Abstract
Most environments where assignment mechanisms (possibly random) are used are such that participants have outside options. For instance private schools and private housing are options that participants in a public choice or public housing assignment problems may have. We postulate that, in cardinal mechanisms, chances inside the assignment process could favor agents with better outside options. By imposing a robustness to outside options condition, we conclude that, on the universal domain of cardinal preferences, any mechanism must be (interim) ordinal.
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Notes
On a different topic, a similar effect is also noticed in Arnosti and Randolf (2021).
The lack of enough supply of public slots in this example could just be offset by the existence of other public schools of very low quality that everyone would like to avoid (e.g. ghetto schools.)
Another example. Consider a student who is talented enough for a scholarship at the school of arts. Yet she would prefer to attend the best public school in the area. Given that she has an extra outside option available, she could better bear the risks of applying for the most popular school than a not so talented student. Differences in talent jointly with selective schools may also generate disruptions in the assignment of public slots. Any other discriminatory criterion, given maybe by other socioeconomics (religion, etnicity etc.) that give some agents higher access to reservation objects, may trigger a similar normative requirement.
We thank a referee who pointed out this observation.
Seemingly, the reader could wonder why the mechanism should not be robust to any modification of the valuation for the outside option, in any quantity and direction. (We thank a referee for this observation.) Such a more stringent condition would lead to identical conclusions as in this paper. We consider our no-regret condition as stated in our paper as a minimal condition regarding outside options that leads to ordinality.
We thank a referee and the associate editor for pointing this simplification of the model.
In what follows, \(1_{|S|+1}\) is a vector of ones with dimension \(|S|+1\).
Notice that two elements ahead of o are not enough. Notice, along the lines of the proof of Lemma 1, that Q(v) must be invariant to \(v^{o}\) as long as ordinal preferences are maintained and no matter how low the valuation of objects worse than o is. Fix any two \(v,v^{\prime }\) consistent with those ordinal preferences. There are affine transformations \(\hat{v},\hat{v}^{\prime }\) of the former vectors such that \(\hat{v}\) and \(\hat{v}^{\prime }\) differ only, if any, on the valuation for the outside option. Hence these types receive equal probability bundles.
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Previous title: “Cardinal Assignment Mechanisms: Money Matters More than It Should.” Caterina Calsamiglia acknowledges financial support by the ERC Starting Grant 638893. Francisco Martínez-Mora gratefully acknowledges funding support from Fundación La Caixa and Fundación Caja Navarra under contract LCF/PR/PR13/51080004. Antonio Miralles thanks the Ortygia Business School and acknowledges financial support from the Spanish Government R&D programs SEV-2015-0563 and ECO2017-83534P, and to the Catalan Government (SGR2017 0711.) We are also grateful to Jordi Massó, Clemens Puppe and to the audiences at both the 13th Matching in Practice workshop and the 2017 UECE Lisbon Meetings, for their comments and help.
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Calsamiglia, C., Martínez-Mora, F. & Miralles, A. Random assignments and outside options. Soc Choice Welf 57, 557–566 (2021). https://doi.org/10.1007/s00355-021-01328-9
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DOI: https://doi.org/10.1007/s00355-021-01328-9