Cumulative voting

Cumulative voting (accumulation voting or weighted voting) is a (usually) multiple-winner voting system intended to promote proportional representation. It is used heavily in corporate governance, where it is mandated by many U.S. states, and it was used to elect the Illinois House of Representatives from 1870 until 1980. It was used in England in the late 19th century to elect school boards.

In the most common implementation of this system, a voter facing multiple choices is given X number of points. The voter can then assign his points to one or more of the choices, thus enabling one to weight one's vote if desired. This could be achieved through a normalized ratings ballot, or through multiple plurality ballots, one per each point allocated. Typically, each voter will have as many votes as there are winners to be selected. If there was only one winner to be selected and voters were given one vote, this is equivalent to the First Past the Post electoral system voting.

A more generalized implementation would simply be to allow the voter to split their vote in whatever proportions they like i.e. a voter might give 67.3% of their vote to one candidate, and the remainder to another.

Tactical voting is the rational response to this system. The strategy of voters should be to balance how strong their preferences for individual candidates are against how close those candidates will be to the critical number of votes needed for election. In general, one should put all of their points behind a single candidate to maximize that candidate's chances of winning i.e. they should use FPTP-like strategy to boost their favorite frontrunner.

Quadratic voting

 * see also: Quadratic voting

A variant of cumulative voting which gained popularity in 2018 is "quadratic voting". Quadratic voting was conducted in an experiment by the Democratic caucus of the Colorado House of Representatives in April 2019. It differs from cumulative voting by altering "the cost" and "the vote" relation from linear to quadratic.

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
Quadratic voting has all the major flaws of cumulative voting such as vote splitting and complexity.

Generalized cumulative voting
Cumulative voting can be further generalized into $$\ell_p$$-norm cumulative voting, where a voter submits a cardinal (Range-style) ballot and that ballot is normalized to have unit $$p$$-norm before it is counted.

If $$p = 1$$, then $$\ell_p$$-norm cumulative voting is standard cumulative voting. If $$p=2$$, then it is quadratic voting, and $$p=\infty$$ results in Range voting (with the highest-rated candidate scaled up to maximum rating).

The optimal tactical vote depends heavily on $$p$$. For instance, vote-splitting is a problem in standard cumulative voting but not in Range voting. On the other hand, the Burr dilemma is a problem in Approval voting (and thus in tactical Range voting) but not in standard cumulative voting.

Clone independence
Every form of $$\ell_p$$-norm cumulative voting except $$p=\infty$$ (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 $$x = \frac{\sqrt[p]{2} - 1}{2\big(\sqrt[p]{2} + 1\big)}$$ and consider the election


 * $$\frac{1}{2}+x-\epsilon$$: A (1) B (0)
 * $$\frac{1}{2}-x+\epsilon$$: A (0) B (1)

for some $$\epsilon$$ where $$x > \epsilon > 0$$. 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 $$\ell_p$$ normalization then leads to the first faction giving each A candidate $$y = \frac{1}{\sqrt[p]{2}}$$ points each. As a result, there's a three-way tie when $$\epsilon=0$$, and B wins for any $$\epsilon > 0$$, which demonstrates the clone failure.

Example
Suppose we want to find a clone failure for $$p=2$$ (quadratic voting). Then $$x = \frac{\sqrt{2}-1}{2+2\sqrt{2}} \approx 0.086$$ and $$y = \frac{1}{\sqrt{2}} \approx 0.707$$. Let $$\epsilon=0.001$$ 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.