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Multidimensional welfare rankings under weight imprecision: a social choice perspective

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Abstract

Ranking alternatives based on multidimensional welfare indices depends, sometimes critically, on how the different dimensions of welfare are weighted. In this paper, a theoretical framework is presented that yields a set of consensus rankings in the presence of such weight imprecision. The main idea is to consider a vector of weights as an imaginary voter submitting preferences over alternatives. With this voting construct in mind, the well-known Kemeny rule from social choice theory is introduced as a means of aggregating the preferences of many plausible choices of weights, suitably weighted by the importance attached to them. The axiomatic characterization of Kemeny’s rule due to Young and Levenglick (1978) and Young (1988) extends to the present context. An analytic solution is derived for an interesting special case of the model corresponding to generalized weighted means and the \(\epsilon \)-contamination framework of Bayesian statistics. The model is applied to the ARWU index of Shanghai University. Graph-theoretic insights are shown to facilitate computation significantly.

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Notes

  1. While conceptually straightforward, the application of Tarjan’s famous circuit-detection algorithm to facilitate the computation of Kemeny optimal rankings has not, to the best of my knowledge, been previously pursued in the literature.

  2. For simplicity, I choose the normalized values to lie in [0,1], though any other interval \([x_{\min }, x_{\max }]\) would also work.

  3. For example, in the case of the HDI, \(f\) could be set in the following manner: ask each country \(c\) to provide its importance function \(f_c\) on \(\Delta ^2\) and then set \(f=\sum _{c=1}^C f_c\) (where \(C\) is the total number of countries).

  4. In the case of discrete \(f\), the integrals in Eqs. (3)–(4) would become summations over the relevant subset of \(\Delta ^{I-1}\) and all of the results naturally extend.

  5. To be clear, in this paper I am using the term partial ranking to refer to a weak ordering: a reflexive, transitive and complete binary relation over the set of alternatives.

  6. In the literature it is also sometimes referred to as the “maximum likelihood” or the “Condorcet” rule. As Young noted in Young (1988), the Marquis de Condorcet was the first to propose this way of deciding an election, even if he did not work out the formal details.

  7. This definition of Kendall-\(\tau \) distance in the presence of partial rankings is consistent with Ailon (2007) and van Zuylen and Williamson (2009). It is appropriate in cases such as the paper’s, where partial input rankings are allowed but the output is constrained to be a full ranking.

  8. Hence, note that while input rankings are allowed to be partial, \(K\) still has to be a set of full rankings (Ailon 2007; van Zuylen and Williamson 2009).

  9. A skew symmetric matrix \(\varvec{Y}\) is a square matrix satisfying \(\varvec{Y}=-\varvec{Y}'\).

  10. In what follows I will suppress dependence of the results on \(\varvec{\bar{w}}\) to avoid cumbersome notation.

  11. While straightforward, I am unaware of other papers that make this connection to facilitate the computation or approximation of Kemeny-optimal rankings. In the numerical example, I use a Matlab implementation of Tarjan’s algorithm due to David F. Gleich, which can be found at http://www.mathworks.it/matlabcentral/fileexchange/24134-gaimc-graph-algorithms-in-matlab-code/content/gaimc/scomponents.m.

  12. Here \(\varvec{e_k}\) denotes the corresponding standard basis vector in \(\mathfrak {R}^{I-1}\).

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Acknowledgments

I am grateful to Jim Lawrence for helpful and clarifying remarks and to Michaela Saisana for introducing me to this interesting area of research. I also would like to thank Andrea Saltelli for his useful comments. The views expressed herein are purely those of the author and may not in any circumstances be regarded as stating an official position of the European Commission.

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Correspondence to Stergios Athanassoglou.

Appendices

Appendix 1: Proofs

Proposition 3

Consider case (a) first. Since \(g\) is invertible,

$$\begin{aligned} W^{\varvec{X}_{\mathcal {A}},u}_{a_ia_j} \cap W^{\varvec{X}_{\mathcal {A}},u}_{a_ja_i}=\left\{ \varvec{w} \in \Delta ^{I-1}: \; \; \sum _{i=1}^I w_i u_i(x_{a_i i})=\sum _{i=1}^I w_i u_i(x_{a_j i})\right\} . \end{aligned}$$

As the functions \(u_i\) are strictly monotone, unless \(\varvec{x_{a_i}}=\varvec{x_{a_j}}\) the above set will be a polytope of dimension \(I-2\). Thus, \(W^{\varvec{X}_{\mathcal {A}},u}_{a_ia_j} \cap W^{\varvec{X}_{\mathcal {A}},u}_{a_ja_i}\) will have zero Lebesgue measure in \(\Delta ^{I-1}\). The result follows.

