Quasitransitivity

Quasitransitivity is a property found in binary relations. A relation R on a set A is said to be quasitransitive if for all elements a, b, and c in A, whenever a R b and b R c, then it follows that a R c, with the exception that a R b and b R c do not imply a R c if b R a and c R b. More formally, a relation R is quasitransitive if for all a, b, c in A, (a R b and b R c) implies a R c, unless it is the case that b R a and c R b.

This property is often encountered in contexts where preferences are being modeled. For instance, if an individual prefers option A to option B, and option B to option C, it is generally assumed that they also prefer option A to option C. However, quasitransitivity allows for a specific type of cyclical preference. If, for example, an individual considers A to be better than B, and B to be better than C, but also considers B to be no worse than A (meaning they are indifferent or prefer A to B), and C to be no worse than B, then the strict preference of A over C is not guaranteed.

Quasitransitivity is weaker than transitivity. A transitive relation is always quasitransitive. However, a quasitransitive relation is not necessarily transitive. The additional condition in quasitransitivity, concerning the reverse relations, is what distinguishes it from strict transitivity. This distinction is important in decision theory and social choice theory, where understanding the precise structure of preferences is crucial.