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?

If we have a 2-candidate election, then, assuming we ignore abstentions, one of the candidates will receive a majority of the votes, or there will be a tie. Since, with a large number of votes cast, the possibility of an exact tie is small, we will ignore it for the purposes of our discussion. Plurality voting (PV) is not bad in this case, altho, as was pointed out in an earlier essay, range voting (RV) has the advantage in that it allows the intensity of preferences to influence the outcome. But if there are more than two candidates, none may be a majority choice. It is a common practice to declare the receiver of the most votes (i.e., a plurality) as the winner, but clearly this can lead to undesirable situations, particularly if the votes are divided among so many candidates, that the plurality winner may get an absurdly low percentage of votes, say 20%.

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?

In addition to eliminating the need for additional elections, IRV also addresses the problem of third party spoiler votes. In US elections other than for local positions, the winner is virtually always a Republican or Democrat. Votes cast for third parties are considered as "wasted", and, in many cases, the third party candidate is considered as taking away votes from the major party candidate closest to that party's position. (For example, Green Party candidates are felt by Democrats to be helping Republicans by draining away votes that would otherwise go mostly to Democrats, and Libertarian Party candidates are similarly regarded by Republicans.) Supporters of third parties face a painful choice between one-handed voting (i.e., vote for a despised major party candidate with one hand while holding one's nose with the other hand) or not doing all that is possible to prevent the election of an even more despised candidate. In Example 2, below, we can see that, in an IRV election, supporters of third party candidate T can rank T first, while still helping to defeat the candidate they think is the worst.

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

So how could any decent, intelligent person not support IRV? One answer is that situations can arise in which IRV results are clearly unreasonable. For starters, what would you think of a system that chose C as the winner in a 3-candidate race where majorities of the voters expressed a preference for A over B and for A over C? In the IRV election of Example 3 below, this is precisely what happens!

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

Now consider Example 6, a variation in which the range votes also conform to those of the IRV example, but where the weights differ enough from those in the previous example to change the outcome. Now C is the clear winner (35-27-52). You might find it interesting to construct another example where RV numbers consistent with the same IRV votes make B the winner.

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

Consider how votes would be tabulated in an IRV election. A fundamental difference with PV, AV and RV, is that IRV results tabulated at the precinct level cannot simply be passed along to be merged at county and state levels. A dramatic illustration of this point (due to Warren Smith) is Example 8 . The IRV winner in both precinct 1 and precinct 2 is B. But if we combine the votes in the two precincts, the IRV winner in the enlarged jurisdiction is A!

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?

After the results of an IRV election have been determined, there remains the issue of how they should be reported. The usual practice in conventional elections is to report, not only the winners, but the actual vote counts in each precinct. This data is publicly available, altho the media usually reports summaries giving overall totals for each candidate. For AV elections there should be no apparent difference in the way this is done. For RV elections the only difference would be that, instead of reporting the number of votes for each candidate, point totals would be given. The total number of votes cast might also be provided. For IRV, the situation is significantly different. In cases where the winner receives a majority of first-place votes, these can be reported in the usual way, along with the first-place votes of the other candidates. But what happens if one or more additional rounds of computation are necessary? It might be mandated that the votes for each round be made available to the public, but it is unlikely that the news media would publish more than minimal information, such as the results of the final round. This would be unfortunate for minor party candidates, whose totals might be not be visible to anybody but those who make an effort to find them. This defeats the whole point of giving at least minimal general publicity to these groups.

Conclusions

Instant runoff voting (IRV) allows voters to do more than choose a single, most preferred, candidate. They can supply additional information, indicating their general ranking of all candidates in a race. One important advantage of this is that it makes it possible for voters to vote for a candidate that they think is the best, when that candidate has little or no chance of winning, without increasing the likelihood that a candidate they feel is the worst will win. But IRV has serious drawbacks. Particularly when there are three or more serious contenders, some very strange things can happen, such as the defeat of a candidate who would have won over each of the other candidates in a 2-person race, or a situation where A is deprived of a victory because several voters changed their first-place votes from B to A.

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

Here is a plausible situation where, in an IRV election, supporters of a fairly strong candidate would feel under great pressure to downgrade the ranking of that candidate in order to prevent what they regard as a very bad candidate from winning. We could imagine this as happening after some third party gains enough strength to be a serious contender. Let T be that candidate, with B, the really bad candidate, and M the "middle" candidate. Assume polls suggest that the votes might be cast as below:

#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
Supporters of T, wishing to prevent B from winning, would find it strategically advantageous to abandon T and make M their first-place pick. This would change the results to

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

This "spoiler" dilemma would not happen with RV or AV, where there is never a good reason not to give the maximum score to your favorite candidate. With reasonable weight assignments corresponding to the rankings in the original situation, M would have won, in both RV and AV elections, with T supporters giving their candidate maximum weight.

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.)


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