## Abstract

The paper deals with voting rules that require voters to rate the candidates on a finite evaluation scale and then elect a candidate whose median grade is maximum. These rules differ by the way they choose among candidates with the same median grade. Call proponents (resp. opponents) of a candidate the voters who rate this candidate strictly above (resp. strictly below) her median grade. A simple rule, called the typical judgment, orders tied candidates by the difference between their share of proponents and opponents. An appealing rule, called the usual judgment, divides this difference by the share of median votes. An alternative rule, called the central judgment, compares the relative shares of proponents and opponents. The usual judgment is continuous with respect to these shares. The majority judgment of Balinski and Laraki (Proce Natl Acad Sci 104(21):8720–8725, 2007) considers the largest of these shares and loses continuity. A result in Balinski and Laraki (Oper Res 62(3):483–511, 2014) aims to characterize the majority judgment and states that only a certain class of functions share some valuable characteristics, like monotonicity. We relativize this result, by emphasizing that it only holds true for non-discrete scales of grades. Properties remaining specific to the majority judgment in the discrete case are idiosyncratic features rather than universally sought criteria, and other median-based rules exist that are both monotonic and continuous.

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## Notes

- 1.
- 2.
We restrict our analysis to finite sets of grades as they cover all practical applications.

- 3.
\(\mathbb {1}_{\mathscr {P}\left( \varvec{x}\right) }\) denotes the indicator function of the property \(\mathscr {P}\) evaluated in \(\varvec{x}\). For example,\(\mathbb {1}_{p_{c}>q_{c}}=1\) if \(p_{c}>q_{c}\) and 0 otherwise.

- 4.
We define tied tuples \(T_{\mu }\) as tuples of choices sharing the

*mj*score \(\mu\): \(T_{\mu }:=\left\{ \left( \alpha _{c},p_{c},q_{c}\right) \,|\,c\in \mathcal{C}\text { and }mj_{c}=\mu \right\}\). - 5.
According to Balinski and Laraki (2011), it was very first proposed by David Gale.

- 6.
The denominator is always positive as we have \(q_{c}<\frac{1}{2}\), \(p_{c}\le \frac{1}{2}\).

- 7.
Admittedly, the \(\left( p^{i}\right) _{i}\) are redundant with the \(\left( -q^{i}\right) _{i}\) for

*d*and*n*as, for any \(i\in \llbracket 1;G-1\rrbracket\), the common complementary score conveys a bijection from \(q^{i}\) to \(p^{i}\). However, this is not the case for \(s^{i}\) when \(p^{i}\cdot q^{i}=0\). In all cases, it is equivalent to order the \(\left( -q^{i}\right) _{i}\) before or after the \(\left( p^{i}\right) _{i}\). We give precedence to the \(\left( -q^{i}\right) _{i}\) only by analogy with MJ. - 8.
Of course, we could have chosen an ultimate tie-breaking rule following dissent instead of consensus, and/or an outermost one. However, as Balinski and Laraki (2014) argue, following consensus is consistent with deciding upon the lower middlemost grade (instead of the upper one, \(\ell ^{\left\lceil \frac{V+1}{2}\right\rceil }\)), in that a majority always grades the winner at least as much as its lower middlemost grade; while being innermost is consistent with deciding upon the middlemost grades. That being said, choosing another ultimate tie-breaking rule would have virtually no consequence on ballots with many voters, as an ultimate tie is highly improbable in such cases.

