Equally Weighted Vote

An Equally Weighted Vote is the concept that every vote should carry equal power or weight. In 1964, Wesberry v. Sanders, The U.S. Supreme Court declared that equality of voting - one person, one vote - means that "the weight and worth of the citizens' votes as nearly as is practicable must be the same."

Votes can be unequally weighted at a number of different stages in the election process. First, a vote can be unequal due to the voting method itself. Any voting method which allows Vote Splitting ensures that voters do not have an equally weighted vote in elections which have more than two candidates. Second, votes for representatives to a larger geographical area who are representing a district within that area can be unequally weighted due to district lines which may bias an election in favor of one faction or another. When district lines are intentionally drawn in order to marginalize specific factions, (reducing the weight of those voters relative to others) it's known as Gerrymandering.

The Electoral College and other mechanisms which use representatives to determine elections rather than directly using the votes cast also violate the Equally Weighted Vote, particularly in cases where electors or representatives are not allocated proportionately to the population. In the case of the Electoral College each state is awarded electors based on the number of members of congress. The House of Representatives is based on population, which would ensure that electoral votes were equally weighted as nearly as is practicable, but each state is also awarded two additional electors per state corresponding to their two Senators. This results in US presidential elections which specifically violate the Equal Vote Criterion.

The 1964, Wesberry v. Sanderscase cited above addressed Gerrymandering. In the case of district lines it's impossible to ensure that elections will not favor one faction or the other over time as populations grow and change, but it is "practicable" to prevent and mitigate this phenomena. However in the case of vote splitting and the Electoral Collage achieving a perfectly Equally Weighted Vote is fully possible.

Equal Vote Criterion

Any voting method or election which passes the Test of Balance passes the Equal Vote Criterion and can be said to guarantee an Equally Weighted Vote.

The Test of Balance

The test of balance is defined as the following "Any way I vote, you should be able to vote in an equal and opposite fashion. Our votes should be able to cancel each other’s out."

Voting methods which ensure an Equally Weighted Vote

Voting Methods which ensure an Equally Weighted Vote with any number of candidates include Approval Voting, Score Voting, STAR Voting, as well as a number of others. In general Cardinal Voting methods ensure an Equally Weighted Vote for each voter. Many Condorcet methods (most that can be calculated only with the pairwise counting matrix, most Condorcet-cardinal hybrids, etc.) also pass the criterion.

Choose One Plurality Voting only satisfies the Equal Vote Criterion in elections with two candidates only. Instant Runoff Voting (often referred to as Ranked Choice Voting) does not satisfy and the Equal Vote Criterion.

Vote unitarity

One generalization of the Equally Weighted Vote for the multi-winner or proportional context is vote unitarity. The basic idea is that the vote should stay equally weighted throughout the election tabulation. A voters influence on electing subsequent winners should directly depend on the amount of support they gave for prior winners. This means that the vote weight is conserved throughout the process.

There is an important nuance to this with regards to Surplus Handling; if, say, every voter gives one of the winners a perfect score, then instead of everyone's vote having no influence on the other winners, vote unitarity tries to ensure some kind of proportional decrease in voting power such that every voter still has a the correct amount of influence on the remaining winners. The simplest implementation of this is with Sequentially Spent Score.

In summary, there is a proportional relationship between how much support the voters give to the winners and the amount of influence that is removed from the voters, to ensure that every voter has a chance to fairly elect someone they prefer. The prominent Reweighted Range Voting method fails vote unitarity.

Notes

Some voting methods which pass the Equal Vote Criterion (which has also been called "Frohnmayer balance" in reference to its creator) don't pass a generalized form which refers to more than two voters being able to cancel each other out.

STAR may or may not pass the generalized criterion depending on how it is defined. Example:

Example modeled off of https://rangevoting.org/TobyCondParadox.html:

3 A:5>B:4>C:0

2 B:5>A:4>C:0

2 B:5>C:4>A:0

2 C:5>A:4>B:0

Scores are A 31, B 32, C 18, with A pairwise beating B and thus being the STAR winner. Removing 6 votes that constitute a cycle and a kind of pairwise tie and a definite scored tie for A, B, and C (2 A:5>B:4>C:0, 2 B:5>C:4>A:0, 2 C:5>A:4>B:0 votes, which give a total of 9 points to A, B, and C, and create a Condorcet cycle between the three where A>B, B>C, and C>A are all matchups of 4 to 2) yields:

1 A>B>C

2 B>A>C

Without even looking at the scores, B must win here, since A and B are unanimously preferred as the top 2 candidates and a majority prefers B>A.

If it is considered a kind of "pairwise tie" for there to be a Condorcet cycle between the three candidates where each candidate's pairwise matchups are either 4 to 2 or 2 to 4, then STAR fails. But if one requires the pairwise tie to be an exact pairwise tie between all candidates, then this example doesn't show a failure for STAR.

The passing or failure of Condorcet methods of this generalization is also similarly dependent on how a pairwise tie is interpreted (shown in https://rangevoting.org/TobyCondParadox.html).

Approval voting and Score voting pass the generalized form of the criterion, since removing any number of votes that constitute a scored tie for all candidates won't change the difference in scores between any candidates, thus since the winner must have originally had more approvals/points than all other candidates, they will still have more and thus still win.