Case (b) is exactly similar, with the only difference that the intersection of \(W^{\varvec{X}_{\mathcal {A}},u}_{a_ia_j}\) and \(W^{\varvec{X}_{\mathcal {A}},u}_{a_ja_i}\) reduces to:

$$\begin{aligned} W^{\varvec{X}_{\mathcal {A}},u}_{a_ia_j} \cap W^{\varvec{X}_{\mathcal {A}},u}_{a_ja_i}=\left\{ \varvec{w} \in \Delta ^{I-1}: \; \; \sum _{i=1}^I w_i \log \left( u_i(x_{a_i i})\right) =\sum _{i=1}^I w_i \log \left( u_i(x_{a_j i})\right) \right\} . \end{aligned}$$

\(\square \)

Theorem 4

  1. (i)

    By Eqs. (4) and (6), it is clear that \(K\) satisfies all six properties.

  2. (ii)

    First, we prove the following lemma that will be needed later on.

Lemma 1

For all subsets \(\mathcal {L}'\) of \(\tilde{\mathcal {L}}\), define the set \(\mathcal {Y}^{\mathcal {L}'}=\{\varvec{Y}^L: \; L\in \mathcal {L}'\}.\) Recall restriction (7) and let \(\tilde{\mathcal {L}}^Q=\{ L \in \tilde{\mathcal {L}}: \; \; Y^L \in \mathcal {Y}^Q\}\). We have \(\mathcal Y^{\tilde{\mathcal {L}}^Q}=\mathcal {Y}^Q=\mathcal Y^{\mathcal {L}^Q}.\)

Proof

First of all, it is clear by the definition of Eq. (3) that

$$\begin{aligned} \mathcal Y^{\tilde{\mathcal {L}}^Q}\subset \mathcal {Y}^Q. \end{aligned}$$

Now, consider the welfare function \(u^*(\varvec{x},\varvec{w})=\sum _{i=1}^I w_i x_i\). Moreover, consider the achievement matrix \(\varvec{X^*}_{\mathcal {A}}=\varvec{I}_A\) where \(\varvec{I}_A\) is the \(A\times A\) identity matrix.

For each \(R\in \mathcal {R}_{\mathcal {A}}\), define the set \(W^*_R=\{ \varvec{w}\in \Delta ^{I-1}: R^{\varvec{X^*}_{\mathcal {A}},u^*}_{\varvec{w}}=R\}\). Letting \(\hat{a}_{ R(i)}=\{a\in \mathcal {A}: R(a)=i\}\), it is easy to see that \(W^*_{R}=\{\varvec{w} \in \Delta ^{I-1}: w_{\hat{a}_{R(1)}}>w_{\hat{a}_{ R(2)}}>\ldots >w_{\hat{a}_{R(A)}} \}\).

The sets \(W^*_R\) are symmetric in \(\Delta ^{I-1}\) and mutually exclusive. Moreover, \(u^*\) satisfies the assumption of Proposition 1. Indeed, we have

$$\begin{aligned} \int _{W^*_R} \mathrm {d}\varvec{w} = \int _{\Delta ^{I-1}} 1\left\{ R^{\varvec{X^*}_{\mathcal {A}},u^*}_{\varvec{w}}=R\right\} \mathrm {d}\varvec{w}=\frac{1}{A!}, \; \; \forall R\in \mathcal {R}_{\mathcal {A}}. \end{aligned}$$
(19)

Now, writing \(\mathcal {R_A}=\{R_1,R_2,\ldots ,R_{A!}\}\), consider the family of importance functions \(\mathcal F^*\), where (\(Q_+^I\) is the set of non-negative rational \(I\)-dimensional vectors)

Define the set of profiles \(\mathcal {L}^* =\{L \in \mathcal {L}: \; \varvec{X}_{\mathcal {A}}=\varvec{X^*}_{\mathcal {A}}, \; f \in \mathcal F^*, \; u=u^* \}.\) By Eq. (19) and the structure of the importance functions in \(\mathcal F^*\), we may conclude

$$\begin{aligned} \mathcal Y^{\mathcal {L}^*}= \mathcal Y^Q. \end{aligned}$$

Thus, since \(\mathcal {L}^*\subset \tilde{\mathcal {L}}^Q\), we have \( \mathcal Y^Q \subset \mathcal Y^{\tilde{\mathcal {L}}^Q}\) implying \(\mathcal Y^Q =\mathcal Y^{\tilde{\mathcal {L}}^Q}.\) That \(\mathcal Y^Q =\mathcal Y^{\mathcal {L}^Q}\) follows from the fact that \(\mathcal Y^{\mathcal {L}^Q}\subset \mathcal Y^Q \) and \(\mathcal Y^Q=\mathcal Y^{\tilde{\mathcal {L}}^Q}\subset \mathcal Y^{\mathcal {L}^Q}\). \(\square \)

