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## Hot answers tagged voting

4

Gibbard and Satterthwaite insist that the social choice function must be defined over all rational preferences over outcomes. That is, if voters' preferences could be anything (subject to the constraint of completeness and transitivity), then we have the Gibbard–Satterthwaite theorem. On the other hand, if preferences were single-peaked, then the ...

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This example comes from Miller (1980) (the paper that introduced the uncovered set definition). Suppose that the majority's preference is defined by: \begin{aligned} x \succ y \\ x \succ z \\ y \succ v \\ y \succ z \\ v \succ x \\ z \succ v \end{aligned} In this example, we have the top cycle set $L_1 = \{x, y, z, v \}$ (you can check this). But notice ...

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Suppose that A={a,b,c,....,z} is a finite set of social alternatives, and let P={>1,>2,....,>N} be a profile of strict preference orders on $A$ (where the set {1,2,...,N} indexes the voters). We say that the profile P is single-peaked if there is some way to order the alternatives in A (e.g. in alphabetical order) such that, for each of the ...

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You ask an excellent question, as it has been partly but not fully discussed in the scientific literature. Your exact proposal (weighting representatives by the amount of votes they got in the election) has been discussed on a blog post (in french) by Jean-François Laslier see here. In practice, representatives are often elected on a geographical basis, ...

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Allowing indifference will not solve the problem of the effect of the independence of irrelevant alternatives (combined with the unanimity and transitivity requirements). You have said $b >_1 c$ and $c >_2 b$ should lead to $b =_s c$ Now consider $a >_1 b$ and $a >_2 b$, which should lead to $a >_s b$ by unanimity so $a >_1 b >_1 c$ ...

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I solved it; here's my solution: Let $f:L^n\to A$ be an incentive compatible, non-dictatorial social choice function and let $F:L^n\to L$ be its extension. To show that $F$ is non-dictatorial, then for any voter $i\in\{1,\dots,n\}$ we need to find $\prec_1,\dots,\prec_n\in L$ such that $\prec_i\,\neq\,\prec$, where $\prec\,=F(\prec_1,\dots,\prec_n)$. So we ...

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Arrow's Theorem concerns a social preference function ---that is, a function that produces a group preference order for every possible profile of individual preferences. The axioms "Nondictatorship", "Independence of Irrelevant Alternatives", etc. must then be satisfied by the function at every profile. (To be more precise, an axiom like IIA involves a ...

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The following recent article might be relevant to your question: Weighted representative democracy, Marcus Pivato and Arnold Soh, Journal of Mathematical Economics 88, pp.52-63, 2020. Abstract: We propose a new system of democratic representation. Any voter can choose any legislator as her representative; thus, different legislators can represent ...

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