(This fact provides an argument in favor a two-party
system that makes most elections straight fights.)N. R. Miller
05/01/97
4th rev. 9/22/04
VOTING TO ELECT A SINGLE CANDIDATE
This discussion focuses on single-winner elections, in which a single candidate is elected from a field of two or more candidates. Thus it pertains to legislative elections within single-member districts (one representative per district) and to the election of executive officials (President, Governor, etc.) under a separation-of-powers (as opposed to a parliamentary) system.
The simplest single-winner election occurs when there are precisely two candidates, producing what the British call a straight fight. In this case, voting by Simple Majority Rule (SMR) strikes most people as fair and reasonable. Each voter votes for one or other candidate (or possibly abstains), and — apart from the possibility of a tie — one candidate must receive an (absolute) majority of votes cast and that candidate is elected. (In the event of a tie, maybe we flip a coin.)
A mathematician by the name of Kenneth May (1952) demonstrated that SMR, and only SMR, meets four conditions that we may want a voting rule to meet in a straight fight between two candidates A and B. These are May's conditions (slightly reformulated).
(a) Anonymity (of votes). We do not need to know who cast which vote to determine the outcome. In other words, all votes (and voters) are treated the same way.
(b) Neutrality (between candidates). If every vote for A becomes a vote for B and vice versa, the winning and losing candidates are reversed. In other words, the two candidates are treated the same way.
(c) Resoluteness. A tie occurs if half the voters vote for A and the other half for B. But otherwise, either A or B wins. In other words, we can’t have a “hung electorate” in the manner of a “hung jury.”
(d) Non-Negative Responsiveness. If A is the winner and then a voter switches his vote from B to A, A is still the winner. In other words, votes count “positively” if they count at all.
May demonstrated that SMR meets these four conditions and is the only voting rule that can do so. Moreover, when SMR is used in a straight fight, no voter ever has reason to consider voting other than for his or her more preferred candidate. That is, we can expect all voting to be sincere and to “honestly” represent voters’ preferences. Put more formally, in a straight fight SMR is strategyproof; no voter can ever improve the outcome with respect to his or her true preferences by misreporting those preferences on the ballot.
But once the number of
candidates expands to three or more, all sorts of problems arise. First of
all, many different apparently fair and reasonable voting rules (including
those discussed below, along with other more esoteric possibilities) are
available (and many are in actual use). Each such procedure reduces to Simple
Majority Rule in the two-candidate case, but different such procedures often
select different winners in the multi-candidate case. Moreover, all such
voting rules have evident flaws. Two important flaws are essentially unavoidable
in elections involving three or more candidates. As noted above, SMR (among
other procedures) is strategyproof in the two-candidate case. But no
voting procedure whatsoever is strategyproof given three or more candidates.
In addition, all voting procedures are vulnerable to spoiler effects
when the field of candidates expands or contracts — that, whether candidate
A or B is elected may depend on whether some third candidate
(the potential “spoiler”) enters the field or not.
(This fact provides an argument in favor a two-party
system that makes most elections straight fights.)
Note that, when we have three or more candidates, a voter’s preferences are not specified simply by listing a most preferred (top ranked) candidate; rather we must specify the voter’s full preference ordering over all candidates in the field, i.e., a first preference, second preference, etc. (We will simplify the discussion by assuming that voters are never indifferent between any candidates.) A collection of preference orderings for all voters is called a preference profile.
Here is an example to focus on. We use British party labels to identify three candidates — Labour, Liberal, and Conservative — one of whom is to be elected. While there are six possible orderings of three candidates, we first consider a simple profile in which only three of these orderings are present and we indicate the popularity of each.
Preference Profile 1
# of voters 46 20 34
1st pref. Labour Liberal Conservative
2nd pref. Liberal Conservative Liberal
3rd pref. Conservative Labour Labour
Under Simple Plurality voting (what the British call "first-past-the-post" or FTPT voting), such as is used in British parliamentary elections and most U.S. elections, each voter votes for exactly one candidate, and the candidate receiving the most votes wins. For the time being, let us assume that, under plurality voting, each voter votes for his or her most preferred candidate, i.e., votes sincerely. Here is the plurality ranking for Profile 1.
