An Iterative Pairwise Voting Procedure
by James Green-Armytage
In
public elections, voting procedures that require multiple rounds are cumbersome
and costly. However, if you have a group small enough to fit in a single room,
the marginal cost of extra rounds of voting is rather insignificant. In
particular, I am thinking of voting within legislatures and councils. So, when
voting in smaller groups, the question is whether multiple-round voting
procedures have advantages over the best single-round procedures.
Definition of the iterative procedure
For
single-winner voting, I prefer methods that are Condorcet efficient. Hence, I’d
like to propose a rather simple multiple-round variation of Condorcet methods.
The procedure is as follows:
1.
Discussion. Ranked vote.
2.
Pairwise tally, using a Condorcet-efficient method such as beatpath or ranked
pairs.
3.
The winner of the tally is announced. Discussion. Yes-no vote on the winner
from the previous ranked vote. If the majority votes yes, then that option is
selected as the final outcome. If the majority votes no, return to step 1.
Note:
At any discussion stage, a particular option can be withdrawn, either by the
sponsor of that option, or by being nominated for withdrawal and confirmed by a
majority. Also, with the approval of a majority, non-members of the Schwartz
set from a previous ranked vote can be removed from further consideration. The
purpose of these measures is to simplify the process by eliminating options
that can be agreed to be irrelevant.
Now
that I have defined the procedure, I return to the question of whether it
offers any benefit at all over single-round applications of beatpath or ranked
pairs. My belief is that it does.
Strategic manipulation in
Condorcet methods
Perhaps
the most important benefit dealing with the possibility of strategic
manipulation in these methods. That is, it seems that any ranked ballot vote
processing rule that is completely Condorcet efficient is also vulnerable to
manipulation using a strategy known as “burying.” Burying is defined by Blake
Cretney as “Insincerely ranking an alternative lower in the hope of defeating
it.” (See http://condorcet.org/emr/defn.shtml)
Let
me try to illustrate this problem using an example. There are 3 candidates: A,
B, and C. There are 100 voters. The sincere preferences of the voters are as
follows:
46: A>B>C
44: B>A>C
5: C>A>B
5: C>B>A
A
is the sincere Condorcet winner, with no cycles present. However, B voters can
“bury” A on their ballots by voting him last, which produces this result:
46: A>B>C
44: B>C>A
5: C>A>B
5: C>B>A
The
pairwise comparisons are as follows:
A:B = 51:49
A:C = 46:54
B:C = 90:10
Minimax
drops A’s defeat over B (which has a magnitude of 51 votes and a margin of 2
votes), leaving B the winner. This strategy has clearly paid off for the B
voters. Beatpath and ranked pairs do the same thing as minimax in this example
and all others in this paper, but for simplicity’s sake let’s assume that the
completion method being used is minimax.
If
the 46 A>B>C voters find out that the B voters are planning to use this
strategy, can they do anything to stop it? Yes and no. If the ballots cast by
the other 54 voters in the second situation above remain the same, there is
nothing that the 46 A>B>C voters can do to get A elected. The only thing
that they can do is to threaten to elect candidate C if the B voters do not
drop their order reversal strategy.
Their
means of carrying out this threat varies depending on whether we are using a
version of minimax that is based on margins (difference between winning and
losing vote totals in pairwise comparison) or winning votes (winning vote
totals in pairwise comparison).
If
we are using a winning votes-based method, then in order for C to win, C must
beat or tie B, or the magnitude of B’s defeat over C must be less than the
magnitude of A’s defeat over B. The A>B>C voters can achieve this if at
least 40 of them truncate their ballots, voting A>B=C. For example:
40: A>B=C
6: A>B>C
44: B>C>A
5: C>A>B
5: C>B>A
A:B = 51:49
A:C = 46:54
B:C = 50:10
If
we are using a margins-based method, then in order for C to win, C must beat or
tie B, or the margin of B’s defeat over C must be less than the margin of A’s
defeat over B. In this case truncation on the part of the A voters will not
suffice, and they are forced to do some order reversal of their own in order to
carry through their threat and prevent the B voters from stealing the election.
At least 34 of the 46 A>B>C voters need to do this for it to work, for
example:
34: A>C>B
12: A>B=C
5: C>A>B
5: C>B>A
A:B = 51:49
A:C = 46:54
B:C = 44:44
So,
these are some of the ways that A can derail the B voters’ burying strategy and
punish them with the election of C. However, the election of C is a very
undesirable result in itself, and it is not clear whether the A voters’ threat
will scare the B voters into voting sincerely, resulting in the election of A.
