Kemeny Preference Ranking (COMSOC)

Definition

Kemeny Preference Ranking ( COMSOC)

A Kemeny preference ranking minimises the Kemeny score: the total Kendall–tau distance to the voters’ rankings.

Let $V$ be the set of voters, $R=({\succ }_i{)}_{i\in V}$ their profile, and $\mathcal{L}(C)$ the set of complete, strict rankings over alternatives $C$. A Kemeny ranking is any

$${\succ }^{\ast }\in \arg \underset{\succ \in \mathcal{L}(C)}{\min }\sum _{i\in V}\mathrm{KT}(\succ ,{\succ }_i).$$

The sum counts one distance per voter; the minimum is taken over all rankings, not just those submitted. Several rankings may tie for the minimum. “Consensus” does not require unanimous agreement.

Fixed Pairs

The order of a fixed pair is fixed!

For a pair of alternatives $\{x,y\}\subseteq C$, if $R(x,y)=R$, i.e, …, then every Kemeny consensus ${\succ }^{\ast }$ has $x{\succ }^{\ast }y$.