Thursday, December 29, 2011

Electoral reform, redux

I had a great set of responses to my recent post about electoral reform. I got emails, Tweets, blog comments, and comments on Google+. Many interesting thoughts and articles, which I thought I'd respond to in this follow-up.

Some folks argued with the decision not to vote. How could that be a good idea? What if everybody did that? This one's easy. Firstly, that'd be great. Secondly, it's irrelevant. Consider: I might do my bit about overpopulation by deciding not to have kids. Sure, if everybody did that then the consequences would be catastrophic but that doesn't disqualify it from being a rational individual choice. Indeed, many people do make that choice with net positive effect.

Nick provided an interesting post on the "fallacy of the deciding vote". It's a well reasoned piece, but premise (2) renders it inapplicable: the argument at hand isn't whether my individual vote unilaterally decides the result (which obviously it doesn't, as the article points out, in all but the most degenerate cases). It's about the probability that adding or subtracting my individual vote has an effect on the outcome. I don't need my vote to be the single deciding vote by any means.

So that we're clear, before we move on let's call a given individual voter democratically impotent in an election if the election has an identical outcome with or without their vote. Conversely the voter is democratically potent if removing their vote changes the outcome.

Dominic commented on the post presenting a syllogism:

Premise: Your vote is the same as everyone else's
Premise: Your vote literally counts for nothing
Conclusion: Everybody's vote counts for nothing

...and yet, we have a result. I suggest that premise 2 is faulty
and he's right. My hyperbolic "I literally count for nothing" should really have been "in the span of my lifetime, with overwhelming probability, I'm democratically impotent in every election in which I'm eligible to vote". There's a mathematical distinction but barely a practical one.

Interestingly, as you remove more and more voters from the electorate the chances of a voter which remains being democratically potent increase significantly. Thomas Pogge, Yale's Professor of Philosophy and International Affairs, showed in a post last year that in a small population of 63 voters, with each voter choosing independently and with equal probability between two electoral candidates, a given voter has a full 10% chance of casting a deciding vote.

I like those odds, but in large populations things look much bleaker.

In the simple equiprobabilistic model which Pogge presents, in an electorate of 100,001 people the probability of a given voter casting a deciding vote is about 1 in 400. Not bad at all! Assume instead, though, a slight general preference in the populace for one candidate over the other, let's say 49% to 51%, and suddenly your 1 in 400 chance of affecting the outcome in Pogge's model sinks to 1 in 193 billion.

Scale up from there to a population of just a million, and you as an individual voter are democratically impotent with a novemvigintillion to 1 probability¹. If you voted in an election of this kind every second for as long as you lived (or even, hey, every nanosecond for the entire lifetime of the universe), chances are—by an inconceivable margin—that you'd not affect the outcome of even a single one.

Little wonder, then, that cynicism about democracy comes so easy. Richard passed along a link to a passionate soi-disant rant including:

I am being sadly sincere when I describe [democracy] as a system which is much better at giving the feeling of participation than actual participation. To me, this is one of the terrible things about democracy (and part of why it is so successful) - because voting lets people feel like they can influence things. Even if they don't vote, they feel like they could have voted.

But any one vote never matters...
…as indeed is demonstrated above.

So we come back to that idea again of selecting the election winner by picking a single ballot at random and going with that. Now one's chances of democratic potency in an electorate of a million people voting between two candidates with a 49%/51% baseline preference is simply a million to one. Still sounds like long odds but it's a trillion trillion trillion trillion trillion trillion trillion times better than before. In this system people are thus vastly more individually empowered, the tyranny of the majority is ameliorated, and the end result remains proportionally representative, albeit with added statistical noise. It's a win-win-win!

At the end of the day, though, as James noted, whoever wins an election it's always a politician. A solution to that problem is left as an exercise for the reader.

