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%[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.

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 out­weigh the polit­ical value.

[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%.

[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.

[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.