A Quantitative Discursive Dilemma

Claussen, Carl Andreas; Røisland, Øistein Working Paper A Quantitative Discursive Dilemma Working Paper, No. 2007/7 Provided in Cooperation with: Norges Bank, Oslo Suggested Citation: Claussen, Carl Andreas; Røisland, Øistein (2009) : A Quantitative Discursive Dilemma, Working Paper, No. 2007/7, ISBN 978-82-7553-401-7, Norges Bank, Oslo, https://hdl.handle.net/11250/2498267 This Version is available at: https://hdl.handle.net/10419/209883 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/deed.no A Quantitative Discursive Dilemma Carl Andreas Claussenyand Øistein Røislandz August 6, 2009 Abstract The typical judgment aggregation problem in economics and other …elds is the following: A group of people has to judge/estimate the value of an uncertain variable ywhich is a function of kother variables, i.e. y=D(x1; :::xk). We analyze when it is possible for the group to arrive at collective judgements on the variables that respect D. We consider aggregators that ful…ll Arrow’s IIA-condition and neutrality. We show how possibility and impossibility depend on the functional form of D, and generalize Pettit’s (2001) binary discursive dilemma to quantitative judgements. Keywords: Judgment aggregation, Dependent variables, Impossibility, Possibility JEL Classi…cation: D71 We thank two anonymous referees for detailed comments and suggestions, and Aanund Hylland and participants at the Judgment Aggregation Workshop in Freudenstadt September 2007 for discussions of this topic. The views presented here are our own and do not necessarily represent those of Norges Bank. yNorges Bank (Central Bank of Norway), P.O. Box 1179, Sentrum, 0107 Oslo, Norway (Phone: +47–22316104, fax: +47–22333568, Email: [email protected] / [email protected] ) zNorges Bank, (Central Bank of Norway), P.O. Box 1179, Sentrum, 0107 Oslo, Norway (Phone: +47–22316739, fax: +47–22333568, Email: [email protected]) 1 1 Introduction The typical decision problem – or judgment aggregation situation –in economics and other …elds is the following: A group of people has to judge or estimate the size of a variable which is a function of some other variables. For example, a monetary policy committee’s interest rate decisions depend on judgments about in‡ationary pressures and …nancial fragility; a cabinet’s judgment of the future budget balance depends on its judgments of future revenues and costs; a corporate board’s investment decisions depend on judgments of future cash ‡ows and cost of capital. In this paper we analyze when a group of people with di¤erent judgments on the variables can make a collective judgment on the variables that respects the dependence between the variables. To illustrate the issue and its importance, consider a corporate board assessing the pro…tability of an investment project. The pro…tability is measured as the project’s expected net present value (NPV ), which per de…nition is the discounted cash ‡ow (DCF) less the investment cost (IC). Thus, the dependence between the variables is given by NPV =DCF IC. Suppose that the corporate board has three members with estimates as in Table 1. Let the members of the board vote on the size of each variTable 1: Example of aggregate inconsistency for a corporate board assessing the net present value of an investment project. Discounted cash ‡ow Investment cost Net present value (DCF) (IC) (NPV ) Member A 10 8 2 Member B 10 11 1 Member C 13 12 1 Board 10 11 1 able, and assume that the outcome of the vote is the median of the individual estimates. Then a vote on the conclusion-variable gives NPV = 1. But, this is not consistent with the majority’s judgments on the two ’premise-variables’, since 10 11 = 1. Thus, the aggregate judgments do not respect the dependence between the variables. As a consequence the board faces a discursive dilemma (Pettit 2001). A premise-based procedure, where they vote on the two premise variables and let the conclusion follow, gives NPV =1. A conclusion-based procedure, where they vote directly on NPV , gives NPV = 1. In this paper we investigate if the example illustrates a general