Perhaps the B voters will carry through their burying plan, without the A
voters following through on their threat. This would result in the election of
B. Perhaps the B voters will carry through their burying plan, and the A voters
will carry through their threat. This would result in the election of C.
Perhaps the 5 C>A>B voters will end the trouble and prevent the danger of
B’s election by voting A>C>B, thus cementing A’s victory. Or perhaps
those 5 voters will prefer to wait and hope that the fight between A and B
throws the election to C. Using a margins-based version of minimax could add in
some more complications, including a situation where A and B throw the race to
C without any insincere intentions, but instead out of a sense of mutual
paranoia that the other group of voters will carry out a burial strategy.
To
make a long story short, the voters have entered into a complicated strategy
game, the outcome of which is unclear. In some ways it is analogous to the game
of chicken. The A voters’ swerving could be not carrying through their threat
and allowing the B voters to successfully use the burial strategy. The B
voters’ swerving could be voting sincerely and allowing A to win. The car crash
would be the election of C.
It
is disturbing that it is possible for elections based on Condorcet’s method to
break down into this sort of situation as a result of the burial strategy, that
is an intense strategy game amongst the voters, with a strong possibility of a
highly unpopular candidate being elected.
Also,
it is disturbing that the burial strategy can be effective in the first place.
Imagine that this example was a Presidential election in a country with
millions of voters, and that the figures represented percentages of the turnout
rather than single voters. The contest between A and B would obviously be the
main focus of the election, as 90% of the voters prefer them to C. The 2% point
difference in the pairwise contest between A and B would represent thousands or
millions of voters. If the B voters pulled off an order reversal strategy under
these conditions, the democratic process would have been completely undermined.
Of
course, the chances of this happening in a public election are not necessarily
very great. Any candidate whose campaign staff called up voters by the
thousands and instructed them to cast an insincere vote might be held up to a
certain amount of public shame. And without a coordinated effort informed by a
rather reliable sense of how other groups are going to vote, the risk of such a
strategy might exceed the possibility of reward.
Well-coordinated
and successful burial strategies might become more likely given a smaller
electorate where it is easier to figure out how other people are voting and
easier to create a strategy covertly. For example, this might be a problem if
Condorcet’s method was being used by a council or legislature to decide on
different versions of a bill or various courses of action.
While
Condorcet efficiency is an extremely desirable property for single winner
voting methods, all Condorcet-efficient methods can, in some circumstances,
offer the incentive to strategically bury a candidate. That is, in Condorcet
methods, if your sincere ordering is A>B>C>D, it is possible to that
making changes in your rankings after A can help to elect A more than a sincere
vote. The specific reason for this is that voting A>C>D>B, for
example, can do more than a sincere vote in terms of hurting B, and hence can be
advantageous if B is the closest rival to A.
By
the way, here is another example that is actually a bit more difficult than the
first example in some ways, because it take fewer altered votes to execute a
successful burial strategy, and because if just a few of the A voters and a few
of the B voters decide to truncate, it will result in the election of C, the
sincere Condorcet loser.
Sincere preferences
27: A>B
25: B>A
24: C>A
24: C>B
A:B = 51 : 49
A:C = 52 : 48
B:C = 52 : 48
A
is a Condorcet winner. But if just a fraction of the B voters reverse their
preferences, they can steal the election for B.
27: A>B
21: B>A
4: B>C
24: C>A
24: C>B
A:B = 51 : 49
A:C = 48 : 52
B:C = 52 : 48
The
C-->A defeat causes a cycle in which the A-->B defeat is dropped. B wins.
Benefits of the iterative
procedure
Now
that I have outlined this strategic problem, I come to the question of whether
an iterative procedure can do anything to cope with it. The answer is yes, I
believe, but not an unqualified yes. That is, the iterative procedure cannot
make this problem go away altogether. Any Condorcet-efficient method will
retain some possibility for burying incentive, and other methods which avoid
this particular problem can be said to have other strategic problems which are
equally grave.
However,
I do think that the iterative procedure can help to mitigate this problem. For
one thing, let’s imagine a situation where A is a sincere Condorcet winner, but
a group of B>A>C voters insincerely vote B>C>A, causing B to win
the initial tally (step 2).
