A review of several systems of voting in symmetric democratic elections

Anderson TW

Department of Secular Ideology, Wivenhoe Metropolitan University, UK

Most of us are familiar with one or more forms of voting for individual candidates, such as the first-past-the-post vote used in the election of British MPs or the two-round system used in electing French deputies. However, other systems exist, and many of them, although unfamiliar, are demonstrably superior to these commonplace examples.

In the example voting, turnout is always 100%, and nobody ever abstains.

The following parties have been used:

I have used a fairly unified right and a fragmented left. This is a fair approximation to politics in some countries, but by no means all. It is not intended to show any political bias. Nor is the fact that left-wing parties win most of the example elections!

First Past The Post (FPtP)

This is an ancient and very straightforward system, used for parliamentary elections in the UK and the USA. Every voter declares his support for one candidate; the candidate supported by the most people is selected. Whilst this system is simple and easy to understand, it does have one major drawback: the elected individual will typically be supported by a minority of the voters. Consider the following example:


In this case, the Republican candiate is elected, even though only 35% of the electorate voted for him, and in fact 65% voted for left-wing parties. Such cases are not uncommon. Tactical voting, where voters vote for candidates who are not their preferred candidates in the hope that they will defeat a better-supported but less preferred alternative, is common, and a particular problem.


This is a modification of the FPtP system. A first round is conducted in the same way as an FPtP election, but rather than determining the winner, this determines which two candidates (the two with the most support) will go through to the next round (a frequent optimisation skips the second round if any candidate gains 50% or more of the vote in the first round; this candidate is automatically elected). The second round is then a simple FPtP, but with only two candidates. Continuing the above example, in the second round we could expect the following


This is clearly preferable to the FPtP case, as the elected candidate was supported by 65% of the electorate, even though he was not necessarily their first choice. The disadvantage is the extra compexity of the extra round, and the fact that the elected candidate does not represent the 35% of voters who voted Republican. In a two-party environment such as the USA, this system is no better than FPtP. Tactical voting is less of a problem, as voters know they will get a second chance to stop a particularly disliked candidate in the second round.


This is a generalisation of the two-round system, whereby the number of rounds is increased, and one candidate is knocked out per round. Thus, if n candidates enter, n-1 rounds are required to produce a winner. In an environment with a large number of evenly-matched parties, this system works well. For example, consider the following election, fought under the two-round system:

PartyRound 1Round 2
Republican 19.0%83.0%
Liberal 16.0%
Socialist 16.5%
Communist 16.0%
Anarchist 15.5%

In this case, despite the fact that 64% voted for a left-wing candidate in the first round, the second round was fought by the Nationalist and Republican candidates. However, an n-round election would probably progress as follows:

PartyRound 1Round 2Round 3Round 4Round 5
Republican 19.0%19.0%27.0%44.0%44.0%
Liberal 16.0%16.0%
Socialist 16.5%16.5%24.5%24.5%
Communist 16.0%31.5%31.5%31.5%56.0%
Anarchist 15.5%

A problem with this system, as illustrated here, is that it once more leads to a significant fraction not being represented (the 44% of voters who supported the Republican candidate in the last round). It would be possible to add a last stage, with only one candidate, and thus claim 100% support; very few people would accept that this demonstrated anything of the sort, and this shows up a more subtle flaw in the system. The 8% of people who supported the Liberal party and ended up voting for the Communist candidate did not do so because they supported him, but because he was the best option available. It is not fair to say that 56% of people supported the Communist candidate, only that 56% preferred him to the Republican candidate. This is reflected in real elections by a considerable fall in turnout for later rounds. A similar optimisation to that used in two-round voting can be used, where the rounds cease if any candidate gets 50% or more of the vote. I am not aware of any real-world uses of this system, probably due to its unwieldiness. Tactical voting is largely eliminated.

