Abstract
We investigate group manipulation by vote exchange in two-tiers elections, where voters are first distributed into districts, each with one delegate. Delegates’ preferences result from aggregating voters’ preferences district-wise by means of some aggregation rule. Final outcomes are sets of alternatives obtained by applying a social choice function to delegate profiles. An aggregation rule together with a social choice function define a constitution. Voters’ preferences over alternatives are extended to partial orders over sets by means of either the Kelly or the Fishburn extension rule. A constitution is Kelly (resp. Fishburn) swapping-proof if no group of voters can get by exchanging their preferences a jointly preferred outcome according to the Kelly (resp. Fishburn) extension. We establish sufficient conditions for swapping-proofness. We characterize Kelly and Fishburn swapping-proofness for Condorcet constitutions, where both the aggregation rule and the social choice function are based on simple majority voting. JEL Class D71, C70.
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Notes
- 1.
Consider for instance the case of 15 voters divided into 5 districts with size 3 each, and where voters 1 to 9 prefer the alternative a to alternative b. Then place voters 1,2,3 in the first district, voters 4,5,6 in the second district, and dispatch each of the three remaining supporters of a in one of the last 3 districts. It follows that b wins. If now voter 3 exchange her location with any of the b supporters, a wins. This example relates to the referendum paradox (see [23, 28]), which holds when indirect majority voting is inconsistent with direct majority voting (i.e. when the choice is made from the voters’ preferences). Example 1 below provides another illustration of gerrymandering.
- 2.
One may consult the website: http://pjmedia.com/zombie/2010/11/11/the-top-ten-most-gerrymandered-congressionaldistricts-in-the-united-states/, where real districts exhibiting weird shapes are shown.
- 3.
On the welcome page of this website, the following text appears: “You should pair vote if either: You want to keep a political party from winning, You don’t feel that there is any point in voting for who you want, as the candidate or party has no chance of getting elected, You are tired of your vote not being represented in Parliament”.
- 4.
Once threatened by the California Secretary of States, the websites voteswap2000.com and votexchange2000.com immediately shut their virtual doors.
- 5.
On 8-6-2007, the 9th U.S. Circuit Court of Appeals ruled that “the websites’ vote-swapping mechanisms as well as the communication and vote swaps they enabled were constitutionally protected. At their core, they amounted to efforts by politically engaged people to support their preferred candidates and to avoid election results that they feared would contravene the preferences of a majority of voters in closely contested states. Whether or not one agrees with these voters’ tactics, such efforts, when conducted honestly and without money changing hands, are at the heart of the liberty safeguarded by the First Amendment.”
- 6.
- 7.
Note that “direct” manipulation by vote swapping cannot occur in the case of anonymous SCFs.
- 8.
Given a profile of district preferences \(p_{D,\theta }=(\theta (p\mid _{D_{1}}),...,\theta (p\mid _{D_{K}}))\), the majority tournament among districts \(T(p_{D,\theta })\) is defined by \(\forall a,b\in A_{m}\), a \(T(p_{D,\theta })\) b iff \(\left| \{k\in \{1,...,k\}:a\theta (p\mid _{D_{k}})b\}\right| >\frac{K}{2}\). The final outcome we consider in this example is known as the Copeland set of \(T(p_{D,\theta })\).
- 9.
(Set-Mon) is actually sufficient for Kelly group strategy-proofness when preferences are transitive, as proved in [12].
- 10.
F satisfies the Aizerman property if \(\forall n,m\in \mathbb {N}\), \(\forall a,b\in A_{m}\), \(\forall P\in \mathcal {Q}(A_{m})^{n}\) with \(F(P)\subseteq B\), then \(F(P/B)\subseteq F(P)\). Moreover, F satisfies the strong superset property if \(F(P/B)=F(P)\).
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Dindar, H., Laffond, G., Lainé, J. (2015). Vote Swapping in Representative Democracy. In: Kamiński, B., Kersten, G., Szapiro, T. (eds) Outlooks and Insights on Group Decision and Negotiation. GDN 2015. Lecture Notes in Business Information Processing, vol 218. Springer, Cham. https://doi.org/10.1007/978-3-319-19515-5_18
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