Why should we only be able to vote for politicians? Why shouldn't we be able to vote against them instead? The evidence suggests that many voters are motivated by antipathy rather than sympathy, by opposition rather than support. So, why not have a system that permits us to express our opposition to a candidate directly? I want to propose a simple reform to the conduct of certain elections that preserves the one person, one vote principle, but also permits this. It would not be a novel reform in mathematical terms, but it differs profoundly in psychological and social terms from mathematically equivalent voting systems. It would therefore produce a different outcome in some cases and communicate different information about voters’ preferences in every case.

In referendums, we can vote positively or negatively: either for or against. But we are only allowed to vote for candidates. The proposal is to bring this restriction to an end, as it does at present, whereas a vote against a candidate would decrease their total number of votes by one. When we vote about propositions, in referendums for example, we can vote positively or negatively: either for or against. But at present, we are only allowed to vote for candidates. The proposal is to bring this restriction to an end. This might not be desirable in every election, but it would certainly be desirable in some. I shall mention a couple of specific cases shortly.

Since the 1970s, there have been several mathematical studies of voting systems that allow votes to be cast either for or against each of the contenders in elections with at least three candidates. But I am interested in the case for permitting votes to be cast against candidates without compromising the one person, one vote principle, including in two-horse races. The usual focus on elections with at least three candidates is telling. The reason for it is the thought (as Dan Felsenthal puts it) that ‘if a voter has only one vote, then it makes no difference if, given there are only two candidates, he decides to vote in favour of one of them or, alternatively, to vote against the other’ (https://onlinelibrary.wiley.com/doi/10.1002/bs.3830340105). But this is not quite true.

In the first place, it makes no difference to the outcome of the election, that is, to who wins and who loses, and by what margin. But voters know that voting for a candidate also communicates support or is at least liable to be interpreted as doing so, so there is a difference between voting for one candidate and voting against the other.

Second, the quoted remark ignores the fact that for this very reason, voters may refuse to vote for a candidate they regard as the lesser of two evils despite being willing to vote against a candidate they regard as the greater of two evils. Some voters may even believe that it is morally impermissible to vote for candidate X, despite believing that electing X is preferable to electing Y. Mathematical studies of voting systems generally ignore these complexities, but the case for the proposed reform in two-horse races depends on them.

There are instrumental, non-instrumental and mixed arguments in favour of the reform, as there are for democracy itself. The main arguments are as follows.

• • It would remove the incentive for the commonest kind of tactical voting. Since voters would be able to cast a vote directly against a candidate, there would be no need for them to decide which of the other candidates is most likely to defeat the one they disfavour.

• • As a result, it would also remove the incentive for candidates to spread misinformation encouraging voters to vote tactically for themselves and remove the need for some complicated and unreliable vote-swapping arrangements.

• • It would enable voters to express a political opinion with their vote which they are currently prevented from expressing in this way.

• • It would enable voters who refuse to vote for any of the candidates running in an election to participate.

• • For both of the last two reasons, it would tend to increase participation.

• • It would enrich the information about voters’ preferences that elections yield, for governments and others.

• • This in turn would increase the influence of voters on the conduct of government, and thereby make government more democratic.

• • It would enhance the democratic process not only by making each vote more meaningful but also by making political debate more meaningful, because it would increase the extent to which each voter's political deliberation can be translated into a vote.

I shall return to these eight points.

It is surprising that this reform has not been widely discussed. Perhaps this is due to the prevalence of the analogy between voters and consumers. If we are choosing between cereals, we may decide to buy shredded wheat, corn flakes or rice crispies. We don't have the option to negatively buy shredded wheat because we dislike it most. But whyever the reform I am proposing has been ignored, the arguments in favour of it are clear. What are the arguments against it?

One argument against the proposed reform is that voters should be encouraged to think positively about the outcome of an election, or at least they should not be encouraged to think negatively about it. Ideally, they should choose which candidates and policies they favour, and express this choice when they vote. But the proposed reform would encourage voters to focus on the candidates and policies they dislike.

This argument is paternalistic and naïve. Even if it is true that we should be encouraged to choose which candidates and policies we favour, it is paternalistic to infer that voting rules should compel us to do so, or rather, confine us to voting as if they this was what we had done. And the argument is naïve because voters do in fact often vote for one candidate or party because they dislike another candidate or party. The voting system we have does not prevent us from doing this. No system can. Indeed, there is no evidence that one kind of system makes voters more or less likely to do this than another.

