our first result provides a necessary and sufficient condition for the invariance of group performance to group expansions. Feld and Grofman (1984) and Paroush and Karotkin (1989) study, respectively, the effect of group expansions on the performance of simple majority rule and on the optimality of restricted majority rules. In contrast to Feld and Grofman (1984), our analysis is not confined to the performance of the simple majority rule which may not be the optimal group decision rule. In contrast to Paroush and Karotkin (1989), our analysis is not confined to the case where the optimal decision rule is a restricted majority rule of order k (a committee of k members applying a simple majority rule). In fact, our first result generalizes Theorem 4 in Paroush and Karotkin (1989). Our analysis of substitutions yields two results establishing sufficient conditions for the inferiority of individually quality-reducing substitutions of order m, m -> 2. The uncertain pairwise choice framework is presented in Section 2. The results are presented in Section 3. A brief summary is given in Section 4.
2. The model
A group of individuals N = {1 . . . . . n) faces two mutually exclusive alternatives, a and b. The individual preferences are identical; however, their independent judgements concerning the alternative that ought to be chosen, the alternative that better suits theircommon objective, may differ. The decisional skill of individual i is represented by his probability Pi of choosing the correct alternative. With no loss of generality, it is assumed that Pi->0.5 and that i < j ~ p i - > p j . The decisional resources of the group are given by p = (P~ . . . . . Pn). An expansion of order m, m >-1, is a transformation of p to
266
D. K a r o t k i n , S. N i t z a n
M a t h e m a t i c a l Social Sciences 30 ( 1 9 9 5 ) 2 6 3 - 2 7 1
(P, q) = ( P l , .
• • , P , , P,+I . . . . Pn+m) = (Pl . . . . . P,, qx, " • , qm)" An individually quality-reducing expansion of order m, m-> 1, is a transformation of p to
(P, q) = (P~ . . . . . P , , P,+I . . . . . P,+m) = (P, . . . . . P , , q~ . . . . , qm) such that qj < p , for j = 1 . . . . . m. An individually quality-reducing substitution of order m, m -> 1, is a transformation of p -- (p~ . . . . . p , ) to ( p l . . . . . P , - ~ , q~ . . . . . qm) such that qj < Pn for j = 1 . . . . . m. An individually quality-reducing substitution of order m might replace the least competent individual in N with less competent individuals whose total optimal weights reflecting their decisional skills is equal to m wq = w,, where wjq = In qj/(1 the optimal weight of individual n, that is, E j=, qj). In other words, the weight assigned to the substituted individual is distributed among the newcomers to the group. An individually quality-reducing substitution satisfying this property will be called an equal-weights substitution o f order m. To study the effect of expansions on quality-reducing substitutions on the decisional quality of the group, namely the probability ~r that the group makes a correct decision, we need to introduce the notion of a collective decision rule. Such a rule is a function f from the set of individual decisions { - 1, 1}" to the set of collective decisions { - 1 , 1}, where - 1 and 1 represent, respectively, decisions in favor of alternative a and alternative b. An element in the domain of the function f is an n-tuple of individual decisions, or a decision profile x = ( x x , . . . , x , ) , xi E { - 1 , 1}. The set of all possible decision rules is denoted F. The group performance 7r corresponding to p is the probability of making a correct collective choice using the optimal decision rule, f. In other words, Ir = I r ( f ( p ) ) =meax 7r(f; p ) .
The solution of this problem (see Nitzan and Paroush, 1982; Shapley and Grofman, 1984) is given by the symmetric weighted majority rule with individual weights that are equal to the logarithms of the individual odds of making a correct choice. That is, f ( p ) = sign
wix i
,
where w i = In p i / ( 1 - p i ) . The weight w~ can serve as a natural measure for the optimal decisional effectivity or political authority of individual i as well as an alternative measure of his decisional skill.
