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Viktor Yushchenko: Runoff voting is inefficient and is founded on two fallacies
Letter 01
10Dec2004
"Hidden in what might appear to be the lower sophistication of the CanadianAmerican procedure, then, might be an underlying wisdom which adds to its high efficiency as an argument for its adoption." � Lubomyr Prytulak
10 December 2004
Viktor Yushchenko
Vul. Prorizna, 9, Office 20
01034 Kyiv34
Ukraine
Dear Mr Yushchenko:
Pinpointing the Inefficiency
Perhaps Ukrainian electoral law can be altered so as to produce the same effect as is produced by the existing procedure, but in a fraction of the time and at a fraction of the cost. At issue is the inefficiency of runoff voting, as can be illustrated in the following simple example.
Imagine five voters, V1 to V5, who vote for three candidates, A to C, with the following results:

Voter 

V1 
V2 
V3 
V4 
V5 
Choice 
A 
A 
B 
B 
C 
As no candidate won at least 50% of the vote, existing procedure requires a runoff vote between the two leading candidates A and B.
However, it may be objected that asking voters V1 to V4 to cast their ballots in a runoff wastes their time, as their initial vote has already expressed which of A and B they prefer. The only new information that is needed is to come from V5 � which information amounts to his being asked how he would have voted if candidate C had been unavailable.
That four out of five voters will have already expressed their opinion concerning the two leading candidates happens to correspond closely to the results of the 31Oct2004 vote, where the two top candidates received 39.87% + 39.32% = 79.19% of the total vote.
Removing the Inefficiency
The solution to this inefficiency comes from recognizing that V5 could have provided the missing information during his initial vote. That is, V5 (and of course all voters) could have been asked to rankorder the candidates, which might have led to the following results, from which it is immediately evident that if C had been unavailable, then V5 would have voted for A. The purpose of the runoff election, then, is to extract the one piece of new information appearing in the yellow cell, and it is inefficient to extract this information by having all five voters appear for a runoff vote when V5 could have supplied that critical piece of information during the initial vote.

Voter 

V1 
V2 
V3 
V4 
V5 
First Choice 
A 
A 
B 
B 
C 
Second Choice 
B 
C 
A 
C 
A 
Third Choice 
C 
B 
C 
A 
B 
The algorithm which the Central Election Commision might apply to a more complex case involving a large number of voters and many candidates is simple:
REPEAT THE FOLLOWING TWO STEPS AS MANY TIMES AS IS NECESSARY TO GIVE ONE CANDIDATE AT LEAST 50% OF THE VOTE
STEP 1 OF THE ALGORITHM: REMOVE THE LEAST SUCCESSFUL CANDIDATE (C has been removed from the table above to produce the table below):

Voter 

V1 
V2 
V3 
V4 
V5 
First Choice 
A 
A 
B 
B 

Second Choice 
B 

A 

A 
Third Choice 

B 

A 
B 
STEP 2 OF THE ALGORITHM: SLIDE ALL CHOICES UP SO AS TO FILL IN THE BLANKS:

Voter 

V1 
V2 
V3 
V4 
V5 
First Choice 
A 
A 
B 
B 
A 
Second Choice 
B 
B 
A 
A 
B 
The First Choice line in the table above, then, is the outcome that interests us. It provides an estimate of the results of a conventional election in which Candidate C had not run. The victory now goes to Candidate A who gets 3/5 = 60% of the vote.
The proposed efficient procedure does not require that all voters rank all candidates (as for example the 24 candidates that had presented themselves on 31Oct2004). Rather, each voter is permitted to rank as many or as few candidates as he wishes � one or two or three, or any number. If a voter decides to rankorder fewer than the total number of available candidates, there is some possibility that all the candidates he does rankorder receive few votes, such that they will all be removed during computation, and the voter will be left voting for no one. However, this is not undesirable. Had all his favored (but hopeless) candidates been absent from the initial vote, he might not have shown up at the polls, or he might have shown up only to indicate "none of the above" � which would have exactly the same effect as we imagine happening in the proposed efficient procedure.
The results from the proposed efficient procedure will not be absolutely identical to the results from the existing inefficient procedure. For example, as things stand now, the two top candidates have an opportunity to campaign some more before the runoff vote, the media have the opportunity to provide more information on the candidates, the people who vote in the original election and the runoff will not be exactly the same, candidates engage in horse trading between initial vote and runoff, and so on. However, it is not obvious that these differences constitute advantages, and the disruption to people's lives and to the economy and to the conduct of government of runoff voting argues for replacing it if some equivalent, and more efficient, alternative presents itself. The loss of opportunity to horse trade, in particular, should not be counted as a disadvantage of the efficient procedure, as it is open to question whether a candidate who fails to make it to the runoff can in fact deliver his votes to any other, or whether his votes will tend to distribute themselves according to the independent decisions of the voters.
Variations of the algorithm proposed above can be considered, but may be so little different that they might alter real election outcomes once in a millenium. For example, instead of removing one leastsuccessful candidate at a time, all but the two mostsuccessful candidates could be removed in one swoop � which comes closer to approximating the existing procedure, but which closer approximation is not obviously desirable.
Too Complicated For The Voter?
Might the proposed method be complex enough to confuse voters? It may be hoped not, as is illustrated in two ballots below, the voter's writing being shown in blue. On the left is a representation of the existing inefficient procedure, where a voter has placed an X opposite his first choice, Candidate C. On the right is the proposed efficient procedure where the voter ranks as many candidates as he wishes, imagining below that he wishes to rank only three. Perhaps a small investment in voter education (at a fraction of the cost of conducting a runoff) would be sufficient to make the proposed procedure understandable to almost everybody:
Existing inefficient procedure:
the voter indicates only his
first choice

