Instant Runoff and Self-Apportionment (IRSA)

Dave Robinson, Apr 2010








Disadvantages in perspective

Other design details

Closing remarks




Common methods for election of a governing body (council, board or legislature) routinely leave large fractions of constituents without satisfactory representation, and large imbalances in mandate among winners. They usually require complex reapportionment (redistricting) schemes that are easily manipulated by pitting arbitrary subsets of voters against each other. An objective method is proposed here that allows voters to naturally reapportion themselves, leaving nearly everyone with equal and satisfactory representation. As a result, even a narrow majority of elected legislators clearly represents a majority of voters, which is rarely true with commonly used election methods. While more complex than a plurality vote, it is less so than current reapportionment procedures. It extends concepts of the well-known Instant Runoff and Single Transferable Vote methods, but it more clearly defines constituencies, is easier to count and audit, and takes full advantage of the mathematical insights behind them.


In this method, voters rank candidates in order of preference. Each voter has several votes, but the procedure uses the rankings to keep them with as few candidates as possible. Votes are counted in a series of rounds. Votes count toward one's highest possible choice, although some candidates may be eliminated, and elected candidates with too many votes may allow voters to apply some of their votes to lower choices, according to prescribed rules. Adjustments are made in every round until the number of candidates equals the number of seats and each has a nearly equal number of votes.


0. Voting

Voters rank candidates in order of preference, with the understanding that if circumstances prevent their vote (or portion thereof) from counting for a higher choice, it will count for a lower choice.


1. Counting votes


Each ballot has a number of votes equal to the number of seats times one more than that. This number is the smallest necessary to arrange so that enough, but not too many, candidates have enough votes to win.


Votes from each ballot are assigned to its highest-ranked candidate, constrained as follows:

A. Each candidate has a prescribed maximum number of votes per ballot, initially equal to the total number per ballot.

B. If there are leftover votes, they are reassigned according to (A) as if they were a separate ballot.

C. Ballots that cannot be assigned are counted for "none of these".


2. Adjustment of maxima


A. Any candidate with more votes than the number of seats times the number of countable ballots is declared elected. All are declared elected if there are exactly enough remaining to fill all seats. Skip step B if all seats are filled but there are more candidates than seats.


B. Maximum reduction

The maximum number of votes that can count for each elected candidate is reduced as far as possible while ensuring that the candidate has enough votes to win. The new maximum is easily determined from a record of how many ballots count how many votes for each winner. If all seats are filled, the target number of votes increases to an equal share: the number of seats plus one, times the number of countable ballots.


C. Elimination

If no maximum reductions are possible, the candidate with the fewest votes is eliminated. Ties encountered here are broken randomly.


3. Repeat steps 1 through 3 until no further changes can be made.


Mid-term vacancies due to resignations or disqualifications can be refilled by recounting the election, considering only the unelected candidates, and using only the ballots counting toward the vacated seats along with the "none of these" ballots.


If a few ballots are to be added or removed, this can often be done without a full recount. In each round, the ballots are added or subtracted from the tally. The number of votes needed to win is corrected by Seats votes for each counted ballot. The new totals are checked to see if there are changes to who is newly elected in the round, if the elimination order has changed, and if there are any changes to maxima. If there are no such changes, a full recount is not necessary.


Rationale behind this method:

This voting method uses lists of preferences made by voters from among a meaningful variety of viable candidates to elect a board, council, or legislative body in which:

- all members have a nearly equal mandate

- nearly all voters have satisfactory representation

- a clear and traceable definition of constituencies exists to connect voters to representatives

- a majority of the legislative body traceably represents the consent of a majority of voters.


This is achieved without the need for a separate redistricting process, or at least a less complicated one that divides a large legislature into geographic groups of roughly 5-10 seats. It is a variation of the "single transferable vote" methods, especially similar to the Warren version. (See the first issue of the online journal "Voting Matters" for details of that method). The modifications to Warren's method are the recounting of exhausted ballots transferred from winners, the use of minimal precision, and the extension of the count to approach equal-sized constituencies.


