Preference Specification

Definition

Usually, preference relations are formalized as strict partial orders (SPO), which means that a preference relation $\succ$ satisfies the following properties:

• irreflexivity: $\forall x\ (x \not\succ x)$,

• transitivity: $\forall x,y,z\ (x\,\succ \,y\,\wedge \,y\,\succ \,z\,\rightarrow \,x\succ z)$.

Additionally, an SPO $\succ$ is a weak order if $\forall x,y,z\ (x \succ y \rightarrow x \succ z \vee z \succ y)$, and a total order if $\forall x,y\ (x \succ y \vee y \succ x \vee x = y)$.

The formal properties of orders capture the nature of preferences in an abstract, application-independent way. It is obvious that irreflexivity should hold: preferring an object over itself seems to violate the basic intuitions behind preference. But transitivity is debatable. On one hand, it captures the rationality of preferences [1, 2]. On the other, transitivity is sometimes violated by preference aggregation in voting scenarios [3]. The...