Besides, it is not true that we should be encouraged to choose which candidates and policies we favour, rather than choosing which ones we disfavour or dislike. Very often it matters more that one candidate should be prevented from exercising power than that another candidate should be enabled to do so, because very often it matters more that the policies of a candidate or party are bad, or even wicked, than that the policies of a candidate or party are good. The vital thing in an election is very often to throw a party or leader out of power, or prevent them from securing power, rather than the opposite.

Hence, if the proposed reform would make voters more likely to think in negative terms, this might be a good thing. But as I have said, there is no evidence that it would, and besides, voting rules probably should not be designed to influence voters in this way. It is arguable (though I shall not argue the case here) that if voters are to be restricted to casting a positive vote or restricted to casting a negative vote, the second restriction is preferable to the first, but in the absence of convincing arguments for either restriction, voters should have the choice.

Another argument against the proposed reform is that voting should be as simple as possible, but the reform would make voting more complicated. The reply to this argument is that in one way, it would make voting more complicated, but in another, more important way, it would make it simpler.

The way in which the reform would make voting more complicated is that voters would have twice as many options. In elections where voters are already faced with dozens of alternatives this argument might carry some weight, although I doubt it. Having twice as many options does not necessarily make it harder to decide what to do, or harder to do it once the decision has been made. But in any case, the argument has no force at all in the case of elections where voters are faced with relatively few candidates, as they generally are in presidential elections and always are in the second (run-off) round of a two-round election, such as the presidential election in Turkey or France.

The way in which the reform would make voting simpler is that although no voting system can eliminate every incentive to vote tactically in elections with three candidates or more, the proposed reform would eliminate the need for the commonest kind of tactical voting, namely, the kind that occurs when one votes for C rather than B because C has a better chance than B of defeating A, even though one would prefer B to be elected. Now it makes sense to vote tactically in this way only if one's vote is primarily intended to reduce a candidate's chances of being elected, and so the proposed reform eliminates the need for this kind of tactical voting because it enables voters to do this – that is, reduce a candidate's chances of being elected – without deciding (sometimes mistakenly) which of the other candidates has the better chance of defeating her. Hence, it makes voting simpler, it removes the need for certain kinds of vote-swapping arrangements, it makes voters less vulnerable to misinformation about the relative support of candidates, and it removes one motive for disseminating misinformation of this kind.

Tactical voting of this kind is confined to cases where there are at least three candidates. But the proposed reform would also benefit voters in two-horse races who find it morally or politically unacceptable to cast their vote in favour of one of two candidates, despite regarding the other candidate as worse. For example, I know from conversations with friends that some French voters in this position refused to vote for Chirac in the run-off against Le Pen and some voters in the US refused to vote for either Trump or Clinton, although many more voted for the candidate they regarded as the lesser of two evils. These voters would have been able to influence the outcome of the election without voting for a candidate they disapproved of, or even loathed, had the reform I am proposing already been introduced in France or the US at that time.

A third argument against the reform is that if it were introduced, a candidate could win an election with an embarrassingly small number of votes, or even with a negative number of votes. There is no procedural difficulty here: if every candidate had more votes cast against her than for her, the winner would be the candidate with the lowest negative number of votes. But it could be argued that this would be an undesirable outcome because it would not give the winning party or candidate a clear mandate to govern.

This argument is unconvincing. To see why, it is helpful to distinguish between two kinds of mandate, which I shall call a legal mandate and a popular mandate. A legal mandate is a legal authorization, whereas a popular mandate is, roughly speaking, a formal sign of a substantial measure of popular support. To keep things simple, consider a presidential election with two candidates. It would be generally accepted that an electoral system that did not normally provide either of the candidates with a clear legal authorization to govern would be unsatisfactory. (This could be the case if, for example, the winning candidate was required to gain a minimum number of votes, or if the validity of the election depended on a threshold of participation being achieved.) But it is obvious that the proposed reform would not have this effect.