3. Results Our first result provides a necessary and sufficient condition for the neutrality of expansions, and in particular individually quality-reducing expansions. The optimal assignment of weights in the original group is w I . . . . . w n. An expansion
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of order m makes a difference, i.e. changes the optimal decision rule and, in turn, increases the performance of the group if the added members alter the decision of the original group in at least one possible profile of decisions. The ability of the added members to affect the group decision depends on the relationship between their optimal weights w q . . . . . w,,q and the optimal weights of the original group members. Specifically, denote by S* a subset of N that solves the following problem:
min/ w, Z w,} ti~-S
]EN\S
such that
w,- jEN\S w,)>0 and let
Proposition 1. A n expansion of order m transforming p to (p, q), such that ~j~_~ w q <- if: is neutral, that is, 7r(f(p)) = 7r(j~(p, q)). Proof. Denote by X the set of all possible decision profiles. With no loss of generality, let alternative b be the correct alternative. Hence, x i = 1 (xi = - 1 ) implies that individual i makes a correct (incorrect) decision. A collective decision rule partitions the set X into two subsets. The first subset consists of all decision profiles resulting in a correct collective decision. A profile x in this subset satisfies ~=~ x~w~ > 0 . The probability of a correct collective decision is equal to the probability of obtaining a decision profile which belongs to this subset, Pr(ETz~x~wi>0). The second subset consists of all profiles resulting in an incorrect collective decision. An expansion of order m multiplies the number of possible decision profiles by 2 m. Since ~j=l wq ~< if, an expansion of a decision profile that originally led to a correct (incorrect) collective decision still leads to a correct (incorrect) collective decision. That is, n
m
sign
x~w~ = sign
x~w~
as #---F~7~_~,mlxiw~>--vL Since the probability of obtaining one of the two possible types of expanded profiles is equal to one, the expansion of order m satisfying E j ~ w q ~< ~ is neutral, that is, the probability of a correct collective decision is unaltered. Q.E.D. Proposition 1 clarifies that the value of expansions hinges on the comparison
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m between r~ and ~j=~ wq and not on the comparison between the mean, median or majority competence in the original and the expanded groups (see Feld and Grofman, 1984, where such comparisons are made assuming that the original and the augmented groups always apply a variant of the simple majority rule). When the original group consists of a single member, i.e. p = (Pl), Proposition m 1 implies that v~ = w 1 --~i=~ wq is the necessary and sufficient condition for the neutrality of an individually quality-reducing expansion of order m, namely for the optimality of the expert rule within the augmented (m + 1)-member group (see the second corollary to the main result in Nitzan and Paroush, 1982). More generally, by Proposition 1, if N consists of an odd number of homogeneous individuals, i.e. p = ( P l . . . . , Pn)= (P* . . . . . p*), then an individually qualityreducing expansion of order m transforming p to (p, q) = (P*,''',P*,q~,''',qm), such that Ejmlw q ~ < ~ = l n p * / ( 1 - p * ) is neutral, that is, ~r(f(p)) = ~r(f(p, q)). This corollary of Proposition 1 is a special case of Theorem 4 in Paroush and Karotkin (1989). In fact, this latter theorem, which assumes that the optimal rule within the original n-member group is a restricted majority rule of order k (a simple majority rule applied within k members of the original n-member group), is also a special case of Proposition 1. If N consists of an even number of homogeneous individuals, then by Proposition 1, ~ = 0 and therefore any expansion improves the performance of the group as long as q~ > 1 / 2 . This is true in particular for an individually quality-reducing expansion of order 1. In other words, when a minimally competent individual joins an even n-member homogeneous group the optimal decision rule for the expanded group becomes the simple majority rule. Regardless of the gap between the decisional skills of the existing members and the decisional quality of the new member, the latter member is assigned decisional effectivity which is equal to that of the other superior decision-makers. The next result provides a sufficient condition for the inferiority of individually quality-reducing substitutions of order m, m -> 2.
Proposition 2.
A n equal-weights substitution o f order m, m >- 2, adversely affects the performance o f the group.
Proof. Suppose that m = 2 . By assumption, then Wlq->w q and w q + w q = w n. Before the replacement of individual n, under the optimal collective decision rule the group makes a correct collective decision if the total weight assigned to the correct alternative exceeds a = Z wi/2. Given the decision of the n - 1 individuals in the subgroup N~n, there are three possible situations: (i) The group makes a correct decision irrespective of individual n's decision, i.e. E N\n wixi ~- a. (ii) The group does not make a correct decision, irrespective of individual n's decision, i.e. ~u\n WiXi < Ct -- Wn.
D. Karotkin, S. Nitzan / Mathematical Social Sciences 30 (1995) 263-271
(iii) Individual n can be an effective pivot, i.e. a > Zx~n wixi Hence, ~r(f(p)) = Pr (,~,
269
>- a - w,.
wixi>-a)+p, P r ( a > v,,, ~ w,x>_a-w,,).
