Ever since I can remember, I have advocated tactical voting. This post tries to make that argument in the context of different motivations for voting. As this is not my area of expertise please let me know of anything I have missed.
Why do people vote? Do we vote in an instrumental way to (help) achieve some outcome, or do we vote in an expressive way to say (often just to ourselves) something about our own views. From my own experience in talking to people, many do vote in an expressive way. Even where the effective choice in a First Past The Post (FPTP) seat is between just two parties (X and Y), and where the voter happily acknowledges that on all counts X is a better choice than Y on all issues they care about (X>Y), you will often hear people say that they still couldn’t vote for X because of their policy on one particular issue. It doesn’t seem to matter that Y has the same policy as X, they nevertheless choose either not to vote at all, or vote for another party that doesn’t have that particular policy, rather than vote for X, even though X>Y and only X or Y can win
This attitude makes little sense (but see below) if voting is instrumental, but does make sense if voting is expressive. There was an even more extreme case of this in the last UK General Election, where some people suggested to me that they couldn’t vote for Labour because Labour was not formally in favour of continuing EU membership, or they didn’t trust Corbyn on Brexit. They would vote LibDem or some other party instead, even if they were in a marginal Lab/Con constituency. It was more extreme because it was almost inevitable that Brexit would happen if Johnson won, while it was far from inevitable that it would happen if Johnson didn’t win, even if, as I suggested here, Labour got an overall majority. In this case people were voting in a way that, from an instrumental point of view, would make them worse off.
Expressive voting is one reason why it makes sense for the opposition parties under FPTP to not field candidates in seats they have no chance of winning in the next election. Even if the parties cooperate informally, there is nothing stopping expressive voters voting for the party closest to their own views even when that party has no chance of winning. I’ve seen it argued that such actions by political parties are bad because it ‘reduces voters choice’, but in reality it only reduces the choice of expressive voters. Unsurprisingly, as most party members are likely to be instrumental in their attitudes, most Labour members (e.g see this poll) would favour not putting up candidates in seats where doing so might split the anti-Tory vote. From an instrumental point of view, this majority is entirely correct.
Instrumental voting is very different from expressive voting. If voting is instrumental, it makes sense for your vote to have some purpose beyond self statement, which usually means voting for party X even if you don’t agree with everything X stands for, simply because the realistic choice is between X and Y and you prefer X to Y. In most cases (but see below) instrumental voters will vote tactically, because that is the most effective way of using their vote.
However instrumental voting is subject to a problem that is often called the ‘paradox of voting’. The problem is the chances that your vote influences the outcome are negligible. To use the language of economics which is quite descriptive, the voter is ‘small’ relative to the electorate in a constituency, so their own vote is hardly ever critical in deciding the result. The ‘paradox of voting’ could also be called the insignificance of the individual voter argument, where insignificance means very small indeed. Note that this paradox only applies to instrumental voting, and not expressive voting.
I remember one General Election trying to persuade someone to vote LibDem in a LibDem/Con marginal. They objected that it was a waste of their time because their vote would be insignificant, but they did it just to please me. The LibDem candidate ended up winning by 10 votes. I triumphantly said to the person I had persuaded to vote, look your vote did matter, to only be told that if they hadn’t voted the LibDem candidate would still have won by 9 votes, so it was they who had been proved right.
Instrumental voting can be rescued from this insignificance paradox by talking about the payoff from your side winning. If the benefit to you of the party you vote for winning is sufficiently large, this can offset in expected benefit terms the very low possibility of your vote being decisive. However that is not the route I want to go down. Instead I have a more fundamental problem with the paradox of voting.
The insignificance of the individual voter argument casts voting in individualistic, expected utility terms. I think that is the wrong way to look at it. Voting is instead a social activity. We can see that quite simply from the fact that large numbers of people do vote in an instrumental way, despite their individual insignificance, so either they are being irrational or there is something else going on. As economists like to think that most people are not irrational I think we can conclude that something else is going on.
You can see why there is a social norm that we should try and vote from imagining a contest between two parties, one of which had rational policies and the other had irrational policies. Rational people prefer the rational party and vice versa. As most people are rational, the rational party should win, but if all rational people decided it was irrational to vote because of the paradox of voting, the irrational party would win, and the rational people would be worse off. Whilst the insignificance of the individual voter argument works for the individual alone, it does not generalise to society.  It may not make sense voting as an individual, but as part of society (or as a member of a class) it certainly does. There is evidence that most people do see it as a social duty to vote.
Seen in this light, as I think it should, expressive voting becomes a rather selfish act when voting is done secretly. You vote to make yourself feel better about yourself, rather than for the good of society as a whole. It also follows that we should vote for the party that would make society (or a group of that society like a class) as a whole better, rather than voting for the party that would make ourselves individually better off. This is because the latter is voting based on the outcome for one individual, where the paradox of voting still applies.