Single Transferable Vote

This system is somewhat different from the previous systems in the manner in which voting is conducted. It attempts to compress the n-round system into one round by gathering more information on the voting papers. Rather than declaring support for one candidate, voters assign an order of preference to the candidates, indicating their first, second, third, etc, choices. Then, this information is used in an analysis which approximates the operation of an n-round election. At first, votes are assigned on the basis of the first candidate in the list of preference. Then, the candidate with the least votes is eliminated, and the votes in his support reassigned to the highest candidate on the list who has not been eliminated. This proceedure is repeated until only one candidate remains. Again, the early-end optimisation may be applied, ending the elimination if any candidate gains over 50% of the votes.

Since a list of six candidates allows 720 arrangements of preference, I have decided not to give an example of this system. It would probably come out quite similarly to the n-round vote, and the same remarks apply. I know this system is used, but not where or for what. Tactical voting is largely eliminated, mostly due to the fact that nobody understands the system well enough to manipulate it.

Approval Voting

Approval voting is a recent development, which is, I believe, used to elect the chairman of the American Dental Association, but little else. The mode of operation of approval voting is very simple; in fact, it is almost a mutant form of FPtP. The voter is presented with a ballot paper listing the candidates, and may indicate of which of them he approves. He may approve of none, any or all of them. In this respect, it is FPtP with unlimited votes per person. The candidate with the most support is thus elected. The result is found in one round and with a simple ballot paper.

The advantages of approval voting are twofold. Firstly, it eliminates tactical voting. Voters will not switch away from one party to support another more likely to win, as they are able to support both! Secondly, it makes it possible for minor parties to win; often, voters dismiss a party because it apparently has little support, and to vote for it would be a tactical blunder. With approval voting, people would not have to make this decision, and so minor parties would be on equal terms with larger ones.

This is an example of how approval voting might work in such a situation:


By 'Support', I mean the fraction of the electorate who wish to see this party win above all others. By 'Approval', I mean the actual fraction of the electorate which approved of a party. In this example, approval for a party is calculated by adding up the support for that party and half the support for each of the two parties either side of it in the political spectrum. In a FPtP election, victory is determined by support, and so the Socialist candidate, with only 42% of the vote, would win. In an approval vote election, victory is determined by approval, and so the Democrat candidate, with 60% approval, would win. In practical terms, the cross-support might be higher or lower; I have absolutely no empirical data on this.

We can see why tactical voting is eliminated if we consider a simple case. The Socialist supporters, predicting that their party will have the second largest vote, decide not to give any support to the Democratic party. Consequently, the Democratic supporters withhold support from the Socialist party. The voting thus proceeds as follows:


Thus, by witholding support from the Democratic party, the Socialist supporters have helped the Republican party to win! The Socialist supporters' interests would be better served if they voted for the Democratic party, so ensuring that the Republican candidate is not elected. I am certain that, even if they did not understand this, their leaders would, and would urge them to approve of the Democratic candidate to deny the Republican party victory. This system is also a relief for floating voters, who no longer have to make a decision between two parties they support equally.

Critics might suggest, quite fairly, that the Socialist supporters, to avoid the Democratic counterattack, might secretly conspire to withold their vote. I have two answers for this. Firstly, on a practical level, this might be possible for electorates of a few tens or even hundreds of people, but on a scale of thousands it is impossible to keep that kind of secret for very long. Secondly, they could not possibly hope to pull this off again, so in the next election the Republican party would win, as outlined above.

Secondly, there is a strong argument against this coming from game theory. An experiment conducted a few years ago by Robert Axelrod, a mathematical biologist, addressed a generalised form of this problem known as the prisoner's dilemma, looking for the best way to handle this sort of situation. This turns out to be a strategy known as 'tit for tat'. Quite simply, the strategy is to cooperate (in this case, to give cross-party support) until he does not cooperate (stealthily withdraws his support), in which case, stop cooperating also. In environments such as an election, where there are a series of encounters over the course of time, as well as several players (parties), the tit for tat strategy beats all comers (such as the strategy espoused by the Socialists in this instance, which was to cooperate for a while, and then to double-cross).