The moot question is whether it would deny the winning candidate the clear formal sign of a substantial measure of popular support, which candidates certainly hope for and, under the current system, often obtain. The answer is that it might sometimes do this, but this speaks in favour of the proposed reform, not against it. It speaks in favour of it because if in fact the winning candidate does not enjoy a substantial measure of popular support, it is desirable that the electoral system should enable voters to express this, and – as it were – place this information in the public domain.

In other words, it counts in favour of the proposed reform that it would enable voters to express a political judgement at the polls – a collective judgement as well as individual ones – which they are currently prevented from expressing. The collective judgement expressed when the winning candidate has a negative number of votes is that she is the lesser of two evils. And if this is the electorate's collective judgement, it is desirable that the electoral system should allow it to be expressed. It is intrinsically desirable, and it would also tend to have desirable effects. For example, it would tend to constrain the winning party's or candidate's exercise of power; and it would tend to impress on them the need to win the popular support they currently lack. Equally, of course, the reform would tend to strengthen the popular mandate of a truly popular government, because a large positive number of votes would be a genuine sign of substantial popular support.

Overall, the reform would add to the information about the political views of voters that an election registers; and to the extent that this information tends to influence the winning party or candidate, it would therefore make government more democratic. This in turn increases the incentive that voters have to participate in elections. In fact, the reform could increase participation in several ways: by enabling voters to express a political judgement at the polls which they are currently prevented from expressing; by enabling them to vote despite being unwilling to vote in favour of any of the candidates; and by increasing the influence of their votes on the conduct of government, after the election. If high levels of participation are considered desirable, these are strong arguments in favour of the proposed reform.

A fourth argument against the proposed reform is this. Suppose an election was held with at least three candidates, with the reform in place. If the majority or near-majority of voters cast negative votes, a candidate whom most voters had ignored – an inveterate loser or an unknown candidate – could be the winner. For example, in the UK in the 1980s, this might have been Screaming Lord Sutch, founder of the Official Monster Raving Loony Party. Electoral systems should be designed to prevent this kind of outcome.

The reply to this objection is as follows. To begin with, we should not assume that this would be a worse outcome than the victory of the candidate who would have won the election under a conventional voting system. That would depend on who the latter was. In fact, it isn't difficult to name winners of conventional elections who proved to be worse MPs or political leaders than Lord Sutch would have been, had he been elected. Nor is it inherently undesirable for a less well-known candidate to beat a better-known candidate, or for the winner of an election to be a candidate who has often been defeated in the past. The more plausible objection this kind of outcome illustrates is that under the reformed voting system, the winning candidate could be the one who would lose a head-to-head contest with each of the other candidates, in other words, the so-called Condorcet loser. But this can happen in a conventional election as well. For example, there is evidence that it happened in the 2016 US Presidential Election, when Donald Trump defeated Hilary Clinton, Gary Johnson and Jill Stein (https://journals.sagepub.com/doi/full/10.1177/1532673X211009499). Besides, the objection only applies to elections with three or more candidates. It does not apply to the second round of run-off elections, which are held in the great majority of countries with an elected Head of State. In fact, the way to ensure that the Condorcet loser cannot win a presidential election is not to oppose the reform I am proposing, but to introduce the run-off system in states such as the US where it does not presently exist.

A fifth argument against the reform is that it would tend to disadvantage ‘marmite’ or divisive candidates, candidates who arouse a stronger antipathy in voters who do not favour them, especially voters who would not vote positively for any of the candidates in the race. Some will regard this as an argument against the reform whereas others will regard it as an argument in its favour – depending perhaps on whether they tend to prefer this kind of candidate themselves, or whether they value or disvalue consensus, or whether they believe it is better for political change to be incremental or disruptive. I shall not attempt to decide which is the correct view.

It is often said that the purpose of elections is to give one group in society the power to rule; and it is sometimes added that they provide a forum to debate policies for the common good. These are certainly among the purposes that elections serve; but they also deny one or more groups the power to rule, they provide information that can influence or constrain the exercise of power by the winners, and they both motivate and crystallize political debate. Especially when we consider these less commonly mentioned purposes of elections, the advantages of introducing the proposed reform are clear. Naturally, those who believe that democratic experimentation is an inherently good thing – or that it is a good thing at times when scepticism about democracy is widespread – will be more disposed to find the case for it persuasive. But it seems hard to deny that the case is strong.