Candidates Votes Received (= First Preferences)
Labour 46 votes (winner)
Conservative 34 votes
Liberal 20 votes
A plurality election with sincere voting takes account of first preferences only — that is, only the top line of the preference profile. The plurality winner is the candidate who has the most first preferences; in the example above, the Labour candidate is the plurality winner (and wins under sincere plurality voting).
A majority winner is a candidate who has an (absolute) majority of first preferences. Clearly a majority winner is also a plurality winner; equally clearly, the reverse is not always true. And if there are three or more candidates and first preferences are dispersed, no candidate will be the first preference of a majority of voters. In Profile 1, there is no majority winner.
In the event that simple plurality does not give one candidate an absolute majority of votes, Plurality Runoff voting prescribes a runoff vote between the top two candidates in the plurality ranking. Thus in Profile 1 there would be a runoff between Labour and Conservative, which Conservative wins because the voters who most prefer the eliminated Liberal candidate all prefer Conservative to Labour and they are sufficient in number to overcome the Labour margin over Conservative with respect to first preferences . (A second trip to the polls can be avoided if voters rank all the candidates on a single ballot. This is called Instant Runoff Voting or IRA.)
Under Approval Voting (Brams and Fishburn, 1983), voters can vote for any number of candidates, and the candidate with the most such “approval votes” wins. In the three candidate case, this means that a voter can vote for just one candidate (as under simple plurality) or for two. (It should be clear that voting for all three is effectively equivalent to abstaining.) While approval voting has some advantages, it can be highly indeterminate. For example, given Profile 1 sincere approval voting can select Labour (if each voter votes for his most preferred candidate only), Conservative (if only voters in the 20-voter bloc cast two approval votes), or Liberal (if only voters in the 34-voter bloc cast two approval votes or if all voters cast two approval votes).
Under Borda Point Voting (proposed by the French philosopher Jean-Charles de Borda), votes rank the candidates on the ballot, and (in a three-candidate contest) candidates are awarded three points for each ballot on which they are ranked first, two points for each ballot on which they are ranked second, and one point for each ballot on which they are ranked third. Here is the Borda ranking for Profile 1:
Candidates Points Received
Liberal 220 points (winner)
Labour 192 points
Conservative 188 points
Finally, suppose we look at all possible pairs of candidates and see which candidate in each pair is supported by a majority of voters. (Apart from “knife-edge” ties, one or other candidate must have majority support.) In other words, let’s examine all possible straight fights. For Profile 1, we see the following:
Liberal vs. Conservative: Liberal wins by 66-34
Conservative vs. Labour: Conservative wins by 54-46
Liberal vs. Labour: Liberal wins by 54-46
Thus we can order candidates in terms of (pairwise) majority preference such that A is ranked over B if and only if a majority of voters prefers A to B. For the example above we get the following majority (or Condorcet) ranking:
Majority Ranking
1st pref. Liberal (Condorcet winner)
2nd pref. Conservative
3rd pref. Labour (Condorcet loser)
Notice that this “majority ranking” is precisely the opposite of the “plurality ranking” based on first preferences only and that it also differs from the “Borda ranking” based on full orderings.
The Marquis de Condorcet, a French philosopher and mathematician, proposed more than two hundred years ago examining pairwise majority preference in this fashion to produce the Condorcet voting rule, under which the candidate at the top of the majority ranking — called the Condorcet winner — is elected. More generally, a Condorcet winner is a candidate who can beat every other candidate is a straight fight.
You should be able to verify the following points, many of which are illustrated in Preference Profile 1. For any preference profile:
(1) a majority winner is always a Condorcet winner, but the reverse is not true;
(2) a plurality winner may not be a Condorcet winner; and
(3) a Condorcet winner may not be a plurality winner — indeed, a Condorcet winner may have the fewest first preferences (e.g., Liberal in Profile 1).
