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On the maximization of menu-dependent interval orders

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Abstract

We study the behavior of a decision maker who prefers alternative x to alternative y in menu A if the utility of x exceeds that of y by at least a threshold associated with y and A. Hence the decision maker’s preferences are given by menu-dependent interval orders. In every menu, her choice set comprises of undominated alternatives according to this preference. We axiomatize this broad model when thresholds are monotone, i.e., at least as large in larger menus. We also obtain novel characterizations in two special cases that have appeared in the literature: the maximization of a fixed interval order where the thresholds depend on the alternative and not on the menu, and the maximization of monotone semiorders where the thresholds are independent of the alternatives but monotonic in menus.

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Notes

  1. Schwartz (1976, Theorem 3) and Aleskerov et al. (2007, Corollary 3.5) characterize the maximization of interval orders. Frick (2015, Theorem 2.4) characterizes the maximization of monotone semiorders.

  2. This version of the weak axiom of revealed preference appears in Arrow (1959).

  3. This axiom appears in Chernoff (1954). It is referred to as \(\alpha \) in Sen (1971) and as Heredity in Aleskerov et al. (2007).

  4. To the best of our knowledge, the concept of anchors is new. Dominant alternatives are related to occasionally optimal alternatives, a concept which appears in Frick (2015). In particular, every occasionally optimal alternative is dominant, but not vice versa. In Sect. 4.2, we will show that occasionally optimal alternatives are precisely those anchor alternatives which are also dominant.

  5. Equivalently P is an interval order if it satisfies asymmetry (if xPy then not yPx) and the following intervality condition: if wPx and yPz, then wPz or yPx. See Fishburn (1970).

  6. This condition also appears as the Bliss Point axiom in Masatlıoğlu and Nakajima (2013).

  7. See, for instance, Plott (1973) who shows that Path Independence and \(\gamma \) together characterize the maximization of strict partial orders.

  8. See also Fishburn (1975). Also recall that Path Independence and \(\gamma \) are equivalent to the maximization of a strict partial order, as shown by Plott (1973). Since every interval order is a strict partial order, interval order representation requires a stronger axiomatic system.

  9. Besides the work we have already cited, other prominent examples of existential axioms include the Reducibility axiom in Manzini and Mariotti (2012) and the WARP-LA axiom in Masatlıoğlu et al. (2012).

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Authors and Affiliations

  1. Institut für Diskrete Mathematik und Geometrie at Vienna University of Technology, Vienna, Austria

    Juan P. Aguilera

  2. Department of Economics, Centro de Investigación Económica (CIE), ITAM, Rio Hondo 1, Ciudad de México, 01080, Mexico

    Levent Ülkü

Authors
  1. Juan P. Aguilera
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  2. Levent Ülkü
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Correspondence to Levent Ülkü.

Additional information

We acknowledge helpful comments by Sean Horan, Romans Pancs and two referees. The first author was partially supported by FWF grants P-26076-N25 and I-1897-N25. The second author gratefully acknowledges financial support from Associación Mexicana de Cultura.

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Aguilera, J.P., Ülkü, L. On the maximization of menu-dependent interval orders. Soc Choice Welf 48, 357–366 (2017). https://doi.org/10.1007/s00355-016-1007-7

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  • DOI: https://doi.org/10.1007/s00355-016-1007-7

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