problem for groups aggregating judgments on dependent variables. We therefore construct a general social choice theoretic model and ask the following question: Under which conditions are there combinations of individual judgments that give aggregate judgments that do not respect the dependence between the variables (impossibility), and under which conditions are there no such combinations (possibility)? By using a general social choice theoretic framework we can treat all aggregation methods ful…lling some general conditions simultaneously. Examples of aggregators ful…lling our conditions are pairwise majority voting over the alternative values for each variable, and an agenda-setting method whereby the aggregate judgment of any two alternatives for one variable is the judgment of the agenda setter (the same each time) unless a supermajority has another judgment. In our model a group of people has to conclude on the value of a dependent variable xk+1 when the value of this variable depends on the value of kindependent variables 2 x1,..., xkby some general ’dependence function’D: xk+1 =D(x1; :::; xk) The dependence function can be a reaction function derived from maximizing an objective function, it can be a rule-of-thumb, a causal relationship between economic variables, a de…nition, or any mapping from values of the independent variables to the dependent variable. The arguments in the dependence function are the variables and parameters on which the members of the group may have di¤erent judgments. Variables and parameters that are relevant for the dependent variable, but which the members of the group always agree on, may be represented by the functional form of D. Suppose, for instance, that y=x is a policy rule where yis a policy instrument (e.g. the central bank’s key interest rate), xis an economic variable (e.g. the rate of underlying in‡ation), and is a parameter that says how much a change in xshould a¤ect y. Then the dependence function is the policy rule (with xas the argument) if all individuals always agree on the value of . Otherwise the dependence function has two arguments: xand . When modelling the judgments on each variable we follow the social choice tradition. We assume that each member holds a strict order over the alternative values for each of the k+ 1 variables: an order over the alternative values of variable 1, an order over the alternative values for variable 2, ..., and an order over the alternative values for variable k+1. The group uses an aggregator that takes pro…les of individual orders (one order for each member) as inputs. The aggregator produces an aggregate relation (not necessarily an order), and ful…lls a set of standard general conditions. The conditions are ’unanimity/Pareto’, ’independence’and a strong and a weaker form of ’neutrality’. We derive characterization results for when there exist non-dictatorial aggregators such that the peaks of the aggregate relations respect the dependence function. It is seen that possibility arises only in the special case when k= 1 and the dependence function is strictly monotonic. The paper has four sections. In Section 2 we present the model. In Section 3 we give the main characterization. We conclude with a discussion of our framework and key assumptions in Section 4. Relation to the literature Considering the aggregation of di¤erent interconnected variables is not new. A variety of aggregation problems has been proposed and solved in production theory, see Blackorby & Schworm (1984) for an overview. In opinion pooling the probability assignments of di¤erent individuals are to be merged into collective probability assignments. Genest & Zidek (1986) give an overview of classical results in opinion pooling. See Mongin (1995) and Dietrich & List (2007b) for more recent results. Rubinstein & Fishburn (1986) consider the problem of aggregating the entries in nrows in an nmmatrix into a summary row, where every entry is an element in an algebraic …eld. They …nd that if the entries always form a hyperplane, then every consistent aggregator is an aggregator whereby the aggregate estimate of