Based on rough estimates of their effectiveness, voting is a very good use of time, but demonstrating far less so – though it may still be worth doing, at least for a very large demonstration on a major issue.


Why bother voting? Your vote will only change the result if it would otherwise be an exact tie; and the chance of that is negligible – one in millions.

But a chance of one in millions is worth taking if the jackpot is billions or trillions. That is, the opportunity for you to select a better rather than worse government, thereby making the country – though not yourself – billions or trillions of dollars better off. So as long as you care at least slightly about the rest of the country, voting is rational; civic duty really is a reason to vote.

(An interesting 2007 paper Voting as a Rational Choice by Edlin et al. models this formally. Though in contrast the Put A Number On It blog argues in part that you have no idea which party will run the country better, so your opinion and hence vote is worthless.)

Does similar reasoning justify taking part in demonstrations (and other forms of collective action), even though they don’t have such direct effects as voting?


Consider the largest current political issue in the UK – Brexit. The background, for non-Britons, is that in 2016 a UK referendum voted by a narrow margin to leave the European Union (EU), on terms still being negotiated. Many ‘Remainers’ who wish to stay in the EU want a second referendum on the matter; recently, hundreds of thousands of them marched through London to call for one.

How worthwhile was it for a Remainer to vote in the 2016 referendum, or to demonstrate for a second one?

Vote response curve

A political action, such as voting or demonstrating, is intended to produce an outcome – in this case, the UK remaining in the EU. We can plot this as a response curve, showing how the number of participants affects the probability of the outcome.

Voting has a response curve like this:

The dashed red line shows that – in principal – if more than 50% of voters vote to remain in the EU, the probability that will happen is 100%, otherwise 0%. So if you knew in advance whether the result would be 50%, you’d know whether to bother voting: only vote if doing so will resolve an exact tie, and push the result up the step from 0% to 100%.

However, when deciding whether to vote, you only have the predicted vote share to go on, which has significant uncertainty in it. Hence the solid red line shows the probability of remaining in the EU given a predicted vote share, which may be wrong by several percent. (Also, because this particular referendum was not legally binding, this line starts and ends a little above 0% and below 100%, in case politicians don't end up implementing the result.)

The slope of the solid line is then proportional to the marginal effect of one extra voter – i.e. the expected effectiveness of your vote. Voting is most worthwhile (i.e. steepest slope) on closely-fought issues – a predicted share of around 50% – because that’s when your chance of breaking an exact tie is highest; whereas when a high or low vote share is predicted, a tie is very unlikely, so your vote has far less chance (i.e. flattish slope) of affecting the result.

Demonstration response curve

Demonstrating has a different response curve. This is how the recent march might have varied in effectiveness, depending on how many demonstrators showed up:

A huge demonstration – say, a million people – might influence politicians enough to make a second referendum somewhat more likely, raising the chance of remaining in the EU by say 1%, from 14%[1] to 15%. Aside from being a big round number, one million is the size of the largest demonstration in British history (against the 2003 Iraq War), hence a sign of strong public feeling.

As it turned out, 700,000 people showed up for the recent march – big, but not historic. More middle-sized marches might not have much effect above baseline: demonstrations routinely occur on any issue, so politicians treat them as background noise. And because ‘smallish’ and ‘largish’ marches have similar political impact, the slope is fairly flat around the middle of the graph; that is to say, adding one marcher has little impact.

A demonstration with a low turnout would either have negligible effect (upper red line on left), or if it had been widely publicized, could harm the cause (lower line) by indicating to politicians that the issue has less popular support than they had thought. (Demonstrations can also be counterproductive in other ways; if one turns violent, it may well have a negative impact.)

Other forms of activism may have other shapes of response curve; e.g. perhaps some are linear.

Issue size

The value of voting or demonstrating clearly depends in part on the size of the issue at stake. Brexit is a very big one.

Of many forecasts for the economic impact of leaving the EU, the median (from NIESR) is around 2.5% reduction in the UK’s GDP. Let’s suppose a typical Remainer believes this figure.[2] Over say 20 years until the UK re-joins the EU or some similar-sized trading bloc, that adds up to half a year of GDP (ignoring growth and discounting, which roughly cancel out). If the total effect on other countries is similar (a guess - I’m no economist), the global damage is roughly 1 year of UK GDP, or about $3 trillion; or conversely, $3 trillion gained by remaining in the EU.