Since w, = Wlq + wg, when individual n is replaced by the two less qualified individuals, a does not change. Given the decisions of the n - 1 individuals in the subgroup N ~ , there are now six possible situations: (1) Under situation (i) the group makes a correct decision irrespective of the decisions made by the two individuals substituting individual n. (2) Under situation (ii) the group makes an incorrect decision independent of the decisions made by the two individuals substituting individual n. Under situation (iii), (3) If the two individuals replacing n make a correct decision, the group makes a correct decision. (4) If the more competent individual among the two substituting n makes a correct decision and the less competent one makes an incorrect decision, the group may reach a correct decision. (5) If the more competent individual among the two substituting n makes an incorrect decision and the less competent one makes a correct decision, the group reaches an incorrect decision. (6) If the two individuals replacing n make an incorrect decision, the group reaches an incorrect decision. Denote by 7r* the probability of a correct collective decision after the equalweights substitution of individual n takes place, assuming that in situations of type (4) the group makes a correct decision. Define rr* = 7r(f(p 1. . . . . Pn-l, ql' q2))" We then have
rr* = Pr(~,~, wixi>-a) + [q'q2 + q l ( 1 - q2)]Pr(a > ,vn ~ wixi>-a - w " ) Since qlq: -q~(1 - q2) = ql < P , , r * < 7r(f(p)), which establishes the inferiority of equal-weights substitutions of order 2. Any equal-weights substitution of order m, m > 2, can be decomposed to a series of equal-weights substitutions of order 2. In each stage the performance of the group is reduced and certainly this is the effect of the equal-weights substitution of order m Q.E.D. If the group N consists of an odd number of homogeneous individuals, then by Proposition 1, an individually quality-reducing expansion of order m can improve group performance provided that Ejm Wq > V~= In p * / ( 1 --p*). However, even in such a case substitution of a member by m = 2 new less competent individuals is disadvantageous, as shown below.
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Proposition 3. L e t p = (Pl . . . . , p . ) , n being odd, satisfy pi = p * , Vi = 1 . . . . . n. A quality-reducing substitution of order 2 transforming p to (Pl . . . . . P.-~, q~, q2) adversely affects group performance, that is, ¢r(f(p I . . . . . p~_l,ql,q2))<
~r(f(p)). Proof. For n = 1, ¢r(f(pl)) =Pl > "/r(?(ql, q2)) = max{ql, q2} since, by assumption, qi < P l , i = 1, 2. For n > 1, the optimal decision rule f(p) is the simple majority rule. By assumption, after the substitution of individual n the optimal weights (wl . . . . , w,_ 1, w q, w q) satisfy the equality w<,+l)/2 + . . . + w,_ I = w I + . . . + W(n_l)/2 and the inequality w q -> w~. Hence, w(,+l)/~ + • • • + w,_ 1 + w 7 -> w 1 + • .. + w(,_~)/z + w q. But this is the sufficient condition ensuring that the restricted majority rule of order n is the optimal rule for the new (n + 1)-member group (see Gradstein, 1986). This means that the optimal rule is a simple majority rule applied within the n more competent individuals whose skills are given by ( P l , ' " ", P,-~, q~). Since the probability of a correct collective decision, given that decisions are made by the simple majority rule, is monotonically increasing in each individual decisional skill and in particular in p , , p~ > q~ implies that ~ ( f ( p ) ) > ~r(f(pl . . . . . P , - 1 , ql, q2)). Q.E.D. When the group consists of an even number n of equally skilled individuals, the optimal decision rule is a restricted majority rule of order n - 1. This implies that one individual, say individual n, is inessential. Substituting him with two less competent individuals is in fact an individually quality-reducing expansion of order 2 transforming the decisional skills of the essential individuals (Pl, • • •, P , - 1 ) to (PI, • • •, P~-~, ql, q2). By Proposition 1, the effect of such an expansion on the performance of the group depends on the relationship between w~ + w~ and w*. Technically, then, when n is even an individually qualityreducing substitution of order 2 (which is in fact an expansion) could be advantageous. Finally, note that according to the definition of an individually quality-reducing substitution of order m, the least competent member n is substituted by m less competent individuals. This definition simplifies the notation in the proofs of Propositions 2 and 3. The propositions remain valid if the substituted member is any m e m b e r t, t ~ n, of the original group.