Although there may be a Condorcet winner in the absence of a majority winner, it is also true that a Condorcet winner does not always exist. It may seem puzzling how this can occur, since — apart from ties —every ranking must have a highest-ranked element. The explanation is that there may be no majority ranking at all. Consider Preference Profile 2.
Preference Profile 2
# of voters 46 20 34
1st pref. Labour Liberal Conservative
2nd pref. Liberal Conservative Labour
3rd pref. Conservative Labour Liberal
Notice that in Profile 2 first preferences are unchanged from Profile 1, so the plurality winner is the same as before and (as before) there is no majority winner. Conservative remains the plurality runoff winner but Labour becomes the Borda point winner, while approval voting remains indeterminate. But another crucial difference is apparent when we look at the straight fights:
Liberal vs. Conservative: Liberal wins by 66-34
Conservative vs. Labour: Conservative wins by 54-46
Labour vs. Liberal: Labour wins by 80-20
It is now impossible to
construct a majority ranking. Instead we have cyclical majority.
Since there is no majority ranking of the three
candidates, there is no Condorcet winner. Thus, we can add a fourth proposition
concerning Condorcet winners:
(4)there may be no Condorcet winner.
There may be a Condorcet winner even in the presence of a majority cycle, provided the cycle does not encompass all candidates. This can occur if there are four or more candidates, as in this example.
Preference Profile 3
# of voters 35 33 32
1st pref. B C D
2nd pref. A A A
3rd pref. C D B
4th pref. D B C
Candidate A is the Condorcet winner, yet there is a cycle including B, C, and D. This example also shows that, with four of more candidates, a Condorcet winner may have no first preferences at all.
A voting rule is Condorcet consistent if, given sincere voting, it always selects the Condorcet winner when one exists. While Condorcet voting is obviously Condorcet consistent, previous examples showed that Liberal may fail to win given Profile 1 under each of the other voting rules discussed, so none of them is Condorcet consistent. But since Condorcet voting does not always select a winner, it cannot be deemed a full-fledged voting rule comparable to the others discussed here. This is especially unfortunate because, in so far as Condorcet voting does select winners, it is (unlike the others) both strategyproof and not subject to spoiler effects.
Of course, to say that majority cycles may exist is not to say that they typically are present. Indeed, if preferences are structured in a simple way by ideology (or otherwise), cycles cannot occur. In British politics, the three major parties are generally perceived to be ideologically ranked from left the right in the following manner:
More leftwing: Labour
Relatively centrist: Liberal
More rightwing: Conservative
If voters commonly perceive this ideological dimension and each ranks candidates according to how “close” they are to the voter's own (most preferred) position on this dimension, voter preference orderings are restricted to the following admissible ordering:
Admissible Orderings Inadmissible
leftwingers centrists rightwingers Orderings
1st pref. Lab Lib Lib Con Con Lab
2nd pref. Lib Lab Con Lib Lab Con
3rd pref. Con Con Lab Lab Lib Lib
If preferences are restricted in this so-called “single-peaked” fashion, regardless of popularity each the admissible orderings is, it is always possible to construct a majority ranking, so a Condorcet winner always exists (Duncan Black, 19/48, 1958). You can check that Profile 1 draws orderings exclusively from the admissible types, while Profile 2 includes an inadmissible type.
Note the strength of the
“centrist” (Liberal) candidate in the admissible orderings. While it may
be that few voters most prefer the centrist, no one likes the centrist least.
The consequence is that the centrist candidate must be the Condorcet winner
unless an (absolute) majority of voters have the leftwing ordering or have
the rightwing ordering. Put otherwise (in the three-candidate case), the
centrist candidate fails to be the Condorcet winner only if one of the extreme
candidates is a majority winner.
We have to this point assumed that voters vote sincerely. But any voting rule with three or more candidates may give voters incentives to vote otherwise than sincerely.
