The post investigates three questions: What is precinct summability? Is it important? And how much should one worry about it being a property of a proposed voting system?
Precinct Summability is Important
The best place to start this discussion is actually with the story of the contentious 2024 Presidential Election in Venezuela. The administration of incumbent president Nicolas Maduro controlled most institutions in the country and repressed the opposition throughout the election. Opposition candidate Edmundo Gonzales is widely believed to have won the election. Despite evidence to the contrary, the Maduro administration announced that Nicolas Maduro had won the election, presenting only percentages and no actual vote tallies. After the opposition candidate fled to asylum in Spain, the Maduro administration got the Supreme Tribunal of Justice to certify the result that the Maduro administration presented, and Maduro was sworn in for a third term in January 2025.
The key question for us is: how did the opposition gather evidence to show that they had won the election, despite the Maduro administration controlling all institutions and media in the country? The radio program This American Life aired a stunning episode in November 2024 called The Official Unofficial Record, which reports on the extraordinary operation organized by the opposition to collect the official voter tallies from voting centers throughout the country. This allowed the opposition to sum the tally sheets from voting centers throughout the country and show convincing evidence that the opposition had won in a landslide, and that the percentages announced by the Maduro administration were almost certainly falsified.
What Exactly is Precinct Summability?
Part of the reason it was quickly determined that the opposition won the Venezuelan election based on the precinct tally sheets is that the election system, plurality voting, satisfies the summability criterion. Roughly speaking, the summability criterion is satisfied if it is both (i) easy to tally results by local precinct, and (ii) the winner of the election can be determined by summing up the tallies for each precinct. More generally in election science, the compilation complexity of a voting system measures the difficulty of vote counting for individual precincts.
Low compilation complexity is desirable in a voting system as it allows for easier verification and audit of results. This was the case in the Venezuelan election. First, the opposition had observers at each voting center throughout the country to collect the sheet of paper that tallied the votes at the voting center. And secondly, the winner of the election could be easily determined by aggregating those tally sheets. This made it possible for the opposition to show such convincing evidence that they had in fact won the election.
Election reform advocates point out the many flaws of a plurality voting system, also known as first-past-the-post. But one advantage of plurality voting is that it has a very low compilation complexity. This is worth paying attention to. While some alternative voting methods may have desirable theoretical properties, it is important to ask if they have a high compilation complexity. If so, they may not be good alternatives in practice.
IRV is not precinct summable
Instant runoff voting, what is often referred to as simply “ranked choice voting,” is not precinct summable. Put another way, it can have a very high compilation complexity.
The reason IRV has a high compilation complexity is because counting ballots in this methods requires knowing the full ranking of each type of ballot. If there are n candidates in an election, then there are n! possible orderings of those candidates (recall from math class that n! refers to the factorial mathematical operator where you multiply n! = n*(n-1)*(n-2)*…*1). Each precinct needs to report the number of votes for each type of ballot. In a race with three candidates, there are 3!=6 possible orderings of the candidates. But this quickly increases with the number of candidates.
3! = 6. 4! = 24. 5! = 120. 6! = 720. 7! = 5,040. 8! = 40,320. 9! = 362,880. 10! = 3,628,800.
An election with seven candidates already has over 5,000 possible vote types. The number of ballots with each vote type needs to be reported by each voting center. With plurality voting, each voting center needs to report only a list of how many votes go to each candidate, so that is a list of n numbers. With IRV, if the number of candidates is large, that list could be unreasonably large. (Actually, the situation is slightly worse than n!, because some ballot will not rank every option. With three candidates and un-ranked options, there are nine possible ballot types. With four candidates, there are 38 ballot types. And with five candidates there are 205 ballot types.)
In an election with only two or three candidates, the compilation complexity for IRV is similar to plurality voting. The compilation complexity really increases as the number of candidates becomes large. One way around this, then, is to use IRV within a voting system that will limit the number of candidates in the instant runoff stage. For example, Katherine Gehl and Michael Porter advocate for a Final Four system in their 2017 report, where there is first stage with an open primary and a multi-winner plurality, followed by IRV in a second stage among only the top four winners. This approach will utilize IRV in a way that ensures lower compilation complexity.
Tweak IRV to be precinct summable: BTR-IRV
Alternatively, instant runoff voting can be tweaked so that it will always have a low compilation complexity. One fantastic method is the Bottom Two Runoff IRV (BTR-IRV), which is very similar to IRV, but with an important difference that make it have more desirable properties. One of these improvements is that it will have a fairly low compilation complexity.
In some ranked systems, a ballot can be represented as a matrix with n rows and n columns, where n is the number of candidates in the election. Because each ballot represents a voter’s pairwise preferences among all the options, the n-by-n matrix can reflect these preferences. If a ballot shows the candidate in row 1 is preferred to a candidate in column 2, then put a “1”, or otherwise put a “0”. Doing this for all preferences on the ballot will yield a matrix of 1’s and 0’s that represents the preferences of the ballot. At each voting center, sum all of those preference matrices to get the preference matrix for the voting center. The final winner of the election can be determined by summing the preference matrices across all the voting centers.
BTR-IRV can be implemented with two pieces of information. The n-by-n aggregate preference matrix, and a list of each candidates’ first-choice votes, which is a list of n numbers. The first-choice votes are needed to determine which are the bottom two candidates in each round that go head-to-head. Then the preference matrix is used to determine the pairwise winner of a head-to-head competition in each round among those two bottom candidates.
This compilation complexity of BTR-IRV is certainly higher than simple plurality voting. Plurality needs only a list of n numbers. BTR-IRV needs a list of n numbers, as well as an n-by-n aggregate preference matrix. But what is nice about BTR-IRV (and some other Condorcet methods) is that it can summarize the relevant preferences in a matrix. With traditional IRV, you need a list of n! numbers. For any number of candidates greater than about n=5, this is simply too much information for an election system to be easy to understand and audit.
Conclusion
Yes, “precinct summability” or “compilation complexity” is an important consideration when comparing voting methods. Low compilation complexity will make an election easier to understand and audit. It can also make calling an election result quicker, as it will be easier to predict the final result based on currently reported information. In practice, it is mostly an issue for instant runoff voting when the number of candidates is large, and so a “Final Four” like what is used in Alaska will limit this issue. Even better, a tweaked ranked voting system like BTR-IRV will have a fairly low compilation complexity even with a larger number of candidates.
Further reading:
Yann Chevaleyre, Jérôme Lang, Nicolas Maudet, and Guillaume Ravilly-Abadie. 2009. Compiling the votes of a subelectorate. In Proceedings of the 21st International Joint Conference on Artificial Intelligence (IJCAI’09). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 97–102.
Xia, L., & Conitzer, V. (2010). Compilation Complexity of Common Voting Rules. Proceedings of the AAAI Conference on Artificial Intelligence, 24(1), 915-920. https://doi.org/10.1609/aaai.v24i1.7627
Electiowiki.org page on summability criterion. https://electowiki.org/wiki/Summability_criterion