Candidate A 

Candidate B 

Candidate C 
X 
Candidate D 

Candidate E 

Proposed efficient procedure:
this particular voter
happened to rankorder
three candidates

Candidate A 

Candidate B 
3 
Candidate C 
1 
Candidate D 

Candidate E 
2 
Too Complicated For The Tabulator?
The proposed efficient procedure also complicates the creation of raw data by the lowestlevel tabulators, but perhaps also to a tolerable degree.
Specifically, the existing procedure has the lowestlevel tabulators do something that is logically equivalent to what is shown on the left below (for an imaginary 20 voters choosing between three candidates AC). What the proposed procedure would have the tabulators do is shown on the right below, and is not that much more complicated � the tabulators now record the frequency with which voters selected different preference sequences, as for example the second line of data showing that 7 voters marked Candidate A as their first choice, C as their second, and B as their third:
RAW DATA from existing inefficient system

Candidate 
Count 
Frequency 
A 
llll llll 
9 
B 
llll 
4 
C 
llll ll 
7 
TOTAL 

20 
RAW DATA from proposed efficient system

Preference 
Count 
Frequency 
ABC 
ll 
2 
ACB 
llll ll 
7 
BAC 

0 
BCA 
llll 
4 
CAB 
llll 
5 
CBA 
ll 
2 
TOTAL 

20 
In either case, the lowestlevel tabulator has no other obligation but to examine a ballot, categorize it, and enter a slash in the appropriate line in the column headed "Count." After all ballots have been processed, the slashes are summed to get the values in the column headed "Frequency."
It is at this point that the proposed procedure becomes complicated enough that if left to the lowestlevel tabulators, substantial errors might be expected; and in any case, the lowestlevel tabulators cannot proceed to subsequent calculations which might be required in the case of no candidate receiving 50% of the vote, as the need to proceed will only be known following processing of all the data for the entire nation, and as in any case such subsequent calculation would be conducted on the national data, and not on the subset of the data available to any given tabulator. For these reasons, then, the proposed procedure terminates the work of the lowestlevel tabulators at this point, and has them submit copies of their raw data, resembling the table on the right above, to the Central Election Commission (CEC) where processing is completed by computer.
If the table on the right above were the complete data from the entire country, then what the computer would start by noticing is that going by first choices only, no candidate receives 50% of the vote, but that if the least successful candidate, B, is removed from consideration, then candidate C gains four votes, and candidate A gains none, such that the terminal tally becomes 9/20 = 45% for A, and 11/20 = 55% for C.
The possibility of error and fraud in the computer analysis of raw data can be completely eliminated as follows. First, it must be recognized that the computer program which performs the analysis according to the proposed algorithm is so elementary that it can readily be taught in any introductory computer programming course. It could become a Ukrainian tradition, then, that all students in introductory computer programming courses write this program as an exercise, and that they run it on the real raw data from the last election, which raw data would be permanently available on the Internet for downloading. Were this the custom, then Central Election Commission computations would be verified thousands of times in every region of the country. An incidental benefit of such a custom might be that performing analyses of raw voting data leaves behind in the student a heightened interest in the democratic process along with a heightened probability of participating in future elections. Even if such a classroom exercise did not develop into a national custom, though, absence of error and fraud would still be guaranteed by the existence of hundreds of thousands of individuals who possessed both elementary programming skills and access to a home computer, and who therefore had it within their power to verify CEC computations.
In short, the proposed efficient procedure does increase the complexity of the categorization required of the lowestlevel tabulators, but leaves the heightened complexity of subsequent computation to computers, while at the same time allowing for such a high degree of verification as to render either error or fraud impossible.
The above example, of course, is highly simplified, and the real data from a real election would require a somewhat elaborated procedure, which elaboration, however, would be straightforward, would easily fall within the competence of the beginning programer, but that need not further add to the length of the instant letter.
Advisability Depends on Comparing Two Sets of Costs
The advisability of the proposed procedure depends on weighing, on the one hand, the total cost in wages, materials, disruption to the economy, annoyance of the voter, and so on, of conducing a runoff vote, and comparing that to the total cost of training ballot tabulators to make sure that they are able to categorize ballots accurately. There cannot be much doubt that a runoff election is prohibitively expensive, and that implementation of the proposed procedure is possible at a fraction of the expense.