a variable is the (normalized) linear sum of the individual estimates. If the entries do not form a hyperplane there is no consistent non-dictatorial aggregator. In an earlier paper on quantitative discursive dilemmas we (Claussen and Røisland 2005) study a situation somewhat similar to the situation studied by Rubinstein & Fishburn (1986), but where we assign one variable the role as a dependent variable and the other variables the role as independent variables. Furthermore we have less strict domain restrictions. In that paper we …nd that if the group aggregates by taking the mean of 3 the individual estimates, then the boundary between possibility and impossibility lies in whether or not the dependence function is linear. If the group aggregates by taking the median of the individual estimates, the boundary lies in whether or not the dependence function is strictly monotonic. In the current paper we step out of the model of our previous paper and the literature on the aggregation of di¤erent interconnected variables by considering the aggregation of k+1 orders rather than aggregation of k+1 estimates. By this move we are able to study the situation when a group aggregates by some voting method, rather than by just combining estimates. The setting of this paper is also somewhat parallel to a setting where a group of people aggregates judgments on interconnected propositions. In such situations an aggregation inconsistency akin to the inconsistency in the example of Table 1 may arise. Pettit (2001) coined that inconsistency the ’discursive dilemma’. Recently, researchers have built general social choice theoretic models to study the aggregation of judgments on propositions. The …rst example is List & Pettit (2002). They also provided the …rst impossibility result which was quickly followed by several stronger impossibility and possibility results. Roughly speaking, the impossibility results say that if the propositions under consideration are interlinked, then there is no aggregator that ful…ls requirements similar to, but not exactly equal to, the Arrovian requirements that aggregate consistent individual judgments on propositions into consistent collective judgments on these propositions. See Dietrich (2007) for a generalized model of judgment aggregation, and List & Puppe (2009) for an overview of the literature. Compared to the judgment aggregation literature the important novelty our current paper is that variables need not be binary. Thus, we introduce a generalization of Pettit’s (2001) discursive dilemma to non-binary and continuous variables. 2 The Model1 We consider a group, where N=f1; :::; ngdenotes the set of members, and n > 2.2 Each member i2Nwill be referred to as a ’member’or an ’individual’depending on the context. The group has to evaluate real-valued variables j= 1; :::; k + 1 where k1 and each variable jtakes values in a non-empty set XjR. This set has at least two elements and might be …nite or in…nite. Examples are Xj=R,Xj= [0;1], and the binary case where Xj=f0;1gas in standard judgment aggregation. The variables 1; :::; k will be denoted ’independent variables’, and variable k+ 1 the ’dependent variable’. Let D:X1::: Xk!Xk+1 be a surjective function, the dependence function, representing how the dependent variable k+ 1 depends on the independent variables 1; :::; k.3 Apreference relation on a set Xjis an arbitrary binary relation on Xj.4Its asymmetric part (representing strict preference) is as usual denoted by and de…ned as the binary relations on Xjgiven by xy,[xyand not yx]for all x; y 2Xj. Its symmetric part (representing indi¤erence) is as usual denoted by and de…ned as the 1This and the following section have bene…tted greatly by detailed comments and suggestions from one of the referees. 2We assume n > 2to make the propositions clear-cut (not contingent on n). If n= 2, systematicity (see section 3) implies that the dependence must be dictatorial, regardless of the functional form of D. 3A function Dis said to be surjective or onto, if its values span its whole codomain; that is, for every xk+1 2Xk+1 , there is at least one vector (x1; ::; xk)2X1::: Xksuch that D(x1; ::; xk) = xk+1. 