Consider Profile 1 again. As we saw, Labour wins under Plurality Voting if voters are sincere. But it is also true that a majority of 54 voters prefers both other candidates to Labour. If they all vote for the same other candidate (either all for Liberal or all for Conservative), that candidate will win — an outcome they all prefer to a Labour victory. But doing this requires some members of this majority of 54 to vote insincerely, i.e., for their second preferences. Thus simple plurality voting (as well as other voting systems) can encourage what the British call tactical voting and most political scientists call strategic voting, i.e., non-sincere voting.
Of course, the problem
remains of how the 54 voter majority will coordinate their votes —
that is, will they vote for Liberal or for Conservative? Notice that, while
all 54 voters prefer to see Labour defeated, they disagree as to how
to defeat him, i.e., by voting Conservative or by voting Liberal. It is generally
believed that, in practice, tactical voting in Britain mostly leads Liberal
supporters to shift their votes “tactically” to their second-preference (Labour
or Conservative) candidate, because they typically observe pre-election polls
showing Liberal trailing well behind both other candidates, and they therefore
conclude that a Liberal vote is “wasted” and that they should vote for the
one of the two leading (non-Liberal) candidates that they prefer.
Under Plurality Runoff, the 46 voters who most prefer Labour would do better by ranking Liberal first, as this assures (in the absence of countermoves by other voters) a Liberal victory without a runoff, which outcome they prefer to the Conservative victory that otherwise results. Given Profile 1, no voters can change their Borda score ballots in a way that improves the outcome for them. Given Profile 2, if the bloc of 20 ranks Conservative first and the bloc of 34 ranks Labour third, then Conservative gets the most Borda points (208 vs. 200 for Liberal and 192 for Labour), an outcome all 54 such voters prefer to victory by the sincere Borda winner Labour. Given some other profiles, the opportunity for strategic manipulation under Borda point voting is far more glaring, as is illustrated by Profile 4.
Preference Profile 4
46 54
1st pref. Labour Conservative
2nd pref. Liberal Labour
3rd pref. Conservative Liberal
Labour wins if voting is sincere (demonstrating that Borda Point Voting can deny victory to a majority [and Condorcet] winner, i.e., Conservative), but the 54 Conservative-preferring voters can elect Conservative if they shove Labour down to third place on their ballots. In turn, the 46 Labour-preferring voters can counteract this by moving Liberal to the top of their ballots (the resulting Liberal victory being preferable to the 46 voters to a Conservative victory). Note that if strategic manipulation stops at this point (though it need not), Liberal is elected even though everyone prefers Labor to Liberal. (An even more perverse example of such strategic manipulation under Borda voting is presented in the Appendix.)
We now examine spoiler effects. Consider an individual who, when given a choice between Conservative and Labour only, chooses Conservative. We would think this voter mighty peculiar if he changed his choice to Labour in the event Liberal is added as a third option. But a sincere electorate using Plurality Voting may do exactly this, as can be verified by checking Profiles 1 and 2 (or thinking about the Bush/Gore/Nader example referred to in footnote 1). So can a sincere electorate using Borda point voting, as can be verified by checking Profile 4. That is, these procedures are subject to spoiler effects.
Plurality Runoff (and especially Instant Runoff Voting) is sometimes advocated on the grounds that it precludes such spoiler effects. It is true that Plurality Runoff is an improvement over Simple Plurality in this respect, in that a third candidate (such as Nader) with little first-preference support cannot act as a spoiler in what is essentially a straight fight between two major candidates, because the runoff will become precisely that straight fight. However, plurality runoff does not eliminate the spoiler problem, as is illustrated by Profile 1. Liberal would win a straight fight with Conservative, but will not even make it into the runoff if Labour enters the field. This is not a distinctive flaw in plurality runoff voting, however; as previously noted, the problem is unavoidable with three or more candidates.