Let us now consider a rule \(\phi \) satisfying the stated properties and recall the constructed electorate \(\mathcal {E}(f)\) with its associated partial input rankings and election matrices. If there are just two alternatives, then \(\phi \) must be simple majority rule and hence equal to \(K\) (Young 1974; footnote 18 in Young 1988). Note that partial input rankings do not create any problems for this result to go through (see page 51 in Young 1974). When there are more than two alternatives, LIIA together with the previous majority rule for all profiles of two alternatives imply that the rule must satisfy the Condorcet property of Young and Levenglick (1978) (see footnote 18 in Young 1988). As a result, Lemma 1 in Young and Levenglick (1978) implies that for every \(L\in \mathcal {L}\), \(\phi (L)\) depends only on the election matrix \(Y^L\). Again, partial input rankings do not affect the proof of this result. Hence, the characterization of \(K\) may be established on the domain \(Y^{\mathcal {L}^Q}\) (instead of \(\mathcal {L}^Q\)), which, by Lemma 1 above, is equal to \(\mathcal Y^{Q}\). The latter is the same domain on which Young and Levenglick (1978) prove their characterization of Kemeny’s rule (see their Lemma 1 on page 292 and the discussion immediately preceding and following it). Hence, \(\phi =K\) follows from Young and Levenglick (1978) and the argument delineated in footnote 18 of Young (1988). \(\square \)

Theorem 5

Let us first concentrate on the denominator of (14). We perform the following two operations on the elements of \(W^{\epsilon }\): (a) we translate them by \(-(1-\epsilon )\varvec{\bar{w}}\), and then (b) multiply them by \(1/\epsilon \). The resulting polytope is \(\Delta ^{I-1}\), the standard \((I-1)\)-simplex. The volume of the standard simplex \(\Delta ^{I-1}\) (which has a side length of \(\sqrt{2}\)) is given by

$$\begin{aligned} \mathrm{{Vol}}\left( \Delta ^{I-1}\right) = \frac{\sqrt{2}^{I-1} \sqrt{I}}{(I-1)!\sqrt{2^{I-1}}}=\frac{\sqrt{I}}{(I-1)!} \end{aligned}$$
(20)

Basic linear algebra (see Lang 1987) implies that

$$\begin{aligned} \mathrm{{Vol}}(W^{\epsilon })=\sqrt{\epsilon ^{2(I-1)}} \mathrm{{Vol}}\left( \Delta ^{I-1}\right) =\epsilon ^{I-1} \frac{\sqrt{I}}{(I-1)!}. \end{aligned}$$
(21)

Now let us focus on the numerator.

Recall the difference vector \(\varvec{d}^{\gamma }\) and consider the constant \(D^{\epsilon ,\gamma }= -\frac{1-\epsilon }{\epsilon } (\varvec{d}^{\gamma })' \varvec{\bar{w}}\), defined in Eq. (15). Performing the same affine transformation as before, namely \(\varvec{w} \leftarrow \frac{\varvec{w}-(1-\epsilon )\varvec{\bar{w}}}{\epsilon }\), the polytope \(W^{\epsilon ,\gamma }_{a_i a_j}\) is transformed into

$$\begin{aligned} \widehat{W}^{\epsilon ,\gamma }_{a_ia_j}=\left\{ \varvec{\widehat{w}} \in \Delta ^{I-1}: \; \; (\varvec{d}^{\gamma })'\varvec{\widehat{w}}\ge D^{\epsilon ,\gamma }\right\} , \end{aligned}$$
(22)

which in turn implies

$$\begin{aligned} \mathrm{{Vol}} \left( W^{\epsilon ,\gamma }_{a_ia_j}\right) =\epsilon ^{I-1} \mathrm{{Vol}} \left( \widehat{W}^{\epsilon ,\gamma }_{a_ia_j}\right) . \end{aligned}$$
(23)

Putting Eqs. (21) and (23) together, we obtain

$$\begin{aligned} V^{\epsilon ,\gamma }_{ij}=\frac{\mathrm{{Vol}}\left( W^{\epsilon ,\gamma }_{a_ia_j}\right) }{\mathrm{{Vol}}\left( W^{\epsilon }\right) }=\frac{(I-1)!}{\sqrt{I} }\mathrm{{Vol}}\left( \widehat{W}^{\epsilon ,\gamma }_{a_ia_j}\right) . \end{aligned}$$
(24)

