Instant Runoff and Self-Apportionment (IRSA)

Dave Robinson, Jan 2008

Contents:

Abstract

Overview

Definitions

Procedure

Rationale

Advantages

Disadvantages in perspective

Other design details

Appendix:

Example 1 (12 ballots)

Example 2 (2001 Cambridge school committee)

Abstract:

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, a narrow majority of elected legislators clearly represents a majority of voters, which is rarely true with the common 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.

Overview:

In this method, voters rank candidates in order of preference. Votes are counted in a series of rounds. A vote counts toward one?s highest possible choice, although some candidates may be eliminated, and elected candidates with too many votes may allow voters to apply a portion of their vote to lower choices. 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.

Definitions:

Voton: Each ballot corresponds to one vote, but each vote can be subdivided into multiples of a fundamental unit, called a voton. The number of votons in a vote is equal to the number of seats multiplied by one more than the number of seats. Votons are the largest units with an important mathematical property: a winner with too many votes can let each voter transfer a whole number of votons, and always end up between Enough and the Ideal Number of votes (which are also whole numbers of votons). If there is only one winner, no such transfers occur, so votes are not subdivided (one voton per ballot).

Enough votes to win: a number of votons equal to one more than the product of the total number of ballots (excluding those counting for "none of these") and the number of seats being elected. If all winners have Enough votes to win, it is impossible for too many candidates to win.

Ideal Number of votes: a number of votons equal to the total number of ballots, (excluding those counting for "none of these") multiplied by one more than the number of seats being elected. If all winners must have more than the Ideal Number of votes, it is impossible for enough candidates to win. If each winner has exactly the Ideal Number, the voting power of members of the legislature is exactly proportional to their number of supportive constituents.

Round: an iteration of steps 1 to 3 of the procedure.

I hope that other terms will be intuitive. The examples below should help clarify how it works.

Procedure:

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

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

A. Each candidate has a prescribed maximum number of votons per ballot (initially equal to the number in a whole ballot).

B. An elected candidate may only accept a ballot's votons if elected in a later round than any higher-ranked winners.

C. Maxima are waived for votons that cannot otherwise count for a candidate. (They count toward the highest-ranked elected candidate.)

D. A ballot that cannot be counted for a candidate is counted for "none of these".

For each candidate, a record (histogram) is kept of how many ballots contribute a given number of votons, sorted in decreasing order of the number of votons per ballot (each pair of numbers is a "bin"). Ballots that exceed the candidate's maximum and bins containing zero ballots are not part of the histogram.

2. Adjustment of maxima

A. If any candidate was eliminated in the previous round, the thresholds (Enough and the Ideal Number) are recomputed.

B. If any candidates have Enough votes to win, they are declared elected. All are declared elected if there are exactly enough remaining to fill all seats.

C. Maximum reduction - open seats

This step is invoked if:

i. There are fewer elected candidates than seats, and either

ii. In this round, any candidate could perform this maximum reduction for the first time, or

iii. In the previous round, any candidate was eliminated.

The maximum number of votons that can count for each elected candidate is reduced as far as possible while ensuring that the candidate has Enough votes.

Procedure: The maximum is adjusted to the number of votons per ballot in successive histogram bins, and the number of votons and ballots above that maximum are each summed until removing those votons would put the candidate at or below Enough votes. If below, the maximum is increased by the difference between Enough and the number of votons that would remain given this adjusted maximum, divided by the number of ballots summed, rounded up. The maximum is set to zero if transfer of all votons in the histogram would not put the candidate below Enough.

D. Maximum reduction - filled seats

This step is invoked if:

i. There are no unelected candidates not yet eliminated, and

ii. No candidate with more than the Ideal Number of votes has a maximum of zero.

The maximum number of votons that can count for each elected candidate is reduced as far as possible while ensuring that the candidate has at least the Ideal Number of votes.? The procedure follows that of 2.C, substituting "the Ideal Number" for "Enough".

E. Elimination

This step is invoked if:

i. There are unelected candidates with nonzero maxima, and

ii. No other maximum reduction occurred in this round.

