[EM] a strategic problem and possible remedy for Condorcet-efficient voting methods
jarmyta at antioch-college.edu
Wed Dec 17 21:24:02 PST 2003
Dear election method fans,
Below is a proposal for a voting procedure, preceded by a rationale for
that proposal, in sort of a paperish format. I hope that you find it
Also, I would greatly appreciate it if people could tell me whether
similar procedures have been proposed before.
You see, I'm kind of thinking about trying to apply to graduate school
for economics in the next year or two. That's my big dream right now, and
they say that it's easier to get in if you have published something. So I
thought that I should try to publish something on voting methods, since
that's sort of related to economics, and I don't know as much about other
branches of economics. I have a few ideas that I want to try publishing,
but this is the first one that I want to pursue... unless it has already
been proposed! So if it has been, please let me know (and be gentle...),
but if not, please give me credit if you repeat the ideas elsewhere. Also,
if it is a viable subject I was wondering if you all could give me
feedback and help get it into some sort of publishable shape (It probably
needs to be much shorter!), and maybe some advice as to where to submit it
So, anyway, I will greatly appreciate your feedback either way.
A Strategic Problem and Possible Remedy for Condorcet-Efficient Voting
by James Green-Armytage
In designing a single-winner voting procedure for the purpose of majority
rule, I take it as axiomatic that it should always select a Condorcet
winner when one exists, since a Condorcet winner is preferred by a
majority over all other candidates.
However, 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, or offensive order reversal.
Let me try to illustrate this strategy using an example. There are 3
candidates: A, B, and C. There are 100 voters. The sincere preferences of
the voters are as follows:
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
The pairwise comparisons are as follows:
A:B = 51:49
A:C = 46:54
B:C = 90:10
Minimax drops As 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. Ranked pairs and beatpath do the same thing as
minimax in this example and all others in this paper, but for simplicitys
sake lets 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 dropping defeats of least
margin (difference between winning and losing vote totals in pairwise
comparison) or least magnitude (winning vote totals in pairwise
If we are using a magnitude-based method, then in order for C to win, C
must beat or tie B, or the magnitude of Bs defeat over C must be less
than the magnitude of As 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:
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 Bs defeat over C must be less than
the margin of As 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:
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 Bs election by voting A>C>B, thus
cementing As 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 Condorcets
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. However, a similar
effect might take place without a grand conspiracy, but as a result of a
simple notion among the voters that there might be some benefit in ranking
their sincere second favorite in last place, if she is the main
competition for their favorite.
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 Condorcets method was being used by a
council or legislature to decide on different versions of a bill or
various courses of action.
In any case the burial strategy can often backfire by leading to the
election of someone you like even less that the second or third-favorite
candidate you are trying to bury. Thus, there are many situations where
the incentive to engage in such a strategy is outweighed by its risk, and
the number of voters who try it will be too small to be decisive.
But again, whether it is likely or not, the fact that a large-scale
burying strategy can conceivably happen is very disturbing, since its
effects can be so negative.
Whether a method offers incentives for burying strategies seems to be
related to whether reversing the order of later preferences on a ballot
can cause an earlier preference to be elected. For example, if a group of
voters rank 5 candidates in the order A>B>C>D>E, and C is elected, are
there any situations where that same group of voters could vote A>B>D>E>C,
and cause B to be elected instead, with all the other votes in the
election remaining constant?
If so, then there will be situations where voters will have incentives to
rank their sincere second favorite in last place, or their sincere third
favorite in fourth place, and so on. That is, in changing the order of the
candidates from the sincere order, voters will insincerely downrank
Given different methods, there are different kinds of strategies that
involve downranking later preferences to help earlier preferences. Some
Condorcet-efficient methods such minimax, ranked pairs, beatpath, find a
completion winner by overruling some majority preferences in favor of
others. Given methods like these, groups of voters can sometimes benefit
by creating an artificial majority against one of their later preferences
which overrules a sincere majority against one of their earlier
preferences, causing the earlier preference to win. This is the burying
strategy as discussed above.
