This website is an extension of The Democracy Amendments (Anthem 2023) -- a book that analyses the fundamental problems in the American Constitution that underlie the massive dysfunction we see in the US federal government today. For the main webpage, see thedemocracyamendments.org.
For each of the 25 main constitutional amendment proposals in the book, there is a subpage on this website with further thoughts and details, which will eventually include responses to reader objections and suggestions as well.
Standard Bottom-Elimination Ranked Choice (RCV) Voting and its Condorcet-based fine-tuning
Voting theory is a complex topic, but we know so much more now than the founders did in 1787. The Democracy Amendments defends using the most common RCV method, in which the candidate with the lowest votes is eliminated on each round, and any ballots cast for them that also named a lower-ranked candidate are transferred to that candidate instead. In most cases, as the first proposal in the book argues, this will work well enough. Sometimes, as in Alaska, the election might allow voters to rank three or four candidates, so that if the voter's second choice is also eliminated, their vote passes to their third-most preferred candidate.
But it is also possible to use a "Condorcet" method, in which any candidate who would win all two-way races (head-to-head matchups with one other candidate at a time) is selected as overall winner. To determine this, for example with three candidates A, B, and C, the voting system must count how many voters ranked A above B or the opposite, how many ranked B over C or the opposite, and how many ranked C over A, or the opposte. The proposal in the book explains a way to combine these methods that allows standard RCV to avoid problems with three or more quite evenly matched candidates (the one sort of scenario in which standard RCV does not do as well).
The Democracy Amendments briefly notes a way of combining the methods, so that (say) one candidate is eliminated first on the basis of being the Condorcet loser (who lose more often than any other candidate does in two-way matchups with each other candidate). This helps in races where three or more candidates are more evenly matched, as opposed to two frontrunners neither of whom has over 50% on the first round.
Still there is no perfect method. Here I would like to introduce interested readers to a bit more detail in Voting Theory, although I claim no particular expertise.
Some Details on Voting Theory
Voting theory is part of the larger science of social choice. Readers may skip this section on Social Choice and go to the voting methods section below. But this background may also be helpful.
Social choice includes any process of trying to make a collective decision out of individual preferences among whatever the relevant options are -- candidates in an election, bills before a legislature, restaurants at which a group might meet, who gets to go on the lifeboat and who does not, etc. The assumption is that reaching a collective choice (at least to pick any one of most options, while a few might be intolerably bad) is better for everyone than making no choice and going our separate ways. In other words, there is a collective gain or payoff. But that gain may not be the same for everyone. Person X may to do best with option A, person Y with option B, and person Z with option C.
In the terminology of collective action theory, this means that there is a fairness problem or distributive equity issue. In many such cases, there may be a rule to pick the fairest option among those on which the group might coordinate. Rules such as standing in line, taking turns, help the neediest first, reward according to merit, equal division etc. may each be more appropriate in certain kinds of circumstances (on the philosophical issues involved in this topic, see my essay on Gauthier and Habermas in Constellations). But in many other cases, we instead need a decision procedure that enables that parties involved to fix on one of their options. Voting methods are such procedures.
Arrow’s Theorem: unfortunately, in 1950, Kenneth Arrow demonstrated that it is impossible to define an optimal collective choice – based only on individuals’ ordinal preference-rankings – that satisfies all four of the following intuitive requirements for a fair combination:
– Note that if this last condition fails in a combination method, individuals may sometimes have strategic reasons to hide their true preferences, e.g. by ranking option A lower than they really think is right in their vote or reported preference. By doing so, they might help A to win. The problem with standard RCV is that it is non-monotonic, which can matter in those cases where three or four candidates are initially nearly equal in voter support as voters' first choices. But Condorcet is monotonic, which is why using that method first in the process can help reduce the likelihood of any problem in an RCV election.
There are also background assumptions in Arrow's proof, such as:
• any individual set of preferences can be input into the social function and output a unique social preference (that will not change as long as the input-preferences remain the same).
Cycling and Agenda Control: One disturbing result of Arrow’s proof that these conditions cannot all be satisfied at the same time is that the order in which options are presented to a group – such as a legislature – in pairwise votes between two options at a time can determine which one wins (or comes out ranked first in the social choice).
In one order of vote: A beats B which beats C;
In another order: B beats C which beats A;
In a third order: C beats A which beats B.
Put another way, which option wins each pairwise competition cycles, depending on which pair competes first: so the collective preference appears to be inconsistent. Thus the old adage that the person controlling the agenda in group discussion has a lot of power.
