Instant Runoff Voting: Looks Good--But Look Again

Instant Runoff Voting: Looks Good--But Look Again

Stephen H. Unger

March 26, 2007

Addenda added 3/28/07 and 1/26/13

There is a significant movement in the US, spearheaded by forward-looking people, to replace the traditional plurality voting (PV) system with the instant runoff voting (IRV) system. The primary motive is to allow supporters of third party candidates with little chance of winning, to vote for these candidates while still helping to defeat whichever of the major party candidates they feel is the worst. A secondary objective is to eliminate the need for costly runoff elections when no candidate receives a majority of the votes. In this article, I will start by showing how IRV works and where its advantages lie over PV.

But the story doesn't end there. While IRV does work as advertised in some important cases, there are other situations in which it produces bizarre results. Furthermore, it will be shown that there are serious problems related to the tabulation and reporting of IRV results. I will argue that both approval and range voting dominate IRV in that they have the same advantages with fewer drawbacks.

(Much of my understanding of the material presented here came from the Range-Voting web-site whose principal founder is Warren Smith. Of course, responsibility for any errors and dumb ideas expressed here is entirely on my shoulders.)

Why Not Stick With Plurality Voting?

The common solution to this problem is a runoff election, usually between the top two vote getters. But even this can lead to unsatisfying results if the top two together received much less than 50% of the votes. This motivated a further refinement, which is to eliminate some number, perhaps just one, of the candidates with the fewest votes and hold the runoff among the survivors. Now we have the risk that the runoff election may also fail to produce a majority winner, setting the stage for a second runoff election. Clearly we are faced here with some painful choices. Being forced to have even one runoff election is a bad thing, for several reasons. One is the monetary cost. Another is that voter participation is likely to fall off considerably. Still another drawback is the difficulty of accommodating absentee voters, such as members of the military.

So what appears to be an elegant solution has been devised in the form of what is known as an instant runoff (IRV) election. The idea here is that voters indicate, not only a first choice, but also, additional ranked choices for the position at stake. If there are four candidates, each voter specifies a first choice, second choice, etc. The winner is determined as follows. A candidate receiving a majority of first-place votes is an immediate winner. If nobody receives such a majority, then the candidate with the fewest first-place votes is eliminated. On the ballots where this candidate was top ranked, the ranks of all the other candidates are upgraded one step. The first-place votes for the remaining candidates are counted again, with the new first-place votes included. As before, a candidate receiving a majority of first-place votes is declared the winner, else the process is iterated. The method is illustrated in Example 1 below.

Example 1. Candidates A, B, C, D are ranked from left to right on each initial set of ballots, shown below in the leftmost column. The integers indicate the number of such ballots in each set (there is a total of 100 ballots). In round-1, A has the most first-place votes, 40, but this is not a majority. So candidate D, with only 10 first-place votes is eliminated and the set of ballots is transformed as shown in the second column. Now B gains 10 first-place votes (from the ballots where D was originally first) so that A and B each have 40 first-place votes, still no majority. C, with only 20 first-place votes, is then eliminated, with the results shown in column-3. Candidate A, inheriting 20 votes from C, now has a majority (60) and is declared the winner.

20 ABCD 20 ABC 20 AB 20 ACBD 20 ACB 20 AB 20 BADC -------> 20 BAC --------> 20 BA ----> A wins 10 BCAD drop D 10 BCA drop C 10 BA 20 CABD 20 CAB 20 AB 10 DBAC 10 BAC 10 BA

What's Good About IRV?

Example 2. Assume T is the third-party candidate, who has almost no chance of winning, and that A and B are the major party candidates. (See below.) Supporters of T can list T as their first choice without contributing to a victory by B (who most of them consider to be the worst choice), because their second-choice will count in the second round. Indeed, in this example, neither A nor B has a majority of first-place votes, so that, when T is eliminated, most of the votes of T-supporters go to A, who thereby wins the second round, and therefore the election, 54-45. (Note that one T-supporter chose not to list second or third choices. This might be to demonstrate extreme unhappiness with both major party candidates.)

30 ABT 30 AB 12 ATB 12 AB 38 BAT --------> 38 BA -----> A wins 5 BTA drop T 5 BA 12 TAB 12 AB 2 TBA 2 BA 1 T

Some Electoral Surprises

Example 3.

