Deterministic voting is just too random

One of the issues that often gets cited as an objection to random ballot is that it’s too random. What happens if a lunatic fringe gets voted in to power? People might get a majority simply by random chance.

This is just a brief note on how unlikely that is.

Suppose there are \(N\) seats, and we have a party with a fraction \(q\) of the vote. Let \(W_i\) be a random variable which is 1 if the party wins seat \(i\) and 0 if it loses. Let \(W = \sum W_i\) be the number of votes won and \(q_i = P(W_i = 1)\).

\(W\) is a sum of independent random variables. I’m going to assume that \(N\) is large enough that we can apply the law of large numbers and approximate it as a normal random variable.

So from now on we’ll pretend \[W \sim \mathrm{Normal}(qN, \sigma^2)\]

Then the probability of getting a majority is
P(W > \frac{1}{2}N) & = P(Z > \frac{(\frac{1}{2} – q)N}{\sigma})\\
&= 1 – \mathrm{erf}\left((\frac{1}{2} – q)\sqrt{\frac{N}{q(1-q)}} \right) \\

erf is monotonic increasing, so this value is maximized by making the argument as small as possible. Everything in there other than \(\sigma\) is fixed, so this is done by maximizing \(\sigma\), or equivalently \(\sigma^2\)

When does this happen?

Well \(W\) is a sum of independent random variables, so its variance is the sum of their variances. So \[\sigma^2 = \sum q_i(1 – q_i)\]

We want to maximize this subject to the constraint that \(\sum q_i = qN\). We can turn this constrained optimisation problem into an unconstrained one by adding a lagrange multiplier t. So we’re optimising the function \[f(q_i, t) = \sum q_i(1 – q_i) + t (\sum q_i – qN)\]

Taking partial derivatives we have

\[\frac{\partial f}{\partial q_i} = 1 – 2 q_i + t\]

Setting this to 0 we get

\[ q_i = \frac{1 – t}{2} \]

The significant part of this being that it’s a constant that doesn’t depend on i. So applying the constraint we have \(q = q_i\), and \(\sigma^2 = N q (1 – q)\).

(I think it’s probably possible to reach this conclusion without the normality assumption, but I haven’t thought through the details).

P(\mathrm{majority}) & = 1 – \mathrm{erf}\left(\frac{(\frac{1}{2} – q)N}{\sigma}\right)\\
& = 1 – \mathrm{erf}\left((\frac{1}{2} – q)\sqrt{\frac{N}{q(1-q)}}\right)\\

At this point we’ll just plug some numbers in. Here’s some python code:

import math
from scipy.special import erf
def probability_of_majority(n,p):
    sd = math.sqrt(n * p * (1 - p))
    deviations_to_majority = n * (0.5 - p) / sd
    return 1 - erf(deviations_to_majority)

At this point let’s just plug some numbers in.

Consider the UK house of commons with its 650 seats. Suppose we have 40% of the vote. What are the chances of us getting a majority?

Plugging them into our python code we get that the answer is 1.84e-13. i.e. if we ran a general election 5 times a year between now and the point when homo sapiens first started to appear, it would still be extremely surprising if any parties with 40% of the vote had ever achieved a majority.

At 45% of the vote we have a probability of 0.00029. If we ran an annual election every year since the birth of Christ, it’d still be more likely than not this would never have happened but it wouldn’t be very surprising.

At 48% of the vote we’re up to 0.15. At this point we expect this to happen every few general elections.

So the lunatic fringe? They’re not going anywhere. The lunatic majority? Well, that’s what you get in a Democracy. But unless someone actually has a majority they’re not getting in.

Compare this to the percentage of the popular vote with which parties typically reach a majority in the UK under a deterministic system.

From this I can only conclude that deterministic voting (in a constituency based system) is simply too random to be practical.

This entry was posted in voting on by .

4 thoughts on “Deterministic voting is just too random

  1. Michael Chermside

    The reasonable argument to make is not ” It’s too random. What happens if a lunatic fringe gets voted in.”, and you demonstrate this quite effectively.

    But making good decisions is only a secondary purpose of government (and, by proxy, of voting systems). A vitally important purpose, but secondary to the core purpose of preventing us from revolting and running around murdering each other as a means of deciding who gets to be in charge. That is why, for instance, a vote-counting process that is observed by representatives from each of the major parties is (in practice) far superior to a vote-counting process that is observed by neutral third parties. Because afterward the losers are far more likely to say “Yeah, I suppose she wins.” instead of “I’ve been Cheated! I’m declaring this election to be invalid and I’m appointing myself dictator!”.

    In a similar vein, the problem (in my opinion) with random voting is that many people who are not well-educated and mathematically savvy will fail to understand the arguments you make here. And it doesn’t matter how correct you are, they will still fail to understand. So there will be more people who feel that the election outcome was unjust (even though it wasn’t) and who will thus be more likely to refuse to accept the outcome — an outcome that leads in the extreme to civil unrest and the failure of government. Counting ballots and seeing who has the bigger number is SO simple that it can be understood by any person with enough intelligence to participate in the political process.

    Not all alternatives to first-past-the-post suffer from this problem. In fact, any approach based on ranking candidates probably translates in the minds of those who are NOT experts on voting as “everyone stated their preferences and then some funky math happened and it spit out the list of winners”. They’ll check with their most trusted mathematically inclined voting expert to ask “did the funky math happen the way it was supposed to?”, and that’s a question that the mathematically inclined voting expert can answer. But mention any hint of randomness and people will have a problem with it. I look at my own experience trying to explain to people WHO ARE EXPERTS ON COMPUTER SECURITY that if we generate 128 cryptographically strong random bits, then we can safely assume that there will never be a collision. It is simple and true, yet even people who ought to be well-qualified (like “security experts” at a bank) don’t understand — we have little hope of getting a solid understanding of probability through to the general public.

    1. david Post author

      Yes, there are definitely education problems with random ballot voting . Fortunately I don’t have to care about that as I’m not actually campaigning for this system in any real sense. :-) It’s also not an obstacle to more small scale applications of this.

      But I will say that the idea that we have to use a broken system because people understand it pisses me right off and is a sign that something, somewhere, is deeply fucked up.

  2. Pingback: The winning strategy for random ballot is not what I thought it was | David R. MacIver

  3. Pingback: Best of | David R. MacIver

Comments are closed.