Extended partial orders: A unifying structure for abstract choice theory

The concept of a strict extended partial order (SEPO) has turned out to be very useful in explaining (resp. rationalizing) non-binary choice functions. The present paper provides a general account of the concept of extended binary relations, i.e, relations between subsets and elements of a given universal set of alternatives. In particular, we define the concept of a weak extended partial order (WEPO) and show how it can be used in order to represent rankings of opportunity sets that display a "preference for opportunities". We also clarify the relationship between SEPOs and WEPOs, which involves a non-trivial condition, called "strict properness". Several characterizations of strict (and weak) properness are provided, based on which we argue for properness as an appropriate condition demarcating "choice based" preference.