- Q: What's wrong with our current system?
- Q: Can we use STAR Voting for Presidential elections?
- Q: Is STAR Voting constitutional? Does it pass One-Person-One-Vote?
- Q: Isn't scoring subjective? What if some voters are "easier graders" than others?
- Q: Is this the same as Ranked Choice Voting?
- Q: Is STAR Voting vulnerable to strategic voting?
- Q: Has STAR Voting been used for elections before?
- Q: Wouldn't I want to "bury" a strong second choice and give a higher score to a weaker opponent to help my favorite win?
- Q: Would STAR Voting cost money or save money?
- Q: How are ties in STAR Voting broken?
- Q: Is STAR Voting committed to open sourced implementation?
- Is STAR Voting auditable?
"Choose One Only" Voting (aka. "Plurality" or "First Past the Post") is universally regarded as the single worst voting system. It works fine if and only if there are two candidates in the race, which is why it leads to a two party system.
If there more than two candidates in the race, Choose One Only Voting is extremely vulnerable to a phenomena called The Spoiler Effect. It consistently results in two party domination, and in order to avoid your vote being wasted there are very strong incentives to vote "Lesser Evil" if you aren't sure your favorite can win.
1. The Spoiler Effect:
Choose One Only Voting is highly vulnerable to a phenomenon called the "Spoiler Effect,” also referred to as “Vote Splitting,” or the “Nader Effect.”
Because of the Spoiler Effect, Choose One Only Voting is wildly inaccurate when there are more than two candidates. Voter blocks who support more than one candidate can end up divided and conquered.
This creates a strong incentive to only vote for the “front-runners.” Voters in a majority can easily lose the election if they don't come together to strategically all vote for one candidate.
- Choose One Only Voting gives a huge advantage to candidates who are deemed "viable" and puts voters who have more candidates on their side at a significant disadvantage.
2. Once we solve the spoiler effect, we don't need to have 2 elections:
- Primaries generally have lower turnouts and unrepresentative voter demographics. In most cases, primaries bias in favor of older, whiter, and more wealthy voters.
- Primary elections are designed to narrow the field, which restricts voter choice in the general election. When people feel like nobody on their ballot represents them, voter turnout suffers.
- For jurisdictions which use a non-partisan primary and a top-two general election, the spoiler effect can be magnified by the large primary field. When this happens the primary election can actually advance two candidates from the minority faction, guaranteeing an unrepresentative winner in the general.
- This two-election process makes for a long campaign season, which is disliked by both voters and candidates. Longer campaign seasons advantage candidates with more money, especially those who can afford to take a year or sometimes more off of work while they campaign.
3. Magnifying the influence of Money in Politics
- To avoid the Spoiler Effect, voters are coerced into voting for the front-runner on their side who is most “viable.”
- The most viable candidate is usually the one who raised the most money and the one with the backing of the media. This gives big money an undue influence over not only voter opinions, but also over voter behavior.
- In order to be seen as viable, or "electable," candidates and politicians have to spend a huge amount of their time fundraising. In many cases this leaves them indebted to their donors, breeding corruption.
4. Wasted Votes and Disenfranchised Voters
- If you know that your favorite is a shoo-in, or that they don’t stand a chance, then it’s a safe bet that your vote won’t make a difference anyway. Together with the other issues listed above, many people choose not to vote at all because voting their conscience would be a wasted vote.
- We are starting with local city and county elections, but this reform has great potential for statewide and national elections too.
- STAR Voting is is a nationally viable and scalable method which could be used for Presidential elections, either with the Electoral College or a National Popular Vote.
- Yes! The Oregon constitution specifically authorizes the use of preferential voting.
- STAR Voting gives every voter an equally weighted vote and equal voting power. This is the legal definition of one-person-one-vote.
The idea that voters are subjectively “grading” the candidates in STAR Voting is not quite correct. In STAR, voters offer an objective level of support from 0 (no support) to 5 (maximum support) to each candidate. These support levels are then added up for all of the candidates to determine the two most broadly supported candidates overall. It is up to each voter to decide how much or how little support to offer each candidate. If a voter strongly prefers one candidate over the rest of the field, this can be shown on the ballot; likewise if a voter wants to support several candidates, that can be shown as well. Every voter has an equal, full range of expression about each candidate.