Eqs. (15) and (22) imply that the volume of \(\widehat{W}^{\epsilon ,\gamma }_{a_ia_j}\) is increasing in \(\epsilon \) if \((\varvec{d}^{\gamma })'\varvec{\bar{w}} <0\), decreasing if \((\varvec{d}^{\gamma })'\varvec{\bar{w}} =0\), and constant if \((\varvec{d}^{\gamma })'\varvec{\bar{w}} =0\). The result now follows from Eq. (24). \(\square \)

Theorem 6

Assume without loss of generality that \(i^*=I\). Consider the polytope \(W^{\epsilon ,\gamma ,*}_{a_ia_j}\) of Eq. (16), obtained by using the equality constraint \(\sum _{i=1}^I w_1=1\) to eliminate variable \(I\) from polytope \(\widehat{W}^{\epsilon ,\gamma }\):

$$\begin{aligned} W^{\epsilon ,\gamma ,*}_{a_ia_j}=\left\{ \varvec{w^*} \in \mathfrak {R}^{I-1}: \; \;\varvec{w^*} \ge 0, \; \;\sum _{i \in \mathcal I^*} w^*_i \le 1, \; \; (\varvec{d^*}-d_{i^*}^{\gamma } \varvec{e})'\varvec{w^*}+d_{i^*}^{\gamma }\ge D^{\epsilon ,\gamma }\right\} . \end{aligned}$$

The affine transformation \(h\) which maps polytope \(W^{\epsilon ,\gamma ,*}_{a_ia_j}\) to \(\widehat{W}^{\epsilon ,\gamma }_{a_ia_j}\) is given by \(h:\mathfrak {R}^{I-1}\rightarrow \mathfrak {R}^I\), satisfying \(h(\varvec{w^*})=T\cdot \varvec{w^*} + [0,0,\ldots ,0,1]'\), where \(T\) is an \(I\times (I-1)\) matrix equal to:

$$\begin{aligned} T = \left[ \begin{array}{ccccc} 1 &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} 1 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} &{} \ddots \\ 0 &{} 0 &{} \cdots &{} &{} 1 \\ -1 &{} -1 &{} -1 &{} \cdots &{} -1\end{array} \right] \Rightarrow T'\cdot T = \left[ \begin{array}{ccccc} 2 &{} 1 &{} 1 &{} \cdots &{} 1 \\ 1 &{} 2 &{} 1 &{} \cdots &{} 1 \\ \vdots &{} &{} \ddots \\ 1 &{} 1 &{} \cdots &{} 2 &{} 1 \\ 1 &{} 1 &{} 1 &{} \cdots &{} 2\end{array} \right] \end{aligned}$$

Thus we have \(\det {[T'\cdot T]}=I\). Once again, basic linear algebra (Lang 1987) implies that

$$\begin{aligned} \mathrm{{Vol}}\left( \widehat{W}^{\epsilon ,\gamma }_{a_ia_j}\right) =\sqrt{\det {[T'\cdot T]}}\cdot \mathrm{{Vol}}\left( W^{\epsilon ,\gamma ,*}_{a_ia_j}\right) =\sqrt{I}\cdot \mathrm{{Vol}}\left( W^{\epsilon ,\gamma ,*}_{a_ia_j}\right) . \end{aligned}$$
(25)

Eqs. (24)–(25) together imply

$$\begin{aligned} V^{\epsilon ,\gamma }_{a_ia_j}=(I-1)!\;\mathrm{{Vol}}\left( W^{\epsilon ,\gamma ,*}_{a_ia_j}\right) . \end{aligned}$$

\(\square \)

Proposition 4

We first identify the vertices of polytope \(W^{\epsilon ,\gamma *}_{a_ia_j}\). In doing so, we divide the set of indicators \(\mathcal {I}^* =\{1,2,\ldots ,I-1\}\) into \(\mathcal {I}^*_1\) and \(\mathcal {I}^*_2\), such that

$$\begin{aligned} \mathcal {I}^*_1=\left\{ i \in \mathcal {I}^* : \; \; \varvec{d}^{\gamma *}_i > D^{\epsilon ,\gamma }\right\} , \; \; \mathcal {I}^*_2=\left\{ i \in \mathcal {I}^* : \; \; \varvec{d}^{\gamma *}_i < D^{\epsilon ,\gamma }\right\} . \end{aligned}$$