Transitivity: we naturally assume that an individual’s preferences are transitive. For example, if Keisha prefers mint chocolate to almond, and almond to coconut, then surely she must prefer mint chocolate to coconut:
M > A > C therefore M > C.
We would think her a little crazy if, in pairwise choices, Keisha ranked
M > A and
A > C but also
C > M ---- thus forming a cycling set of preferences (compare Condorcet ties in voting below).
So why should it be any different with a group of people making collective pairwise choices? But it can be: that is par tof Arrow's discovery. Ever since, political scientists and mathematicians have been searching for practical ways of limiting individual preferences that would avoid Arrow’s impossibility result. There are promising directions in this regard (but they are too technical to introduce here).
Criteria for Voting Methods
Because voting is a way of constructing a collective decision out of individual party’s preferences (which may or may not reflect deeper ethical convictions) among relevant options before them, Arrow’s theorem and related findings in social choice theory have shows that there is no perfectly fair method of voting. But there are better and worse options.
In this type of social choice procedure, we have to distinguish between
• kinds of ballots and aggregation methods, and
• intuitive rules of fairness for such ways of giving voters decision powers.
Of course, what fairness requires will differ by the type of matter being voted on, and who the relevant parties are. As my parents used to maintain, for example, 3 kids vs 2 adults does not mean kids carry the vote on family decisions (“this isn’t a democracy!”). If it is a group of investors, or an NGO serving stakeholders with very different levels of stake in outcomes, a weighted vote giving various parties different numbers of votes to cast might be fair. Instead, for most democratic political contexts, we assume the one-person one-vote rule.
Fairness Conditions. Here are some other intuitively appealing criteria for fair procedures.
Election Methods, Illustrations, and Problems with Each (also see Fairvote's comparison chart)
I. Traditional Plurality method (one round) or “first past the post:” the candidate with the most votes wins, however small a percentage of the total votes they get (which may be less than 50%).
Alice wins 49%; Irfan wins 45%; and Karl wins 6% —> Alice Wins
Benefits: this method meets the majority favorite and later-no-harm criteria, and is monotonic.
Problems: low plurality outcomes. Consider Maine’s 2010 race for governor:
| Candidates | Paul LePage (Republican) |
Elliot Cutler (Independent) |
Libby Mitchell (Democrat) |
Shawn Moody (Independent) |
| Voters | 37.6% | 35.9% | 18.8% | 5.0% |
We see here just how bad our traditional voting method in almost all federal and state elections really is.
• Big “spoiler” problems (compare Nader in the presidential election of 2000).
• This system can also produce unrepresentative results: given the votes, it is possible that more than 50% of voters would not have ranked LePage in the top three if given that chance.
• More strongly, this method can elect a Condorcet loser (who would lose to each other candidate in a two-way race between just these two).
II. Standard Ranked Choice Voting (RCV) a.k.a. Instant Runoff: In Round 1, eliminate the candidate L1 (first loser) with the fewest first place votes, and transfer votes for L1 to their caster’s second choice candidate (if any). In Round 2, eliminate the candidate L2 with the lowest total votes (first-choice picks + second choice picks transferred in Round 1); then transfer ballots for them to their caster’s second choices (if any) – or to their caster’s third choice (if any) if their second favorite was L1, who is already eliminated. Repeat until only two candidates remain: the candidate with more votes (which could be a plurality, but often an absolute majority) wins.
To illustrate this, I offer two cases. The first is the imaginary instant runoff race in Florida during the presidential election of 2000. Here is the table from the book illustrating a likely scenario:
And here are the actual results in Alaska’s 2022 Senate election: Murkowski wins with 53.7% of the vote because most of those voting first for the Democrat chose moderate Republican Murkowski as their second pick. The four candidates in the general election for a Senate seat are those with the top four numbers of votes in the primary election.
| Candidates | Round One | Transfers | Round Two | Transfers | End Result |
| Chesboro (D) | 28,233 | +901 | 29,134 | eliminated | |
| Kelley (R) | 8,575 | eliminated | |||
| Murkowski (R) | 114,118 | +1,641 | 115,759 |
+20,571 | 136,330 (win) |
| Tshibaka (R) | 112,101 | +3209 | 115,310 | +2,224 | 117,534 |
This case illustrates why standard RCV (instant runoff) works perfectly in most cases: the voters for the third finisher get to be the tie-breakers between two clear front-runners (in this case, both Republicans). The system favors more moderate candidates, which is precisely what the US needs now.
