Essentiality and convexity in the ranking of opportunity sets

This paper studies a class of binary relations on opportunity sets which we call opportunity relations (ORs). These are reflexive and transitive (pre-orders) and further satisfy a monotonicity and desirability condition. Associated with each OR is an essential element operator (Puppe, J Econ Theory 68:174–199, 1996). Our main results axiomatically characterise three 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; those whose essential element operator is the extreme point operator for some closure space (Ando, Discrete Math 306:3181–3188, 2006)—the closed ORs; and those whose essential element operator is the extreme point operator for some abstract convex geometry (Edelman and Jamison, Geometriae Dedicata 19:247–270, 1985)—the convex ORs. Our characterisation of convex ORs generalises the analysis of Klemisch-Ahlert (Soc Choice Welf 10:189–207, 1993). Our results also provide complementary perspectives on the well-known characterisation of closure operators by Kreps (Econometrica 47:565–577, 1979), as well as the recent work of Danilov and Koshevoy (Order 26:69–94, 2009; Soc Choice Welf 45:51–69, 2015).