- Ranked Choice Voting (RCV) technically refers to a family of voting systems that use voter's rankings to determine the winner. Instant Runoff Voting is the most common RCV system in use today. Because STAR Voting actually uses the ranking derived from scores to make the final runoff decision, it is arguably an RCV system, but it is not at all the same as Instant Runoff Voting. STAR is similar to IRV in that voters can show relative preferences between candidates, but there are important differences. For example:
- With STAR Voting you can show that you like two candidates equally; with IRV you would have to rank one over the other.
- In STAR Voting everyone's full ballot is counted; in IRV your down ballot rankings may never be counted, depending on the order of elimination.
- In STAR Voting there is only one automatic or "instant" runoff; with Instant Runoff Voting there can potentially be many instant runoffs.
- With STAR Voting honesty is the best policy. The best strategy is to give your favorite or favorites a full 5 stars and to use your scores to show your preferences between the other candidates.
- While there are some hypothetical scenarios where you might get an edge by putting down higher or lower scores for various candidates, in practice there is no way to know when this might help and when it would hurt. This kind of dishonest voting is more likely to backfire so it’s not a good strategy.
- You can read a more technical comparison of strategic incentives in various systems here.
- STAR Voting was used in the 2020 Independent Party of Oregon primary for Secretary of State and State Treasurer, as well as in a presidential preference poll.
- The Democratic Party of Oregon is using it to elect Oregon's presidential delegates to the Democratic National Convention.
- It is being used for all internal elections for the Multnomah County Democrats.
For more information, check out our Elections Case Studies page.
- Additionally, STAR Voting has been tested in small groups and computer simulations of various election scenarios and has performed very well. http://star.vote lets you easily set up a STAR Voting election or vote in existing polls. Check it out!
- We see this as a pivotal moment for Oregon to pioneer a new path in voting reform, just as Oregon was a pioneer in adopting the initiative and referendum a hundred years ago, and more recently in nation-leading reforms like vote-by-mail and automatic voter registration.
Q: Wouldn't I want to "bury" a strong second choice and give a higher score to a weaker opponent to help my favorite win?
"Burying" is where you promote an electorally weak third choice candidate over a electorally strong second choice, in the hopes that will help your first choice candidate win against your weak third choice.
Here’s the problem with that in STAR Voting: if you think your first choice can’t win head-to-head in the runoff against your second choice, that means you think your favorite is vying for the second seat in the runoff. In that case the worst thing you can do is add points to a candidate you like less than both your first and second favorites. If you aren’t sure your favorite has a shot at the runoff, you’re very likely to give your strong second choice at least one point. Adding any support to a less preferred candidate in this scenario simply increases the likelihood your own favorite will be squeezed out and that your runoff vote will go to someone you really don't like.
Fundamentally, for the "burying" tactic to work, voters from opposing factions have to gang up on a well-liked consensus candidate by supporting the opponent they really don't like higher on the ballot than their true second choice. But if either of those factions actually believe that the opposing faction is going to adopt this strategy, their own best play is to simply vote honestly in order to give their own favorite the best chance of winning against the (now diminished) consensus choice. For "burying" to work in STAR, voters of true opponents must work together to be dishonest on their ballots, yet if one faction decides to be honest instead, the honest faction will gain the significant upper hand.
Conclusion: "burying" is not a viable tactic in STAR Voting.
- Both. There would be some initial costs associated with educating voters and reprogramming vote tabulation computers, but in the long run STAR Voting would save money by eliminating the primary election for local offices.
- The shorter election season would reduce the amount of money required for candidates to run a successful campaign. This should make running for office more accessible for candidates without big money backers.
Tie votes in STAR Voting are rare -over 10 times less common than with choose-one voting- but as with any voting method they can occur, especially in small demos or elections without many voters.
In most cases, what appear to be ties in STAR voting can be broken by referring back to the ballots themselves:
- Ties in the Runoff round should be broken in favor of the candidate who was scored higher.
- Ties in the Scoring round should be broken in favor of the candidate who was preferred head-to-head by more voters. Multi-candidate ties are broken in favor of the Condorcet winner if one exists, and can also be narrowed down by eliminating any Condorcet losers. The info needed to do this is present in the preference matrix for the election. (See below.)
- Ties which can not be broken as above are considered a "True Tie."
Q: What is a preference matrix and how do I read one?
A: A preference matrix is a chart which shows all the data from a given election. Unless you are doing a hand count, a matrix is generated automatically and will be available with your election results.