The Ultimate Simplification May Enhance Fairness
The best election procedure might be the one that happens to be most familiar to Canadians and Americans � which is to give the victory to whichever candidate wins the most votes, no matter how far his support falls below 50%. This procedure can be defended by noting two fallacies on which the Ukrainian procedure is based, fallacies which are made palpable with the help of the following extreme example.
Imagine two out of many voters (V1 and V2) in the 31Oct2004 election who are asked to rank all 24 candidates (AX), and who provide the following data:
VOTER 
RANKING OF ALL 24 CANDIDATES 
V1 
ABCDEFGHIJKLMNOPQRSTUVWX 
V2 
CDEFGHIJKLMNOPQRSTUVWXBA 
Suppose further, that Candidates A and B win the largest number of votes, making their rank of paramount interest (which is why they are shown in red above), but with neither candidate reaching 50%.
Now according to the existing Ukrainian procedure, a runoff vote would be held between Candidates A and B, in which case the runoff votes of V1 and V2 would cancel each other out, V1 continuing to prefer A as we saw him doing above in the initial election, and V2 continuing to prefer B, as we also saw him doing above. However, is such a cancellation of one vote by another fair? It is unfair, for the reason that V1's preference for A over B is probably strong, whereas V2's preference for B over A is probably weak � and it is unfair for a weak preference to cancel a strong preference.
To elaborate somewhat, if voters indicated the strength of their preference by assigning to each candidate a number from 1 (very much dislike) to 100 (very much like), then the typical result might be something like their giving a 90 to their first choice, and a 40 to their second choice � in other words, a very large gap might be expected. And what sort of ratings might be expected for their 23rd and 24th choices? Undoubtedly two ratings that were very low, and that differed little, possibly ratings in the order of 2 and 1. Thus, it is unfair to have the vote of V2, who essentially fails to distinguish between candidates A and B, cancel out the vote of V1, who strongly prefers A.
More generally, a plausible hypothesis is that the intensity of the voter's firstchoice preference so distances itself from all his weaker preferences, that it alone should be permitted to determine the outcome of the election. To express the same principle graphically � the Ukrainian model assumes that the function between likeability and a rank order of candidates is linear (the likeability difference between any two adjacent candidates is equal), whereas it is more plausibly a descending function that is concave upward (the likability difference between two higherranking candidates is greater). Hidden in what might appear to be the lower sophistication of the CanadianAmerican procedure, then, might be an underlying wisdom which adds to its high efficiency as an argument for its adoption. The first Ukrainian fallacy which needs to be renounced, then, is the equation of a weak preference with a strong one. Asking a voter who he likes best from a full slate of candidates gives an answer of much greater importance than asking a voter who he likes best from a slate of candidates which excludes his favorite candidate, or which might even exclude every candidate that he can bring himself to tolerate.
It may be further expected that V2, having been told that his first 22 choices will not be offered in the runoff, will have weak motivation to show up to vote in the runoff, and it may be expected more generally that the very people from whom the runoff seeks new information are the ones who in fact tend to absent themselves from runoff voting because their favored candidates have been withdrawn. The expectation that V2 will show up for a runoff vote to indicate which of two candidates he prefers when he in fact may care little about either, or may despise both almost equally, is an expectation unsupported by evidence. Perhaps he is more likely to feel disenchanted with, and to begin to boycott, a system that removes from consideration every candidate in whom he sees any virtue. The same can be said of V5 in the table at the top of this letter � who knows if he will show up for a runoff vote when his favored Candidate C � to whom he may have strong loyalty, and on whom he may have pinned all his hopes � has been removed from consideration? More generally, it may be supposed that the very people whose information the runoff vote solicits are the people with the weakest motivation to trouble themselves to supply it.
In Short
A runoff vote is an inefficient way to gather the information on how voters rankorder candidates that could be collected during the original vote.
A runoff vote, furthermore, is founded on two fallacies: that it is legitimate to allow a weak preference to cancel a strong preference, and that voter turnout will not suffer for voters whose preferred candidates have been excluded from consideration.
The alternative voting procedure which brings with it high efficiency, and which avoids the two above fallacies, is to give the victory to whichever candidate receives the most votes in the original election, no matter what proportion of all votes this may be.
Lubomyr Prytulak
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