Assumption 1 ensures that \(\left\{ \mathcal {I}^*_1,\mathcal {I}^*_2\right\} \) is a partition of \(\mathcal {I}^*\), and we let \(I_1^*=\left| \mathcal {I}^*_1\right| \) and \(I_2^*=\left| \mathcal {I}^*_2\right| \) . It will be useful to express polytope \(W^{\epsilon ,\gamma *}_{a_ia_j}\) in the following way:

$$\begin{aligned} W^{\epsilon ,\gamma ,*}_{a_ia_j}=\{ w^* \in \mathfrak {R}^{I-1}:\; \; \varvec{y_k}'\varvec{w^*} \le \varvec{b}\}, \end{aligned}$$
(26)

where the \((I-1)\)-dimensional vectors \(\varvec{y_k}\), \(k=1,2,\ldots ,I+1\), and \(\varvec{b}\) satisfy (a) \(\varvec{y_k}=-\varvec{e_k}\) Footnote 12 and \(\varvec{b}_k=0\) for \(k=1,\ldots ,I-1\), (b) \(\varvec{y_I}=[1,1,1,\ldots ,1]'\) and \(\varvec{b}_I=1\), and (c) \(\varvec{y_{I+1}}=-\varvec{d}^{\gamma *}+d_{i^*}^{\gamma }\varvec{e}\) and \(\varvec{b}_{I+1}=-D^{\epsilon ,\gamma }+d_{i^*}^{\gamma }\).

With representation (26) in mind, a vector \(\varvec{v}\) is a vertex of \(W^{\epsilon ,\gamma *}_{a_ia_j}\) if it satisfies \(I-1\) linearly independent inequality constraints with equality (Bertsimas and Tsitsiklis 1997). The structure of vectors \(\varvec{y_k}\) for \(k=1,2,\ldots ,I+1\) and \(\varvec{b}\) imply that a vertex of \(W^{\epsilon ,\gamma *}_{a_ia_j}\) can have at most two nonzero entries. Furthermore, Assumption 1 ensures primal nondegeneracy so that every vertex \(\varvec{v}\) will correspond to a unique basis matrix \(B_v\), i.e., a unique set of linearly independent constraints satisfied with equality.

We may distinguish between four kinds of vertices and their corresponding bases:

  1. (V1)

    \(\varvec{v_0}=\varvec{0}\). \(B_{0} =\{\varvec{y_k}': k=1,2,..,I-1\}\)

  2. (V2)

    \(\varvec{v_i}=\varvec{e_i}\) for all \(i\in \mathcal {I}^*_1\). Here \(B_{i} =\{\varvec{y_k}': k=1,\ldots ,i-1,i+1,..,I-1, I\}\), for all \(i\in \mathcal {I}^*_1\).

  3. (V3)

    \(\varvec{v_j}=\pi _j \varvec{e}_j\), where \(\pi _j=\frac{d_{i^*}^{\gamma }-D^{\epsilon ,\gamma }}{d_{i^*}^{\gamma }-\varvec{d}^{\gamma *}_j}\), for all \(j \in \mathcal {I}^*_2\). Here \(B_{j} =\{\varvec{y_k}': k=1,\ldots ,j-1,j+1,..,I-1,I+1\}\), for all \(j \in \mathcal {I}^*_2\).

  4. (V4)

    \(\varvec{v_{ij}}=\pi _{ij} \varvec{e}_i + (1-\pi _{ij}) \varvec{e}_j\), where \(\pi _{ij}=\frac{D^{\epsilon ,\gamma }-\varvec{d}^{\gamma *}_j}{\varvec{d}^{\gamma *}_i-\varvec{d}^{\gamma *}_j}\), for all \(i\in \mathcal {I}^*_1\) and \(j \in \mathcal {I}^*_2\). Here \(B_{ij} =\{\varvec{y_k}': k=1,\ldots ,i-1,i+1,..,j-1,j+1,\ldots ,I,I+1\}\), for all \(i\in \mathcal {I}^*_1\) and \(j \in \mathcal {I}^*_2\).

Two vertices are connected by an edge if they share \(I-2\) common linearly independent active constraints (Bertsimas and Tsitsiklis 1997). An examination of the preceding expressions for the vertices of \(W^{\epsilon ,\gamma ,*}_{a_ia_j}\) and their bases, implies that we may identify the following eight kinds of undirected edges:

  1. (E1)

    \(\left( \varvec{v_0},\varvec{v_i}\right) \), for all \(i\in \mathcal {I}^*_1\).

  2. (E2)

    \(\left( \varvec{v_0},\varvec{v_j}\right) \), for all \(j\in \mathcal {I}^*_2\).

  3. (E3)

    \(\left( \varvec{v_i},\varvec{v_k}\right) \) for all pairs \((i,k)\) where \(i,k\in \mathcal {I}^*_1\).

  4. (E4)

    \(\left( \varvec{v_j},\varvec{v_k}\right) \) for all pairs \((j,k)\) where \(j,k\in \mathcal {I}^*_2\).

  5. (E5)

    \(\left( \varvec{v_i},\varvec{v_{ij}}\right) \) for all pairs \((i,j)\) where \(i\in \mathcal {I}^*_1\) and \(j\in \mathcal {I}^*_2\).