A preference matrix is a great reference point for looking at the additional data which can be gleaned in STAR elections. In most elections a full matrix isn't needed. All that is needed to select the winner is to determine the preferences between the two highest scoring candidates. When an election is tabulated electronically the full preference matrix is generated automatically anyways.
Q: When might I need a matrix and why?
In the event of ties, the full set of voter preferences shown in the matrix can often be used to break ties in favor of the more preferred candidate.
When ballots are not all tallied centrally, creating of matrix for each sub-set of ballots allows each set to be fully tabulated on site and then be compiled later. This is a feature known as precinct summability, and it means that with STAR Voting local audits and recounts are possible if needed. Summability is an important requirement for election security and integrity. STAR Voting and most voting methods are summable, but Instant Runoff voting, the type of Ranked Choice widely used around the world is not.
Preference matrices provide a lot more information beyond who won and lost, so they are often used in data analysis. One advantage of STAR Voting over choose-one is that all of this information is available.
Creating a preference matrix by hand is just like tallying a STAR election, but with an extra step at the end:
- Total the scores given to each candidate in the election.
- Just like in the STAR runoff, the two highest scoring candidates are selected. Sort the ballots to find how many voters preferred each of those finalists. Ballots are sorted into three stacks: Ballots preferring one finalist, ballots preferring the other, and ballots who gave both the same score and thus have no preference between those two. If you are doing a hand count you will have found your winner and can stop here. In the example below Alison won with 89 points. She was preferred by 8 out of 10 voters, or 80%.
- To create a full preference matrix, repeat the step above for each pair of candidates.
Q: What if two or more candidates are tied for 2nd highest scoring candidate in the scoring round?
A: Take the tied candidates and compare them head to head with eachother. If one is preferred by more voters then they should advance to the runoff. If needed you can also compare them each head to hear against the highest scoring candidate.
In this election Bill and Carmen are tied for 2nd highest scoring candidate with 67 stars each. Looking at the preference matix we can determine that Bill is preferred over Carmen, so this is not a true tie. Bill advances to the runoff.
In the runoff, we find that Allison and Bill are both preferred by the same number of voters, 5 each, but looking at the scores we find that Allison was scored higher overall. Allison wins the election.
Q: What if three or more candidates are tied for highest scoring candidate in the scoring round?
A: Take each candidate and compare them head-to-head with the others in the tie. This is known as a Condorcet method. (See below.) The two most preferred candidates advance to the runoff and the candidate preferred by more voters wins. If none of these candidates are preferred by more voters then it's a true tie.
Q: What is a Condorcet winner?
A: A Condorcet winner is one who in head-to-head match-ups was preferred over all others candidates. Finding Condorcet winners is helpful for breaking 3 way ties or ties with even more candidates if needed. If one candidate was preferred over all others they win the tiebreaker.
In the context of a 0-5 star ballot we determine which candidate was preferred by counting which candidate was scored higher on more ballots, just like in the STAR runoff. It's also essentially the same as a choose-one plurality election if there are only two candidates. Each ballot is one vote and the candidate with more votes wins.
Q: What is a Condorcet loser?
A: A Condorcet loser is a candidate who was not preferred over any of the others. In the event that you are trying to find two candidates to advance to the runoff and there is no Condorcet winner, you can at least eliminate any Condorcet losers.
Q: How do I compare candidates head-to-head and find Condorcet winners or losers?
A: You can use a preference matrix to quickly compare any two candidates head-to-head.
In this election Allison is preferred head-to-head over all other candidates, which makes her the Condocet winner. Doug is not preferred over any of the others, so he is the Condorcet loser. If Doug was eliminated, then Bill would become the new Condorcet loser.
Note: STAR Voting usually elects the Condorcet winner if there is one. If STAR elects a different winner, it's because a Condorcet winner only takes into account preference order but doesn't take into account the strength of support (total score) for the candidates.
STAR Voting finds winners by maximizing both strength of support and number of supporters.
Q: What do you do if there is a true tie?
A: True ties can happen in any voting method, so it is critical to set up a protocol for this in advance and agree upon it. Most organizations which run elections have a protocol in place in their bylaws or charter.
True ties can be broken by random selection. Alternately, if complexity is not an issue they can sometimes be broken using another voting method such as Minimax voting. Minimax itself was developed as a tiebreaker system for 3 way ties in Condorcet elections.
