Suppose we have a population of voters, each of them casting a vote in some (not necessarily deterministic) voting system.

Let w be the winning candidate.

Consider the following experiment:

Fix some number N. Draw a sample of N voters with replacement (i.e. you may draw the same voter more than once). Run the election again. What is the probability of the result of the vote being different from the vote on the whole population (if the voting system is non-deterministic, rerun the system on the whole population too)?

Call this number S(N).

I suspect S(N) might reveal interesting behaviours of voting systems, but I haven’t yet analyzed this in any great detail or thought about it much.

It does however have the interesting property that it isn’t dependent on the type of voting system used – it works equally well for ranked, graded or scored votes, and for deterministic and non-deterministic systems, so it provides an interesting way of comparing potentially very different voting systems.

Some interesting questions one may reasonably ask:

What is the limiting behaviour of S(N) as \(N \to \infty\)?

Is S(N) monotonic increasing?

What is the behaviour like for very small N? (I don’t have a precise formulation of this yet)

Is there anything special about S(original number of voters)?

Two examples:

For first past the post voting, \(S(1)\) is the fraction of the population who voted for the candidate, and \(S(N) \to 1\) as \(N \to \infty\), because as \(N \to \infty\) the fractions of the samples who vote for a given candidate concentrate on the fractions in the original sample. I think \(S(N)\) is monotone increasing but have only proven it for the two candidate case. It’s intuitively plausible though.

For random ballot, \(S(N) = \sum_{i = 1}^m p_i^2 \), where we have candidates \(1 \ldots m\) and \(p_i\) is the fraction of the population who voted for candidate \(i\). Note that this is independent of N. This is because picking N candidates then picking a random one of them is exactly the same as just picking a random candidate in the first place, so \(S(N)\) is just the probability of running the election twice and getting the same result.

I haven’t worked out the answers for other, more complicated, systems. I would be interested to do so, and may well at some point, but if someone wants to do it for me or tell me if it’s already been done that’d be cool too.