IS A MAJORITY SUFFICIENT?

The purpose of a voting system is to combine the wishes, or preferences

of individuals or voters into a decision for society or a group of people. In

assessing the desirability of different voting systems in single-winner

elections two basic questions need to be asked. First, what criteria can

we use to judge which candidate is the most representative of the

electorate given individual voter's preferences? And second, which voting

system is most likely to insure the election of that candidate? In the case

of a two man election in which each voter casts one vote the answer

seems quite clear. The candidate with the majority support is the most

representative and any voting system that chooses that candidate, such

as our system of first-past-the-post or plurality voting, is considered

legitimate. But in some cases of multicandidate elections, elections

with three or more candidates, the appearance of a majority winner may

be misleading in terms of electing the most representative candidate.

Such elections can be classified as pseudomajority elections rather

than real majority elections.

One example of pseudomajority elections can be found in the top-two

runoff system adopted in California in 2010. Under this system, if no

candidate wins a majority in the first round, a second election is held

between the top two candidates with, sans a tie, the candidate having

the most votes winning. The problem with this system, which is also

labeled the double ballot majority system, is that it can result in the

election of a candidate that does not in reality have credible majority

support. The 2012 Egyptian presidential election is an example of

this. In that election you had the Muslim Brotherhood's candidate

Morsi in the runoff against Shafik who was the last Egyptian Prime

Minister appointed by Mubarak. Despite the fact that Shafik was a

representative of the then very unpopular Egyptian political

establishment, he lost to Morsi by a "narrow margin" according to

Wikipedia. That and data on the Wikipedia entry tell an interesting

story. The data for the elections are:

Morsi 24.78% 51.73%

Shafik 23.66% 48.27%

Sabahi 20.72%

Fotouh 17.47%

Moussa 11.13%

In the Wikipedia piece on the Egyptian election there were data on a

poll that Al Ahram did on some pairwise runoffs which from the above

and what is known about that election seems to be quite credible.

They are as follows:

Moussa 77.6%

Morsi 22.4%

Fotouh 74.7%

Morsi 25.3%

Even if these polls might be seriously skewed, they are so lopsided

that they clearly suggest that in a head to head (pairwise) competition

against either of the two lowest top five candidates in the first round

Morsi would have lost handily.

Instant runoff voting (IRV) is an example of another voting system that

is vulnerable to pseudomajority elections. Under instant runoff voting,

voters list their preferences for candidates one by one. If no candidate

had a majority of first place votes, the candidate with the least first

place votes is dropped and that candidate's supporters' second place

votes are reallocated for a second count. If that gives no majority to

any candidate the process is repeated until a majority in votes counted

is reached or the process runs out of candidates to drop for a recount.

Such a process can easily lead to the election of a pseudomajority

candidate as happened in the 2009 IRV election for mayor in Burlington,

VT. In that election, in the penultimate round of vote counting the

Republican candidate had 3,297 votes, the Democrat 2,554 votes and

the Progressive 2,982. Under the IRV system the Democrat candidate

was dropped and the Progressive candidate won by 4,314 votes

to the Republican's 4,064 votes or a margin of 250 votes. Had the

Republican candidate been dropped instead and his voters' second

choices been reallocated, the vote would have been 4,067 for the

Democrat and 3,477 for the Progressive for a margin in favor of the

Democrat of 590 votes, more than twice the original margin of 250

votes.

Under our first-past-the-post voting system in which a voter only casts

one vote, a plurality, just more votes than any other candidate has

where none have a majority, is sufficient to elect a candidate. The

argument here being that in such a situation a plurality is the closest

to getting a majority and therefore presumably the closest to choosing

the most representative candidate. But as the data on the 2012 Egyptian

presidential election above show had it been conducted as a simple

plurality election, Morsi would still have won though clearly he was not

a very representative candidate. The problem here was already recognized

in the 18th Century by two Frenchmen the Marquis de Condorcet and

Jean-Charles de Borda.