The number of votes for all candidates is sorted from least to most.? Candidates' votes are summed in increasing order, omitting from the sum a number of top candidates equal to the number of seats. The highest point at which the sum is less than the votes for the next candidate is identified. Candidates below that point are eliminated, and the maxima for those candidates are set to zero.

If the candidates with the fewest nonzero votes are tied and the tie must be resolved to choose a candidate to eliminate, a candidate is chosen 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 Enough and Ideal thresholds are corrected by Seats or Seats+1 votons for each counted ballot.? The new totals are checked to see if there are changes to who is newly elected in the round, and to see if the gap in the elimination sum has changed, if applicable.? For each elected candidate, the remainders in maximum calculations are corrected for the threshold change, and by the number of ballots added or removed from the affected bins, in addition to the votons contributed by those ballots.? If there are no changes to who is elected or eliminated and no changes in maxima in any round, a full recount is not necessary.

If the election is counted without the assistance of a computer, it would be merciful to cause 2.D to terminate after a round with no new candidate able to reduce.?

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, drawing from the Meek and Gregory versions.

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 discrete votons. 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.

2. It is simple and deterministic: who one's vote counts for can be determined at any point knowing only one's rankings, the maximum number of votons per ballot countable toward a candidate, and the order in which candidates have been elected. 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 vote 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 only occur when everyone in a histogram bin can contribute one voton.

4. It scales well down to a small number of votes, making the method usable by small organizations or communities. A single vote with a sufficient number of rankings can produce a meaningful outcome. Many methods sacrifice this feature by using thresholds rounded to the nearest ballot, a facade for the complexity of floating-point ballot fractions.? When there are few votes, this causes the Enough threshold to exceed the Ideal threshold, causing strange results.? Discrete votons allow whole-number thresholds but avoid this problem.

5. It approaches the ideal outcome: each winner has about (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 converging on the wrong threshold, shoveling huge numbers of perfectly good votes into the garbage can 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. It shows moderation and equity in the slicing up of ballots, 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 vote. In the process, it does a good job of honoring higher rankings.

8. It is Pareto optimal: ballots ranked AB and BA do not each end up giving votons to both A and B, making them wish they could trade votons.? Traditional STV methods have not paid attention to this, but it is a valuable feature in an apportionment method.

9. 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 votons, 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 and the Ideal Number. 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.

The use of discrete votons is a much more benign simplification than limiting the number of rankings that voters can list, which causes many ballots to be involuntarily exhausted. Regardless, the latter seems to be a tradeoff between functionality and simplicity of implementation (SF bay area, London) that has been crucial to making ranked ballot methods politically viable.? This may sound like a straw-man argument, but it is important to be grounded by practicality. If simplicity is of paramount concern, discrete votons are a good step toward that.

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 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.

3. Rule 1B (transfer only to later winners) makes counting each ballot more difficult, and in other versions is criticized for allowing a type of strategic voting in which one votes for an obvious first-round winner and then surfs along the surpluses of several other candidates, helping several of them win. This does not occur here, because surplus votons usually have low priority for subsequent transfers, and they end up in the same place as they would if the strategy were not used. Voting for the obvious winner does not guarantee that one can transfer more votons, because maxima are also affected by the number of recycled ballots. In other methods, voting for an obvious loser can also allow surplus surfing, but with this method, these ballots are indistinguishable from those that don't attempt the strategy after the loser is eliminated.

Usually, without a rule like 1B, if all of winner A's second choices are winner B, and all of B's second choices are A, everyone will end up with their second choices, and each ballot must be recounted many times. The Meek method mitigates this by bleeding ballots into the exhausted pile or to lower choices. If that approach is not acceptable, it is hard to avoid 1B.

If we are using our counting method to achieve apportionment, we care about Pareto optimality. This means that we do not want to have situations where ballots ranked AB and BA each end up giving votons to both A and B. We do not want to leave voters wishing they could trade votons so that they have more votons counting toward their first choice. Meek, which essentially always gives weight to lower-ranked winners, is normally suboptimal. By allowing transfers from winners only to those elected later, suboptimality can be avoided.