In this election, Allison, Bill, Carmen, and Doug are all tied for highest scoring with 78 stars each. Looking at the preference matrix we find that there is a Condorcet cycle as well! Allison is preferred to Bill, Bill to Carmen, and Carmen to Allison.
A close look also reveals that Doug is not preferred by anyone, Doug is a Condorcet loser and can be eliminated. This election is a three way tie between candidates Allison, Bill, and Carmen.
True ties in elections should be resolved by a tie-breaker chosen and agreed to in advance. Many organizations use a random method like a coin toss. If complexity is not an issue we recommend using another voting method which can process 0-5 star ballots such as Minimax to break the ties.
If you are running an election and have additional questions or would like guidance please email us at email@example.com
The STAR Voting project, STAR Elections, and the Equal Vote Coalition are 100% committed to open sourced elections, and all of our implementation and tabulation tools are open sourced. As largely volunteer driven organizations our resources are all designed and built collaboratively, transparently, and accessibly.
If you are interested in contributing to any aspect of the STAR Voting project please reach out to firstname.lastname@example.org, join our Slack, plug in with the tech team, and get involved.
Yes. Any election can be audited, fully, regardless of the voting method. STAR Voting is not only auditable, but unlike some other voting methods it is also compatible with best practices in auditing and election integrity.
Is STAR Voting precinct summable?
Yes, in STAR Voting, any subset of ballots can be independently tallied fully. This means that if an election was run statewide, any precinct within that state could independently process and tally their own ballots. This also means that vote tabulation can begin and can proceed unobstructed as soon as votes start to come in.
While the process is a bit more involved than tallying a Choose-One Plurality election there is no need to wait until all ballots are in hand or until the Scoring Round tally is complete before beginning to tabulate the Automatic Runoff.
On election day, precinct summability is important because it means that preliminary results can be shared as soon as they are available, during the tally, in real time, just like they are with Choose-One Plurality voting. To see how STAR Voting results can update in real-time, click the "show results" button on any live poll on the star.vote website.
For STAR Voting a precinct sum or tally includes the total score for each candidate, and also the number of voters who preferred each candidate. Head-to-head pairwise preferences are displayed as a preference matrix.
Most voting methods are precinct summable, including Choose-One Plurality voting, Score Voting, and Approval voting, but it's worth noting that Ranked Choice Voting (Instant Runoff) is not. In Ranked Choice Voting a preference matrix is not sufficient for summing ballots due to the fact that not all rankings will ultimately be tallied.
A note on Risk-Limiting-Audits and recounts:
For small scale non-governmental elections, full recounts are a simple option. When paired with other election integrity best practices, full recounts are always the most thorough way to verify the integrity of an election, especially if an error or foul-play is expected.
Audits and recounts are an important part of election integrity best practices, and every election should have a plan in place for this, but full recounts can be time consuming and expensive, especially for large, governmental elections. Risk limiting audits for STAR Voting are a sufficiently accurate and reliable method for doing partial recounts as needed to confirm an election's validity.
Risk limiting audits, or (RLA)s, prescribe a number of ballots to be recounted depending on the margin of victory. If a race is won decisively, then an audit will look at a small fraction of ballots, but if the margin is smaller a larger recount is prescribed. If the RLA finds that the audit results are consistent with the reported election outcome, or if the margin of error is within expected limits, the election is certified. If the evidence from the initial sample does not provide enough evidence to meet the risk limit, the sample size is expanded until it does.
Risk Limiting Audits for STAR Voting can be done using the same tools and similar protocols as are used for plurality voting.
Risk limiting audits are possible for most voting methods, including Choose-One Plurality voting, Score Voting, and Approval voting, but it's worth noting that while it is possible for Ranked Choice Voting (Instant Runoff) the complexity of the process, the existence of exhausted ballots, and the fact that not all ballot data is counted in RCV may present serious barriers for the real world use of RCV RLA's in practice. Procedures for efficient audits of Single Transferable Vote are still in the research phase.
How does a Risk Limiting Audit For STAR Voting work?
All risk-limiting audits start by defining some amount of allowable risk, called "alpha". This is set arbitrarily by statute, and is a percentage risk such as "5%" or "10%". For instance, if a jurisdiction sets the allowable risk at 5%, that means that, under the pessimistic assumption that the election is NOT valid, there can be no more than a 5% risk of incorrectly validating the tabulation of the election.