How altruistic are people?

The value of voting or demonstrating also depends on altruism – as people care how much an issue affects others, not just themselves. Indeed, as mentioned, without altruism, voting behaviour is fairly inexplicable. But how much do people value others?

Charity donations are 2.1% of GDP in the US, 2.0% in the UK, [3] and mostly used to help others in the same country. People spend money on what they value, so it’s plausible that they spend it on others in proportion to how much they value them (or more precisely, to how much they value the prospect of helping them). This suggests that donors value others in their country collectively at about 2% of themselves (plus family & friends), on whom they spend the rest. This seems low; but if they value their compatriots more, why don’t they give them more?

One reason may be that tax is effectively a compulsory donation to other people, in the absence of which they would voluntarily give rather more – perhaps even most of what they currently pay in tax. So to correct for this I will increase this 2% to 10%; a guesstimate, but fine for an illustrative calculation. (And incidentally, several religions traditionally expect members to donate 10% of their income to charity.)

Benefit of voting

We can now estimate what it was worth a Remain supporter to vote in the EU referendum. The Edlin paper estimates the probability of casting a decisive vote in a close election as 10 / turnout, = 1 in 3 million for UK national votes. So for a $3 trillion issue like Brexit, the expected benefit to you of voting is $3 trillion / 3 million x 10% altruism (since your benefit is almost entirely the vicarious effect of helping others) = about $100,000 – well worth bothering. And for lower turnouts, it's worth even more.

This also means voting would even be worthwhile for an almost entirely selfish person with an altruism of 0.01%. Whereas the expected benefit of voting to you alone is the Brexit harm to you / 3 million, = $3 trillion / 2 (effect on UK only) / 65 million (UK population) / 3 million = 0.7 cents – illustrating why voting needs at least a tiny bit of altruism to be rational.

Benefit of demonstrating

As the second graph supposes a million-person march would make remaining in the EU 1% more likely, the expected value of a march that big is 1% x $3 trillion = $30 billion. Per marcher, that is $30 billion / 1 million people x 10% altruism = $3,000.

That said, the graph is not a straight line to 1 million, and you don’t know how many people will show up. Your expected benefit from attending is a distribution over turnouts, but about proportional to the slope in the most likely middle region. If this is say one-tenth the slope of a straight line to 1 million, that’s $300 expected benefit to an extra marcher. (And should a million people show up, the benefit might be $10,000+ per extra marcher, as the slope gets very steep at high turnouts.)

This is extremely rough, but suggests you may well be justified in attending. The detail depends on the actual response curve (e.g. the likelihood of the march influencing politicians), the marcher’s own altruism, their opportunity cost (i.e. whether they have anything better to do instead), and the travel cost.

(I am not considering the value of organizing a demonstration or other campaign, or persuading others to come; though doing so is clearly more effective than merely participating yourself, and it seems likely that an organizer’s net benefit can be much higher than a demonstrator’s, too.)

How does one extra demonstrator help?

In the case of voting, it’s clear how one extra vote directly affects the result, albeit in an extremely unlikely tied situation. But with a demonstration, it’s less clear how an extra person helps. This seems akin to the Sorites paradox, in which piling up individual grains of sand eventually produces a heap, even though at no point does adding one grain to a non-heap turn it into a heap. Similarly, adding just one person to an ordinary demonstration does not suddenly make it important, and it’s hard to see how they add any efficacy at all – how does the +1 feed through to politicians? Via the attendance count, even though it’s extremely approximate, heavily rounded, and often exaggerated?

Actually, adding one person to an ordinary demonstration can suddenly make it important – if it’s a very important person (e.g. Barack Obama). And though ordinary people lack this level of influence, if they have a particularly striking placard or costume, it could attract media attention, perhaps as the hook for a front-page photo or headline which otherwise wouldn’t have appeared. Thus turning a negligible +1 into a visible step up in the response. (For instance, a friend of a friend of mine attracted national press photos in the recent march by posing in an EU-flagged cape in front of Big Ben.)

This is another difference from voting: given which way you want to vote, you can’t control how effective your vote is; but you can make demonstrating more effective.

Non-political benefits of demonstrating

Even if it isn’t worth their while on political grounds, people will still attend a demonstration if they expect to get enough personally from the spectacle, company, and/or social esteem (‘virtue signalling’). Presumably this is one reason why demonstrations often include celebrity speakers, music, etc. – to entice those who wouldn’t turn up just on altruistic grounds.