4. Summary This study is concerned with the effect on group performance of two types of changes in the distribution of decisional inputs in which the number of decisionmakers is increased. In the first case the group of decision-makers is expanded. In
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the second case some individual is replaced by m less competent ones ( m - 2). These two types of changes are, respectively, referred to as expansions and individually quality-reducing substitutions of order m. The accuracy of majority decisions in groups with added members is studied in Feld and Grofman (1984). Notice that in the current work we assume that the group utilizes the optimal decision rule which is usually not the simple majority rule. This latter assumption might be less realistic. Nevertheless it renders possible the extension of the normative analysis of group decision-making which focuses on the trade-off between group size and quality. The effect of expansions on the performance of the group (the quality of collective decision-making) is studied for any m, m -> 1. Our first result (Proposition 1) provides a necessary and sufficient condition for the invariance of group performance to expansions. The effect on collective performance of individually quality-reducing substitutions of order m is studied for m >- 2. The second result provides a sufficient condition for the inferiority of individually quality-reducing substitutions of order m, m -> 2 (Proposition 2). The third result establishes that, for a homogeneous group with an odd number of members, individually qualityreducing substitutions of order 2 are disadvantageous, i.e. they always reduce the decisional performance of the group (Proposition 3).
Acknowledgment We are indebted to an anonymous referee of this journal for his useful suggestions.
References S.L. Feld and B. Grofman, The accuracy of group majority decisions in groups with added members, Public Choice 42 (1984) 273-285. M. Gradstein, Necessary and sufficient conditions for the optimality of simple majority and restricted majority decisions rules, Theory and Decision 21 (1986) 181-187. M. Gradstein, S. Nitzan and J. Paroush, Collective decision making and the limits on the organization's size, Public Choice 66 (1990) 279-291. S. Nitzan and J. Paroush, Investment in human capital and social self protection under uncertainty, Int. Econ. Rev. 21 (1980) 547-557. S. Nitzan, and J. Paroush, Optimal decision rules in uncertain dichotomous choice situations. Int. Econ. Rev. 23 (1982) 289-297. S. Nitzan and J. Paroush, Are qualified majority rules special?, Public Choice 42 (1984) 257-272. J. Paroush and D. Karotkin, Robustness of optimal majority rules over teams with changing size. Social Choice and Welfare 6(2) (1989) 127-138. R.K. Sah, and J.E. Stiglitz, Human fallibility in economic organizations, Amer. Econ. Rev., Papers and Proceedings 75 (1985) 292-297. R.K. Sah and J.E. Stiglitz, Committees, hierarchies and polyarchies, The Econ. J. 98 (1988) 451-470. L. Shapley and B. Grofman, Optimizing group judgemental accuracy in the presence of interdependence, Public Choice 43 (1984) 329-343.
mathematical
social sciences
EI.SEVIER
Mathematical Social Sciences 30 (1995) 263-271
The effect of expansions and substitutions on group decision-making D r o r a Karotkin, Shmuel Nitzan* Department of Economics, Bar-llan University. 52900 Ramat-Gan, Israel Received July 1994; revised January 1995
Abstract In the current paper we analyze the effect of expansions and substitutions on the (optimal) performance of a group that makes decisions in pairwise choice situations. Within our framework expansions cannot be detrimental to group performance. Proposition 1 provides a necessary and sufficient condition for the invariance of group performance to expansions of order m. This condition implies that group performance is never invariant to such expansions if the original group consists of an even number of homogeneous members. In such a case any expansion always favorably affects group performance. Substituting an individual with at least one more skillful individual obviously results in increased group performance. Our search for conditions ensuring the invariance or the inferiority of group performance to substitutions is therefore naturally confined to individually quality-reducing substitutions, i.e. some group member is replaced by individuals whose decisional competencies are inferior to his. Proposition 2 provides a sufficient condition for the inferiority of individually quality-reducing substitutions of order m, rn -> 2. Proposition 3 establishes that in a homogeneous group with an odd number of members, individually quality-reducing substitutions of order 2 always adversely affect group performance. Keywords: Group decision-making; Performance
Dichotomous choice; Expansions; Substitutions;
1. Introduction When a production function is monotonically increasing in inputs, there is a natural trade-off between any two inputs, the marginal rate of substitution * Corresponding author. 0165-4896/95/$09.50 ~ 1995 - Elsevier Science B.V. All rights reserved SSDI 0165-4896(95)00790-3
264
D. Karotkin, S. Nitzan / Mathematical Social Sciences 30 (1995) 263-271
