Borda Count Doesn’t Have to Care whether You Complete Your Ballot

In my past posts on voting (here, here, here, and here), Borda Count has come out looking quite strong compared to the other ordinal voting methods I’ve compared it to. (Ordinal voting methods have voters rank candidates relative to each other, rather than, say, give them scores.) It has outperformed Plurality, Plurality Runoff, and Ranked Choice (IRV) in most metrics in almost every circumstance. However, Borda Count can have a particular weakness to perhaps the simplest form of strategic voting — not listing candidates you don’t like anywhere on your ballot. So far in my simulations, I have assumed that voters are voting in honest accord with their preferences, not only by ranking candidates in order of how much they want them to win, but also by filling out their ballots completely. Can we make Borda Count resilient against this simple form of strategic voting? Yes. Let’s see how it works.

A brief, obligatory description of Borda Count:

Voters rank candidates. Candidates receive points from each voter based on where they are ranked. For instance, the last place candidate on my ballot receives zero points from me, the second-to-last place candidate on my ballot receives one point from me, the third-to-last candidate on my ballot receives two points from me, and so on, until my top ranked candidate gets the most points from me. The candidate with the most total points from all ballots wins.

Borda Count’s Incomplete Ballot Conundrum

In an election using the simplest version of Borda Count, you should not completely fill out your ballot.

Candidates are given points based on where they rank on a ballot. For instance, in a six-candidate election, a complete ballot would be scored as:

Rank	1	2	3	4	5	6
Score	5	4	3	2	1	0

So the first ranked candidate gets 5 points, the second ranked candidate gets 4 points, and so on. But why give points to candidates you don’t like? Instead, we could not rank candidates we don’t like and leave them with zero points from our ballots instead.

Rank	1	2
Score	5	4	0	0	0	0

This means that we can give our preferred candidates more of an edge by not putting the candidates we don’t want to win anywhere on our ballots. That would be a pretty easy way to game Borda Count to our advantage! Surely any voter in an election using Borda Count would be able to understand the implications of this and vote strategically by only partially completing their ballot, violating the assumption I’ve made in past comparisons of Borda Count to other voting methods, where voter completely filled out their ballots and Borda Count outperformed several other voting methods most of the time under the assumption that voters fully listed all candidates on their ballots.

Does this mean that we should dismiss Borda Count’s great performances in my past simulations? Not yet. There is no reason to have Borda Count score incomplete ballots in a way that rewards such a simple voting strategy. In fact, there are three distinct yet intuitive ways we can have Borda Count score incomplete ballots. (Here is a good paper talking about another strategic voting issue in these different variants of Borda Count.)

The three different methods are:

• BordaUp:

This is the method we saw above. We score the top ranked candidates as though all candidates are ranked on the ballot, and then give the unranked candidates zero points. Essentially, we “round up” the points for the ranked candidates while giving unranked candidates zero points.

• BordaDown:

Score the ranked candidates as though they are the only candidates in the race, starting with 1 point for the last ranked candidate and increasing by 1 point for each spot higher in the rankings. Give the unranked candidates 0 points. This decreases the gap in points received by ranked versus unranked candidates when compared to BordaUp, since the ranked candidates do not receive the full amount of points they would have received had all candidates been ranked on the ballot.

• BordaAverage:

Score the ranked candidates as we did in BordaUp, but instead of giving unranked candidates 0 points, distribute the remaining points would have been given to candidates if all candidates were ranked to the unranked candidates evenly, so that all unranked candidates get the average points that would have been assigned to them had they all been ranked on the ballot. This ensures that all ballots reward the same total number of points to candidates, regardless of how completely they are filled out.

Strategic Voting via Incomplete Ballots

Now let’s see how well strategic voting works in these different versions of Borda Count.