These side-benefits may make demonstrations more effective overall than some other campaign methods, even when the latter are more effective per person. For example, supposing that instead of going on the recent march, the same 700,000 people had each composed and hand-written a letter to their member of parliament (MP) calling for a second referendum. That’s over 1000 letters per MP - a huge amount to receive in a short time, which might have at least as much impact as a big demonstration, for rather less effort & cost. But as writing letters is much less fun than demonstrating, persuading 700,000 people to do it may be much harder. (And easier shortcuts such as tweets, emails and standard letters get ignored.)

Voting/demonstrating on lesser issues

Not many political issues are as large as Brexit. Climate change is bigger still, but being a global issue, harder to influence politically.

Let’s consider instead a more normal-sized issue – a UK general election. To estimate its size: if you think your party will make the country better off by 10% of central government spending, that works out to $220 billion over an average 4-year term, or about 7% of the size of Brexit. This is still plenty big enough to justify voting, but only borderline worth demonstrating about – $300 x 7% = $21 – assuming the response curve is similar to Brexit’s.

Local issues will have less impact still, but a smaller turnout will be required to win a local election or get the attention of local media and politicians; so they are probably still worth voting about, and may be worth demonstrating about.


  • Rough estimates show that voting is very worthwhile, and demonstrating may be worthwhile, at least for major issues.
  • But this requires people to be somewhat altruistic, as their participation almost entirely benefits others. Voting requires minimal altruism, but demonstrating requires significant altruism.
  • The case for demonstrating is unclear because most demonstrations have little political effect, unless the turnout is unusually high.
  • In contrast, voting is most worthwhile at low turnout levels, and on closely-fought issues.
  • Unlike voters, demonstrators can increase the expected value of their own participation via attention-grabbing placards, costumes, etc. They also get significant non-political value from demonstrations (e.g. enjoyment), which may be the main motivation for some demonstrators.

[1] Bookmakers put the chance of a second referendum at around 28%; with about 50-50 odds of that producing a Remain result (and supposing it wouldn’t happen without a referendum), that makes the chance of remaining in the EU 14%. [Added: these figures are unchanged as at 16 Mar 2019, though the odds of a Remain result on a simple Remain/Leave referendum were and are slightly better than 50-50.]

[2] A Leave supporter might believe a different forecast which predicts a GDP gain instead; or even if not, there are non-economic issues at stake for many Leavers, such as UK self-determination, and the consequences for UK democracy of holding a second referendum, particularly if the result contradicts the first referendum.

[3] GDP includes businesses, and many businesses make donations too - though perhaps they have less altruistic motives than individuals, e.g. public relations (though individuals also of course indulge in 'virtue-signalling' donations). We could focus solely on individuals by using a ratio such as household charitable donations / household income, but this produces similar figures anyway, e.g. 2.7% for the US.

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21 comments, sorted by Click to highlight new comments since: Today at 11:33 AM

You're making a big assumption in your analysis of the value of voting, that one's preferred policy bundle is on the ballot to vote for in the first place. Anyone who feels more aligned with a 3rd party (in the us at least) has to a nearest approximation 0% chance of having their party's candidate elected in even small local elections, much less national ones. And that candidate still likely doesn't align 100% with their policy preferences. Recent polling shows reps and dems increasingly don't like their party's candidates either, they just really hate the other party's, resulting in a lose/lose spiral of fielding 'lesser of two evils' candidates in lowest-common-denominator contests to drink each other's tears. It hardly seems altruistic to me to perpetuate or participate in such a system, even if you have a small (basically 0) chance of moving one or two policies marginally in a less terrible direction (and remember: there's no guarantee your candidate actually does move policy as you'd hoped once in office).

Incidentally, I did vote today; it's a wonderful signal to my peers that I am a good and responsible member of society who doesn't need any lectures on civic duty. Quite rational, though next year I might just order a roll of 'I voted' sticker on Amazon instead.

Yes, I didn't get into more detailed arguments about the pros/cons of voting & voting systems; the Put A Number On It post I linked to has a quite good discussion of these, and I didn't mention other reservations of my own (in particular I'm suspicious of multiplying very small probabilities by very large benefits).

