Cumulative voting: Difference between revisions
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[[Category:Multi-winner voting methods]]
==
{{wikipedia|Quadratic voting}}
:''see also: [[Quadratic voting]]''
A variant of cumulative voting which gained popularity in 2018 is "[[
Quadratic voting is characterized by the optimal zero-information strategic vote (using Myerson-Weber strategy) being to vote honestly -- that is, to vote according to any affine scaling of one's honest utilities.
=== Criticism ===
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Cumulative voting can be further generalized into <math>\ell_p</math>-norm cumulative voting, where a voter submits a cardinal (Range-style) ballot and that ballot is normalized to have unit <math>p</math>-norm before it is counted.
If <math>p = 1</math>, then <math>\ell_p</math>-norm cumulative voting is standard cumulative voting. If <math>p=2</math>, then it is quadratic voting, and <math>p=\infty</math> results in
The optimal tactical vote depends heavily on <math>p</math>. For instance, vote-splitting is a problem in standard cumulative voting but
=== Clone independence ===
Every form of <math>\ell_p</math>-norm cumulative voting except <math>p=\infty</math> (plain [[Range voting]]) is vulnerable to vote-splitting.
For continuous cumulative voting, suppose each candidate can be given any score between 0 and 1. Then let <math>x = \frac{\sqrt[p]{2} - 1}{2\big(\sqrt[p]{2} + 1\big)}</math> and consider the election
* <math>\frac{1}{2}+x-\epsilon</math>: A (1) B (0)
* <math>\frac{1}{2}-x+\epsilon</math>: A (0) B (1)
for some <math>\epsilon</math> where <math>x > \epsilon > 0</math>. A wins by majority rule no matter the value of p.
Clone A into A1 and A2 so that everybody rates the two clones equal. The <math>\ell_p</math> normalization then leads to the first faction giving each A candidate <math>y = \frac{1}{\sqrt[p]{2}}</math> points each. As a result, there's a three-way tie when <math>\epsilon=0</math>, and B wins for any <math>\epsilon > 0</math>, which demonstrates the clone failure.
==== Example ====
Suppose we want to find a clone failure for <math>p=2</math> (quadratic voting). Then <math>x = \frac{\sqrt{2}-1}{2+2\sqrt{2}} \approx 0.086</math> and <math>y = \frac{1}{\sqrt{2}} \approx 0.707</math>. Let <math>\epsilon=0.001</math> to compensate for the roundoff error. Before cloning, the ballots are
* 0.585: A (1) B (0)
* 0.415: A (0) B (1)
where A wins 0.585 points to 0.415.
After cloning:
* 0.585: A1 (0.707) A2 (0.707) B (0)
* 0.415: A1 (0) A2 (0) B (1)
A1 gets 0.413 points, A2 the same, and B gets 0.415 points, thus making B the winner.
▲The optimal tactical vote depends heavily on <math>p</math>. For instance, vote-splitting is a problem in standard cumulative voting but is not in Approval voting. On the other hand, the Burr dilemma is a problem in Approval voting but not in standard cumulative voting.
== Notes ==
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