From that time on Caesar managed all the affairs of state alone and after his own pleasure; so that sundry witty fellows, pretending by way of jest to sign and seal testamentary documents, wrote “Done in the consulship of Julius and Caesar.”
Suetonius, Lives of the Caesars. Chapter XX.
Abstract
The Gibbard–Satterthwaite theorem is a cornerstone of social choice theory, stating that an onto social choice function cannot be both strategy-proof and non-dictatorial if the number of alternatives is at least three. The Duggan–Schwartz theorem proves an analogue in the case of set-valued elections: if the function is onto with respect to singletons, and can be manipulated by neither an optimist nor a pessimist, it must have a weak dictator. However, the assumption that the function is onto with respect to singletons makes the Duggan–Schwartz theorem inapplicable to elections which necessarily select multiple winners. In this paper we make a start on this problem by considering rules which always elect exactly two winners (such as the consulship of ancient Rome). We establish that if such a consular election rule cannot be expressed as the union of two disjoint social choice functions, then strategy-proofness implies the existence of a dictator. Although we suspect that a similar result holds for k-winner rules for \(k>2\), there appear to be many obstacles to proving it, which we discuss in detail.
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Notes
The original Duggan–Schwartz result thus applies to regular correspondences where the smallest winner set is a singleton.
Gaius Marius held the consulship of Rome an unprecedented seven times. There was nothing in the Roman constitution that prevented a pair without Marius being elected, so strictly speaking this wasn’t a range restriction. However the high probability of Marius winning made this an appropriate name for this degenerate case of a consular rule.
Among other problems. If we take \(\succ _i^O\cup \succ _i^P\) to represent a voter’s strict preferences, we have to posit that a voter is able to at once strictly prefer X to Y and Y to X—what is sometimes known as conflicted preferences in the literature.
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Supported by the NZ Marsden fund, grant UOA3706352. The authors acknowledge useful conversations with Arkadii Slinko.
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Ianovski, E., Wilson, M.C. Manipulability of consular election rules. Soc Choice Welf 52, 363–393 (2019). https://doi.org/10.1007/s00355-018-1152-2
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DOI: https://doi.org/10.1007/s00355-018-1152-2