But on your particular point, my brief thought is that not participating in a voting system doesn't make it change (though organizing a mass boycott of it could do). And on my estimated numbers, in the same way that even a tiny bit of altruism makes it worth voting, if you have even a tiny preference between the two lead parties, and even if it's very uncertain they will implement their policies, it is probably still worth voting to keep out the worse one.

If you're locking yourself in to voting for one of the two major party candidates to even have a chance at moving the needle on policy, then I would argue you're not multiplying a small probability by a large benefit because the major parties actually agree on almost everything and only clash on marginal execution issues (the occasional brexit type referendum notwithstanding).

For all the anti-war protests in the Bush years, Obama kept them rolling at pace for almost a decade; for all the tea party protests in the Obama years, republicans had no fiscal restraint when given the purse strings. These were just the noisiest issues on each side (and may be interesting to explore deeper in relation to the value of demonstration) but the list of policy more-or-less agreement between them is nearly endless. So the benefit in normal (non-brexit) elections is marginal at best.

Yes, but since on my numbers the benefits of voting are so huge, a tiny difference between parties can still justify it. E.g. near the end of the article I calculate that in a UK general election, if the difference between the two main parties equals 10% of government spending (in benefit to the country, not necessarily actual spend), that equals 7% of Brexit or about $7,000 to a marginal voter.

So even if it's only worth 0.1% of government spending (e.g. a small confidence that one party will make a small execution improvement on a few policies), that's $70 - enough to justify voting.

By most accounts Nixon was a really horrible president. At the same time Nixon went to China which might have been on of the decisions of a US president that produced the most counterfactual wealth creation ever.

It would have likely not been as easy for a Democratic president to do the same as Nixon even if the Democratic president would have wanted to do so.

It would have been harder for a Republican attorney general to go after the Associated Press the was the Obama's Eric Holder did.

It's easier for a Democratic politician to push through Republican policy and easier for a Republican to push through Democratic policies.

I.e. hence you can't tell how effective a president will be from their party's policies, because sometimes their most effective actions are following their opponents' policies.

Yes, could be. It's in line with the Putanumonit arguments that you just can't tell which party will be better for the country.

I can't think of particular instances of this in the UK, so I don't know if this is more of a US thing. What quite often happens in the UK (particularly since Tony Blair) is parties stealing each others' policies, even sometimes in stronger form than the other party. But presumably that's just them trying to tempt voters across from the other side with occasional juicy little morsels. I.e. both parties converging on the median voter. [ADDED] Though similar to your point that the other party may implement your party's policies, and perhaps more effectively, which makes it harder to predict which party would run the country better.

Thinking about this more in the shower the following occurred to me:

3rd party voting meets both conditions. As a vote, the potential gains are much larger both on nominal alignment and execution (as non incumbent parties are (I imagine) more likely to take action to disrupt the status quo if elected).

But 3rd party voting is also a form of demonstration, enticing mainstream parties to align more closely with your highest value positions (at least nominally) to win proven (and numerable) voters. E.g., if you're a Democrat with special consideration for environmental issues, a vote for the green party probably has a bigger influence on democratic environmental policy than a vote for a moderate democrat (with no included signal that it's environmental justice motivating your vote) paired with attending an earthday rally. Maybe?

I suspect that the chances of a 3rd party winning are orders of magnitude lower than a 1st or 2nd, so the expected value from you having the deciding vote would be too small. But in terms of policy influence, if the 3rd party does unusually well (without winning), I agree that can be significant. Indeed I recall an example of this happening in the UK in the 1990s, when in one national election the Green party (then the 4th or 5th party) did unexpectedly well, albeit still only getting a few % of the vote, which immediately made the major parties start saying how important the environment was and announcing new policies.

This part from Putanumonit does not make sense:

A few months ago all the business papers and TV channels were intensely covering Twitter’s CEO search. Twitter is a simple company of 3,900 employees, it’s financial and operational information is public, and its CEO has a single mission: increase share price. Surely an educated person watching CNBC can easily predict the effect of each CEO candidate on Twitter’s share price and make a gajillion dollars! And yet, no one does.
The American government is unfathomably complex, employs 22 million citizens and impacts 300 million others, the president is judged by a thousand different measures and knows thousands of things the public doesn’t. Thinking that I can predict which candidate is objectively better for America when I can’t begin to do the same for Twitter is simply delusional.

