Rationality of demonstrating & voting
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% 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.
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. 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,  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.
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.
 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.]
 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.
 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.