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Essentiality and convexity in the ranking of opportunity sets

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  • Matthew Ryan

    (Auckland University of Technology)

Abstract

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).

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  • Matthew Ryan, 2016. "Essentiality and convexity in the ranking of opportunity sets," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(4), pages 853-877, December.
    • Handle: RePEc:spr:sochwe:v:47:y:2016:i:4:d:10.1007_s00355-016-0994-8
    DOI: 10.1007/s00355-016-0994-8
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    2. Jimena Galindo & Levent Ülkü, 2020. "Diversity relations over menus," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 55(2), pages 229-242, August.

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