Modifications to the Voting System

Continuous Voting

The above systems specify the existence of discrete votes; that is, a voter either votes for a party or does not, whether he does this once, as in PR, twice, as in two-round voting, or as many times as he likes on the same sheet, as in approval voting. However, these systems are all readily adapted to use continuous votes. This does not refer to repeatedly casting votes, but to the ability to cast a non-integer vote (hence, some wits wise in the ways of computer programming call this 'float voting' or 'real voting'). For example, in continuous FPtP, the voter still has one vote, but rather than giving it all to one party, he may split it between as many as he likes (eg 50% for the Republicans, 30% for the Democrats, 20% for the Socialists).

For approval voting, voters would indicate to what extent they approved of each candidate, without any constraint on the total vote. I do not see any way to construct a system of single transferable continuous votes; the complexity in these two systems is perpendicular, and any attempt to combine them would yield a system of such baroque - no, grotesque - complexity and difficulty that it would be certain to be introduced by the European Union in a flash. The world is safer not knowing.

The effect of this is to increase the accuracy of the voting; rather than rely on statistics to generate a 5:3:2 ratio of votes from a dithering electorate, why not let them express their real wishes? The system is somewhat more complicated to use, and to process, but most people are capable of adding up to 100 (if voting is carried out in percent), and a simple full-support tick box could be provided for those not comfortable with the system. Computerised voting systems would make this even easier, providing a visual guide, and verifying the input at source.

Negative Voting

A common feature of elections everywhere is the protest vote, which also manifests itself as the tactical vote in many systems. Some voters do not want to get any particular party into power, but rather to stop one particular party getting into power. The system should thus be changed to allow them to exercise their democratic franchise in the way they see fit - they should be given the ability to use their vote, not for a party, but against a party.

This is simple to add to FPtP; rather than voting for a party, it is possible to vote against one. Votes against are subtracted from the party's total positive vote. This extends to two-round and n-round voting. It could also be added to the approval vote, although in this system a non-vote is just as significant as a vote, and so it becomes somewhat redundant. Furthermore, the same effect can be gained by voting for all the parties except the opposed one, although this does give the impression that the voter actively approves of those parties, rather than simply being neutral.

It is not obvious how to add negative voting to the single transferable vote. One way would be to allow for two types of vote, negative and positive. Positive votes work as usual; negative votes are applied in the same way as positive votes, but count against the total vote rather than towards it; I am not convinced that this would work, although it might well do.

One effect of the negative vote would be that support for parties would be much lower across the board than it is at present. However, this would accurately reflect the way most people feel, and might lead to a little humilty amongst the parties (when a candidate is elected for having a -10% vote rather than -20%, it is no ringing endorsement). The total net vote could provide a useful indicator of the current feeling towards the establishment parties. A high turnout is usually thought to indicate enthusiasm for the political process; if the net vote was negative, this might be questioned!

Negative voting may be combined with continuous voting; in this case, the mathematics of the continuous vote are disrupted. Ordinarily, the total vote must sum to 100%, but if negative voting is used the following vote would be legal:


Since ( 500 + 0 + -400 ) = 100! Rather, the rule must be that the absolute values must sum to 100%, so that positive and negative votes count equally.

Adaptation of Systems

These systems allow the election of an individual by a set of voters. Often, an individual is not required; rather, a set of people is required, for instance in a legislative assembly. In this case, the set of voters should be partitioned into subsets, and each subset should elect a representative using one of the systems above (note that ordinarily, all subsets use the same mechanism; if they do not, they should be able to vote for which mechanism they wish to use, and this leads inevitably to an infinite regress). This provides a set of representatives.

This approach may be used to allow a set of voters to elect a single representative in a more complex way. They elect a set of representatives as outlined above; this is called the electoral college. In turn, this college then acts as a smaller set of voters to elect an individual. Whilst this may seem unnecessary and absurd, it is in fact how the president of the USA is elected, which is perhaps a fitting point on which to terminate this paper.