Some Condorcet-efficient methods find a completion winner by reverting to
a different method that is not Condorcet-efficient, such as single
transferable vote or Borda. Given methods like these, groups of voters can
sometimes benefit by downranking one of their later preferences who is a
sincere Condorcet winner, so that the tally finds no Condorcet winner, and
the other method that is reverted to finds one of their earlier
preferences as the winner. In addition to giving voters truncation and
burying incentives in order to prevent a Condorcet winner from emerging,
these methods will also reintroduce the strategic incentives inherent in
their given completion method.
Unfortunately, I think that all Condorcet efficient methods give some
strategic incentives for further downranking later preferences in order to
help earlier preferences. That is, take any ranked ballot voting method
that satisfies universal domain, anonymity, Pareto, non-dictatorship. If
it is a method where a group of voters reversing the order of options
ranked after some candidate B cant change the result to B under any
circumstances, this implies that those rankings cant be looked at while B
is still in consideration. That is, B must be eliminated before they are
looked at. If this voting method eliminates candidates before all the
rankings are looked at, then it will not be able to avoid eliminating a
Thus, it seems that all Condorcet-efficient methods can be undermined by
voter strategy. However, I dont think that the correct response in light
of this is to give up on the Condorcet principle and stick with other
methods that dont offer these particular incentives. For one thing, the
Condorcet criterion is a highly desirable one, and secondly, these methods
have strategic problems of their own.
My thought is that we might be able to use a procedure which selects a
Condorcet winner when one exists, and yet which gives people some
opportunity to undo the effects of a burying strategy if one occurs.
I propose that since no fully deterministic vote-processing rule can
satisfy both Condorcet efficiency and resistance to the burying strategy,
we should consider systems that incorporate further human choice and
judgement after the initial balloting. I will propose such a procedure for
use within legislative bodies, and then I will propose a slightly modified
procedure for use in public elections where a single representative is
Proposal for use within legislative bodies:
A. Discussion. Ranked vote. Go to B.
B. Discussion. Yes-no vote on the winner from the previous ranked vote,
whether a Condorcet winner or the winner based on a chosen completion
method, such as ranked pairs or beatpath. If the relative majority votes
yes, then that option is selected as the final outcome. If the relative
majority votes no, return to A.
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 relative majority. Also, with the approval of a
relative majority, non-members of the Schwartz set from a previous ranked
balloting 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.
This procedure gives legislators a chance to discuss the winner given by
a completion method, and make an attempt to determine whether a burying
strategy has taken place. For example, they might look over the ranked
votes cast by other legislators and see if two very similar options are
placed suspiciously far apart on the ballot. They may fail to detect a
burial strategy if it exists, but they at least have an opportunity.
In general, the fact that the final outcome must be approved by a
relative majority ensures majority rule and prevents any strange surprises
from getting locked into place before people see 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 given the best 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.
Proposal for use in public elections:
A. Ranked vote. If a Condorcet winner exists, then this candidate is
selected as the final outcome. If no Condorcet winner exists, go to B.
B. Yes-no vote on completion method winner from previous ranked vote. If
the relative majority votes yes, then this candidate is selected as the
final outcome. If the relative majority votes no, go to C.
C. Ranked vote on candidates already included in the process. Return to B.
Note: Stages B and C should be combined in a single balloting. If the
relative 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
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 A and C in the public elections version is
that a Condorcet winner in stage A is automatically selected, but a
Condorcet winner in stage C must be confirmed by a relative 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 isnt 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.
If a Condorcet winner exists in the initial vote, it is a fairly
trustworthy option to pick. Note that a group of voters cant change a
candidate B from a non Condorcet winner to a Condorcet winner by changing
the order of candidates that they have ranked after candidate B. For
example, if a group of voters lists B as their first choice and B is still
not a Condorcet winner, there is nothing further that they can do to make
B a Condorcet winner.
Obviously no effective burying strategy has taking place if there is a
Condorcet winner, because this strategy depends on a fabricated majority
overruling a genuine majority through a cycle.
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
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 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
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.
I would recommend the legislative bodies procedure to any legislature or
council that can acquire the resources to process ranked ballots according
to a Condorcet completion method. I would recommend the public elections
procedure to any government that can acquire the resources to process
ranked ballots and to hold successive ballotings.
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