However, Plurality Runoff does have another flaw that is distinctive (and avoidable). We wouldn’t expect a “reasonable” voting rule to respond negatively when a candidate’s position in a preference profile becomes more favorable — put otherwise, increased support in the electorate should never hurt a candidate. (This notion generalizes May's Non-Negative Responsiveness.) But Plurality Runoff can fail on this score.
Suppose we have three candidates A, B, and C, among whom first preference are fairly equally divided. Suppose that A and B go into the runoff, which is therefore decided by the second preferences of the voters who most prefer C. Suppose enough of these second preferences are for A that A wins the runoff. Now suppose the preference profile is revised in a way that makes “public opinion” even more favorable to A (without changing anyone's preferences between B and C). In particular, suppose that some voters who previously most preferred B now move A up to their first preference (but A still is not a majority winner). The result of this change may be that the number of first preferences for B falls below the number of first preferences for C, with the result that A and C are paired in the runoff, which is decided by the second preferences of the remaining voters who most preferred B. And it may be the enough of these second preferences are for C that C rather than A wins the runoff. Thus this added support costs A electoral victory. Here is a specific example.
.
Original Preference Profile 5 Revised Preference Profile 5
35 10 25 30 35 10 25 30
A B B C A A B C
B A C A B B C A
C C A B C C A B
Here is a related peculiarity of Plurality Runoff voting.
Preference Profile 6
5 6 4 [2]
B C A [A]
C B B [B]
A A C [C]
The preference profile is as shown above, but the two individuals with the bracketed preference orderings fail to vote. Thus the election outcome is determined by the remaining 15 voters. Candidates B and C are paired in a runoff, which B wins. This is a somewhat disappointing outcome for the two individuals who failed to vote, in that their second preference won. They regret their failure to get to the polls, since they wonder whether their first preference A might have won if they had not failed to vote. But it can be checked that, if they had gotten to the polls and voted according to their preferences, the outcome would have been worse, not better, for them. (Candidates A and C would be paired for a runoff, which C would have won.)
Appendix: Voting Rules, “Clone” Candidates, and “Turkey Raising”
Consider the following
preference profile, in which a Republican minority is united behind a single
candidate R but the Democratic majority is split between the two “clone”
candidates D1 and D2.
Democrats Republicans
35% 25% 25% 15%
D1 D2 R R
D2 D1 D1 D2
R R D2 D1
Simple Plurality voting is notorious for penalizing clone candidates. In this case, the Republican candidate would win due to the Democratic split, even though R is at the bottom of the majority ranking. (R is the Condorcet loser, beaten by both D1 and D2 in straight fights.) Of course, it is precisely the expectation of such outcomes under Simple Plurality voting that leads to party formation and party discipline, i.e., the Democrats have a huge incentive to hold a prior nominating convention or primary to choose between D1 and D2 and then send just one of the two clones forward against the Republican. Given the preference profile above, D1 would win the nomination and then the general election.
The question arises of whether there are other voting rules that can reduce, eliminate, or even reverse the self-defeating effect of running clone candidates.
First we may note that, given the profile above, Plurality Runoff (instant or otherwise) solves the clone problem. In effect, the first-round election functions as the (Democratic) primary and the runoff as the general election in which the Democratic majority gets its way. But if there are four or more candidates, Plurality Runoff does not treat clones so well and, as we have seen, it is subject to other problems in addition.
As noted previously, Steven Brams and Peter Fishburn advocate Approval Voting as a desirable voting rule that (among other things) does not punish clones. In the profile above, presumably (almost all) Democrats would vote for both D1 and D2, one of whom would be elected. Of course, by not penalizing clones, AV does not encourage party formation or party unity. For this reason, many political scientists are more inclined to support AV for primary elections and non-partisan elections than for partisan general elections.
A variation of one type of party-list PR (Proportional Representation) system presents another voting method that does not penalize clones who have the same party affiliation. Each voter votes for a single candidate, as under Simple Plurality, but this vote counts in two ways: first, as a party vote to determine which party wins the election and, second, as a candidate vote to determine which candidate of the winning party is elected. In the profile above, D1 would be elected.