4The term ’preference’should not be taken literally and is not meant as a restriction. The model applies in many contexts; see Section 4 for examples and a discussion. 4 Figure 1: Illustration with k= 1 and concave dependence function. binary relations on Xjgiven by xy,[xyand yx]for all x; y 2Xj. Let G Xj be the set of complete and anti-symmetric preference relations on Xj(i.e. all preference relations on Xjthat satisfy [xyor yx]for all distinct x; y 2Xjand xxfor all x2Xj). Let GXjbe the set of complete, anti-symmetric and transitive preference relations on Xj, (i.e. complete and anti-symmetric preference relations that also satisfy [x<yand y<z])x<zfor all x; y; z 2Xj). An element in GXjis called a (strict) order. Note that GXj(G Xjas the relations in G Xjneed not be transitive. A value x2Xjis the peak of variable junder <Xj2 G Xjif xXjyfor all y2Xjnx. Peaks will sometimes, depending on the context, be called estimates. A sequence of relations (<X1; :::; <Xk+1 )2 G X1:::  G Xk+1 is said to respect the dependence function Dif xk+1 =D(x1; :::; xk)whenever x1; :::; xk+1 are peaks of <X1; :::; <Xk+1 , respectively. We will also use a (rationality) requirement for alternatives ranked lower than the peaks. The requirement will rule out sequences that respect Dbut where the preference relations are otherwise somewhat arbitrary. To give an illustration, put k= 1,X1= fv; x; y; zg;and suppose Dis concave as illustrated in Figure 1. Let the preference relation on X1be vxyz. Our requirement will then allow for individual sequences with orders over the corresponding alternatives in X2like D(v)D(x)D(y)D(z) or D(v)D(z)D(y)D(x); but it will rule out arbitrary individual sequences like sequences where the order on X2 is D(v)D(y)D(x)D(z) or D(v)D(x)D(z)D(y). Formally, a sequence of relations <X1; :::; <Xk+1 2 G X1:::G Xk+1 is called arbitrary if 5 there exists x0;x00;x000 2X1::: Xkand m2 f1; :::; kgwith x0 m; x00 m; x000 mthree pairwise distinct elements in Xmand x0 j=x00 j=x000 jfor all j6=m, such that (i) Dis strictly monotonic for x0;x00;x000, (ii) x0 mXmx00 mXmx000 mand (iii) <Xk+1 ranks D(x00)strictly above or below both D(x000)and D(x0). A sequence is non-arbitrary if it is not arbitrary. Let Gbe the set of all non-arbitrary sequences (<X1; :::; <Xk+1 )2 GX1:::  GXk+1 that respect D. A pro…le (of sequences), denoted g, is an (k+ 1)n-tuple in Gnwith one sequence for each member. An aggregator fis a mapping f:Gn! G X1:::  G Xk+1 : Notice that the aggregator takes (pro…les of) orders as inputs but produce relations. Thus, we do not require the outcome of the aggregation to be transitive. Furthermore, we do not require the outcome of the aggregation to be non-arbitrary. The aggregator respects the dependence function Dif, for every pro…le g2 Gn,f(g)respects D. Denote individual orders by <i;Xjand aggregate relations by <N;Xj. We say that the aggregator is non-dictatorial if there is no i2Nsuch that for all pro…les g2 Gn, f(g) = (<i;Xj)j=1;:::;k+1. 3 The Characterization We will now see when there is a g2 Gnsuch that f(g)does not respect the dependence function (impossibility), and when f(g)respects the dependence function for all g2 Gn (possibility). We will consider aggregators that ful…ll a set of standard conditions. The …rst condition is the unanimity principle which says that if every member of the group …nds that x2Xjis better than y2Xj, then the collective view should also be that xis better than y. The second condition is Arrow’s independence condition (IIA). This condition says that the aggregator obtains aggregate relations by comparing two alternatives at a time taken in isolation from the other alternatives. Thus, the aggregate preference of any pair of alternatives for a variable will depend exclusively on the individual preferences over that pair. Consequently, the aggregation is independent between variables and independent for each variable seen in isolation. Formally, the conditions are as follows. Unanimity principle/Pareto: For all (<i;Xj)i2N;j=1;:::;k+1 2 Gn, all j2 f1; :::; k + 1g and all x; y 