STV has been around for more than 100 years, so why do we need another variation? Because it is still not used as widely as many of us think it should. A main reason for this is that it is difficult to implement and understand, and the advantages are not always clear to people. Improvements to the method that build on past successes and failures can reduce the magnitude of these obstacles.


IRSA has the following advantages over older Single Transferable Vote methods:


1. It is reasonably easy to hand count and audit, using whole numbers of votes. There are no worries about floating point errors or rounding precision. The reliance of the Meek method on taking many infinitesimal slices off each ballot is absent. However, IRSA allows the principles of the Meek method to be satisfied without the assumed requirement of a complex computer program to perform the count.


2. As with the Meek method, it is simple and deterministic: who one's ballot counts for can be determined at any point knowing only one's rankings and the maximum number of votes per ballot countable toward a candidate. You do not need to know who the ballot counted for previously. Randomness is used only to break ties, and this is usually unnecessary. Independence on detailed ballot history greatly simplifies audits.


3. With most other versions, if a single ballot needs to be added or removed from the count, or if a mistake was made in an early round, it is necessary to recount the entire election in order to have an exact result. With IRSA, this is rarely the case, because surplus transfers occur in coarser steps.


4. It scales well down to a small number of ballots, making the method usable by small organizations or communities. A single ballot with a sufficient number of rankings can produce a meaningful outcome. Many methods sacrifice this feature by using election thresholds rounded to the nearest ballot, a facade for the complexity of floating-point ballot fractions. When there are few votes, this causes strange results where not enough candidates can meet the threshold. Discrete votes allow whole-number thresholds but avoid this problem.


5. It approaches the ideal outcome: the winners are close to (100/seats)% of the vote, except for a small pile of ballots that rank no winners. This is in contrast to the Meek method, which is very good at giving winners a mandate that looks equal, but which does so by discarding many countable votes in the process.


6. It captures or closely approximates desirable properties of traditional STV methods. Voters in a constituency with enough votes for two seats can all rank the same favorite and second favorite and elect both. Surplus transfers are fair and deterministic, and strategic voting is not worthwhile in practice.


7. As with the original Warren method, it shows moderation and equity in the division of ballots among candidates, helping to provide a clear connection between voters and their representatives. Nearly everyone ends up with one or two representatives, both receiving a large fraction of their votes. In the process, it does a good job of honoring higher rankings.


8. It provides a clear measure of unsatisfied voters. Only ballots that do not rank any winning candidates appear in the "None of these" pile.


Disadvantages in perspective:


1. With discrete votes, surplus transfers are not always complete, but they are very close to that in practical cases. With fully ranked ballots, the worst-case surplus transfer error is (% of vote for winner)/(seats*(seats+1)) - always leaving the winner between enough to win and a share of votes. Recycling of exhausted ballots also reveals a type of incomplete transfer that is present but masked in most methods that do not recycle. Here, most elected candidates have more than one chance to transfer a surplus, greatly improving completeness.


2. Fault can be found with any solution to the Solomonic problem of dividing a winner's ballots into some that transfer and some that do not. Most other approaches either involve either randomness or more ballot slicing. In others, most of the ballots can be left unsliced, at the expense of a small number that become sliced finely. The use of maxima, as proposed by Warren, gives transfer priority to previously unsliced ballots. This way, every ballot has an equal opportunity to be sliced, but is unlikely to be sliced more than once. Such equal treatment removes opportunities for strategic voting. Limiting repeated slices more clearly establishes one or two specific winning candidates as one's representative, and makes it easy to audit the election and determine who one's vote counted for.


Another dilemma is how to transfer ballots that do not list a lower choice. Resolving it requires a standardized interpretation of a voter's intent. If a voter only lists one ranking, a common-sense interpretation is that the voter wants the ballot to count unconditionally for that candidate. Some people strongly feel that these voters should have their surpluses sent to the "none of these" pile and their vote should count less than others; I. D. Hill et al. claim that this is fairer to remaining candidates. However, the original candidate is a remaining candidate, and it is by no means fair that he or she should be singled out for exclusion. Every voter should have the right to direct a surplus to any remaining candidate. Exhausting surpluses does not honor the voter's intent to count the ballot for less than its full value, and contradicts our goal of maximizing the number of votes counted toward winners. The approach also causes the enough threshold to decrease more than necessary, which can elect candidates who have less of a mandate, and induce surplus transfers from candidates the voter does not support.