between the inputs being negative. In the context of group or organizational decision-making, the individual decisional quality and the number of decisionmakers can be viewed as inputs to the collective decisional quality-the performance of the group. Employing the uncertain dichotomous choice framework, Nitzan and Paroush (1980) have shown that in a group of homogeneous individuals who make collective decisions by resorting to the simple majority rule, the marginal product of individual quality (the common individual decisional skill) and the marginal productivity of quantity (the number of decision-makers) are positive. There exists therefore a natural trade-off between quality and quantity of decision-makers. This natural trade-off is carried over to the asymmetric dichotomous choice setting where qualified majority rules are the decision rules applied by the group (Sah and Stiglitz, 1988). Nitzan and Paroush's (1984) study of this asymmetric setting contains an analysis of the trade-off between the common individual decisional skill and the number of decision-makers along iso-optimal qualified majority rule curves (the quality-quantity combinations generating a particular optimal qualified majority rule). Along such curves quantity and quality are always inversely related. More recently, Gradstein et al. (1990) studied a different trade-off, namely the sensitivity of the optimal quantity (the number of individuals that ought to participate in the collective decision-making process) to the common individual decisional quality. Their comparative statics analysis established that the optimal quantity is not necessarily inversely related to individual quality. The current study is also concerned with the quality vs. quantity issue within the symmetric uncertain pairwise choice setting (Nitzan and Paroush, 1982; Sah and Stiglitz, 1985, 1988; Shapley and Grofman, 1984), but not necessarily under the assumption of homogeneous individuals. Specifically, we study the effect on group performance of two types of changes in the distribution of decisional inputs in which the number of decision-makers is increased. In the first case the group of decision-makers is expanded. When the decisional skills of the rn individuals added to the group are lower than those of the existing group members, we refer to the expansion as an individually quality-reducing expansion of order rn. In the second case some group member (for notational convenience we choose the least competent member) is replaced by other individuals. Substituting a group member with at least one more skillful individual obviously results in increased group performance. Our search for conditions ensuring the invariance or the inferiority of group performance to substitutions is therefore confined to individually quality-reducing substitutions of order rn (rn t> 2), i.e. some group member is replaced by rn individuals whose decisional competencies are inferior to his. There are many instances in which voting or decision rights are opened up to include additional decision-makers. In an economic organizational context,
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augmenting the size of an existing group of decision-makers seems a natural possibility that ought to be considered if the organization is committed to efficiency. The need to substitute a decision-maker also arises quite frequently. For example, when a group member resigns, is fired or, simply, is unable to fulfill his decision-making task (e.g. absence due to health problems). Notice that decision-making groups are often formed on the basis of quality or expertise. That is, the candidates of higher individual decisional quality are selected to the group. When the selection is confined to a fixed set of candidates (such as members of the board of directors), this implies that added members will be of a lower quality. This further justifies our interest in individually quality-reducing substitutions. An expansion cannot adversely affect the performance of the group, regardless of whether it is individually quality-reducing or not. The possibility of enhancing group performance certainly exists when the group is augmented by more skillful individuals. However, it may also exist even when less skillful individuals are added to the group. Our first result provides a necessary and sufficient condition for the invariance of group performance to group expansions. Feld and Grofman (1984) and Paroush and Karotkin (1989) study, respectively, the effect of group expansions on the performance of simple majority rule and on the optimality of restricted majority rules. In contrast to Feld and Grofman (1984), our analysis is not confined to the performance of the simple majority rule which may not be the optimal group decision rule. In contrast to Paroush and Karotkin (1989), our analysis is not confined to the case where the optimal decision rule is a restricted majority rule of order k (a committee of k members applying a simple majority rule). In fact, our first result generalizes Theorem 4 in Paroush and Karotkin (1989). Our analysis of substitutions yields two results establishing sufficient conditions for the inferiority of individually quality-reducing substitutions of order m, m -> 2. The uncertain pairwise choice framework is presented in Section 2. The results are presented in Section 3. A brief summary is given in Section 4.
