An axiomatic characterization of the Slater rule

Abstract

Despite being a well-studied decision rule, the Slater rule has not been analyzed axiomatically. In this paper, we show that it is the only rule which is unbiased, monotone, tournamental, tie-breaking, and gradual. Thereby we provide a characterization of it for the first time. We also show these axioms to be logically independent.

Appendices

Appendix

Monotonicity

This appendix is on remarks related to update monotonicity.

A tournament choice correspondence, say $\Psi ,$ assigns to every tournament $T \in {\mathbb {T}}$ a subset $\Psi (T)$ of the set of alternatives A. For instance consider the Copeland tournament choice correspondence $\Psi ^{C}$ defined for an arbitrary tournament T as follows

$$\begin{aligned} \Psi ^{C}(T)=\{x\in A:Cscore(x,T)\ge Cscore(y,T)\text { for all }y\in A\}, \end{aligned}$$

where $Cscore(x,T)=|\{z\in A:xz\in T\}|.$

In Brandt et al. (2016), the following monotonicity for such correspondences is introduced: a tournament choice correspondence is said to be monotone if for all $a\in A$ and all T and $T^{\prime }$ in ${\mathbb {T}}$, with both (1) T $\cap$ $(A\backslash \{a\})^{2}=T^{\prime }\cap$ $(A\backslash \{a\})^{2}$ and (2) $ay\in T$ implies $ay\in T^{\prime }$ for all $y\in A\backslash \{a\}$

$$\begin{aligned} a\in \Psi (T)\text { implies }a\in \Psi (T^{\prime })\text {.} \end{aligned}$$

As the ranges of tournament preference correspondences differ from those of tournament choice correspondences, a direct comparison between update monotonicity and the monotonicity defined above is not possible. For a special subclass of tournament choice correspondences it is possible to translate the monotonicity condition to the setting of tournament preference correspondences. This translated condition appears to be weaker than update monotonicity. By that, for this subclass, the difference between monotonicity and update monotonicity can partially be revealed. Let $\Phi$ be a tournament preference correspondence. To $\Phi$ we can associate a tournament choice correspondence $\Psi ^{\Phi }$ in the following way

$$\begin{aligned} \Psi ^{\Phi }(T)=\{x\in A:x\in best(R)\text { for some }R\in \Phi (T)\}. \end{aligned}$$

Here $best(R)=\{y\in A:$ for all $z\in A$ we have $yz\in R\}.$ So, $\Psi ^{\Phi }(T)$ chooses all best alternatives from the outcome of $\Phi$ at tournament T. It is obvious that $\Psi ^{C}=\Psi ^{\Phi ^{C}}$.

Translating the monotonicity condition for tournament choice correspondences to tournament preference correspondences via such a best choice association yields the following. Let $\Phi$ be a tournament preference correspondence. We say that $\Phi$ is weakly monotone if for all tournament pairs $(T,T^{\prime })$ forming an elementary change from ab to ba,  with $b\in best(R)$ for some $R\in \Phi (T),$ there are $R^{\prime }\in \Phi (T^{\prime })$ with $b\in best(R^{\prime }).$

Proposition 1

Let $\Phi$ be a tournament preference correspondence and $\Psi ^{\Phi }$ its associated tournament choice correspondence. Then

$$\begin{aligned} \Phi \text { is weakly monotone if and only if }\Psi ^{\Phi }\text { is monotone.} \end{aligned}$$

Proof

(if part) Let $\Psi ^{\Phi }$ be monotone. Let tournament pair $(T,T^{\prime })$ forms an elementary change from ab to ba,  with $b\in best(R)$ for some $R\in \Phi (T).$ It is sufficient to prove that $b\in best(R^{\prime })$ for some $R^{\prime }\in \Phi (T^{\prime }).$ As $b\in best(R)$ for some $R\in \Phi (T),$ it follows that $b\in$ $\Psi ^{\Phi }(T).$ Monotonicity of $\Psi ^{\Phi }$ implies that $b\in$ $\Psi ^{\Phi }(T').$ Meaning that for some $R^{\prime }\in \Phi (T^{\prime })$ we have that $b\in best(R^{\prime }).$

(only if part) Let $\Phi$ be weakly monotone. Let tournament pair $(T,T^{\prime })$ forms an elementary change from ab to ba,  with $b\in \Psi ^{\Phi }(T).$ In order to prove that $\Psi ^{\Phi }$ is monotone it is sufficient to show that $b\in \Psi ^{\Phi }(T^{\prime }).$ As $b\in \Psi ^{\Phi }(T),$ there are R $\in \Phi (T)$ with $b\in best(R).$ Weak monotonicity of $\Phi$ guarantees the existence of $R^{\prime }\in \Phi (T^{\prime })$ with $b\in best(R^{\prime }).$ So, $b\in \Psi ^{\Phi }(T^{\prime }).$ $\square$

To see that update monotonicity implies weak monotonicity take notations like in the definition of the latter notion. So, $R\in \Phi (T)_{ba}.$ Update monotonicity implies that $\Phi (T)_{ba}\subseteq \Phi (T^{\prime })\subseteq \Phi (T)$. In particular $R\in$ $\Phi (T^{\prime }).$ So, we may take $R^{\prime }=R$ to conclude that $b\in best(R^{\prime }).$ So, a consequence of Proposition 1 is that update monotonicity of $\Phi$ implies that $\Psi ^{\Phi }$ is monotone. The Copeland preference correspondence $\varphi ^C$ is not update monotone. Therefore, the Copeland tournament correspondence $\Phi ^C$ is not T-monotone. The Copeland tournament correspondence $\Psi ^C$ is monotone. Therefore, by Proposition 1, $\Phi ^C$ is weakly monotone, but not T-monotone.