2Xj, if xi;Xjyfor all i2N, then xN;Xjy. Independence (of Irrelevant Alternatives): For any two pro…les in Gn,ga= (<a i;Xj )i2N;j=1;:::;k+1; gb= (<b i;Xj)i2N;j=1;:::;k+1, any variable jand any alternatives x0; x00 2 Xj, if for all individuals i[x0<a i;Xjx00 ,x0<b i;Xjx00], then [x0<a N;Xjx00 , x0<b N;Xjx00]. In addition to ful…lling independence, we require the aggregator to be neutral in two respects. First, if the aggregate preference over two alternatives for one variable is determined by some method, for example a pair-wise majority vote, then the aggregate preference on any other two alternatives for the same variable shall be determined by the same method. Second, if the aggregate preference relation on one variable is determined by some method, then the aggregate preference relation on any other variable shall 6 be determined by the same method. Neutrality and independence give the following condition:5 Systematicity:For any two pro…les in Gn,ga= (<a i;Xj)i2N;j=1;:::;k+1; gb= (<b i;Xj )i2N;j=1;:::;k+1, any two variables jand m, and any alternatives x0 j; x00 j2Xjand x0 m; x00 m2Xm, if for all individuals i[x0 j<a i;Xjx00 j,x0 m<b i;Xmx00 m]then [x0 j<a N;Xj x00 j,x0 m<b N;Xmx00 m]. The following proposition holds. Proposition 1 A non-dictatorial aggregator f:Gn! G X1:::G Xk+1 that satis…es the Unanimity principle and Systematicity respects the dependence function Dif and only if k= 1 and Dis strictly monotonic. Proof. The proof is in the appendix. It might come as a surprise that the boundary between possibility and impossibility is somewhat simpler in our framework where the group aggregates orders than in a model of aggregating just estimates, c.f. Rubinstein & Fishburn (1986). In the latter case, a crucial question is whether the dependence function is linear. The reason why we get the simpler boundary is that in our case only the relative ranking of the estimates for a variable matter in the aggregation. In the literature on the aggregation of estimates the relative size of the individuals’estimates matters for the aggregate estimate. This is because the authors assume that the aggregate estimate is some linear or non-linear combination of the individual estimates. An exception is Claussen and Røisland (2005) where we assume that the aggregate estimate is the median of the individual estimates. With this aggregator, the crucial question is whether the dependence function is strictly monotonic or not, as it is in the case with aggregating orders. 4 Discussion We will conclude by a discussion of our framework and some key assumptions. Preference relations We use the term ’preference relation’and not e.g. ’judgment relation’, as ’preference relation’is the well established term in the literature. The term ’preference’should not be taken literally and is not meant as a restriction. The model applies in many contexts. To see this, remember that the de…nition of a preference relation only says that each member can, for any two distinct alternatives x; y 2Xj, say that she ’prefers’xto y(or yto x). The de…nition does not say anything about why she ’prefers’xto y. Member icould, for instance, prefer xto ybecause she …nds that xgives her higher utility than y, she could prefer xto ybecause she believes that xis closer to the true value of the variable than y(it is a "better estimate"), or – if variable jis a policy variable –she could prefer xto ybecause she …nds that xgives higher social welfare than y. Another question is if the members actually hold preference relations. Empirically it is clear that members of many groups do. In monetary policy, for instance, the minutes of the meetings of the monetary policy committees reveal that the members disagree and 5The condition is inspired by a similar concept from the literature on the aggregation of judgments on propositions where it was …rst introduced by List & Pettit (2002). It will be relaxed somewhat in Section 4. 