2. The model
A group of individuals N = {1 . . . . . n) faces two mutually exclusive alternatives, a and b. The individual preferences are identical; however, their independent judgements concerning the alternative that ought to be chosen, the alternative that better suits theircommon objective, may differ. The decisional skill of individual i is represented by his probability Pi of choosing the correct alternative. With no loss of generality, it is assumed that Pi->0.5 and that i < j ~ p i - > p j . The decisional resources of the group are given by p = (P~ . . . . . Pn). An expansion of order m, m >-1, is a transformation of p to
266
D. K a r o t k i n , S. N i t z a n
M a t h e m a t i c a l Social Sciences 30 ( 1 9 9 5 ) 2 6 3 - 2 7 1
(P, q) = ( P l , .
• • , P , , P,+I . . . . Pn+m) = (Pl . . . . . P,, qx, " • , qm)" An individually quality-reducing expansion of order m, m-> 1, is a transformation of p to
(P, q) = (P~ . . . . . P , , P,+I . . . . . P,+m) = (P, . . . . . P , , q~ . . . . , qm) such that qj < p , for j = 1 . . . . . m. An individually quality-reducing substitution of order m, m -> 1, is a transformation of p -- (p~ . . . . . p , ) to ( p l . . . . . P , - ~ , q~ . . . . . qm) such that qj < Pn for j = 1 . . . . . m. An individually quality-reducing substitution of order m might replace the least competent individual in N with less competent individuals whose total optimal weights reflecting their decisional skills is equal to m wq = w,, where wjq = In qj/(1 the optimal weight of individual n, that is, E j=, qj). In other words, the weight assigned to the substituted individual is distributed among the newcomers to the group. An individually quality-reducing substitution satisfying this property will be called an equal-weights substitution o f order m. To study the effect of expansions on quality-reducing substitutions on the decisional quality of the group, namely the probability ~r that the group makes a correct decision, we need to introduce the notion of a collective decision rule. Such a rule is a function f from the set of individual decisions { - 1, 1}" to the set of collective decisions { - 1 , 1}, where - 1 and 1 represent, respectively, decisions in favor of alternative a and alternative b. An element in the domain of the function f is an n-tuple of individual decisions, or a decision profile x = ( x x , . . . , x , ) , xi E { - 1 , 1}. The set of all possible decision rules is denoted F. The group performance 7r corresponding to p is the probability of making a correct collective choice using the optimal decision rule, f. In other words, Ir = I r ( f ( p ) ) =meax 7r(f; p ) .
The solution of this problem (see Nitzan and Paroush, 1982; Shapley and Grofman, 1984) is given by the symmetric weighted majority rule with individual weights that are equal to the logarithms of the individual odds of making a correct choice. That is, f ( p ) = sign
wix i
,
where w i = In p i / ( 1 - p i ) . The weight w~ can serve as a natural measure for the optimal decisional effectivity or political authority of individual i as well as an alternative measure of his decisional skill.
3. Results Our first result provides a necessary and sufficient condition for the neutrality of expansions, and in particular individually quality-reducing expansions. The optimal assignment of weights in the original group is w I . . . . . w n. An expansion
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of order m makes a difference, i.e. changes the optimal decision rule and, in turn, increases the performance of the group if the added members alter the decision of the original group in at least one possible profile of decisions. The ability of the added members to affect the group decision depends on the relationship between their optimal weights w q . . . . . w,,q and the optimal weights of the original group members. Specifically, denote by S* a subset of N that solves the following problem:
min/ w, Z w,} ti~-S
]EN\S
such that
w,- jEN\S w,)>0 and let
Proposition 1. A n expansion of order m transforming p to (p, q), such that ~j~_~ w q <- if: is neutral, that is, 7r(f(p)) = 7r(j~(p, q)). Proof. Denote by X the set of all possible decision profiles. With no loss of generality, let alternative b be the correct alternative. Hence, x i = 1 (xi = - 1 ) implies that individual i makes a correct (incorrect) decision. A collective decision rule partitions the set X into two subsets. The first subset consists of all decision profiles resulting in a correct collective decision. A profile x in this subset satisfies ~=~ x~w~ > 0 . The probability of a correct collective decision is equal to the probability of obtaining a decision profile which belongs to this subset, Pr(ETz~x~wi>0). The second subset consists of all profiles resulting in an incorrect collective decision. An expansion of order m multiplies the number of possible decision profiles by 2 m. Since ~j=l wq ~< if, an expansion of a decision profile that originally led to a correct (incorrect) collective decision still leads to a correct (incorrect) collective decision. That is, n
m
sign
x~w~ = sign
x~w~
as #---F~7~_~,mlxiw~>--vL Since the probability of obtaining one of the two possible types of expanded profiles is equal to one, the expansion of order m satisfying E j ~ w q ~< ~ is neutral, that is, the probability of a correct collective decision is unaltered. Q.E.D. Proposition 1 clarifies that the value of expansions hinges on the comparison
