An axiomatic characterization of the Slater rule

See Kemeny (1959) and Can and Storcken (2018).

Copeland (1951).

The rule introduced in Miller (1980) takes all linear extensions of the cover relation. Here at a profile p an alternative a covers alternative b,  if for all alternatives x we have that b beats x in pairwise comparison at p implies that a beats x in pairwise comparison.

Muller and Satterthwaite (1977) show the equivalence of strategy proofness and strong positive association hence making the connection to the impossibility results of Gibbard (1973) and Satterthwaite (1975). Brandt et al. (2016) also introduces a monotonicity condition for tournaments only. We provide a detailed comparison between this condition and our framework in the Appendix.

We take linear orders to be irreflexive without theoretical consequences in our approach. It has however notational advantage here as now linear orders are partial tournaments.

For every two distinct alternatives a and b we can partition \({\mathbb {T}}\) in \({\mathbb {T}}_{ab}\), \({\mathbb {T}}_{ba}\) and \({\mathbb {T}}_{\{a,b\}}=\) \(\mathbb { \ T\backslash (T}_{ab}\cup \) \({\mathbb {T}}_{ba}).\) Note that complete tournaments are in \({\mathbb {T}}_{ab}\cup \) \({\mathbb {T}}_{ba}\) and that \( {\mathbb {T}}_{\{a,b\}}\) consists of the incomplete tournaments at which a and b are incomparable. Every elementary change from ab to ba or from ba to ab is now a connecting relation (edge) between an element in \( {\mathbb {T}}_{ab}\cup \) \({\mathbb {T}}_{ba}\) and one in \({\mathbb {T}}_{\{a,b\}}.\) Further, an elementary change from xy to yx does not effect the preference between a and b in case \(\{x,y\}\ne \{a,b\}.\) Call an elementary change between a and b if it is either from ab to ba or from ba to ab. So, a path of elementary changes with an odd number of changes between a and b starting in \({\mathbb {T}}_{ab}\cup \) \( {\mathbb {T}}_{ba}\) ends in \({\mathbb {T}}_{\{a,b\}}.\) As \({\mathbb {T}}_{\{a,b\}}\) consists of incomplete tournaments and both R an \(R^{\prime }\) are complete it follows that on every path of elementary changes from R to \( R^{\prime }\) there is an even number of elementary changes between a and b. As this holds for every two distinct alternatives it follows that such a path is of even length.

For instance, it is possible to deduce the same results as presented here with only marginal changes based on a condition like \(\varphi \) is gradual if for all pairs of profiles (p, q) forming an elementary change from ab to ba in \({\mathbb {L}}^{N}\) there are non-negative integers k such thatMeaning that if at such an elementary change the sets of outcomes \(\varphi (p)\) and \(\varphi (q)\) are disjoint then graduality imposes that for some (large) replica \(k\cdot p\) of p the influence of q is limited. Meaning that the outcome at \(k\cdot p+q\) has something in common with the outcome at p. This condition is related to Young’s continuity condition for scoring rules. See Young (1975).

This resembles the Fishburn’s C2 condition for choice correspondences.

See footnote 5 in Baigent and Klamler (2003), which attributes the clarification of the issue to Hannu Nurmi.

Instead of representing \(\succ \) by an additive weight function \(\omega \) as above one may choose \(\succ \) such that