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. This method does not smoothly accommodate ballots with equal rankings. Some other approaches do by splitting votes equally among equal-ranked candidates, constrained by surpluses and eliminations. When using discrete votons, such splitting can leave a remainder that counts for no one, but must be kept track of, because the leftover votons may be countable later. Also, if a ballot counts recycled votons toward a candidate (due to 1C), the ballot must not change when the candidate?s maximum changes, and this doesn?t work if those votons are split among multiple candidates. There are ways to count equal rankings, such as dividing them equally without regard to maxima, but they do not fit seamlessly with these rules. Liberal use of slicing to process equal rankings is contrary to this method?s goal of creating a clearly defined relationship between a voter and a small number of representatives.

Other design details:

Number of actions per round ? If transfers from winners and losers 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 the Ideal Number 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.

Where to recycle ballots - when votons cannot count toward lower choices, it is perhaps more intuitive to leave votons with the low-ranked winning candidate for whom they most recently counted. However, giving votons to their highest-ranked winner is respectful of the voter's preferences. Furthermore, a major function of recycled votons is to help candidates achieve the ideal number of votes. It is difficult for candidates elected near the end of the count to get rid of ballots above the ideal number. If ballots can recycle to highest-ranked winners, the votons have a place to go.

If the term "voton" invokes daunting thoughts of nuclear physics, a plainer term such as "pieces" could be used.

The major features that distinguish IRSA from earlier STV versions enable each other, or provide added benefits by their combination. Votons enable exact thresholds and histogram-based surplus transfers that minimize repeated slicing and simplify incremental ballot additions.? Recycling of exhausted pieces enables convergence on the Ideal Number, and is also necessary for exact thresholds and Pareto optimality.? Allowing transfers only to later winners enables Pareto optimality as well as independence of a ballot on its specific history, while also enabling convergence on the Ideal Number.? Half-baked versions of IRSA are conceivable, but they would have less than half the benefits.

Most election reformers are ironically conservative about the methods they support. This is actually 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.

Appendix: Two example counts (many more are available at demochoice.org)

Example 1

This artificial example demonstrates most of the major features of the method. It should be possible to follow along by counting the ballots yourself as the maxima change.

Candidates: 6 (A through F)

Seats: 3

Votons per ballot: 12

Ballots: 12

Votons: 144

Ballots:

A B

A C

A D

A E

C B

C D

D B

F C A

Initial maximum for each candidate is 12.

Enough votes to win = 144/4 + 1 = 37

Ideal number of votes = 144/3 = 48

Round 1 count:

A B C D E F

60 24 24 12 12 12

Ballots count as (listing number of votons immediately after candidate; no number = no votons):

A12 B

A12 C

A12 D

A12 E

B12

C12 B

C12 D

D12 B

E12

F12 C A

A has Enough votes and is elected.

A's histogram is 12: 5 (5 ballots that count 12 votons for A)

A's maximum can be adjusted to 8 by inspection (8*5=40 votons; 7*5=35 which is below Enough, so 8 is as low as possible). The maximum reduction procedure defined in the rules works for more complicated histograms, and is shown below.

Round 2 count:

Ballots in A's histogram are recounted, using the reduced maximum for A.

A B C D E F

40 32 28 16 16 12

Ballots count as:

A8 B4

A8 C4

A8 D4

A8 E4

B12

C12 B

C12 D

D12 B

E12

F12 C A

A's histogram is now 8: 5.

A's maximum cannot be adjusted further.

F is eliminated.

Round 3 count:

A B C D E F

40 32 40 16 16 --

Ballots count as:

A8 B4

A8 C4

A8 D4

A8 E4

B12

C12 B

C12 D

D12 B

E12

F C12 A

C has Enough votes and is elected.

C's histogram is 12: 3, 4: 1.

C's maximum can be adjusted to 11, leaving C with Enough votes.

To go through the detailed procedure to determine the maximum:

The maximum is set to the number of votons in the first bin (12). This would not reduce the number of votons counting for C.

The maximum is set to the number of votons in the second bin (4). This would leave C with 16 votons. This is 21 votons fewer than Enough votes. There are 3 ballots in higher bins, and we need each to make up the difference, so the maximum must be 4+21/3 = 11.

Round 4 count:

A B C D E F

40 33 38 17 16 --

Ballots count as:

A8 B4

A8 C4

A8 D4

A8 E4

B12

C11 B1

C11 D1

D12 B

E12

F C12 A

A voton from ballot F C A was returned to C. It cannot be counted for A due to rule 1B.