But it is still a non-monotonic method that may not work well when there are too many candidates, resulting in more than two who start out nearly equal in the polls. For example, consider adapted from a Burlington VT mayoral election from the past:
| Number of Voters: | Ranked candidate 1st | Ranked candidate 2nd | Ranked candidate 3rd | Candidates and their votes: | Round 1 | Round 2 |
| 34 voters rank | K | M | W | K | 34 | 49 (win) |
| 37 voters rank | W | M | K | M | 29 | eliminated |
| 15 voters rank | M | K | W |
W | 37 | 46 |
| 9 voters rank | M | W | K | |||
| 5 voters rank | M |
• M is eliminated in Round 1; 15 M-first-ballots go to K, and 9 go to W.
• As a result, K wins the election.
• But look at M in the first four rows (in bold).
48 voters put M ahead of W compared to only 37 for whom W > M
51 voters put M ahead of K compared to only 34 for whom K > M
Thus M is the Condorcet winner, but still loses the election.
But suppose that 10 of the 37 voters who ranked W first switched to vote for K first. So their ranking would be K > W > M instead (think of this as a new row between 37 and 15).
– Then in Round 1, K would have 44, W would have only 27 instead, with M still at 29.
– Now it is W who is eliminated first: the 27 W-first ballots would then go to M.
– As a result, in Round 2, M would win with 46 votes to K’s 44.
Two responses:
1. [from Fairvote]: it is not just because 10 more people voted for K that K loses in this alternative scenario. Rather, it is the decline in support for W from 37 to 27 that changes who is eliminated first. That alters who K faces in the last round, in which M picks up more votes from the W-first voters. If instead we just took the actual vote in the table and added a row with 10 new (extra) voters who only list K first and no one second, K would still win.
2. This problem -- that a candidate C who a strong majority (say 70%) have ranked second may get eliminated first (due to not being enough people’s first choice), although they would have won Murkowski-style if only they had survived Round 1 -- is most likely in cases where the first-round vote is split a bit more evenly between three or more candidates than is shown in this illustration. Note that the very same problem can arise in actual runoff election systems.
For example, suppose Murkowski received 110,000 votes, Chesboro 111,000, and Tshibaka received 112,000. Then Murkowski would have been eliminated, even if all Chesboro candidates ranked Murkowski second – and if they could foresee the outcome, at least 2001 of them would have switched to rank Murkowski first in order to block Tshibaka from winning. In other words, we need the moderate candidate to survive Round 1.
This explains why I recommend that an RCV system first eliminate the Condorcet loser among the two candidates finishing lowest in the first round, and repeat this until there are only three candidates remaining, before deciding among these by the straigh instant runoff method.
The main goal remains to require all federal and state elections to choose an alternative to the traditional plurality method, which fails more criteria of fairness than either the Condorcet or straight instant runoff methods.
III. Borda count method: Assign points for the position each candidate finishes on each ballot; 0 points for last place, 1 for second-to-last place, 2 for third-to-last, etc. Whoever receives the most of these Borda points is the winner. There are many variations of this method. The simplest is this:
Unlike standard RCV or instant runoff, in which a second-place vote weighs as much as a first-place vote when the former is transferred, this method weights the amount each candidate gets from different rankings on voters' ballots.
So for example, imagine there are five voters who each rank four candidates differently:
| Voters: | First Pick | Second Pick | Third Pick | Fourth Pick |
| V1 | A 4 points | B 3 points | C 2 points | D 1 point |
| V2 | B 4 points | A 3 points | D 2 points | C 1 point |
| V3 | C 4 points | B 3 points | D 2 points |
A 1 point |
| V4 | D 4 points | C 3 points | B 2 points | A 1 point |
| V5 | C 4 points | A 3 points | D 2 points | B 1 point |
Totals:
A gets 12
B gets 13
C gets 14 -- winner
D gets 11
Multiple Problems
• Resistance to spoilers is low because (for example) two Democratic candidates could divide Democratic support while Republicans unite around one candidate.
• Immediately invites strategic voting: if you are voter V1 and want A to win, then don’t rank C at all (denying them 2 points). Or if you have to rank every option, then rank the most likely rival to your favorite lower than you actually think they should be ranked.
• So Borda fails the later-no-harm criterion: how you rank candidates after your first can deny your first choice the win.
• As this change illustrates, Borda voting can also yield ties.
• And the Borda method only works well if most voters rank most candidates.
This explains why Borda makes best sense for cases like judging a competition, in which (a) the judges have little personally invested in any contestant and so have no incentive not to communicate openly how the judge rates them, and (b) each judge will rank every contestant.
It also explains why I do not follow the recommendation of Christoph Borgers in his "Beyond Instant Runoff" article on The Conversation. But it is well worth a read.
Copyright © John Davenport 2023