Abstract
This paper covers the theory of the uncovered set used in the literatures on tournaments and spatial voting. I discern three main extant definitions, and I introduce two concepts that bound existing sets from above and below: the deep uncovered set and the shallow uncovered set. In a general topological setting, I provide relationships to other solutions and give results on existence and external stability for all of the covering concepts, and I establish continuity properties of the two new uncovered sets. Of note, I characterize each of the uncovered sets in terms of a decomposition into choices from externally stable sets, and I define the minimal generalized covering solution, a nonempty refinement of the deep uncovered set that employs both of the new relations.
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Duggan, J. Uncovered sets. Soc Choice Welf 41, 489–535 (2013). https://doi.org/10.1007/s00355-012-0696-9
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DOI: https://doi.org/10.1007/s00355-012-0696-9