7 have preference relations over the relevant alternatives for the key interest rate and the premise-variables. Similarly, minutes and reports of other expert panels reveal that the members have preference relations over relevant alternatives for the relevant variables. In formal models any cardinal utility function embodies a preference relation. Strict preference relations The assumption of strict preference relations may at …rst glance seem strong. But, it is usually a reasonable assumption, in particular if variables are continuous. If a variable is continuous, it is in practice impossible for the group to consider all possible alternatives in the aggregation. What groups normally do is to perform an aggregation –the ’ote’–over a limited set of alternatives. These alternatives will typically be each member’s preferred value, i.e. the peaks of the individual preference relations. The combination of a continuous variable and aggregation over peaks only imply that it is reasonable to assume strict preferences. To see this, suppose …rst that Zj(6=;)is a convex subset of Rrepresenting all possible values variable jcan take. Let fx1; :::; xng 2 Zn j be the set of peaks with one peak for each member of N(orders may be weak). As the group only consider the set of peaks when aggregating we put Xj=fx1; :::; xng, i.e. the set of alternatives that is up for a vote is Xj(and not Zj). Let member i’s preferences over the alternatives in Zjbe described by the ’preference’ function u= (xxi)2, where xi2Zjis the most preferred alternative of member i. Notice that the function imply a single peaked (weak) preference relation. Suppose that xiis drawn from a distribution described by a continuous density function hi(x)over Zj(allowing for di¤erent continuous density functions for each member). Now, suppose for contradiction that member iis indi¤erent between two distinct alternatives xs; xm2Xjnxi. It then follows from the preference function that xs= 2xixm. However, as all elements in Xjare drawn from continuous distributions, we have that per de…nition Rxm xmhs(x)dx = Rxs xshm(x)dx = 0. Thus, there is zero probability that xs= 2xixm, i.e. there is zero probability that one of the members of Nis indi¤erent between two alternatives in Xj. Discrete variables are often discrete for practical reasons (e.g. rounding), not because the variable is discrete in nature. An argument similar to the argument above therefore applies. The interest rate decisions of monetary policy committees are illustrative. The key policy interest rate of a central bank is a continuous variable that in theory can take any value in R. However, in practice monetary committees only consider alternatives in the set Q=f1 4;1 2;3 4;1;11 4; :::g. Suppose that the members’preferences are described by u=(xxi)2as above. Suppose that a member of a monetary policy committee …nd 2to be the best level of the key rate among the elements in Q. Would she then be indi¤erent between 13 4and 21 2? If xi= 2 she would. However the preference relation over Rwould typically have its peak at another value than 2. When considering the true set of values that the key rate may take, the member’s best estimate would be, say, 1:90, but the preferred rate of the alternatives in Qis 2(the closest feasible one). But then, with a symmetric preference relation, the member would not be indi¤erent between 13 4 and 21 2. She would prefer 13 4to 21 2. The aggregator The aggregator used in the model takes the individual orders as inputs, and produces an aggregate relation for each variable. We think it is relevant to study the properties of this aggregator for at least three reasons: First, if the members of the group cannot agree, but have to reach a decision, they have to use some aggregation method. Many groups resort to majority voting or some other method that implicitly take ordinal preferences as the input and output of the aggregation. They do this even though the primary 8 Using claim 1 for the neglected parts we then have that f(g) = yN;X1xN;X1zN;X1:::; D(z)N;X2D(x)N;X2D(y)N;X2::: , a sequence that does not respect D. Case 3. k > 1, and Dis non-monotonic in one or more j2 f1; :::; kg. Let Dbe non-monotonic in some variable m2 f1; :::; kg. Let <0 Xjbe any order on variable j. Let Y Gnbe the set of pro…les where <i;Xj=<0 Xjfor all