  6. (E6)

    \(\left( \varvec{v_j},\varvec{v_{ij}}\right) \) for all pairs \((i,j)\) where \(i\in \mathcal {I}^*_1\) and \(j\in \mathcal {I}^*_2\).

  7. (E7)

    \(\left( \varvec{v_{ij}},\varvec{v_{ik}}\right) \) for all triplets \((i,j,k)\) where \(i\in \mathcal {I}^*_1\) and \(j,k\in \mathcal {I}^*_2\).

  8. (E8)

    \(\left( \varvec{v_{ij}},\varvec{v_{kj}}\right) \) for all triplets \((i,j,k)\) where \(i,k\in \mathcal {I}^*_1\) and \(j\in \mathcal {I}^*_2\).

Recall that we wish to exhibit a vector \(\varvec{c}\in \mathfrak {R}^{I-1}\) such that the function \(h(\varvec{w^*})=\varvec{c}'\varvec{w^*}\) is non-constant on each edge of \(W^{\epsilon ,\gamma *}_{a_i a_j}\). To this end, recall the vertices of \(W^{\epsilon ,\gamma *}_{a_ia_j}\) enumerated above as (V1)–(V4) and the values of \(\pi _j\) for \(j\in \mathcal {I}^*_2\) and \(\pi _{ij}\) for all \(i\in \mathcal {I}^*_1\) and \(j \in \mathcal {I}^*_2\). Define the following two quantities

$$\begin{aligned} \displaystyle \delta _i&=\min _{i \in \mathcal {I}^*_1, \; \; j,k \in \mathcal {I}^*_2}\{|\pi _{ij}-\pi _{ik}|: \pi _{ij} \ne \pi _{ik}\}.\quad \hbox { If undefined, set }\delta _i=1.\\ \displaystyle \delta _j&=\min _{i,k \in \mathcal {I}^*_1, \; \; j \in \mathcal {I}^*_2}\{|\pi _{ij}-\pi _{kj}|: \pi _{ij} \ne \pi _{kj}\}.\quad \hbox { If undefined, set }\delta _j=1. \end{aligned}$$

Consequently let \(\delta =\min \{\delta _i,\delta _j\}\) and define \( C=\frac{2}{\delta }\).

Finally, relabel the elements of set \(\mathcal {I}^*_2=\{j_1,j_2,\ldots ,j_{I^*_2}\}\) so that \(\pi _{j_1}\le \pi _{j_2} \le \ldots \le \pi _{j_{I^*_2}}\). Now, define the vector \(\varvec{c}\) satisfying

$$\begin{aligned} c_{l}&= \left\{ \begin{array}{ll} C+\frac{k}{(I^*_1+1)}, &{} \quad \text {if} \; \; l=i_k \in \mathcal {I}^*_1, \\ \frac{k}{I^*_2 (I^*_1+1)}, &{} \quad \text {if} \; \; l=j_k \in \mathcal {I}^*_2. \end{array} \right. \end{aligned}$$
(27)

With this choice of \(\varvec{c}\) we can check all eight kinds of edges (E1)–(E8) and verify that \(\varvec{c}'\varvec{v} \ne \varvec{c}'\varvec{u}\) for all pairs of adjacent vertices \((\varvec{v},\varvec{u})\). Thus the function \(f(\varvec{w^*})=\varvec{c}'\varvec{w^*}\) is non-constant on each edge of \(W^{\epsilon ,\gamma *}_{a_i a_j}\). Hence, in conjunction with Assumption 1, we may apply Theorem 1 in Lawrence (1991) to conclude

$$\begin{aligned} \mathrm{{Vol}}\left( W^{\epsilon ,\gamma ,*}_{a_ia_j}\right) =\displaystyle \sum _{\begin{array}{c} \text {vertices } \varvec{v} \\ \text {of } W^{\epsilon ,\gamma *}_{a_ia_j} \end{array} }\frac{\left( \varvec{c}'\varvec{v}\right) ^{I-1}}{(I-1)!\left| \det \left( B_{\varvec{v}}\right) \right| \displaystyle \prod _{i=1}^{I-1} \left[ \left( B_{\varvec{v}}'\right) ^{-1}\varvec{c}\right] _i}. \end{aligned}$$
(28)

\(\square \)

Appendix 2: Computational issues

Equation (6) summarizes the Kemeny-optimal ranking we wish to find. We would at this point be done, if not for the fact that the associated optimization problem is NP-hard (Bartholdi et al. 1989).

