Essentiality and convexity in the ranking of opportunity sets

This paper studies the essential elements (Puppe, 1996) associated with binary relations over opportunity sets. We restrict attention to binary relations which are reflexive and transitive (pre-orders) and which further satisfy a monotonicity and desirability condition. These are called opportunity relations (ORs). Our main results axiomatically characterise two important classes of ORs: those for which any opportunity set lies in the same indifference class as its set of essential elements — the essential ORs; and those whose essential element operator is the extreme point operator for some abstract convex geometry (Edelman and Jamison, 1985) —the convex ORs. Our characterisation of convex ORs generalises the analysis in Klemisch-Ahlert (1993), who restricts attention to a particular subclass of ACGs known as convex shellings. We present an example which suggests that this latter class is restrictive —there are ACGs which are not convex shellings but which are associated with plausible ORs.