No further maximum adjustments are possible. E is eliminated.

Suppose that E had 17 votes. In this case, E would be tied with D. Their histograms are hypothetically:

D 12: 1, 4: 1, 1: 1

E 12: 1, 5: 1

The tie-breaking procedure would resolve this by dropping the smallest bin, leaving D with fewer votes than E, and D would be eliminated instead.

Round 5 count:

A B C D E F None

44 33 38 17 -- -- 12

Ballots count as:

A8 B4

A8 C4

A8 D4

A12 E

B12

C11 B1

C11 D1

D12 B

F C12 A

One ballot went to None of these, so the thresholds must be recomputed:

Enough = 132/4 + 1 = 34

Ideal = 132/3 = 44

A's histogram is 8: 4 (a ballot that exceeded the maximum was removed).

A's maximum is reduced to 6.

C's histogram is 11: 2, 4: 1 (a ballot that exceeded the maximum was removed).

C's maximum is reduced to 9.

Round 6 count:

A B C D E F None

36 39 36 21 -- -- 12

Ballots count as:

A6 B6

A6 C6

A6 D6

A12 E

B12

C9 B3

C9 D3

D12 B

F C12 A

B has Enough votes and is elected.

All seats are filled, so D is eliminated.

Round 7 count:

A B C D E F None

42 51 39 -- -- -- 12

Ballots count as:

A6 B6

A6 C6

A12 D

A12 E

B12

C9 B3

C12 D

D B12

F C12 A

B has more than the Ideal.

B's histogram is 12: 3, 6: 2, 3: 1. To shed 6 votons, the maximum is reduced to 10.

Round 8 count:

A B C D E F None

42 51 39 -- -- -- 12

Ballots count as:

A6 B6

A6 C6

A12 D

A12 E

B12

C9 B3

C12 D

D B12

F C12 A

This didn't work; B can only shed votons by recycling to higher winners.

A second try is allowed.

B's histogram is 6: 2, 3: 1. The maximum is reduced to 3.

Round 9 count:

A B C D E F None

48 45 39 -- -- -- 12

Ballots count as:

A9 B3

A6 C6

A12 D

A12 E

B12

C9 B3

C12 D

D B12

F C12 A

A now has more than the Ideal.

Histogram: 6: 1; new maximum is 2.

B is 3: 3; no change is possible.

Round 10 count:

A B C D E F None

45 45 42 -- -- -- 12

Ballots count as:

A9 B3

A3 C9

A12 D

A12 E

B12

C9 B3

C12 D

D B12

F C12 A

No further changes are allowed. Final maxima are 2, 3, 9, 0, 0, 0.

Example 2

A version of the Cambridge, MA 2001 school committee election was counted using these rules on a spreadsheet. For convenience, I used a sample of 20% of the ballots. There are 11 candidates for 6 seats. Tallies are in terms of votons. Note that recounts between rounds require consultation of only those ballots from histogram bins that are above newly changed maxima.

Candidates: 11

Seats: 6

Votons per ballot: 42

Ballots: 3298

Votons: 138516

Round 1

w/i Baker Delan Erlien Fantini Grassi Hardin Price Segat Turkel Walser

378 2352 1806 10458 23268 18396 13986 16296 12684 24024 14868

Enough: 19789

Ideal: 23086 (these numbers change only after an elimination)

Fantini and Turkel are newly elected.

Surpluses, histograms, and new maxima of elected candidates:

Fantini (3479) 42: 554. Max = 36

Turkel (4235) 42: 572. Max = 35

Surpluses are in parentheses.

In histograms, the number before a colon is a number of votons on a ballot counting toward candidate. Following the colon is the number of ballots counting that many votons.

Maxima are 42 at the beginning of the count, and only changed maxima are noted; these apply to the next round.

Round 2

w/i Baker Delan Erlien Fantini Grassi Hardin Price Segat Turkel Walser

384 2539 1962 10814 20556 20418 14315 16704 14113 20594 16117

Grassi is newly elected.

