The Story Your Ballot Doesn’t Tell: The Mathematics of Single-Winner Elections
Anna HaenschUniversity of Wisconsin - Madison
Introduction: Who’s the Spoiler?
The great majority of national and state-level elections in the United States are contests between two candidates representing our two major political parties. Democrats and Republicans exist on the left and right end of the political spectrum, respectively, and while it is a well-studied fact that candidates tend to creep as close as they can towards the center to ensure the broadest base possible, they typically stay on their side of the spectrum. Because of this, the results of elections often come down to what’s happening in the margins.
Occasionally, however, it will happen that a third candidate gains traction in the race and this can have election-upending consequences. In these cases, the third candidate who takes their own bite out of the voter base is called the “spoiler." The weird thing though, is that whether or not this spoiler actually spoils, and for whom they spoil things is all a matter of how votes are cast and counted.
In August 2022, Alaska held a special election for its single U.S. House seat, and in the process, it became a case study in how a voting method can shape an outcome. The race featured three candidates: Democrat Mary Peltola, and Republicans Nick Begich and Sarah Palin. Alaska was using a voting system called “Instant Runoff Voting," in which each voter cast a single ballot with candidates ranked from favorite to least favorite, with possible elimination rounds. Begich, who was closer to the center than Palin, was considered the compromise candidate, and according to most predictions, should have won the election. But that’s not what happened. Even though Begich had the necessary broadness of support, he lacked that intensity of first-choice preference, and Peltola won.
The choice of voting method (a choice made by a vote of its own) often determines what kind of support is the most valuable. Common methods like plurality and instant runoff both prioritize first-choice dominance, but other methods, like Borda count give more value to broad if only mildly-enthusiastic support. Given this range of possibilities when it comes to setting up a voting regime, it’s reasonable to wonder: is the candidate really the spoiler, or is the spoiler democracy itself? I know, provocative. To understand this better, let’s strip away some of the specifics and get down to the geometry of the problem.
What Do Voters Look Like? Modeling Preferences with Gaussian Mixtures
Although most people’s beliefs are multidimensional, to keep things simple let’s assume that voters live in a one-dimensional space of political ideology, with Democrats on the left and Republicans on the right. Many polls exist that try to get at the precise distribution of voters along this left-right spectrum, but we’ll use the distribution from the Pew Research Foundation’s National Public Opinion Reference Survey. The 2025 results from that survey, shown below, place respondents along a 5-point spectrum from Democrat ($−2$), Lean Democrat ($−1$), No Lean / Refused ($0$), Lean Republican ($1$), and Republican ($2$). These results are plotted and fit to a 5-component Gaussian Mixture Model with standard deviation $\sigma = 0.5$, and probability density function, $f(x)$ (also shown below).
We can imagine political candidates as being situated along the left-right spectrum. For example, imagine a 3-candidate election where a left candidate is placed at $\ell = −1.8$, a center candidate is placed at $c = 0.4$, and right candidate is placed at $r = 1.6$. These candidates are shown as colored triangles in the figure below.
Assuming that voters prefer the candidate closest to them on the political spectrum and in descending order according to proximity (also situation known as a single-peaked preference) we can begin to carve up the electorate into regions of candidate support. Any voter to the left of the midpoint between $\ell$ and $c$ (shown as the left-hand dashed black line in Figure 2) will prefer the left candidate, and we’ll call this region $r_1$. On the other end of the spectrum, any voter to the right of the midpoint between $r$ and $c$ (shown as the right-hand dashed black line in Figure 2) will prefer the right candidate, and we’ll call this region $r_4$. In the remaining region between $r_1$ and $r_4$, voters will always prefer the center candidate, since they are the closest one. But for reasons that will become clear below we’ll split this into two regions, $r_2$ and $r_3$ divided at the midpoint between the left and right candidates (shown as a solid black line in Figure 2). Now on to the election!
Plurality: The Simplest System
Under plurality voting, each voter gets to vote for one candidate and the candidate with the most votes wins the election. Because of the careful work we did above, a candidate’s vote share can be easily computed as a sum of definite integrals. The left candidate vote share as a function of each candidate’s position is $$\begin{aligned} S_{\text{L}}^\text{Plurality}(\ell, c, r) & = & \int_{R_1}f(x) \,\,dx, \end{aligned}$$ where $f(x)$ is the probability density function of the voter distribution, possibly the Gaussian mixture model as described above. The vote shares for the center and right candidates can be computed similarly as $$\begin{aligned} S_C^\text{Plurality}(\ell, c, r) = \int_{R_2\cup R_3}f(x) \,\,dx \,\,\,\,\text{ and } \,\,\,\,S_R^\text{Plurality}(\ell, c, r) = \int_{R_4} f(x) \,\,dx. \end{aligned}$$ Using the values of $\ell, c,$ and $r$ from the previous section, we can see that the left candidate wins the election (see Figure 3. Despite the fact that the voter distribution skews right and the right candidate has even moved slightly towards center to appeal to more voters, the presence of the slightly right-of-center candidate has taken a big enough bite from the right candidate’s base to cost them the election. In this way, the center candidate is acting as the spoiler, since they lost the election and their removal from the election would have changed the outcome.