- 9.
Balinski and Laraki (2011) define a related notion (p. 204),

*monotonicity*, equivalent to*choice monotonicity*as long as the social-ranking function is antisymmetric (i.e. \(A\succeq B\text { and }B\succeq A\Rightarrow A=B\)). All tie-breaking rules dealt with in this paper are antisymmetric and thus monotonic, as each specifies an order. Hence, we sometimes use*monotonicity*as a shortcut for*choice monotonicity*. - 10.
Let us precise that in absence of any explicit definition from BL, we understand

*transformation*in its usual sense of a function from a set to itself. Indeed, it would make little sense to take a codomain of \(\phi\) larger than its domain, because then some new grades could not be used. - 11.
When

*V*is odd, the domain \(\mathcal{S}\) of \(\pi _{MJ}\) is \(\llbracket 0;V-1\rrbracket\), and when*V*is even, \(\pi _{MJ}\) is instead defined on \(\llbracket 1;V\rrbracket .\) - 12.
The assumption on \(\mathscr {G}\) (which amounts to take \(G\ge 4\)) is made to simplify the argument, but a similar proof exists for \(G=3\). Take \(\Phi _{1}=\left( \begin{array}{ccccc} -1 &{} 1 &{} 1 &{} 1 &{} 1\\ 0 &{} 0 &{} 1 &{} 1 &{} 1 \end{array}\right)\), \(\Phi _{2}=\left( \begin{array}{ccccc} 0 &{} 0 &{} 0 &{} 0 &{} 1\\ -1 &{} -1 &{} 0 &{} 1 &{} 1 \end{array}\right)\), \(\Phi _{3}=\left( \begin{array}{ccccc} -1 &{} -1 &{} -1 &{} 0 &{} 0\\ -1 &{} -1 &{} -1 &{} -1 &{} 1 \end{array}\right)\) and \(\Phi _{4}=\left( \begin{array}{ccccc} -1 &{} -1 &{} 0 &{} 1 &{} 1\\ -1 &{} 0 &{} 0 &{} 0 &{} 0 \end{array}\right)\). In each case, \(A\succ B\). Using \(\Phi _{1}\), this ranking implies that quintile 2 should be compared before quintile 1, which we denote \(2\vartriangleleft 1\). Combining this with the condition given by \(\Phi _{2}\): \(2\vartriangleleft 4\text { or }1\vartriangleleft 4\), we obtain that \(2\vartriangleleft 4\). Similarly, \(4\vartriangleleft 5\) (\(\Phi _{3}\)), and \(4\vartriangleleft 2\text { or }5\vartriangleleft 2\) (\(\Phi _{4}\)) imply that \(4\vartriangleleft 2\), which yields a contradiction.

- 13.
We choose the uniform distribution since we have no good prior on the real-world distribution of grades of an ordinary choice.

- 14.
Drawing the variation in

*q*independently from that in*p*(from a uniform distribution in \(\left[ -0.02;0.02\right]\)) yields equivalent results. - 15.
BL prefer to say that a language is sufficiently rich if “a voter who gives the same grade to two candidates has no preference between them”.

- 16.
To see this, let us define \(M=\left\{ v\,|\,g_{A,v}>g_{B,v}\right\}\), \(\hat{g}=\underset{v\in M}{\min }\left\{ g_{A,v}\right\}\) and \(\hat{M}=\left\{ v\,|\,g_{A,v}=\hat{g}\right\}\). Take \(k\in \hat{M}\). \(\forall j\in M\smallsetminus \hat{M}\), \(g_{A,j}>\hat{g}=g_{A,k}\). Thus, as

*A*and*B*are polarized, \(g_{B,j}\le g_{B,k}<g_{A,k}=\hat{g}\). In addition, \(\forall i\in \hat{M}\), \(g_{B,i}<\hat{g}\). Hence, \(\forall v\in M\), \(g_{B,v}<\hat{g}\le g_{A,v}\). Finally, we deduce that \(\alpha _{B}<\hat{g}\le \alpha _{A}\) from \(\left| M\right| >\frac{V}{2}\). - 17.
One could also argue that in practice, there are often more choices than grades, so that the universally sought properties do not apply. However, even with a large number of choices, one must acknowledge that it is unlikely that the number of choices that are tied together exceeds the number of grades.

- 18.
The Kendall distance counts the number of pairwise disagreements between two rankings. In other words, it gives the minimal number of swaps between neighboring choices required to transform one ranking into the other.

