Monometrics and their role in the rationalisation of ranking rules

Pérez‐Fernández, Raúl; Rademaker, Michaël; Baets, Bernard De

doi:10.1016/j.inffus.2016.06.001

Cited by 38 publications

(16 citation statements)

References 39 publications

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“…Similarly to the method of Kemeny where we compute the Kemeny score for each ranking

≻

, here we have the corresponding cost associated with a closest profile of rankings for which

≻

is the acclaimed ranking (measured by means of the chosen monometric

M : scriptL {(scriptC)}^{r} \times scriptL {(scriptC)}^{r} \to double-struckR

). Formally, given the profile

scriptR

r

rankings on

scriptC

given by the voters, we have the following cost for each ranking

≻

C_{M} (≻) = \min_{\begin{matrix} R' \in L {(C)}^{r} s.t. \\ ≻ is acclaimed ranking \end{matrix}} M (R, R') .

In particular, we propose to consider the monometric

double-struckK : scriptL {(scriptC)}^{r} \times scriptL {(scriptC)}^{r} \to double-struckR

defined by the sum of Kendall distances (as proposed by Pérez‐Fernández et al), that is,

double-struckK (scriptR, scriptR') = \sum_{i = 1}^{r} K (≻_{i}, ≻_{i}')

for any

scriptR, scriptR' \in scriptL {(scriptC)}^{r}

. This leads to the introduction of a Kemeny‐like method where instead of unanimity we search for acclamation.…”

Section: The Ranking Rulementioning

confidence: 99%

The acclamation consensus state and an associated ranking rule

Pérez‐Fernández

Baets

2019

Int J Intell Syst

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The study of conditions, under which the existence of an “absolute” best winner can be assured, is a hot topic in the field of social choice. Unanimity is an evident example of a condition under which the winner is obvious. However, many more properties weaker than unanimity have been analysed in literature: the presence of a Condorcet winner, strong stochastic transitivity, the presence of a candidate that Borda dominates all other candidates, etc. Unfortunately, one could easily find a prominent ranking rule, for which the outcome does not agree with these relaxed conditions. In this study, we aim to identify a condition weaker than unanimity, but under which the social outcome is still obvious. This condition, defined as the conjunction of three properties already studied by the present authors and hereinafter referred to as acclamation, will be proven to be a meeting point for the most prominent ranking rules in social choice theory, and will be used for introducing an intuitively appealing ranking rule.

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“…Similarly to the method of Kemeny where we compute the Kemeny score for each ranking

≻

, here we have the corresponding cost associated with a closest profile of rankings for which

≻

is the acclaimed ranking (measured by means of the chosen monometric

M : scriptL {(scriptC)}^{r} \times scriptL {(scriptC)}^{r} \to double-struckR

). Formally, given the profile

scriptR

r

rankings on

scriptC

given by the voters, we have the following cost for each ranking

≻

C_{M} (≻) = \min_{\begin{matrix} R' \in L {(C)}^{r} s.t. \\ ≻ is acclaimed ranking \end{matrix}} M (R, R') .

In particular, we propose to consider the monometric

double-struckK : scriptL {(scriptC)}^{r} \times scriptL {(scriptC)}^{r} \to double-struckR

defined by the sum of Kendall distances (as proposed by Pérez‐Fernández et al), that is,

double-struckK (scriptR, scriptR') = \sum_{i = 1}^{r} K (≻_{i}, ≻_{i}')

for any

scriptR, scriptR' \in scriptL {(scriptC)}^{r}

. This leads to the introduction of a Kemeny‐like method where instead of unanimity we search for acclamation.…”

Section: The Ranking Rulementioning

confidence: 99%

The acclamation consensus state and an associated ranking rule

Pérez‐Fernández

Baets

2019

Int J Intell Syst

Self Cite

View full text Add to dashboard Cite

show abstract

“…In this section, we introduce the concept of a monometric [126], which is closely related to that of a distance function or metric. Like a distance function, a monometric satisfies the axioms of non-negativity and coincidence, but a monometric requires compatibility with a given betweenness relation [111] and does not impose symmetry nor the triangle inequality.…”

Section: Monometricsmentioning

confidence: 99%

“…Obviously, the corresponding adaptation of these problems to the search for monotonicity of a representation of votes is computationally harder. In [126], we proposed an algorithm for identifying the optimal ranking running in factorial time. We recall this algorithm throughout this section.…”

Section: Changes In the Profile Of Rankingsmentioning

confidence: 99%

“…Although many betweenness relations on L(C ) r may be considered, as discussed in [126], the most interesting betweenness relation in the field of social choice is the one given by Kemeny [80]. A profile of rankings R = ( j ) r j=1 is said to be in between two other profiles of rankings R = ( j ) r j=1 and R = ( j ) r j=1 , denoted by [R, R , R ], if it holds that…”

Section: Changes In the Profile Of Rankingsmentioning

confidence: 99%

“…As discussed by Pérez-Fernández et al [126] for a more general problem setting, the search for a closest profile of rankings satisfying some desired property can be addressed as a transportation problem [101] in case the property can be expressed as a constraint of an Integer Linear Programming problem. Indeed, monotonicity of all previously discussed representations of votes can be expressed as a transportation problem.…”

Section: Changes In the Profile Of Rankingsmentioning

confidence: 99%

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