Fantini (767) 36: 452. Max = 35

Grassi (629) 42: 438, 6: 331. Max = 41

Turkel (805) 36: 490. Max = 34

Round 3

w/i Baker Delan Erlien Fantini Grassi Hardin Price Segat Turkel Walser

386 2595 2018 10879 20104 20499 14429 16782 14387 20104 16333

write-in, Baker, and Delaney are eliminated (their maxima become zero).

Round 4

Erlien Fantini Grassi Hardin Price Segat Turkel Walser None

11227 21511 21229 14741 17199 14543 20228 16704 1134

Enough: 19627

Ideal: 22897

Fantini (1884) 35: 467. Max = 31

Grassi (1602) 41: 224, 8: 40, 7: 325. Max = 34

Turkel (601) 34: 485. Max = 33

Round 5

Erlien Fantini Grassi Hardin Price Segat Turkel Walser None

11457 19643 21001 15412 17494 15403 19743 17229 1134

Erlien is eliminated (max = 0).

Round 6

Fantini Grassi Hardin Price Segat Turkel Walser None

19970 21365 17857 19504 17458 22038 18560 1764

Enough: 19537

Ideal: 22792

Fantini (433) 31: 464.

Grassi (1828) 34: 218, 11: 330, 9: 41. Max = 26

Turkel (2501) 33: 538. Max = 29

Round 7

Fantini Grassi Hardin Price Segat Turkel Walser None

19970 19785 18605 20000 18814 19886 19692 1764

Price and Walser are newly elected.

Fantini (433) 31: 464.

Grassi (248) 26: 218, 13: 41, 11: 330. Max = 25

Price (463) 42: 440, 16: 29, 13: 66, 11: 18. Max = 41

Turkel (349) 29: 538. Max = 29

Walser (155) 42: 386, 16: 43, 13: 197, 11: 21.

Round 8

Fantini Grassi Hardin Price Segat Turkel Walser None

19970 19567 18829 19706 19059 19886 19735 1764

Harding is eliminated (max = 0).

Round 9

Fantini Grassi Price Segat Turkel Walser None

21339 22674 24093 21757 21677 21264 5712

Enough: 18973

Ideal: 22134

Segat is newly elected; all seats are filled.

Grassi and Price approach Ideal for the first time.

Grassi (540) 25: 200, 13: 51, 11: 359. Max = 23

Price (1959) 41: 284, 17: 52, 13: 81, 11: 24. Max = 35

Round 10

Fantini Grassi Price Segat Turkel Walser None

21339 22274 22493 23651 21677 21370 5712

Segat approaches Ideal for the first time.

Grassi (140) 23: 200, 13: 51, 11: 359.

Price (359) 35: 284, 19: 52, 13: 81, 11: 24. Max = 34

Segat (1517) 42: 388, 19: 95, 13: 219, 11: 65, 7: 284. Max = 39

Round 11

Fantini Grassi Price Segat Turkel Walser None

21339 22274 22209 23935 21677 21370 5712

Grassi (140) 23: 200, 13: 51, 11: 359.

Price (75) 34: 284, 19: 52, 13: 81, 11: 24.

Segat (1801) 19: 95, 13: 219, 11: 65, 8: 284. Max = 10

Round 12

Fantini Grassi Price Segat Turkel Walser None

21404 23129 22209 22358 22334 21370 5712

Turkel is above Ideal for the first time, but can't reduce maximum.

Grassi (995) 23: 105, 13: 51, 11: 359. Max = 14

Price (75) 34: 284, 19: 52, 13: 81, 11: 24.

Segat (1801) 10: 379, 8: 284.

Turkel (200) 29: 346.

The standard deviation from the Ideal Number of votes is 664 votons (15.8 ballots). This is 3% of the Ideal Number, 0.5% of the total, and about 28% of the difference between Enough and the Ideal Number.

Slicing: At the end, 1636 ballots counted toward a single candidate, 1526 counted toward two, and 136 ballots did not rank a winner.

Depth is the percentage of votons counting toward a certain position among the rankings.

Cumulative depth is the percentage counting toward that ranking or higher.

Depth Cumulative

1st 71.6 71.6

2nd 19.6 91.3

3rd 6.0 97.3

4th 1.7 98.9

5th 0.7 99.7

6th 0.2 99.9

7th 0.1 100

8th 0.0 100