The reason you can't make a killing on the stock market over Twitter's CEO is because everyone else has the same information that you do. An unfathomably complex institution is exactly what we expect would make it easy to get an information advantage, and this is the case in my opinion: only a few hours' investment is necessary to get a leg up on virtually all voters for a given issue.

The reason this doesn't let voters pile up huge amounts of utility is because you only get one chance every 4-6 years to add your information for a given office, and it requires a separate evaluation to even determine what the payoff is. By contrast, the stock market is effectively real-time and very clear about payoffs.

This is a bad foundation for a modesty argument.

I'm inclined to agree with you - the Twitter comparison doesn't make sense.

I think he half has a point when saying you can't tell which party will do a better job, inasmuch as there is an information asymmetry between government and opposition. In the UK anyway, opposition parties don't have access to the civil service who understand the detailed workings of government and have internal information, which means that opposition policies are somewhat speculative, and may have to be modified (or not) when they get into government. Particularly so on matters of national security, where it's almost impossible for a voter to assess whether a government policy (based on secret information) is better/worse than an opposition policy (based only on public information). (Aside from your personal preferences about these policies - e.g. if you have an absolute moral opposition to nuclear weapons, or something.)

But that doesn't apply to all policies, and in any case, the expected benefit of voting is so huge that you only need a minimal amount of information about which party is better (on some objective measure like GDP, or relative to your preferences). E.g. it's fine to be only 1% confident.

It's interesting that you chose Brexit as the best example of a tight vote on a single issue with clear and massive advantage to one side - the best possible case for voting. And yet, even with Brexit it's easy to make the case for discounting the apparent impact of voting.

$3 trillion gained by remaining in the EU.

$3 trillion gained to whom? A lot of people who voted "Leave" would probably agree that Brexit will lower GDP, but they clearly think that the benefits of leaving (in social cohesion, security, whatever) are greater than the GDP hit. You could say that they're voting selfishly and that the lost GDP accrues to all Europeans and the narrow benefits only to them, but that's a problematic argument - why didn't the Europeans selflessly vote to safeguard the interests of rural Britons? As you noted, even pure altruism often (and unfortunately) stops at a nation's borders.

The half of Britain who voted to leave think that the benefits of leaving to them, and thus to half of Britain, is greater than $3 trillion. If we could sum up actual utils accrued to people, rather than dollars accrued to some national accounts, how certain can you be that the number is even positive given that half the nation disagrees with you?

Many ‘Remainers’ who wish to stay in the EU want a second referendum on the matter; recently, hundreds of thousands of them marched through London to call for one.

So even in the cleanest possible case of a single-issue referendum, the decision that was voted on is yet to be implemented. And when you vote for candidates with a whole list of proposals, the chance that electing the candidate will get the actual proposal implemented is much lower still.

My argument is that doing the math on voting makes it look a lot worse than you'd naively assume, not that the math comes out against voting in every possible case. But this also means that just because the math may barely favor voting on Brexit it doesn't mean that it will for other, dirtier cases.

[Response substantially edited:]

If I understand you right, you're saying that if Remain were to happen then Leavers would incur a large actual loss (relative to the Leave scenario), because they reckon the benefits of leaving in terms of social cohesion, security etc. will not occur.

Perhaps those aren't the best examples, as arguably those are matters of fact, so Leavers could find out they were wrong if it turns out there is no loss in social cohesion & security by remaining; so they wouldn't necessarily lose utils. A better example might be national self-determination, which a Leave supporter would value come what may, and a Remain supporter might put little value on. That is, Leavers aren't merely predicting that leaving the EU would make things better for the UK, they are expressing a (non-falsifiable) preference for being out of the EU.

I haven't thought of that, and that could be so - or perhaps more likely it's a mixture of prediction and preference. In which case Leavers would only get some negative utils, still leaving tens of thousands of $ per extra Remain voter. (And still plenty enough to justify voting even after major shrinkage by the uncertainty that policies will turn out/be implemented as expected.)

Complicated by the fact that if Remain happens, Leave supporters would always feel things would have been better if Leave had happened, even if their predictions were unknowingly false, because they never get to try out & compare both scenarios. I.e. Leavers will never be satisfied if the UK remains, and Remainers will never be satisfied if the UK leaves, regardless of how the other possible world would have been. (Maybe that's your main point here.) But I reckon that dissatisfaction is small compared with the economic harm caused by leaving (if the median GDP predictions are true).

