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Public Choice

Erschienen in:

02.05.2017

The relationship between the normalized gradient addition mechanism and quadratic voting

verfasst von: Daniel Benjamin, Ori Heffetz, Miles Kimball, Derek Lougee

Erschienen in: Public Choice | Ausgabe 1-2/2017

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Abstract

Quadratic voting and the normalized gradient addition mechanism are both social choice mechanisms that confront individuals with quadratic budget constraints, but they are applicable in different contexts. Adapting one or both to apply to the same context, this paper explores the relationship between these two mechanisms in three contexts: marginal adjustments of continuous policies, simultaneous voting on many public choices, and voting on a single public choice accompanied by private monetary consequences. In the process, we provide some formal analysis of quadratic voting when (instead of money) votes are paid for with abstract tokens that are equally distributed by the mechanism designer.

Vorheriger Artikel Equality, legitimacy, interests, and preferences: historical notes on Quadratic Voting in a political context

Nächster Artikel Quadratic election law

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Using this setup, it is straightforward to calculate the gradients and indifference curves for each individual. Setting the right angle of the triangle as the origin, the horizontal movement of the mechanism is \(x_{1}=\frac{\sqrt{2}}{2}(w_{2}-w_{1})\) and the vertical movement is \(x_{2}=\frac{-\sqrt{2}}{2}(w_{1}+w_{2})\), where \(w_{1}\) and \(w_{2}\) are the wealth for person 1 and person 2, respectively. Then, we have the money-metric utility for person 1 as \(U_{1}=\frac{u_{1}}{c}-\frac{\sqrt{2}}{2}x_{1}+\frac{\sqrt{2}}{2}(\frac{2u_{1}}{c}-1)x_{2}\), where \(u_{1}\) is her valuation of the public good, \(x_{1}\) is the horizontal distance, and \(x_{2}\) is the vertical distance. From this utility function, the gradient is \(\nabla {\varvec{U}}_{1}=\frac{\sqrt{2}}{2}(-1,\frac{2u_{1}}{c}-1)\), and the indifference curve is \(x_{2}=\frac{\sqrt{2}U_{1}-\frac{2u_{1}}{c}+x_{1}}{\frac{2u_{1}}{c}-1}\).

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Titel: The relationship between the normalized gradient addition mechanism and quadratic voting
verfasst von: Daniel Benjamin
Ori Heffetz
Miles Kimball
Derek Lougee
Publikationsdatum: 02.05.2017
Verlag: Springer US
Erschienen in: Public Choice / Ausgabe 1-2/2017
Print ISSN: 0048-5829
Elektronische ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-017-0414-3

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