Abstract
We study three classes of diversity relations over menus arising from an unobserved categorization of alternatives in the form of a partition. A basic diversity relation declares a menu to be more diverse than another if and only if for every alternative in the latter there is an alternative in the former which belongs to the same category. An extension of a basic diversity relation preserves its weak and strict parts and possibly makes additional diversity judgements between hitherto incomparable menus. A cardinality-based extension is an extension which ranks menus on the basis of the number of categories that exist in each menu. We characterize each class axiomatically. Two axioms satisfied by each of the three classes are Monotonicity, which says that larger menus are at least as diverse, and No Complements, which rules out certain complementarities between alternatives in generating diversity.
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Notes
See, for instance, Tversky and Russo (1969).
Similar remarks apply to van Hees (2004), Dutta and Sen (1996), Bavetta and Del Seta (2001) and Peragine and Romero-Medina (2006). In all these papers, the axiomatic approach explicitly links binary relations over sets (freedom rankings) with a binary relation over alternatives, which usually (but not always) gives an objective measure of similarity between alternatives. A slightly different approach is taken by Klemisch-Ahlert (1993) where menus are subsets of \({\mathbb {R}}^{n}\). This allows for taking convex hulls of menus and transforming menus by adding vectors from alternatives.
A preorder is a binary relation \(\succsim \) which is reflexive (\(A\succsim A\) for every \(A\in Z\)) and transitive \((A\succsim C\) whenever \(A\succsim B\) and \(B\succsim C\)). A weak order is a preorder \(\succsim \) which is also complete (\(A\succsim B\) or \(B\succsim A\) for any A and B).
Letting \(E^{\succsim }(A)=\{x\in A:A\succ A{\setminus }\{x\})\) for every A as in Puppe (1996), we can rewrite A2 as follows: \(x\in E^{\succsim }(A\cup \{x\})\Leftrightarrow x\in E^{\succsim }(B\cup \{x\})\) for every \(B\subset A\). This is equivalent to saying that the map \(E^{\succsim }\), taken as a possibly empty-valued choice correspondence, satisfies the classical \(\alpha \) axiom and a weakening of the classical \(\gamma \) axiom.
A4 is a weakening of the Independence axiom in Pattanaik and Xu (1990), which says that \(A\succsim B\) if and only if \(A\cup \{x\}\succsim B\cup \{x\}\) whenever \(x\notin A\cup B\). The difference between our formulation and theirs is that in the models of diversity relations we are interested in, adding a new alternative to a menu does not necessarily lead to an improvement in diversity. In contrast, in the pure cardinality ranking over menus which Pattanaik and Xu (1990) study, adding a new alternative to a menu always leads to an improvement. Hence we need to make sure that adding x improves diversity in both A and B in our version of this property.
We should explicitly thank Clemens Puppe for suggesting A6 to us.
For instance, if language and subject matter both define attributes of books, a Spanish book on calculus would have two attributes.
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Helpful comments from Matthew Kovach, Clemens Puppe, Matthew Ryan and anonymous referees are gratefully acknowledged. Levent Ülkü was financially supported by the Asociación Mexicana de Cultura.
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Galindo, J., Ülkü, L. Diversity relations over menus. Soc Choice Welf 55, 229–242 (2020). https://doi.org/10.1007/s00355-020-01237-3
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DOI: https://doi.org/10.1007/s00355-020-01237-3