By the way, I'm not convinced voting is rational (hence I have never voted in my life), and believed that it wasn't, until the altruism calculation occurred to me a year or so ago. My current suspicion is about the validity of multiplying a very small probability by a very large benefit to get a justification; but I haven't yet read/thought of a strong argument against this.

(PS Ah, you're Jacob F - good to meet you! I enjoy your blog.)

I don't get the sentiment of "your vote only matters if it would be an exact tie otherwise." By that logic, if the outcome of a US presidential election would either save 10,000 lives or not, then the altruistic thing to do is to get the vote as close 50/50 as possible, so that every voter can save 10,000 lives each for a total of 3 trillion lives saved, as opposed to the normal outcome, where no lives are saved at all.

Hmm, I see your point; but if each vote is independent, then given how all the other voters voted, my vote really does decide the election. E.g. if I go into the ballot box, what I write on my poll slip does not cause and is not caused by what's written on all the other slips (as I don't see them and they don't see mine).

How about this thought experiment: I am the very last person in the country to vote. Unknown to anyone, all the votes made before mine constitute a tie, so my vote will be the deciding vote. Then it really is the case that if I vote one way, 10,000 lives are saved, and the other way, none are. And it is also the case that, given how I voted, if my neighbour had voted the other way, he would have changed the outcome too. (Incidentally it seems only people who vote the same way as me have the power to decide the outcome, given how I voted.)

I do sense the counterfactual complications. Is your argument that the 10,000 lives saved should be apportioned among all the voters in the case of a tie-break, and hence it still isn't worth anyone's while voting? What is the argument for apportioning?


Here's a further hand-wavy argument:

You're saying that in the case of a tie-break, everyone who voted for the winning party each gets to save 10,000 lives (overcounting the benefit). But in a normal outcome with no tie-break, none of them do, even though 10,000 lives are still saved (undercounting the benefit). If we account differently, with only the final voter getting the 10,000-life benefit in the tie-break case, and all voters for the winning party (or all after a majority was reached?) sharing the 10,000 in the normal case, so that in every winning scenario the benefit adds up to exactly 10,000 lives (more intuitively), doesn't it all work out the same in terms of expected benefit per voter? (I wonder, without thinking/calculating further.)

Actually I was saying that if only tie breaks matter, then in the case of a tie-break, everyone who voted for the losing party also saves 10,000 lives each. Because if they didn't vote against saving lives, then it wouldn't be a tie break, and then no individual would save any lives at all.

Of course, I don't actually believe that - I think voting matters even if it wouldn't break a tie.

In the scenario where you vote last in the tie-break, it is true everything depends on you. But everything depended on everyone else too, even though they already voted. In terms of expected utility of your decision, you get to tell your utility function that you saved 10,000 lives. In terms of moral credit though, everyone who voted SAVE still gets a fraction of the credit, because without their vote you couldn't have done it.

Let's consider a smaller scale vote, like a supreme court decision. Even if you know exactly how other justices will vote, and you know it's going to be 5-3 not including you, well your decision still matters, and not just for signalling reasons. 6-3 is different from 5-4 because in the former case, two of the five would have to flip in order for it to change and in the latter case, only one of the five would have to flip for it to change. Even if everyone knew how everyone else was going to vote ahead of time, still each justice's preference for one ruling over another still matters.

Also, I don't think it makes sense to say that one ballot matters more than another ballot, based solely on the order that they are counted. Votes are votes.

Yes I follow your argument, though I'm a bit doubtful about a result that produces a large difference between utility function and moral credit.

Re your Supreme Court example (and I agree this is a clearer way of thinking about it), I don't quite follow the argument. It's true that if the other justices had voted differently, more of them would have had to vote differently ('flip') had you done so, but as it's a given that you knew how everyone else was going to vote, flipping is ruled out - their votes are set in stone.

And re 'still each justice's preference... matters', I wasn't clear if this is the same point or a separate point - i.e. a signalling or similar argument that the size of the majority matters, e.g. politically.

I think this EA forum post explaining Shapley Values encapsulates my current opinion better than my comments above.

Why bother voting? Your vote will only change the result if it would otherwise be an exact tie; and the chance of that is negligible – one in millions.
But a chance of one in millions is worth taking if the jackpot is billions or trillions. That is, the opportunity for you to select a better rather than worse government, thereby making the country – though not yourself – billions or trillions of dollars better off. So as long as you care at least slightly about the rest of the country, voting is rational; civic duty really is a reason to vote.