A few parameters determine the size of audit needed. First, the smaller the allowable risk you set, the more ballots you will end up having to check.
The next parameter is the win margin. The closer the race, the more ballots you will have to check.
The third parameter for risk-limiting audits is known as “gamma”. This is a safety factor, not for the final outcome of the audit, but for the chances you might have to check multiple ballot sets in sequential audit rounds, or even fall back to a full ballot count. The lower gamma is, the lower the number of ballots you will re-check in the initial round of the audit, but the higher the chances you might have to go back and re-check more. For STAR voting, you should use 1.1 if it is easy to add ballots to the audit (“pull new ballots”), and 1.2 if it is harder.
The formula for all risk limiting audits is complex (regardless of the voting method,) and auditing for governmental elections is handled by professionals, but for those running non-governmental elections, or for those interested in the matter, there is an online audit calculator for running Plurality RLAs, which can also be employed for auditing STAR Voting.
The main difference between auditing Plurality and STAR using the online audit calculator is that you will run the calculations twice, once for the Scoring Round, and again for the Runoff Round. Your audit's result — the number of ballots you will need to check — will be whichever round's output number is larger. Checking those ballots will allow you to limit the risk for both rounds.
(Note: the procedure used here makes a few “conservative” simplifications in order to keep the formulas involved relatively straightforward. That is to say, in practice, it will generally ask you to check a few more ballots than would be strictly necessary, and thus result in a risk even lower than the minimum you set — a better audit than the one you asked for. If necessary, a qualified statistician could design a procedure that required checking fewer ballots and didn’t have this extra safety margin, but the formulas for deciding how many to check would be substantially more complex. In particular, the procedure here for dividing the risk between the two rounds is conservative, as is the tallying step where you round up the one-vote and two-vote discrepancies for the score round.)
Instructions for using the Online Audit Calculator for STAR Voting:
Before you begin:
- Decide on the overall Allowable Risk for your election. (See above for more information.) This number will be referred to as "alpha" for your audit. The calculator will need this as a fraction or decimal.
For example, if the overall alpha is 5% you would divide this into two parts, as described below; 3.33% for the scoring round, which would be input as “0.0333” in the calculator, and 1.67% for the runoff round, which would be input as “0.0167” in the calculator.
- Set the “gamma”, to either 1.1 or 1.2, as explained above.
- Determine the total number of votes cast in the race to be audited.
Calculate Size for Scoring Round Audit:
Step 1: Calculate the margin of victory for the Scoring Round, m:
a.) Find the second- and third-highest scoring candidates. Let's call those B and C, respectively.
b.) Find c, the difference in total scores between B and C.
c.) Calculate the minimum possible margin of victory, m, by dividing c by 5, and then dividing that result by t, the total number of ballots. (c/5)/t
d.) On the RLA calculator, you will use the second form (“Risk-limiting audit parameters: comparison audits”) to plan your audit. Input your m margin in the first field, where it says: "Margin of victory: the closest margin between a winner and a loser as a fraction of the total number of ballots for the given contest."
Step 2: Multiply your Allowable Risk (alpha) by 2/3 and input this number into the calculator where it calls for "Risk limit (alpha) as a fraction."
For example, if the overall alpha is 5% (0.05), the allowable risk of error for this round is 3.3% (0.033 in the calculator.)
Step 3: Set overstatement and understatement rates. The default values (.001, .0001, .001, and .0001) are appropriate for machine-counted ballots; these correspond to error rates of 1 in 1,000 for mistaking valid votes for undervotes in either direction, and 1 in 10,000 for mistaking a vote for one candidate with a vote for another. If you are hand-counting ballots, you might want to allow for higher error rates, such as 0.01, 0.005, 0.01, 0.005 respectively.
Step 4: Click "Calculate" to determine the number of ballots needed to audit the Scoring Round.
Calculate Size for Runoff Round Audit:
Step 5: Calculate the margin of victory for the runoff round, n. This is calculated the same way as a "plurality-style" margin - the winning margin in the Runoff Round, divided by the total number of all ballots:
a.) Find d, the difference in total number of votes for the runoff winner, Candidate W, and the runner up, Candidate R.
b.) Calculate the minimum possible runoff margin of victory, n, by dividing d by t, the total number of ballots.
Step 6: Take the overall alpha for the RLA, and multiply by 1/3 for the alpha in the calculator.