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m between r~ and ~j=~ wq and not on the comparison between the mean, median or majority competence in the original and the expanded groups (see Feld and Grofman, 1984, where such comparisons are made assuming that the original and the augmented groups always apply a variant of the simple majority rule). When the original group consists of a single member, i.e. p = (Pl), Proposition m 1 implies that v~ = w 1 --~i=~ wq is the necessary and sufficient condition for the neutrality of an individually quality-reducing expansion of order m, namely for the optimality of the expert rule within the augmented (m + 1)-member group (see the second corollary to the main result in Nitzan and Paroush, 1982). More generally, by Proposition 1, if N consists of an odd number of homogeneous individuals, i.e. p = ( P l . . . . , Pn)= (P* . . . . . p*), then an individually qualityreducing expansion of order m transforming p to (p, q) = (P*,''',P*,q~,''',qm), such that Ejmlw q ~ < ~ = l n p * / ( 1 - p * ) is neutral, that is, ~r(f(p)) = ~r(f(p, q)). This corollary of Proposition 1 is a special case of Theorem 4 in Paroush and Karotkin (1989). In fact, this latter theorem, which assumes that the optimal rule within the original n-member group is a restricted majority rule of order k (a simple majority rule applied within k members of the original n-member group), is also a special case of Proposition 1. If N consists of an even number of homogeneous individuals, then by Proposition 1, ~ = 0 and therefore any expansion improves the performance of the group as long as q~ > 1 / 2 . This is true in particular for an individually quality-reducing expansion of order 1. In other words, when a minimally competent individual joins an even n-member homogeneous group the optimal decision rule for the expanded group becomes the simple majority rule. Regardless of the gap between the decisional skills of the existing members and the decisional quality of the new member, the latter member is assigned decisional effectivity which is equal to that of the other superior decision-makers. The next result provides a sufficient condition for the inferiority of individually quality-reducing substitutions of order m, m -> 2.
Proposition 2.
A n equal-weights substitution o f order m, m >- 2, adversely affects the performance o f the group.
Proof. Suppose that m = 2 . By assumption, then Wlq->w q and w q + w q = w n. Before the replacement of individual n, under the optimal collective decision rule the group makes a correct collective decision if the total weight assigned to the correct alternative exceeds a = Z wi/2. Given the decision of the n - 1 individuals in the subgroup N~n, there are three possible situations: (i) The group makes a correct decision irrespective of individual n's decision, i.e. E N\n wixi ~- a. (ii) The group does not make a correct decision, irrespective of individual n's decision, i.e. ~u\n WiXi < Ct -- Wn.
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(iii) Individual n can be an effective pivot, i.e. a > Zx~n wixi Hence, ~r(f(p)) = Pr (,~,
269
>- a - w,.
wixi>-a)+p, P r ( a > v,,, ~ w,x>_a-w,,).
Since w, = Wlq + wg, when individual n is replaced by the two less qualified individuals, a does not change. Given the decisions of the n - 1 individuals in the subgroup N ~ , there are now six possible situations: (1) Under situation (i) the group makes a correct decision irrespective of the decisions made by the two individuals substituting individual n. (2) Under situation (ii) the group makes an incorrect decision independent of the decisions made by the two individuals substituting individual n. Under situation (iii), (3) If the two individuals replacing n make a correct decision, the group makes a correct decision. (4) If the more competent individual among the two substituting n makes a correct decision and the less competent one makes an incorrect decision, the group may reach a correct decision. (5) If the more competent individual among the two substituting n makes an incorrect decision and the less competent one makes a correct decision, the group reaches an incorrect decision. (6) If the two individuals replacing n make an incorrect decision, the group reaches an incorrect decision. Denote by 7r* the probability of a correct collective decision after the equalweights substitution of individual n takes place, assuming that in situations of type (4) the group makes a correct decision. Define rr* = 7r(f(p 1. . . . . Pn-l, ql' q2))" We then have
rr* = Pr(~,~, wixi>-a) + [q'q2 + q l ( 1 - q2)]Pr(a > ,vn ~ wixi>-a - w " ) Since qlq: -q~(1 - q2) = ql < P , , r * < 7r(f(p)), which establishes the inferiority of equal-weights substitutions of order 2. Any equal-weights substitution of order m, m > 2, can be decomposed to a series of equal-weights substitutions of order 2. In each stage the performance of the group is reduced and certainly this is the effect of the equal-weights substitution of order m Q.E.D. If the group N consists of an odd number of homogeneous individuals, then by Proposition 1, an individually quality-reducing expansion of order m can improve group performance provided that Ejm Wq > V~= In p * / ( 1 --p*). However, even in such a case substitution of a member by m = 2 new less competent individuals is disadvantageous, as shown below.