In such single winner elections with multiple candidates, Condorcet

argued that the most representative candidate is the one who could

beat all the other candidates in all the pairwise, one-to-one, contests

possible from the field of candidates. The number of pairwise contests

possible in such multicandidate elections is given by the formula

p = n(n-1)/2 where p represents the required number of pairwise contests

and n represents the number of candidates in a multicandidate election.

Thus if we have three candidates we would require three pairwise

contests, for four candidates it would require six pairwise and for five,

ten pairwise contests.

Prior to the advent of computers conducting a Condorcet election was

not very practical. But with computers one could have voters list their

preferences for candidates one by one and calculate from that all the

possible pairwise elections from the field of candidates to determine

a Condorcet winner. However there is just one little, but rather serious,

problem here, the Condorcet paradox. The paradox is that while

preferences for individual voters are transitive; that is if a voter prefers

A to B to C (in standard notation A > B > C), he does not prefer

C to A; preferences may not be transitive for voters taken collectively

and a Condorcet leader cannot be determined.

And example of the Condorcet paradox can be seen from the simple

case of three voters, X, Y, and Z and three candidates A, B and C.

If the preferences of the voters in descending order are as follows:

X Y Z

A B C

B C A

C A B

We get for each pairwise contest possible from the field of

candidates the following for:

A vs B a win for A by 2:1

B vs C a win for B by 2:1

C vs A a win for C by 2:1

Or stated in more standard notation we have A > B > C > A. Various

complex fixes have be proposed for this which in reality are not so

transparent.

Borda's answer to the question of which candidate is the most

representative in a single-winner multicandidate election is that

candidate who has the highest "Borda count." The Borda count

is calculated by asking voters to list one by one all candidates in

their order of preference and assigning numerical scores according

to the listings. In the classic Borda count, if one has ten candidates

the candidate that a voter lists first is given a score of 9, or the

number of candidates minus one, the second 8 points, the number

of candidates minus two and so on to the last candidate who is

assigned zero points. The count is summed over the preference

listing of all the voters and the candidate with the highest Borda

count is considered the most representative candidate and the one

who should be chosen. As Sir Michael Dummett points out in his

book "Principles of Electoral Reform" the count represents the

total number of votes each candidate would get in all the pairwise

contests possible from the field of candidates in which voters

vote consistent with their rankings of the candidates.

Simulations on the Internet by the mathematician Ka-Ping Yee

suggest that Condorcet and Borda voting systems tend towards

similar results, but with one notable exception, Borda results are

not always of a majoritarian outcome as are Condorcet results

where there is a clear Condorcet leader. Sir Michael Dummett in

his book "Principles of Electoral Reform" presents an interesting

hypothetical example of this which raises the issue of whether a

majority is sufficient to identify the most representative candidate.

His example consists of 56,000 electors and four candidates A,

B, C, and D. In the example A is a highly polarized or divisive

candidate who is supported by 29,000 of the electors but is

considered the least desirable by 25,000 of them. The rankings

and their support levels used in Dummett's example are as

follows:

Voters 29,000 24,000 2,000 1,000

A B C D

B C A C

C D B B

D A D A

In this example A would clearly win in a regular plurality or IRV election

having an absolute majority of first place votes. And he is clearly the

Condorcet leader as Dummett would label him. But in this hypothetical

election B's Borda count of 133,000 is higher that A's of 91,000. In

this case Dummett points out that:

There is undoubtedly a case to be made for saying that B

is a more representative candidate than A. All of A's

supporters reckon him [B] the second best candidate, but no

elector thinks him the worst: he would surely represent

opinion in the constituency better than A.

In short, a polarized or divisive candidate may have a majority, but may

not be the most representative candidate.

Heretofore we have tacitly been assuming that voters are voting sincerely

and not tactically or strategically. In the case of our two candidate election

that surely would be the case. But as the Gibbard-Satterthwaite theorem

establishes, tactical voting can happen in any multicandidate election.