If we should vote instrumentally to make society (in our personal view) better, then it almost follows automatically that tactical voting under FPTP is the only sensible way to vote. By the same logic, you will often, under FPTP, end up voting for the lesser evil, because even if there is a party you could vote for that is better than the lesser evil, voting for it will be a wasted vote because that party cannot win. If you don’t like voting for the lesser evil that probably means either you prefer expressive voting or you don’t like FPTP systems..
Why do I say that it almost follows automatically? There are two common arguments against voting tactically, which in particular (and I would say pretty exceptional) circumstances are valid. The first arises from voting being a repeated activity (a repeated game). This brings in the possibility that voting one way today may influence our choice in the next election. Suppose X>Y, but X is still pretty awful, and if X lost this time it might be replaced by Z, where Z>X>Y and Z was a lot better than X. In those circumstances the best social choice might be to allow X to lose, because it would allow Z to emerge next time (see here).
This is a way you can try and rationalise why some of those on the right of the Labour party, for example, might have worked to enable Corbyn to lose in 2019, because they believed that would allow a much better Labour party to emerge from the ashes of his defeat that wouldn’t have emerged if Labour had won.
The problems with this argument come from both the many ifs involved and from the present sacrifice made (party Y gets to rule badly for 5 years). The many ifs include whether defeat for X does lead to Z, and whether Z would have happened even if X had won, and that Z will win against Y in 5 years time. (The problems are intensified if during that 5 years party Y changes the rules of the game to make a future victory for Z less likely.) The argument has some similarities to accelerationism, which argues that the best way to achieve a 'good society' in the future is to intensify rather than try to ameliorate the current bad features of society. In my view it’s an argument that is not always wrong, but most of the time it is. On the particular example of the Labour right and Corbyn, I think the cost of a hard Brexit should pretty well dominate any perceived advantages that the Labour right might have imagined getting their party back might involve.
A second argument against tactical voting goes like this. If voters make their vote for party X conditional on X adopting some policy, then voters have some power over policy. If instead they always vote for X, they have no power. However, for this power to have any credible influence on party X, individuals have to be prepared not to vote for X if X doesn’t adopt their policy. It then becomes an argument like the one above but related to individual policies rather than party leadership or complete parties. (Because people didn’t vote for X because of some policy, X does badly and is forced to adopt the policy in the next election.)
The main problem with this argument is credibility: on most issues most voters for X will not be prepared to carry out the threat of not voting for X if they don’t get their way. For that reason most of the time those that do make this threat will be ignored. Equally there are very few issues where enough people can make common cause to pose a sufficient threat to X. But just occasionally the argument might work if the issue is central enough, voters feel strongly enough and they have a means to demonstrate the threat. The run up to the 2019 election provides one possible example, where the policy was Labour’s position on a second referendum. Voters who would normally vote for Labour voted against them in local and EU elections, and threatened to not vote for them in general election polls, and this did change Labour’s policy on a second referendum, for better or worse.
Putting these two exceptions against tactical voting aside, I want to end by saying something about responsibility. If voting is a social norm or duty, then with that goes the idea of responsibility for outcomes. (I’m indebted to @spinninghugo for thinking about this, although he has no responsibility for my current views!) If we go back to my example of people who voted LibDem (say) in a Lab/Con marginal because they didn’t trust Corbyn on Brexit, in what sense if any are they responsible for what Johnson’s government subsequently did?
The first and obvious point is that they share that responsibility with everyone who didn’t vote tactically against the Conservatives in 2019, and those who voted for the Conservatives have a greater responsibility. However pointing this out does not avoid the issue of responsibility. To see why, imagine someone was murdered by 100 people, each of whom struck once with a dagger. You would certainly say that each of the 100 was responsible for the death of the victim, and the fact that ‘he would have died anyway even if I hadn’t been one of the 100’ doesn’t absolve anyone of their responsibility for the murder. Suppose you were not one of the 100, but had the opportunity of trying to prevent the murder, but had chosen not to. You too would have some responsibility for allowing the murder to happen. It remains the case that anyone who didn’t vote tactically against the Conservatives in 2019 remains in some small way partly responsible for things the Conservative government did that Labour would not have done.
In contrast, as tactical voting is the choice of the lesser evil, it is not the case that the voter is responsible (among millions) for everything the winning party does. Instead they are just responsible for what the winning party does differently from the only credible alternative party (X-Y in implemented policy space, if that helps).
What the only credible alternative party would have done is a counterfactual, so it allows people to try and avoid responsibility by making claims about that counterfactual. It is why, for example, those who supported the Conservatives in 2019 like to make the unlikely claim that a Corbyn led government would have been even worse than a Johnson et al government, because otherwise they would bear some responsibility for the mess we are currently in. The fact that people make such incredible claims shows that the idea that voting is a choice we should be responsible for has some force.
 To drill down a little more, the paradox of voting assumes that your action of not voting is peculiar to you, and will not be followed by anyone else. If instead you assume that other people will make the same calculation as you do, then you would want to vote to avoid the irrational party always winning. More generally social norms that voting is part of being a good citizen can be seen as a social response to the paradox of voting.