The main difficulty arises from the potential presence of Condorcet cycles, which imply intransitive majority pairwise preferences. Formally, a set of alternatives \(\{a_{i_1}, a_{i_2},\ldots ,a_{i_M}\}\) forms a Condorcet cycle if \(\min \{Y^L_{12}, Y^L_{23},\ldots ,Y^L_{M-1 M}, Y^L_{M 1}\}\ge 0\). If no such cycles existed, finding a unique Kemeny-optimal ranking would be a straightforward affair (see Theorem 7 below). Thus, it is important to identify and, in some fashion, resolve these cycles.

1.1 Graph-theoretic preliminaries

In this section, I briefly introduce some relevant graph-theoretic concepts that will be useful later. A directed graph \(G\) is a pair \(G=(\mathcal V,E)\), where \(\mathcal V\) is a set of vertices and \(E\) a set of ordered pairs of vertices, referred to as edges.

Given two subsets \(\mathcal {V}_1\), \(\mathcal V_2\) of \(\mathcal {V}\) such that \(\mathcal V_1 \cap \mathcal V_2 =\emptyset \), let

$$\begin{aligned} \mathrm{{in}}(\mathcal V_1,\mathcal V_2)= \left| \{e \in E: e=(u,v), \; \; u \in \mathcal V_2, \; v \in \mathcal V_1\} \right| . \end{aligned}$$
(29)

Thus Eq. (29) denotes the number of edges leading to vertices in \(\mathcal V_1\), originating from vertices in \(\mathcal V_2\).

A subgraph \(G'=\left( \mathcal {V'}, E'\right) \) of a directed graph \(G=(\mathcal V,E)\) is a graph such that \(\mathcal {V}'\subset \mathcal {V}\) and \(E'=\left\{ (v,u) \in E: \; \; v, u \in \mathcal {V}'\right\} \). A directed graph is called strongly connected if there is a path from each vertex to every other vertex (in both directions). The strongly connected components (SCCs) of a directed graph G are its maximal-size strongly connected subgraphs.

1.2 \(L\)-dominance graphs and Kemeny rankings

Given a profile \(L\), define the directed graph \(G^{L}=\left( \mathcal {A},E^{L}\right) \), whose vertices correspond to alternatives in \(\mathcal {A}\) and whose edges satisfy \(E^{L}\!=\!\left\{ \!(a_i,a_j)\!\in \! \mathcal {A}\! \times \! \mathcal {A}: \;\! Y_{a_ia_j}^{L}\!\ge \! 0, \; i\!\ne \! j\! \right\} \). Thus, a pair \((a_i,a_j)\) belongs to \(G^L\)’s edge set if and only if a weak net majority of the voters in the electorate \(\mathcal {E}(f)\) assign to \(a_i\) a higher welfare than \(a_j\), as measured by \(u\) applied to \(\varvec{x_{a_i}}\) and \(\varvec{x_{a_j}}\)—with no self-edges \((a_i,a_i)\) allowed. We refer to \(G^{L}\) as an \(L\) -dominance graph, as it captures the above dominance relationship between all pairs of alternatives. Clearly, a cycle in \(G^{L}\) corresponds to a Condorcet cycle in the underlying social choice problem. Moreover, alternatives forming such a cycle must, by definition, belong to the same SCC of \(G^L\). As a result, if no Condorcet cycles existed, then all strongly connected components of \(G^L\) would be singletons.

The following Theorem exploits the special structure of \(G^{L}\) to connect it to the Kemeny-optimal ranking \(K^L\) of Eq. (6).

Theorem 7

Consider a profile \(L=(\varvec{X}_{\mathcal {A}}, f, u) \in \mathcal {L}\) and its \(L\)-dominance graph \(G^{L}=\left( \mathcal {A}, E^{L}\right) \). Let \(G'=\left( \mathcal {A}', E' \right) \) denote a SCC of \(G^L\) and define the profile \(L_{\mathcal {A}'} =(\varvec{X}_{\mathcal {A}'},f,u)\), where \(\varvec{X}_{\mathcal {A}'}\) is the restriction of \(\varvec{X}_{\mathcal {A}}\) to alternatives in \(\mathcal {A}'\). We have the following holding (here \(K^L_{\mathcal {A}'}\) is the restriction of \(K^L\) to alternatives in \(\mathcal {A}'\)):

$$\begin{aligned}&K^L_{\mathcal {A}'}=\frac{\mathrm{{in}}\left( \mathcal {A}',\mathcal {A}\backslash \mathcal {A}' \right) }{|\mathcal {A}'|}\cdot \varvec{\mathcal {E}} + K^{L_{\mathcal {A}'}} , \end{aligned}$$
(30)

where \(\varvec{\mathcal {E}}\) is a matrix of ones of appropriate dimension.