Next in this series: Electoral reform, epilogue

¹ a novemvigintillion is a billion billion billion billion billion billion billion billion billion billion; about ten billion times the number of atoms in the universe.


Dominic Sayers said...

OK then, consider an election where one candidate is massively more popular than the others. She polls at 49% and the other 5 candidates at around 10.2% each.

Your system is likely (p=0.51) to result in a candidate with minimal support winning. Boo!

The head of Hepworth Electoral Reform Society would be strung up from the nearest lamppost.

1. I don't think we can do entirely without politicians and government (I'm not Pyotr Kropotkin). But we can look to gradually circumscribe their terms of reference to minimize the harm they can do.

2. My counterexample is stolen from the comments here:

Stephen Johnson said...

The tombola approach to electoral reform may provide PR for the country, but it doesn’t work at the constituency level. Good Candidates would lose by the luck of the draw. Bad candidates might win. At the constituency level the result is unfair and undemocratic.

For a PR system where every vote makes a difference to the result of the election, regardless of where the vote is cast, but also allows the best MP for the constituency to be elected, regardless of party, look at Direct Party and Representative Voting (DPR Voting).

Isaac said...

Stephen, similar to the current popularity contest, no guarantees are made about candidate quality. "Bad candidates might win" is a criticism which applies equally to the current system, except there we have empirical evidence that they actually do. I call it a wash on that front.

DPR Voting suffers from the exact same problems as the current system with respect to democratic impotence. That's the main thing I'm trying to solve for.

Isaac said...

Dominic, you're good at this :-)

I don't find your counterexample convincing, though. What if you change the numbers so that the popular candidate has 10% of the vote compared to 45 others having 2%. Should she always win? Even though 90% of people would prefer someone else?

I think it's fine that someone with 10% of the vote should have a 10% chance of winning... and back to your example it doesn't seem unreasonable that someone with 49% of the vote should have a 49% chance of winning... a full 4.8x the chance of a candidate with 10.2%.

Also—because my system has financial incentives for voting we'll see greater turnout, actually increasing the fairness with respect to the full electorate (turnout for general elections is ~65% in the UK, 57% in the US).

Stephen Johnson said...

I agree there are problems with the FPTP system which conflates the vote for the candidate with the vote for party, and this can and does result in candidates getting more votes than they deserve. ‘Tombola’ is worse because it can elect a bad candidate who didn’t even get many votes. Or do you argue that if the votes are not counted there would be no evidence for anyone to argue about democratic legitimacy at the constituency level, and thus no basis for anyone to feel aggrieved?

The Voter in a DPR Voting election has a 100% chance of making a difference to the overall election result. Judging the system in terms of democratic impotence, this is surely superior to a ‘non-zero’ chance of your vote affecting the final outcome of the constituency result.

Isaac said...

Stephen, "the voter in a DPR Voting election has a 100% chance of making a difference to the overall election result" isn't true at all.

If I understand the explanation at, I have two votes in this system: a representative vote and a party vote. Both are simple "majority wins" outcomes. So if I vote for a losing option, and the losing margin isn't 1, my vote is irrelevant.

Most of the time, the losing margin won't be 1. Hence, most of the time, you can subtract my vote with no impact to the result: the very definition of democratic impotence. I didn't count.

Stephen Johnson said...


The Party vote isn't a 'majotitarian winner takes all' vote. It is the totals of votes for each party that is important.

In DPR Voting every party vote makes a mathematical difference to the ‘result’ -the result being the number of votes each political party has in the Commons, which directly determines which party or parties can form the Government. After all, isn’t that why we vote – to determine which party rules us for the next five years?
Yes, of course each vote only makes a very small difference, but it’s the same for each voter. Also, in DPR Voting every vote makes a difference regardless of the constituency where the vote is cast.
I argue that the certainty that your vote will make a difference to the result is more powerful in democratic terms than a miniscule chance that your vote might decide the election, regardless of the weight of (uncounted) votes who voted against you.