4 CAB 4 CB 3 BAC ----------> 3 BC ----> C wins 2 ACB drop A 2 CB

In the first round, A is eliminated. C, second choice of A supporters, gets 2 more votes in round-2 and therefore beats B 6-3. But notice that 6 of the 9 voters placed A ahead of B and 5 voters placed A ahead of C. So, altho A would have beaten both rivals in 2-candidate elections, C comes out on top in this 3-candidate race. Putting it another way, if there had been a 2-candidate election between A and C, A would have won, but the entry of B into the race mysteriously makes C the winner. Not good!

Now consider another strange situation, illustrated in Example 4.

Example 4. In this contest, B suffers a first round knock-out, leading to C gaining 6 more first-place votes and victory, 13-10. But suppose 2 A-supporters change their votes from ABC to BCA, reducing A's first-place votes from 10 to 8. Treachery? No--fiendishly clever strategy! Now C is knocked out in round-1, and A gets 7 more first-place votes and the victory! A case where a vote for your candidate can be a kiss of death.

10 ABC 10 AC 7 CAB ----------> 7 CA -----> C wins 6 BCA drop B 6 CA

A basic cause of such bizarre outcomes is that the counting process does not consider the overall strength of candidates. Rather, it sequentially considers top ranking, then second ranking, etc., so that a candidate receiving many second-place rankings may be eliminated before this can be taken into consideration. Another factor is that voters cannot express the intensities of their preferences. For example, if a voter ranks the candidates as ABC, it might be that A is most preferred, B is considered as almost as good, and that C is considered as terrible. Or, the voter might consider B only slightly less terrible than C. Range voting allows such distinctions to be made, (with degrees of precision varying with the number of permitted weights).

Consider Example 5 below, which shows the results of a range vote (with weights ranging from 0 thru 9) consistent with the Example 3 data (repeated in the first column). The winner is clearly A, by a large margin (74-27-42).

Example 5

#voters A B C 4 CAB 4 8 0 9 3 BAC 3 8 9 0 -----> A wins 2 ACB 2 9 0 3

Example 6.

#voters A B C 4 CAB 4 2 0 9 3 BAC 3 3 9 0 -----> C wins 2 ACB 2 9 0 8

Now let's take another look at the Example 1 election, in which IRV did a good job. How would approval voting (AV) and range voting (RV) handle this situation? This is shown in Example 7 below.

Example 7. The IRV votes are repeated in the first column. The second and third columns show plausible versions of RV and AV votes assuming the same voter rankings.

IRV RV AV #voters A B T #voters 30 ABT 30 9 4 0 30 A 12 ATB 6 9 0 5 6 AT 6 9 0 8 6 A 38 BAT 38 2 9 0 38 B 5 BTA 5 0 9 5 5 BT 12 TAB 10 8 0 9 10 AT 2 5 0 9 2 T 2 TBA 2 0 8 9 2 BT 1 T 1 0 0 9 1 T A Wins A Wins A Wins

A was the winner in all three cases. In the RV election, voters had many more options. In the example, of those who ranked the candidates in the IRV election as ATB, half of them decided to give T 5 points, and half of them gave T 8 points. Of the 12 voters whose rankings were TAB, 10 gave A 8 points, while 2 gave A only 5 points. In the AV election, half of the ATB voters approved T as well as A. Other examples of such choices can be seen by examining the examples carefully. Since, in an RV or AV election, voting for Y can never help X win, voters can make their choices in a more straightforward manner. They should always choose their favorite candidate in AV and give that candidate the maximum score in RV. Then they can decide what to do about other candidates. In particular, if, among the likely winners there is a candidate that they feel it is very important to defeat, they can give full support to a less disliked front runner.

The examples illustrate that IRV elections are capable of producing strange results that cannot occur for RV and AV elections. There is controversy over the extent of this problem. My current assessment is that IRV would not do too badly with respect to third parties as long as they are not serious contenders for actual election victories. Supporters of such parties would be less hesitant to give them top rankings, with their "lesser evil" choice in second place. This would lead to a significant increase in votes cast for third party candidates, stimulating their growth, giving their viewpoints on issues more public exposure, and increasing their influence on the behavior of major party candidates. However, should one or more third parties grow to the point where they become serious contenders for actual election victories, the likelihood of anomalous IRV election scenarios would greatly increase. In such situations, we would be much safer with RV or AV type elections.

The Need for Central Counting

Example 8.