Such a
group can only be successful if there is a majority rule cycle in the final
result. That is, B>A>C voters cannot alter the fact that A beats B in
pairwise comparison, but they might be able to create a fake C-->A defeat
which overrules the A-->B defeat. Hence, the existence of a majority rule
cycle should be a cause for concern, and in step 3, the voters should try to
discuss and understand the meaning of the cycle. (On the other hand, if step 2
indicates a clear Condorcet winner, I think that acceptance of that option in
step 3 should be fairly automatic.) In their discussion, the voters should ask
whether it is a sincere cycle or one that is caused by strategic manipulation.
It is possible, of course, that the voters will not be able to answer this
question correctly, but at least they will have an opportunity to look into it
before making a final decision on the issue. In small groups it should be
easier to investigate this matter, since the voters should be relatively aware
of each other’s ideas on the issue. Also, in some situations there might be
something clearly illogical about a B>C>A vote, for example if A has much
more in common with B than C does. If people get suspicious, they can ask the
B>C>A voters about their specific logic in voting C>A... if they are
unable to present a convincing answer, then it might be clear that A is in fact
the logical winner. If a majority accepts this idea, then they can easily get A
elected, simply by voting “no” on B, raising A up to first place on their
ballots in the subsequent ranked vote, and then voting “yes” on A to finalize
the decision.
Of
course, it shouldn’t be expected to work so smoothly every time, but the point
is that the procedure provides a line of defense against strategic incursion, a
line of defense which is, in my opinion, fairly formidable. When you consider
the very low marginal cost of additional rounds of voting within a legislature,
it seems to be rather clearly worth it for this extra degree of integrity.
When
dealing with multiple options in a small group, it seems unnecessarily hasty to
have a final binding vote right away. Why not take the extra time to be sure
that a majority is comfortable with the outcome, before locking it into place?
Indeed, I think that this procedure more fully satisfies the idea of majority
rule than a single-round procedure, since a majority needs to indicate very
specifically that a given option is acceptable in order for it to pass.
Although
it is not particularly likely, it is possible that strategic voting can be used
to overrule some considerably strong majorities. For example
Sincere preferences
29: A>B>C
26: B>A>C
36: C>A>B
9: C>B>A
A
wins
Altered preferences
29: A>B>C
26: B>C>A (these are strategically
altered)
36: C>A>B
9: C>B>A
B
wins
In
this example, A is a sincere Condorcet winner preferred over B by a sincere
majority of 65:35, but the B>A>C voters are able to change the outcome to
B by burying A. Such a result would be unlikely to stand given an iterative
procedure, but if it was locked into place using a single-round procedure, it
could be a real disaster. Hence, the stability concerns in single-round
Condorcet are not insignificant.
In
general, the iterative procedure prevents any strange surprises from getting
locked into place before everyone sees them coming. It is possible that
legislators will wrestle with a variety of strategies and counter strategies,
drawing the process into several repetitions. However, they have been a very
strong set of tools available for building a majority decision. If the process
goes into a deadlock where the amount of repetitions exceed the patience of the
legislature and the issue is dropped, this is arguably a natural deadlock which
could not be given a truly satisfactory resolution by another method.
Iterative procedures for use in
public elections
It
is not clear to me that an iterative procedure would be worth its cost in a
large public election. Actually, in such a situation I would ideally prefer the
use of the weighted pairwise comparison method
that I have proposed elsewhere,
which I think would reduce the severity of strategic incursion and decrease the
likelihood of severely unstable disequilibria. However, I’ll include the public
elections procedures here for the sake of completeness.
Multiple round version:
1.
Ranked vote. If a Condorcet winner exists, then this candidate is selected as
the final outcome. If no Condorcet winner exists, go to 2.
2.
Yes-no vote on completion method winner from previous ranked vote. If the
majority votes yes, then this candidate is selected as the final outcome. If
the majority votes no, go to 3.
C.
Ranked vote on candidates already included in the process. Return to 2.
Note:
Stages 2 and 3 should be combined in a single balloting. If the majority votes
yes on the option presented by the previous ranked vote, then the subsequent
ranked vote is of course irrelevant. However, in order to save time and
resources (and keep turnout high) it is better to perform the subsequent ranked
vote at the same time as the yes-no vote. The gap between the ballotings is a
matter of preference. I imagine gaps of a week or so.
Note:
Any candidate is free to withdraw in between ballotings, but no candidates can
enter beyond the initial vote. Thus, the number of candidates can only decrease
given subsequent rounds, simplifying the process.
The
discussion that is an important part of this process would hopefully still take
place, but since it is a public situation with a large number of voters, the
discussion would rely on some types of media, and hence the quality of
deliberation would rely on the structure of public media.