In this model, both voters and candidates are uniformly distributed in a two-dimensional ideological space. All voters in the second, third, and fourth quadrant vote normally, by filling out their ballots completely and in honest accordance with their preferences. All voters in the first quadrant, however, vote strategically by only listing their favorite half of the candidates (rounded down) on their ballots.

The more that the strategic voters shift the election outcomes into the first quadrant, the more successful strategic voting by omission is. We can measure the effectiveness of strategic voting in each of these versions of Borda Count by taking the dot product of the vector going from the origin to the position of the average winning candidate with the vector (1,1). This gives us a single value measuring how effective strategic voting was in each of our Borda Count variants. Since the average winning candidate in a Borda Count election where all voters completely fill out their ballots will be located at (0,0), the larger the value of this dot product, the more effective the strategic voting was, the larger the dot product will be, the smaller (more negative) the value of this dot product, the more counterproductive the strategic voting was, and the close the dot product is to 0, the less the results were affected by the strategic voting. For the sake of giving this value a name, I’ll call it the Strategic Effectiveness Score (SES).

Results

I ran simulations using different values for Beta, a parameter that determines how much voters prefer candidates who are ideologically similar to them versus candidates that will push the country (or state, city, etc.) in their preferred direction the most. (For more on Beta and what it means, I describe it here or, even better, you can pick up Merrill and Grofman’s book on the topic.) Graphs of the average winning candidate positions are shown below; the farther up and to the right the average winning candidate is, the more effective the strategic voting was, and the farther down and to the left the average winning candidate is, the more the strategic voting backfired.

For a more complete picture of how these different versions of Borda Count fare, here are the SESes for each of the Borda Count variants for different values of ß. The higher the SES value, the more effective strategic voting by leaving ballots incomplete is. The lower the value, the more counter-productive this strategic voting is. A value near zero means that the strategic voting was irrelevant. BordaComplete is just Borda Count where voters completely filled out their ballots.

As we might expect, strategic voting by omission is effective in BordaUp but counter-productive in BordaDown. For most values of ß, strategic voting by omission is mostly irrelevant in BordaAvg, though in situations where ß is low (voters are much more concerned about how much candidates will push governance in their preferred direction instead of how ideologically similar candidates are to them), this kind of strategic voting backfires significantly. This occurs because in situations where voters can be modeled using low ß values, they prefer to vote for more extreme candidates, so extreme candidates in the third quadrant benefit from getting high scores from voters who are in the same left/down ideological direction while also receiving the same number of points from voters in the first quadrant as candidates who are near the center.

Which Version of Borda Count Should We Use?

If we want to replicate the results of Borda Count with complete ballots, BordaAvg does the closest job, though it does diverge significantly in situations where ß is very low, though I don’t think we should be too concerned about this for two reasons. First, punishing ballot incompletion is preferable to rewarding it, since voters can respond by ranking more candidates. Second, BordaAvg punishes incomplete ballots in low ß value situations for the same reasons that low values for ß don’t strike me as being very realistic. I believe we have plenty of evidence that moderate voters don’t prefer the most extreme candidates on their side of the political center, but according to a model of voters with a very low value of ß, they do.

In situations where we want to incentivize ballot completion, we may want to use BordaDown. I don’t think this is a great solution for political elections — punishing incomplete ballots may be preferable to rewarding them, but I think we should avoid punishing voters for not ranking candidates they don’t feel comfortable ranking as much as possible, especially since Borda Count works well for elections with large numbers of candidates and might promote large fields of candidates. Punishing voters for not listing candidates they know nothing about, for instance, does not strike me as a good idea. But for other kinds of elections where voters are expected to be able to rank all candidates, incentivizing voters to rank all candidates could be a good idea.

This is only one kind of strategic voting, but, importantly, it is a kind of strategic voting that could be particularly effective in Borda Count compared to other voting methods like Ranked Choice. That it can be so easily neutralized should be considered to be a good sign for how Borda Count could work in real elections when compared to other voting methods, though Borda Count is still susceptible to other strategies.