Precinct 1 Precinct 2 Combined Votes

6 ACB 6 CAB 6 ACB

4 BAC 4 BAC 8 BAC

3 CBA 3 ABC 3 CBA

B wins B wins 6 CAB

3 ABC

A wins



Apart from being another instance of strange IRV behavior, this highlights a serious practical problem with IRV. We cannot decentralize the ballot counting process. For example, if a ballot includes a race for a seat in the House of Representatives, then the ballots from all precincts in that congressional district must be sent to one central place to determine the IRV winner. We cannot count the votes in each precinct and forward the totals to a merging point. If the election is for a statewide position (or issue), the ballots for the entire state must go to one point. If there is even one statewide IRV race on the ballot then all ballots in the state must be processed at one location. This introduces a number of problems.

Should DRE machines be used, it would be necessary to forward to the central processing point electronic images of the portions of the ballots dealing with IRV elections. These would then be processed by a different computer at the center. If optical scan systems are used, then the paper ballots might be scanned locally and numbers corresponding to each ballot transmitted. Or ballot images could be faxed to the center. Or local scanners could be omitted and the paper ballots themselves sent to the center. In all these cases, the already formidable problem of guarding against malfunction or fraud becomes even more difficult. Parallel testing and random checking via manual recounts would have to be re-considered. Greatly increased use of transmission channels becomes another feature vulnerable to error, breakdowns, and fraud. Manual counts and recounts become slower and more costly.

How Will Election Results be Reported?

Conclusions

The complexity of IRV also mandates central counting of votes and this, in turn, provides increased opportunities for wholesale fraud or malfunction. Hand counting and recounting becomes slower and more expensive.

A lesser problem is that the reporting of election results to the general public is likely in many cases to omit significant information, such as local data and support for minor party candidates.

Both range voting (RV) and approval voting (AV), which is a special case of RV, have the same advantages as IRV with respect to voting for minor party candidates. Neither of these is subject to the strange effects mentioned above. The counting process for RV and AV can both be decentralized as in the case of conventional elections. The counting process for RV is a bit more complex than for conventional PV, but nowhere near as complicated as for IRV. Counting votes in an AV election is essentially the same as for PV elections, except that there are no over-votes.

It therefore seems clear that changing over to RV or at least to AV would be a much better move than a switch to IRV from every point of view.

Unfortunately, because RV and AV are not well known, many good people concerned about voting reform are strongly advocating IRV, which is actually being adopted in several US jurisdictions. I hope that they will reconsider in the light of the advantages of RV and AV and the drawbacks of IRV that they might not have been aware of.

Addendum-1 (3-28-07): A Spoiler Situation for IRV

#voters 3 T M B 3 T B 2 M B T ---------------> 2 B T ---> B wins 4 B M T M dropped 4 B T

#voters 3 M T B 2 M B T ---------------> M wins 4 B M T

Addendum-2 (1/26/13): A Truly Weird IRV Pathology

Imagine that, in an election involving at least two candidates, we want to know, not only who won the election, but also which candidate came in last. This certainly poses no problem in conventional plurality or approval type elections. We simply note which candidate received the fewest votes. Similarly, in a range (score) type election, the last place candidate is the one with the lowest total score.

Now consider an IRV election. The computation generating the winner does not automatically identify the big loser. The logical way to find the most unpopular candidate is to reverse the rankings assigned by each voter and then to apply the IRV procedure to find the candidate "elected" as the worst of the lot. So, e.g., if a voter ranks the candidates as A>B>C>D in the original election, we reverse that vote to get D>C>B>A to get the vote for the worst candidate. The result of this process can be very surprising.

Consider the following simple example involving 3 candidates and 5 voters. (It would work the same if there were 5 thousand or 5 million voters.)

2 A>B>C 2 B>C>A 1 C>A>B

C is knocked out in round-1, and A wins.

Now consider the election of the most unpopular candidate. We reverse all the votes to get

2 C>B>A 2 A>C>B 1 B>A>C

Now B is knocked out in round-1 and the winner is A!

No, this is not a misprint. For this IRV election, the clear winner is also the clear loser!

Nor is this a rare singularity. The relative values of the numbers can be varied over a large range without changing the results. E.g., the results are the same for the following variation:

251 A>B>C 238 B>C>A 137 C>A>B

All that is needed is that the smallest number be for line 3, that the sum of the line-3 and line-2 numbers exceed the line-1 number, and that the sum of the line-3 and line-1 numbers exceed the line-2 number. The likelihood of such an election outcome is not miniscule.

(Warren Smith first identified this problem.)