Step 7: Leave other fields untouched for now. Overstatements and understatements should be set to 0. The field for "gamma" should be set to 1.1.
Step 8: Click "Calculate" to determine the number of ballots needed to audit the Runoff Round.
Perform Audit for Both Rounds Simultaneously:
Step 9: Identify ballots to audit:
a.) Compare the numbers calculated in steps 4 and 8. The larger of these two numbers is the total number of ballots needed for your initial Risk Limiting Audit.
b.) Select a random sampling of ballots to audit.
Step 10: Evaluate the selected ballots for the scoring round, comparing scores for candidates B and C on each ballot to their scores on the cast vote record. Document any discrepancies you may find. Discrepancies should be recorded as overstatements and understatements. For STAR Voting in the Scoring Round an understatement is the number of points a candidate was given in error which made their "margin of victory appear smaller than it really was," and an overstatement is the number of points a candidate was given in error which "make the margin of victory appear larger than it really was."
The overstatements and understatements can be kept in 4 running tallies:
- The total score discrepancy of the overstatements on ballots where the overstatement is less than or equal to 5 points. This total, divided by 5 and rounded up to the next whole number, is the number you’ll use for “1-vote overstatements” in step 12.
- The total score discrepancy of the overstatements on ballots where the overstatement is greater than 5 points. This total, divided by 10 and rounded up to the next whole number, is the number you’ll use for “2-vote overstatements” in step 12.
- The total score discrepancy of the understatements on ballots where the understatement is less than or equal to 5 points. This total, divided by 5 and rounded up to the next whole number, is the number you’ll use for “1-vote understatements” in step 12.
- The total score discrepancy of the understatements on ballots where the understatement is greater than 5 points. This total, divided by 10 and rounded up to the next whole number, is the number you’ll use for “2-vote understatements” in step 12.
For example: If a ballot had originally been counted as a score of 4 for B, but is in fact a score of 2 for B, then that would be counted as an overstatement of 2 for B because the difference between the original count and the audit count is 2 points, and because the error increased the margin of victory between B and C. This would go in tally 1 because 2 is less than 5. If this same exact error was found 11 times, the total number you would put in the calculator for 1-vote overstatements would be (2*11)/5=4.4, rounded up to 5.
Step 11: Evaluate the selected ballots for the runoff round, comparing the number of ballots with a preference for candidate W (the winner) vs R (the runner-up,) to the corresponding preferences on the cast ballot record. Document any discrepancies you may find.
As above, discrepancies should be recorded as overstatements and understatements. Overstatements and understatements in the Runoff Round are tallied as 1-vote or 2-vote over/understatements and this number indicates how incorrect the original error was. In the runoff there are three options. A vote may be a vote for the winning candidate, Candidate W, a vote of No-Preference, or a vote for the runner up, Candidate R.
For example, a vote for Candidate W which was erroneously recorded as a vote of no preference represents a 1-vote change. A preference for Candidate W which was erroneously recorded as a preference for Candidate R is recorded as a 2-vote change. If the error increased the lead for A over B, then that is an overstatement. If the error decreased the lead for W then that is an understatement.
Check if audit is complete:
Step 12: Repeat steps 1-8, but use the upper block in the calculator (“Risk-limiting audit parameters”). In step 3, instead of setting approximate over- and under-statement rates, input the total numbers calculated in step 10 into the fields for 1- and 2-vote over- and under-statements. Similarly, in step 6, input the numbers from step 11.
Results: If the number of ballots the calculator tells you to recount for both rounds is not higher than the number you have already recounted, your audit is complete.
If not, then repeat steps 9-12, drawing only any new ballots needed. For instance, if you had previously recounted 100 ballots, and the new highest number needed was 150, then draw only 50.
A note on the audit parameters above:
The procedure used above makes a few “conservative” simplifications in order to keep the formulas involved relatively straightforward. That is to say, in practice, it will generally ask you to check a few more ballots than would be strictly necessary, and thus result in a risk even lower than the minimum you set — a better audit than the one you asked for. If necessary, a qualified statistician could design a procedure that required checking fewer ballots and didn’t have this extra safety margin, but the formulas for deciding how many to check would be substantially more complex. In particular, the procedure here for dividing the risk between the two rounds is conservative, as is the tallying step where you round up the one-vote and two-vote discrepancies for the score round.