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Proposition 3. L e t p = (Pl . . . . , p . ) , n being odd, satisfy pi = p * , Vi = 1 . . . . . n. A quality-reducing substitution of order 2 transforming p to (Pl . . . . . P.-~, q~, q2) adversely affects group performance, that is, ¢r(f(p I . . . . . p~_l,ql,q2))<
~r(f(p)). Proof. For n = 1, ¢r(f(pl)) =Pl > "/r(?(ql, q2)) = max{ql, q2} since, by assumption, qi < P l , i = 1, 2. For n > 1, the optimal decision rule f(p) is the simple majority rule. By assumption, after the substitution of individual n the optimal weights (wl . . . . , w,_ 1, w q, w q) satisfy the equality w<,+l)/2 + . . . + w,_ I = w I + . . . + W(n_l)/2 and the inequality w q -> w~. Hence, w(,+l)/~ + • • • + w,_ 1 + w 7 -> w 1 + • .. + w(,_~)/z + w q. But this is the sufficient condition ensuring that the restricted majority rule of order n is the optimal rule for the new (n + 1)-member group (see Gradstein, 1986). This means that the optimal rule is a simple majority rule applied within the n more competent individuals whose skills are given by ( P l , ' " ", P,-~, q~). Since the probability of a correct collective decision, given that decisions are made by the simple majority rule, is monotonically increasing in each individual decisional skill and in particular in p , , p~ > q~ implies that ~ ( f ( p ) ) > ~r(f(pl . . . . . P , - 1 , ql, q2)). Q.E.D. When the group consists of an even number n of equally skilled individuals, the optimal decision rule is a restricted majority rule of order n - 1. This implies that one individual, say individual n, is inessential. Substituting him with two less competent individuals is in fact an individually quality-reducing expansion of order 2 transforming the decisional skills of the essential individuals (Pl, • • •, P , - 1 ) to (PI, • • •, P~-~, ql, q2). By Proposition 1, the effect of such an expansion on the performance of the group depends on the relationship between w~ + w~ and w*. Technically, then, when n is even an individually qualityreducing substitution of order 2 (which is in fact an expansion) could be advantageous. Finally, note that according to the definition of an individually quality-reducing substitution of order m, the least competent member n is substituted by m less competent individuals. This definition simplifies the notation in the proofs of Propositions 2 and 3. The propositions remain valid if the substituted member is any m e m b e r t, t ~ n, of the original group.
4. Summary This study is concerned with the effect on group performance of two types of changes in the distribution of decisional inputs in which the number of decisionmakers is increased. In the first case the group of decision-makers is expanded. In
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the second case some individual is replaced by m less competent ones ( m - 2). These two types of changes are, respectively, referred to as expansions and individually quality-reducing substitutions of order m. The accuracy of majority decisions in groups with added members is studied in Feld and Grofman (1984). Notice that in the current work we assume that the group utilizes the optimal decision rule which is usually not the simple majority rule. This latter assumption might be less realistic. Nevertheless it renders possible the extension of the normative analysis of group decision-making which focuses on the trade-off between group size and quality. The effect of expansions on the performance of the group (the quality of collective decision-making) is studied for any m, m -> 1. Our first result (Proposition 1) provides a necessary and sufficient condition for the invariance of group performance to expansions. The effect on collective performance of individually quality-reducing substitutions of order m is studied for m >- 2. The second result provides a sufficient condition for the inferiority of individually quality-reducing substitutions of order m, m -> 2 (Proposition 2). The third result establishes that, for a homogeneous group with an odd number of members, individually qualityreducing substitutions of order 2 are disadvantageous, i.e. they always reduce the decisional performance of the group (Proposition 3).
Acknowledgment We are indebted to an anonymous referee of this journal for his useful suggestions.
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