As Michael Dummett put it in his book on electoral reform,

Under all electoral systems, some voters may vote tactically,

that is to say, in a way that does not conform to their true

preferences. It can be mathematically demonstrated that no

system can avoid this. That is to say, there can be no system

under which, given his preferences, every voter will always have

only one way of voting that will bring about an outcome as

desirable as possible from his point of view, however the others

choose to cast their votes. Nevertheless, different systems

vary greatly in the degree of incentive they give for tactical

voting. A system that debars a voter even from expressing

all his preferences (or takes minimal account of most of

them) gives the most potent of all incentives for it. This is

why tactical voting plays so large a role in elections under

the 'First Past the Post' system.

Tactical voting generally involves either feedback on the preferences

of other voters (e.g., it would be a wasted vote to vote for a third

party candidate like a Ralph Nader) or some a priori assumption of

how a given voting system can be gamed to one's advantage in

voting. In the case of ranking voting systems, like one has in an

IRV or Borda election, feedback on the preferences of other voters is

generally a daunting task. In the case of a five candidate election one

is dealing with 120 different rankings (5! in mathematical notation)

with differing frequencies of occurrences. Given the inherent complexity

of such a situation Samuel Merrill's assertion that IRV is relatively

resistant to manipulation does not come as a surprise at least on

the individual level. But on a collective level that does not seem to

be the case as the wide spread use of "how to vote cards" in

Australia attests in its IRV elections. Based on information from at

least one survey, half of the electorate in Australia may rely upon

such cards in voting. Such tactical voting is surely pernicious and

undesirable. In the voting process of determining a winner it is

certainly fair to weigh one vote against the others in the algorithm

determining a winner or the most representative candidate. But

to engage in a voting pattern on the part of one group that affects

how the preferences of other groups are or are not expressed surely

cannot give credible outcomes in terms of the voting system

identifying the representative candidate. The erratic way in which

Hare based systems like IRV behave with shifts in preferences

that Dummett's analysis suggests, surely makes such manipulation

of serious concern.

In terms of feedback on others' preferences, Borda elections face

similar problems but have a different response in terms of tactical

voting strategy. As Dummett and others have pointed out the Borda

system offers a fair incentive to tactical voting. Generally this takes

the form of voters ranking the candidate who is most perceived as

threatening their first choice low in their rankings even though

otherwise that candidate is closer to theirs in terms of preferences.

If a great many voters engage in such behavior, the outcomes are

hardly likely to bear any relation to their actual preferences, i.e.,

they are not representative. Given that there is a lot of anecdotal

evidence of this problem, especially in Borda elections held in

academic departments, it is not surprising that a Michael Dummett

would conclude that "the Borda system is arguably too vulnerable

to tactical voting to be adopted" in single-winner elections.

Approval voting, in which in multicandidate elections voters are

allowed to give one vote each to the candidate or candidates they

support with the candidate having the most votes winning, may be

a much more benign voting system when it come to the issue of

tactical voting. Given the similarities of outcomes in Ka-Ping Yee's

simulations of approval, Condorcet and Borda voting systems, it may

well be that an approval voting election may better reflect a sincere

Borda election than an actual Borda election. As far I can see, an

approval voting election's tactical voting would not involve the

manipulation of the expression of other's preferences nor

misrepresentation of one's own preferences thereby contributing to

unintended outcomes.

In an approval election, if one's favorite candidate has little chance

and a voter does have a definite preference among the credible

contenders, it is not a misrepresentation to vote accordingly (e.g.,

to give both a Ralph Nader and an Al Gore an approval vote in a

Florida-style election). And certainly under approval voting the

information needed to make intelligent choices given the constraints

of other voters' preferences is rather simple and straight forward. For

instance, in a five candidate contest the relevant information that one

needs in considering the tactical aspects of voting is simply the

percentage support levels for each candidate, five pieces of information

as opposed to 120 pieces. Given modern polling techniques, such

information is likely to be fairly available with reasonable accuracy.

Since approval voting does not always choose the Condorcet leader

but rather the candidate that is acceptable to the largest fraction

of the electorate, it well resist choosing a highly polarized and

divisive candidate as in the case of the Borda leader in Dummett's

example above. On this score, approval voting may well be a very

good approach to electing the most representative candidate and for

opening up our elections to independent and third party candidates.