The
only difference between 1 and 3 in the public elections version is that a
Condorcet winner in stage 1 is automatically selected, but a Condorcet winner
in stage 3 must be confirmed by a majority.
The
fact that a Condorcet winner from the initial vote is automatically selected is
a trouble-saving device which I have put into the public elections version but
not the legislative body version. It isn’t much extra trouble for a legislature
to take an extra vote to confirm a Condorcet winner, but in a public election
the cost and trouble of an extra balloting would be significant.
The
possibility of a large number of repetitions of this process would be more of a
problem for a public election than for a legislative decision, because of the
larger cost of subsequent votes, and the possibility of term limits. Hence, a
question remains about whether to limit the number of repetitions, and if so,
how to do so.
One
could go on repeating the process indefinitely until a relative majority
approved the outcome, taking majority “no” votes as an endorsement of the
status quo. At the end of a term limit, one would have to ask the
representative in question to step down in favor of a substitute such as a Vice
President, who would hold the office until the conclusion of the ranked vote.
However, this would be awkward, the repeated ballotings might be expensive, and
the instability of a temporary office holder might be undesirable.
One
could place a specific the number of repetitions ahead of time, for example
declaring that the results of the fifth ranked vote were final and binding.
However, all of the strategic concerns relevant to Condorcet-efficient method
would apply here once again.
Perhaps
the nearest solution is to declare a candidate to be the final selection once
they have been the winner of a certain number of ranked votes, whether a clear
Condorcet winner or based on a completion method. For example, if a candidate A
wins three separate ranked votes, candidate A is elected.
Hopefully,
however, these kinds of rule will never come into play. Even if no Condorcet
winner is found in the initial vote, one can hope that the majority will
approve whatever completion method winner is given, and hence only one
additional balloting will be necessary. The primary purpose of the subsequent
votes is to serve as a safeguard against burial strategies, and if the majority
is not convinced that such a strategy has affected the outcome, they should
approve the completion method winner. Even if they do not approve the first
winner that comes forward, I imagine that the cycle should collapse into a
Condorcet winner within a couple rounds, through the withdrawal of other
candidates in the cycle, or through the consolidation of voters who were split
between two candidates to support a single candidate.
1. Ranked vote. If a Condorcet winner exists, then this
candidate is selected as the final outcome. If no Condorcet winner exists, go
to 2.
2. Non-members of the Schwartz set (union of minimal
undominated sets) based on the initial vote are eliminated. There is an interim
period where candidates who are members of the Schwartz set are also given the
opportunity to drop out of the race and have their named deleted from the
second ballot.
3. Ranked vote between remaining candidates. If a
Condorcet winner exists, then this candidate is selected as the final outcome.
If no Condorcet winner exists, the outcome is determined by a given Condorcet
completion method, such as beatpath or ranked pairs.
Note: Another optional feature of the two round
procedure would be to include on the second ballot an up-down vote on the
winner of the first ranked vote according to the given completion method. If
more people vote yes than no, then the completed result of the first ranked
vote is the final outcome. If more people vote no than yes, the result of the
second ballot is the final outcome.
This two round-procedure, although it would not allow
such a drawn-out equilibrium-building process as the multiple-round version,
would at least give people some opportunity to identify strategic abuses and
coordinate a response. The withdrawal of candidates might play a more important
role in the two-round version than in the multiple-round version.
Incorporating weighted pairwise
into an iterative procedure
Again,
for public elections, I’m currently advocating a single round of the weighted
pairwise method. However, for small group elections I strongly prefer an
iterative procedure. In that case, the method that is iterated could be
weighted pairwise, rather than standard beatpath or ranked pairs. I don’t feel
strongly that weighted pairwise should be used in this situation, but if the
marginal cost is not too substantial, I would recommend it.
Using an iterative procedure with
proportional representation methods:
A similar iterative procedure can also be used with multiple-winner proportional representation methods, such as CPO-STV. Applying the principle should be straightforward. For example, you can simply take a provisional CPO-STV vote, and then hold a yes/no vote on the full outcome, where once again a majority “yes” vote is required for it to pass. Granted, it is a slightly strange mixture of majoritarianism and proportional representation, but it seems that the ability of the majority to choke out minority options in the final outcome would be rather limited. That is, they can vote down an outcome which includes minority viewpoints, but they cannot create an outcome that excludes them; hence if they want to pass any outcome at all they will have to pass an inclusive one.