You don’t have democracy unless your voting system gives the same amount of influence over the outcome (same amount of political power) to every voter regardless of the voter’s opinion on the candidates.

A voting system is undemocratic if and only if elections are possible in which the voting system privileges one voter (P) at the expense of another voter (S).

Influence on the outcome can be measured by the ability to shift the outcome.

Consider an election in which the tally if applied to the ballots from all the other voters except P and S would produce one outcome, call it Outcome-sub-one. I am discussing primarily single-winner elections here, so obviously part of the outcome is the fingering of a particular candidate as having won. The outcome can also for some voting systems include secondary information about how well the candidates did relative to each other.

Now suppose we take into account P’s ballot, and the outcome shifts to a different outcome from Outcome-sub-one.

Now let S vote. What happens? Can S swing the election outcome back to Outcome-sub-one, thus exactly canceling the effect of P’s vote?

If not, then we have a concrete case that proves that the election has accorded more power to P than it has to S. In this scenario, P was able to move the needle, but S was not accorded the power to move it back. Therefore, the election was not democratic. It Privileged P and Screwed Over S. And a voting system that can produce an undemocratic election is an undemocratic voting system.

Conversely, a voting system that provably cannot produce an undemocratic election in the kind of scenario I just described, is a democratic voting system.

For a counterexample, let’s look at “First-Past-The-Post” Voting (FPTP). This is the system used for almost all elections in the US, where voters can only choose one candidate, and the most-chosen candidate wins.

FPTP is not democratic by the test I describe above.

To apply the test, we have to understand the form of the outcome of an FPTP election. I described the test in terms of changed outcome. To apply the test, we have to know how to compare two outcomes to determine whether they have identical political meaning. And to understand the form of an outcome, we have to ask, does the outcome include any meaningful component that indicates something politically significant about how well the candidates did relative to one another, beyond the simple indication of who is the winner and who are the losers?

I contend that the outcome of FPTP does signify politically beyond the mere fingering of who wins. A candidate who does poorly is less likely to run again, and people are less likely to run again on the same ideas as that candidate, than a candidate who receives almost so many votes as the winner received.

One of the functions of elections is to function as a vehicle whereby voters inform one another of some partial information about how popular candidates and their ideas are with the voters, even if in a sense this delivery of information is of secondary importance to the primary result, the fingering of the winner of office.

For FPTP with a fixed number of votes, the results can be expressed simply by pairing with each candidate the count of votes that candidate received. Two results have the same meaning if and only if, for each candidate, the count of votes in favor of that candidate’s winning is the same in the two statements of results. If we allow abstentions, then I guess the percentages express the meaning, normalizing to some scale. But to prove the case, I don’t need an abstention.

So let’s apply the test. To prove that FPTP is democratic, I would have to prove that the scenario of unfairness I described above could not happen. On the other hand, if I am going to successfully contend that FPTP is undemocratic, all I have to do is present one example where it can fulfill the scenario.

Let’s say candidate A has 30 votes, candidate B has 30 votes, and candidate C also has 30 votes. If no one else votes, the election is a three-way tie.

Now along comes P and votes for A. The outcome has shifted, from a tie, to a win for A.

How is S going to vote to cancel P’s vote? It’s not possible. The only possible votes for S are A, B, and C. If S votes for C for example, the election will be a tie between C and A, which is different from a three-way tie.

S cannot cancel P’s vote, therefore FPTP is undemocratic.

These ideas are not original with me. I paraphrase from Mark Frohnmayer and his father, both of Oregon, and perhaps I build on their thought a bit.

[Edit Mon Nov 14 02:04:11 UTC 2016]

Someone commented (on Facebook):

…

If, after everyone has voted, the result is A=31, B=31 and C=30, then that is the correct result (under the given voting system). I don’t think it makes too much sense to penalize the system because its final result is two votes away from the intermediate state of A=30, B=30, C=30.

….

My response:

One of the possible reasons for P to vote the way she did is that she actually approves A and disapproves B and C. Let us suppose that S has the opposite opinion. He would be happy if B or C were to win, but disapproves A. Now note that there could be an election with a million people who share the opinion (and valuations) of P. But it would take **two** million voters who share the opinion of S, to counter the effect on the electoral outcome wielded by those who agree with P. Therefore, the amount of power accorded to each member of the S-agreeing faction is only half that accorded to each member of the P-agreeing faction. No doubt you can recognize this as an example of “vote splitting”. And I hope you see that counting a million voters’ votes at only 50% strength is numerically significant. In moral terms, we might about as well decide that blue-eyed peoples’ votes should only count half.

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