The Borda Count: Asking Voters to Say More
Another well-known voting method is the Borda count which requires voters to order candidates from most-preferred to least-preferred. For a race with $n$ candidates, votes are tabulated by assigning $n − 1$ points to a first place ranking, $n − 2$ to a second place ranking and so on, until the last place candidate who receives 0 points. Any candidates not ranked will receive 0 points, the so-called pessimistic Borda count (although there are other options for what to do with incomplete ballots). Assuming again that a voter will order their candidates according to their proximity on the political spectrum, we can compute the left candidate’s vote share as $$\begin{aligned} S_L^\text{Borda}(\ell, c, r) & = & 2 \cdot \int_{R_1}f(x) \,\,dx + 1 \cdot \int_{R_2}f(x) \,\,dx. \end{aligned}$$ The left candidate gets first choice ranks from voters in $r_1$, and second choice ranks in $r_2$, but as soon as we pass over the midpoint line between $\ell$ and $r$, the left candidate becomes the third choice. For the remaining candidates, the vote shares are computed as $$\begin{aligned} S_R^\text{Borda}(\ell, c, r) & = & 2 \cdot \int_{R_4}f(x) \,\,dx + 1 \cdot \int_{R_3}f(x) \,\,dx, \end{aligned}$$ and $$\begin{aligned} S_C^\text{Borda}(\ell, c, r) & = & 2 \cdot \int_{R_2\cup R_3}f(x) \,\,dx + 1 \cdot \int_{R_1\cup R_4}f(x) \,\,dx. \end{aligned}$$ Once again using the values of $\ell, c,$ and $r$ from above, we now see that the center candidate wins (see Figure 4). Even though the center candidate is sitting in the valley of the distribution and has fewer first-choice votes than the other candidates, the fact that they are nobody’s least favorite candidate gives them the boost they need to win handily. Who’s the spoiler now?
Instant Runoff Voting: Elimination as a Strategy
In instant runoff voting, each voter casts a ballot ranking candidates from first choice to last choice (as in the Borda count), but now votes are tabulated in multiple rounds. In the first round, each candidate’s share of first-choice votes are counted; if any candidate crosses the $50\%$ threshold, they are declared the winner and the election is over. In the more likely scenario that no single candidate crosses the threshold, the candidate with the lowest share of first-choice votes is eliminated, and their ballots are reassigned to their second-choice candidates. At this point, first-choice vote shares are counted, and in the case of our three-candidate election, the candidate with the highest number of votes is elected. For each individual round, votes shares are computed exactly as in plurality voting, with just the limits of integration depending on which candidates are still in the race.
Under instant runoff voting, the center candidate is eliminated after the first round (all of those second choice votes do nothing to help them here). In the second round of voting, the left candidate gets all of the votes from regions $r_1$ and $r_2$ and the right candidate gets all of the votes from regions $r_3$ and $r_4$. Now the fact that the right candidate is slightly closer towards center means that they are able to capture enough of the votes from the eliminated candidate to bring them to victory (see the figure below).
A Quick Detour Into Candidate Strategy
This is an interesting story in the context of Hotelling’s Law from economics which says that in any market it is rational for producers to make their products as similar as possible. In the world of politics, this law specializes to the Median Voter Theorem which says that for a single-peaked electorate, candidates will converge to the median voter. At first it looks like we’ve spotted a violation of the Median Voter Theorem since the left and right candidates are able to win in some cases, but of course in a real-life scenario neither the candidates nor the electorate are completely stable. You can imagine (and we’ve modeled!) what happens when candidates move in opportunistic ways. There’s also the question of whether voters actually turn out to vote, which can be impacted by all kinds of things including strategic voting. Put another way, the Median Voter Theorem tells us where the equilibrium should be, but the dynamics of getting there (or not) becomes more interesting with the addition of movement and a little bit of reality.
Conclusion: Mathematics as Democratic Infrastructure
Returning now to the Alaska special election. Begich was the center-most candidate and seemed like the good compromise and likely winner. While he didn’t have the strength of first-choice votes, he also didn’t have the anchor of last-choice votes. Functionally, he was a lot like the center candidate in the previous sections. Under a voting method like Borda count, Begich almost certainly would have won, but under plurality or instant runoff, he would (and did) lose. The specifics of the left and right candidates in the worked-out example above are slightly different than what happened in Alaska (notably, Peltola would have won both plurality and instant runoff), but they demonstrate the brittleness of the electoral process.
Mathematics is an important piece of the democracy, and this is not a purely academic exercise. All over the world governments have been thinking about what it would look like to adopt different voting methods. In the immediate aftermath of the Alaska special election, states were all over the place in terms of ranked choice voting. Some states like Florida and Tennessee banned ranked choice voting, while others had active campaigns to adopt it. In Alaska, polls showed that $54\%$ of Alaskans supported repealing ranked choice voting, although when put to a vote the repeal ultimately failed. As of March 2026, nineteen states have banned ranked-choice voting.
While mathematics can’t tell us which system is right, or which candidate is right, it can tell us what a voting system prioritizes. Every ballot quietly asks a question, and depending on the voting method we use we choose to listen to or discard little bits of the answer. That is why the voting system, which does so much to shape our democracy and governance, deserves at least as much scrutiny as the candidates themselves.