4. In general, the rule set is not as simple as possible, but it is simpler than most other proposals. Some rules make actual counting easier, create a clearer connection between winners and constituents (fewer repeatedly sliced ballots), and allow a clearer measure of unsatisfied voters.


5. The use of discrete votes makes it difficult to accommodate voters who want to give the same rank to several candidates. This can be done by splitting votes evenly among candidates given the same rank, constrained by maxima. When using discrete votes, there can be a remainder that is not counted, but the leftover votes may be countable later. Perhaps this is a useful disincentive; spreading votes among many candidates is contrary to this method's goal of creating a clearly defined relationship between a voter and a small number of representatives.


6. If there are two winners A and B, there may be votes counting for B from ballots that list A followed by B, and votes counting for A that list B followed by A. Economics professors would argue that the voters should trade representatives. This could be arranged after the winners are determined, or even prevented during the count, but it would add complexity. No other STV methods address this problem, and plurality methods cannot even draw attention to the problem because they do not collect enough information from voters.


Other design details:


Number of actions per round - If transfers from winners are handled one at a time instead of allowing several simultaneous transfers, a method can slightly improve the efficacy of surplus transfers in electing their next choices, and of convergence on equal shares of votes. However, the benefit is tiny, and the paperwork is more complicated, making hand counts tedious and results more verbose. Ties also occur more frequently.


The total number of rounds may appear disturbingly larger than some other counting methods. Even Meek elections show fewer rounds because the many intermediate steps used to determine candidate weights (the equivalent of maxima) are not reported, and methods like ERS97 include substages. Analysis of several hundred test elections (a combination of artificial and real data) shows a consistent increase in number of reported rounds of 50% over Meek, with about 10% associated with convergence on the ideal number of votes. Giving voters 100 times more votes (increasing precision) increases the number of rounds by less than a factor of two. The number of reported rounds could be reduced by lumping together neighboring rounds that transfer from the same set of winners, and combining elimination of obvious losers whose votes could not make another candidate win or whose combined votes are still lower than any other candidate. The latter could also reduce the amount of counting at the expense of creating additional rules that are hard to explain.


Where to recycle ballots - when votes cannot count toward lower choices, it is perhaps more intuitive to leave votes with the low-ranked winning candidate for whom they most recently counted. However, giving votes to their highest-ranked winner is respectful of the voter's preferences. Furthermore, a major function of recycled votes is to help candidates achieve an equal share of votes. It is difficult for candidates elected near the end of the count to get rid of ballots above an equal share. If ballots can recycle to highest-ranked winners, the votes have a place to go. Treating recycled ballots as new ballots is a simple way to resolve this dilemma. The use of maxima and low precision are necessary to do this; otherwise such ballots could be sliced into a long series of tiny fragments.


More generally, the major features that distinguish IRSA from earlier STV versions enable each other, or provide added benefits by their combination. Multiple votes enable exact thresholds, and simplify surplus transfers that minimize repeated slicing and simplify incremental ballot additions. Recycling of exhausted pieces enables convergence on equal shares, and is also necessary for exact thresholds. Half-baked versions of IRSA are conceivable, but they would have less than half the benefits.


Closing remarks:


Most election reformers are ironically conservative about the methods they support, resisting new proposal such as this one and favoring those that have been around longer. This is sensible, given the plethora of ideas and approaches out there. The fact remains, though, that STV does not have the balance between performance and complexity to get across the "valley of death" that technologies must bridge to get from a few specialty uses to widespread adoption, and significant conceptual changes are needed to achieve this. I believe that the approach proposed here offers major improvements to both performance and simplicity of implementation that will allow obstacles to adoption to be overcome.