Proof

Recall that \(L=(\varvec{X}_{\mathcal {A}}, f,u)\). Suppose graph \(G^L\) has \(M\) strongly connected components denoted by \(G_1=(\mathcal {A}_1,E_1), G_2=(\mathcal {A}_j, E_2), ,\ldots ,G_M=(\mathcal {A}_M, E_M)\).

For each \(m=1,2,\ldots ,M\) define \(g_m\equiv \frac{\mathrm{{in}}\left( \mathcal {A}_m,\mathcal {A}\backslash \mathcal {A}_m \right) }{|\mathcal {A}_m|}\). Now for all pairs \((G_i, G_j)\) either all vertices in \(G_i\) are pointing to all vertices in \(G_j\), or vice versa. For, if this were not true, it would be possible to form a cycle that included vertices in both \(G_i\) and \(G_j\) contradicting the fact that they are separate strongly connected components. If all vertices in \(G_i\) are pointing to all vertices in \(G_j\), then they are also pointing to all of the vertices \(G_j\) is pointing to (if not there would exist a cycle), implying \(g_i < g_j\). Conversely, if all vertices in \(G_j\) are pointing to all vertices in \(G_i\) then \(g_i > g_j\). Thus, we may deduce that \(g_i < g_j\) if and only if all vertices in \(G_i\) are pointing to all vertices in \(G_j\).

The above implies that it is possible to order the \(g_i\)’s in strictly ascending order and we may assume, without loss of generality, that \(g_1 < g_2 <\cdots .<g_M\). With this ordering in mind, define the sets \(\mathcal {A}_m^-=\mathcal {A}_1 \cup \mathcal {A}_2 \cup \cdots \cup \mathcal {A}_m\) and \(\mathcal {A}_m^+=\mathcal {A}_{m} \cup \mathcal {A}_{m+1} \cup \cdots \cup \mathcal {A}_M\), for all \(m=1,2,\ldots ,M\).

We proceed by proving that \(K^L\) must be such that, for all \(m=1,2,\ldots M-1\), all alternatives in \(\mathcal {A}_m^-\) are ranked before all alternatives in \(\mathcal {A}_{m+1}^+\). To wit, consider the alternatives in \(\mathcal {A}_{m}^-\). Since \(g_i < g_j\) for all \((i,j)\) such that \(i \le m\) and \(j >m\), we know that all alternatives in \(\mathcal {A}_{m}^-\) are pointing to all alternatives in \(\mathcal {A}_{m+1}^+\). Hence \(Y^L_{ij}>0\) for all pairs \((a_i,a_j) \in \mathcal {A}_{m}^- \times \mathcal {A}_{m+1}^+\). Thus, since \(K\) satisfies the extended-Condorcet property, any Kemeny-optimal ranking will have to rank all alternatives in \(\mathcal {A}_{m}^-\) before all alternatives in \(\mathcal {A}_{m+1}^+\).

Now, note that by the definition of \(G^L\) we have \(\left| \mathcal {A}^-_{m}\right| = \frac{in(\mathcal {A}_{m+1},\mathcal {A} \backslash \mathcal {A}_{m+1})}{|\mathcal {A}_{m+1}|}\). Thus, we obtain the following relation for all \(R\in K^L\):

$$\begin{aligned}&R(a) \in \left\{ \frac{in(\mathcal {A}_m,\mathcal {A} \backslash \mathcal {A}_m)}{|\mathcal {A}_m|} +1, \; \; \frac{in(\mathcal {A}_m,\mathcal {A} \backslash \mathcal {A}_m)}{|\mathcal {A}_m|}+\left| \mathcal {A}_m \right| \right\} , \; \; \forall a \in \mathcal {A}_m, \; \; m=1,2,\ldots ,M. \quad \quad \end{aligned}$$
(31)

To determine the precise rank of the alternatives in each \(\mathcal {A}_m\) within the ranges given by Eq. (31), it is sufficient, by LIIA, to solve for \(K^{L_{\mathcal {A}_m}}\), and update accordingly. \(\square \)

The SCCs of the graph \(G^L\) can be efficiently identified with Tarjan’s algorithm (Tarjan 1972). Theorem 7 suggests that, since the relative ranking of alternatives belonging to different SCCs can be easily determined, problem (6) may be efficiently reduced to a series of smaller sub-problems (i.e., the computation of Kemeny sub-rankings), corresponding to the SCCs of graph \(G^L\). If these SCCs are small enough (in practice, \(|\mathcal {A}'|\le 10\)), then these sub-problems can be swiftly dealt with by simple enumeration. Otherwise, approximation algorithms are needed.

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Athanassoglou, S. Multidimensional welfare rankings under weight imprecision: a social choice perspective. Soc Choice Welf 44, 719–744 (2015). https://doi.org/10.1007/s00355-014-0858-z

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