Abstract
I refine the multi-utility model and identify the minimal number W of total orders that is sufficient to rationalize a path independent choice function. This identification invokes the well-known pigeonhole principle: any dataset of size \(W+1\) that is rationalized by W rankings must contain at least two distinct observations where the same ranking is maximized. In general, the index W can be huge even for reasonable choice functions, such as top-ten rules. If W is constrained, then minimal rationalizations can be found in polynomial time via an explicit focal algorithm. The axiom of Expansion (Sen’s \(\gamma \)) describes a special case where the index W must equal the capacity—the largest number of elements that may be selected together in a menu.
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I thank Larry Epstein, Itzhak Gilboa, Ran Spiegler, Michael Richter, Asen Kochov, Bart Lipman, Chris Chambers, Erya Yang, Nicholas Yannelis, Federico Echenique, Jean-Claude Falmagne, Don Saari, Luca Anderlini, and anonymous referees for their suggestions. I received valuable feedback at seminars in Boston University, Caltech, Paris School of Economics, Georgetown Unirversity, and IMBS.
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Kopylov, I. Minimal rationalizations. Econ Theory 73, 859–879 (2022). https://doi.org/10.1007/s00199-021-01345-w
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DOI: https://doi.org/10.1007/s00199-021-01345-w