That's an incredibly spurious premise right from the bat. Personally, I don't care all that much if the country is billions or trillions better off... That's ranging from single-digit dollar amounts to a couple hundreds. Also that's supposing the government has this kind of influence (esp. if you counter the last by positing bigger amounts). Also as long as people are not going into poverty, I still mostly care about myself.

People hate to hear this, and I usually don't bring it up because it's counterproductive, but: voting is not rational except in very small elections. The problem is that if everyone thinks this, you have a serious problem. Yep, that's the tragedy of the commons.

A possible way to solve the issue is to make the vote legally mandatory (which is the case in my country - Belgium). This might lead to more uninformed ballots being cast, but I'm not entirely sure (most of the ballots are uninformed regardless).

A little bit of altruism still seems to make it rational even if you care almost entirely about yourself - see the example calculations.

I used to think that making voting mandatory was a good solution, but nowadays I think it's a draconian measure. Because what if you disapprove for example of the particular voting system (First Past the Post in the UK/US)? Then forcing you to comply with it, perhaps only symbolically (as you can discomply in other ways like spoiling your ballot paper - unless that will be criminalized too) is a waste of everyone's time.

Similarly if you don't want to vote because you are indifferent between the candidates, or think you don't know enough about the issues to choose a candidate, etc.

Something somewhat similar to, but less draconian than, compulsory voting would be to pay people to vote, e.g. £5 / $5 in cash or vouchers as you exit the polling station. Which would also somewhat correct the current skew in turnout - poorer people are currently less likely to vote.

Indeed, as mentioned, without altruism, voting behaviour is fairly inexplicable.

I vote to reward or penalize politicians based on their previous choices, rather than to create better outcomes. That is, I look back, not forward.

There are some exceptions, e.g. when a candidate before assuming office is sending unusually credible signals, e.g. glorifying torture or some such. Other than that, I mostly ignore promises, and instead implement reciprocity for past decisions.

Edited after more reflection:

Whereas the expected benefit of voting to you alone is the Brexit harm to you / 3 million, = $3 trillion / 2 (effect on UK only) / 65 million (UK population) / 3 million = 0.7 cents – illustrating why voting needs at least a tiny bit of altruism to be rational.

This is interesting. I do expect for things like marginal tax rates, my emotions are scope-insensitive and my reciprocity mostly symbolic/psychological.

However, if I share interests with many other voters who voted for those interests, all of their votes benefited my interests and I can reciprocate not just for/against politicians, but also for/against all these other voters. If I like low tax rates, I can benefit every voter who's voted for low tax rates by voting for low tax rates.

More importantly, some issues have much higher impact on my utility than marginal tax rates. If I could choose between $1 billion personal purchasing power, and the liberty to buy a deadly dose of pentobarbital if/when I choose to die peacefully, I'd take the pentobarbital. Which means that politicians who've reduced the probability that this liberty is legal for me have forced an opportunity cost of over $1 billion on me. Perhaps voting is still not the best way to implement reciprocity in such a case, but outside of direct attacks on ex-politicians, e.g. what the Christians did to Els Borst, it's one of the remaining ways to get back at them and therefore still well worth doing.

Yes, interesting points. I haven't really given any thought to voting as a reward/punishment, but many voters do this. Though of course it's mixed up with forward-looking voting, since (for many people) you vote against a politician who did something bad so that they won't be around to do more bad things.

And politicians anticipate punishment-voting as a deterrent to them doing bad things, since there isn't much other deterrent (except the law).

Also an interesting point re voting as reciprocation to similar voters as a kind of solidarity group. (Parties are themselves solidarity groups, but so of course are special interest groups and other supporters of particular policies.)

I'm not sure whether or how all this affects the calculus. Eliezer wrote an article on voting a while back in which if I recall his line was something like 'it's all too complicated to model, so just stick to simple reasoning'.

Re your pentobarbital example, this could be something where the 0.7 cents direct effect on you is bigger - though it would indeed have to be something approaching a $1 billion effect to count (since the expected benefit to you is this / 3 million, in the UK). Though that said almost all issues like this affect quite a few other people too, so altruism makes it worthwhile anyway.