Perhaps surprisingly, Borda Point Voing actually rewards the running of clones. Suppose that there are two candidates and Republicans are again in the minority.
60 voters 40 voters
D R1
R1 D
With just two candidates, the Borda point rule is identical to Plurality Voting (and Majority Rule), so the Republican candidate R1 loses. But, if Borda Voting is in use, the Republicans can reverse the outcome by nominating an additional clone candidate R2 whom every one sees as identical to R1 with respect to issues and ideology but inferior with respect to (let’s say) personal qualities.
60 voters 40 voters
D R1
R1 R2
R2 D
Now R1 wins with 60×2 + 40×3 = 120 + 120 = 240 points, while D gets 60×3 + 40×1 = 180 + 40 = 220 points and R2 gets 60×1 + 40×2 = 60 + 80 = 140 points. Of course Democrats can counteract this by strategically ranking R2 above R2, thereby reducing R1 to 60×1 + 40×3 = 60 + 120 = 180 points and raising R2 to 60×2 + 40×2 = 120 + 80 = 200 points, allowing D to win with the unchanged 220 points. Alternatively, they can counteract by running their own clone. Though it has strong advocates, the Borda scoring system evidently is highly susceptible to strategic maneuvers of this sort (which, moreover, have the effect of expanding the candidate field rather than winnowing it down in the manner of Plurality Rule).
Here is a considerably
worse thing that the Borda point rule can do.
Suppose are three candidates: a more or less
reasonable Democrat D, a more or less reasonable Republican R,
and a real “turkey” T (e.g., a Lyndon LaRouche type). Everyone one
ranks T last, except two deranged supporters. The profile is:
50 voters 48 voters 1 voter 1 voter
D R T T
R D D R
T T R D
Voting is by the Borda rule. It easy to see right off that, if everyone votes sincerely, D wins (the same outcome as under Simple Plurality). Doing the arithmetic, the point totals would be D = 249, R = 247, and T = 104. Anticipating this defeat, Republican voters caucus and notice an interesting feature of Borda Point Voting — it can pay voters to engage in “turkey raising,” i.e., to strategically raise the “turkey” in their ballot rankings, so as to push the rival “serious” candidate down in their rankings. Suppose the Republicans strategically modify all their ballots so as to produce the following ballot profile:
50 voters 48 voters 1 voter 1 voter
D R T T
R T D R
T D R D
The point totals would now be D = 201, R = 247, and T = 152, producing a clear R victory. But suppose that before the actual balloting takes place, Democrats also notice this feature of Borda Voting and, concerned that Republicans may engage in turkey raising, they determine that they must engage in some turkey raising of their own in order to counteract the anticipated Republican stratagem. So the final ballot profile is:
50 voters 48 voters 1 voter 1 voter
D R T T
T T D R
R D R D
And the final point scores are D = 201, R = 197, and T = 202. May the best turkey win!
References
Kenneth Arrow. Social Choice and Individual Values. New York: Wiley, 1951.
Duncan Black. “On the Rationale of Group Decision-Making,” Journal of Political Economy, 1948
Duncan Black. The Theory of Committees and Elections. Cambridge, England: Cambridge University Press, 1958.
Jean-Charles de Borda, “On Elections by Ballot,” 1784.
Steven Brams and Peter Fishburn. Approval Voting: Birkhauser, 1983.
Marqis de Condorcet. “An Essay on the Application of Analysis to the Probability of Decisions Rendered by a Plurality of Voting,” 1785.
Kenneth May. “A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision,” Econometrica, 1952.
Burt Monroe. “Raising Turkeys: An Extension and Devastating Application of Myerson-Weber Voting Equilibrium.” Presentation to 2001 Annual Meeting of the American Political Science Association (http://accuratedemocracy.com/archive/condorcet/Monroe/ 004004MonroeBurt.pdf)
T. Nicolas Tideman, “Independence of Clones as a Criterion for Voting Rules,” Social Choice and Welfare, 1987.