Strict Preference Ranking (COMSOC)

Definition

Strict Preference Ranking ( COMSOC)

A strict preference ranking over alternatives $C$ orders every pair of distinct alternatives without ties. It is a relation $\succ \subseteq C\times C$ satisfying

$$\begin{array}{rlrl}& a\nsucc a& & \text{for every }a\in C,\\ & a\succ b\wedge b\succ c\Rightarrow a\succ c& & \text{for every }a,b,c\in C,\\ & \text{exactly one of }a\succ b\text{ or }b\succ a& & \text{for every distinct }a,b\in C.\end{array}$$

Thus it is irreflexive and transitive. “Exactly one direction” combines asymmetry (not both directions) with comparability (at least one direction for distinct alternatives). The statement $a\succ b$ expresses a strict preference for $a$ over $b$.

These are the rankings in $\mathcal{L}(C)$: the preference ranking note uses “ranking” to mean this strict, no-ties case. A strict relation alone need not compare every pair; a strict ranking does.