- 19.
For example, assume that among four voters, all attribute a grade to

*E*: \(\left( g_{E,v}\right) =\left( \begin{array}{cccc} -1&0&1&1\end{array}\right)\), but only two attribute a grade to*F*: \(\left( g_{F,v}\right) _{v\,|\,g_{F,v}\in \mathcal{G}}=\left( \begin{array}{cc} 0&0\end{array}\right)\). As*E*and*F*share the same lower middlemost grade 0, the lexi-order characterization of MJ requires that we compare another rank, say a lower one (the example should be adapted if we were to compare a higher one). One could naively think that comparing the first quartile would be natural, as \(V=4\). However, as \(\ell ^{\frac {1}{4}}\left( E\right) =-1<0=\ell ^{\frac {1}{4}}\left( F\right)\), this would lead to elect*F*, against the spirit of MJ and the ranking of*mj*scores: \(mj_{E}=\frac{1}{2}>0=mj_{F}\). - 20.
In our example, as \(\Pi =4\), the step \(\frac{k}{2\Pi }\) between each (same side) rank used in the comparison is one eighth (and not one quarter), and one can check that this leads to electing

*E*, as it should be.

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## Acknowledgements

I am grateful to Jean-François Laslier, Rida Laraki as well as three anonymous reviewers for their comments and suggestions. I am thankful to François Durand for the proofreading. I thank Maria Sarmiento for the grammar check.

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## Appendices

### Appendix

### A Applying the tie-breaking rules on real examples

### B Allowing for partial abstention

As some voters may not express an opinion over all choices, for example because there are plenty of choices, it is useful to allow for partial abstention. In this Appendix, we extend our setting in such a way, and show that all previous results hold. The formalization simply needs some adjustments.

The set of grades becomes \(\mathscr {G}\cup \left\{ \emptyset \right\}\), where \(g_{c,v}=\emptyset\) indicates that voter* v* does not attribute any grade to *c*. The number of expressed grades for *c* is \(E_{c}:=\left| \left\{ v\in \mathscr {V}\,|\,g_{c,v}\in \mathscr {G}\right\} \right|\). We then define \(\alpha _{c}\) as the lower middlemost grade among expressed grades to *c*, and we define the shares of proponents and opponents to a choice relative to its number of expressed grades: \(p_{c}^{n}:=\frac{1}{E_{c}}\left| \left\{ v\in \mathscr {V}\,|\,g_{c,v}\ge \alpha _{c}+n\right\} \right|\) (resp. \(q_{c}^{n}:=\frac{1}{E_{c}}\left| \left\{ v\in \mathscr {V}\,|\,g_{c,v}\le \alpha _{c}-n\right\} \right|\)), for \(n\in \left( 0;G\right)\). We also redefine each order function \(\ell ^{j}\left( c\right)\), as the *j*th (lowest) quantile of expressed grades of *c*, and we adopt the convention that when this quantile falls between two grades of *c*, then \(\ell ^{j}\left( c\right)\) equals the lowest of the two. For example, assuming that there are two voters and that the grades of *E* are: \(\left( g_{E,v}\right) =\left( \begin{array}{cc} 0&1\end{array}\right)\), we have \(\ell ^{j}\left( E\right) =0\) for \(j\le \frac{1}{2}\) and \(\ell ^{j}\left( E\right) =1\) for \(j>\frac{1}{2}\); and in particular, \(\alpha _{E}=\ell^{\frac {1}{2}}\left( E\right) =0\). Then, the characterization of MJ as a lexi-order social-ranking function (see Sect. 3.1 and Example 2) requires that, when comparing grades of same rank, we move away from the median with a step small enough to capture any change in grade “at the right quantile”.^{Footnote 19} Thus, we introduce \(\Pi\), the least common multiple of \(\left( E_{c}\right) _{c\in \mathcal{C}}\) (or more simply, \(\Pi :=\prod _{c\in \mathcal{C}}E_{c}\)), and redefine the bijection \(\pi _{MJ}\) that characterizes MJ as a lexi-order social-ranking function as follows: \(\pi _{MJ}\left( 2k+1\right) =\frac{1}{2}-\frac{k}{2\Pi }-\frac{1}{4\Pi }\) and \(\pi _{MJ}\left( 2k\right) =\frac{1}{2}+\frac{k}{2\Pi }-\frac{1}{4\Pi }\) for \(k\in \llbracket 0;\Pi -1\rrbracket\).^{Footnote 20}

With the appropriate formalization just described, all the results of the paper can be as easily derived when allowing for partial abstention as for the restrictive case used in the main text.