i2Nand all j2 f1; :::; kgnm. By Claim 1 we then have that for all pro…les g2Y,<N;Xj=<0 Xj for all j2 f1; :::; kgnm. For all g2Ywe may then plug the peaks (estimates) of <N;Xjj=f1; :::; kgnminto Dand consider Da function of xmonly. It then follows from case 1 and 2 that fdoes not respect the dependence function. Case 4.k > 1, and Dis strictly monotonic in all j2 f1; :::; kgand non-injective. We consider the case when k= 2:The extention to cases when k > 2is straight forward (c.f. case 3). As Dis strictly monotonic and non-injective, there exsists two distinct elements x0 1; x00 1in X1and two distinct elements x0 2; x00 2in X2, such that x0 3=D(x0 1; x0 2),x00 3=D(x0 1; x00 2) = D(x00 1; x0 2)and x000 3=D(x00 1; x00 2)are the three corresponding pairwise distinct alternatives in X3. Consider the pro…le g2 Gnwhere x0 1i;X1x00 1i;X1:::; x0 2i;X2x00 2i;X2::: and x0 3i;X3x00 3i;X3x000 3i;X3::: if i2C1, x0 1i;X1x00 1i;X1:::; x00 2i;X2x0 2i;X2::: and x00 3i;X3x0 3i;X3x000 3i;X3::: if i2C2, x00 1i;X1x0 1i;X1:::; x0 2i;X2x00 2i;X2::: and x00 3i;X3x000 3i;X3x0 3i;X3::: if i2C3. Using claim 1 for the neglected parts we then have that f(g) = 0 @ x0 1N;X1x00 1N;X1:::; x0 2N;X2x00 2N;X2:::; x00 3N;X3x0 3N;X3x000 3N;X3::: 1 A, a sequence that does not respect D. Case 5.k > 1, and Dis strictly monotonic in all j2 f1; :::; kgand injective. We consider the case when k= 2:The extention to cases when k > 2is straight forward (c.f. case 3). As Dis strictly monotonic and injective, there exsists two distinct elements x0 1; x00 1in X1 and two distinct elements x0 2; x00 2in X2, such that x0 3=D(x0 1; x0 2),x00 3=D(x0 1; x00 2),x000 3= D(x00 1; x0 2)and x0000 3=D(x00 1; x00 2)are the four corresponding pairwise distinct alternatives in X3. Put x0 3< x00 3< x000 3< x0000 3(The analysis of the other cases is similar when Dis monotonic.). Consider a pro…le g2 Gnwhere x0 1i;X1x00 1i;X1:::; x0 2i;X2x00 2i;X2::: and x0 3i;X3x00 3i;X3x000 3i;X3x0000 3i;X3::: if i2C1; x0 1i;X1x00 1i;X1:::; x00 2i;X2x0 2i;X2::: and x00 3i;X3x000 3i;X3x0 3i;X3x0000 3i;X3::: if i2C2; x00 1i;X1x0 1i;X1:::; x0 2i;X2x00 2i;X2::: and x000 3i;X3x0000 3i;X3x00 3i;X3x0 3i;X3::: if i2C3: Using claim 1 for the neglected parts we then have that f(g) = 0 @ x0 1N;X1x00 1N;X1:::; x0 2N;X2x00 2N;X2:::; x00 3N;X3x000 3N;X3x0 3N;X3x0000 3N;X3::: 1 A, a sequence that does not respect D. 15 Proof of Corollary As the set of aggregators satisfying Systematicity is a subset of the set of aggregators satisfying Weak systematicity it follows from our proposition that no non-dictatorial aggregator f:Gn! G X1:::  G Xk+1 that satis…es the Unanimity principle and Weak systematicity respects the dependence function when k > 1or Dis non-monotonic. Thus, it su¢ ces for the proof to consider the case when k= 1 and Dis strictly monotonic. Put k= 1 and let Dbe strictly monotonic. Then there are two distinct elements x; y 2X1such D(x)6=D(y). Let fAand fBbe two di¤erent aggregators satisfying the Unanimity principle and Systematicity. Let g2 Gnbe a pro…le where fA(g)6=fB(g). Denote the set of winning coalitions under fAby CfA, and the set of winning coalitions under fBby CfB(’winning coalition’is de…ned as in the proof of the proposition). As fA(g)6=fB(g)we have that there is a coalition CAsuch that that CA2 CfAand CA=2 CfB. Put jX1j>2. As fA and fBsatisfy the Unanimity principle and Systematicity claim 1 and 2 in the proof of our proposition also apply to fAand fB. By claim 1, CA2 CfA)NnCA=2 CfA. By claim 1 and 2, NnCA2 CfB. Thus, we have that there is a partition of Nsuch that N=fCA; NnCAgand CA2 CfAand NnCA2 CfB. Let fCbe an aggregator where the aggregate relation on variable 1is determined by the same aggregation method that is used in fAand the aggregate relation on variable 2is determined by the same aggregation method that is used in fB. Let g2 Gnbe a pro…le where all members of CArank x strictly above all other alternatives in X1and D(x)strictly above all other alternatives in X2, and the members of NnCArank ystrictly above all other alternatives for X1and D(y)strictly above all other alternatives in X2. Then fC(g)have peaks x; D(y). 16