### C Other properties specific to the majority judgment

*Resists manipulability* In Theorem 13 of Balinski and Laraki (2014) and Theorem 13.5 of Balinski and Laraki (2011)), BL show that electing the choice with the highest median grade minimizes manipulability. This result applies also to alternative tie-breaking rules. However, among social-ranking functions, MJ is the least manipulable because ties are resolved using middlemost grades. That being said, it is not clear if this theoretical nuance would have any behavioral implication in practice, as the different voting systems differ only by their tie-breaking rules, and elect the same choice in most of cases (see Sect. 6).

*Other features* The following characteristics of MJ consists more in idiosyncratic features than in universally sought criteria for a rule.

### Definition 10

(Balinski and Laraki 2007)* Decisive for the center grades*: the ranking between *A* and *B* is the ranking determined by the middlemost grades unless that ranking is a tie; in that case, the ranking is determined by the residual grades.

### Example 3

Majority judgment is decisive for the center grades (Balinski and Laraki 2007), while *D, S* and* N* are not. Indeed, with the latter rules, choices with distinct middlemost grades can share the same primary score, in which case the tie is resolved using secondary score instead of comparing the middlemost grades. For example, take \(\mathcal{C}=\left\{ E;F\right\}\), \(\Phi =\left( \begin{array}{c} g_{E,v}\\ g_{F,v} \end{array}\right) _{v\in \mathcal{V}}=\left( \begin{array}{ccccc} -1 &{} 0 &{} 0 &{} 1 &{} 1\\ 0 &{} 0 &{} 0 &{} 0 &{} 2 \end{array}\right)\), and consider the rule *D*. We have \(\alpha _{E}=\alpha _{F}=0\) and \(\Delta _{E}=\Delta _{F}=\frac{1}{5}\), so that *F *is the winner for *D* as \(\Delta _{F}^{2}=\frac{1}{5}>0=\Delta _{E}^{2}\). Conversely, MJ decides with the middlemost grades and elects *E*.

### Definition 11

(Balinski and Laraki 2014) Suppose the first of the *j*th-middlemost intervals (\(j\ge 0\)) where *A*’s and *B*’s grades differ is the *k*th. A social-ranking function *rewards consensus* when all of *A*’s grades strictly belong to the *k*th-middlemost interval of *B*’s grades implies that *A* is ranked above *B*.

### Example 4

Majority judgment rewards consensus (Balinski and Laraki 2014), while *D, S* and *N* do not. For example, take \(\mathcal{C}=\left\{ E;F\right\}\) and \(\Phi =\left( \begin{array}{c} g_{E,v}\\ g_{F,v} \end{array}\right) _{v\in \mathcal{V}}=\left( \begin{array}{ccc} -1 &{} 1 &{} 2\\ 0 &{} 1 &{} 1 \end{array}\right)\): MJ elects *F* (as lowest grades decide the ranking) while the alternatives elect *E* (because \(\Delta _{F}=-\frac{1}{3}<0=\Delta _{E}\)).

### Proposition 4

(Theorem 17 in Balinski and Laraki (2007), Theorem 15 in Balinski and Laraki (2014))) The majority-ranking is the unique monotone social-ranking function that is decisive for the center and rewards consensus.

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Fabre, A. Tie-breaking the highest median: alternatives to the majority judgment.
*Soc Choice Welf* **56, **101–124 (2021). https://doi.org/10.1007/s00355-020-01269-9

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