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What is Strategy?

Strategy comes in two forms, strategic voting and strategic nominating.

What is Strategic Voting?

For most voting methods there is a generally assumed sincere way of voting.  For example, in a ranked method, it is assumed that the sincere vote is to list the candidates in order of preference.  A strategic vote is one where the voter casts other than a sincere vote in the hopes of getting a more favourable outcome.  For example, in a plurality election the outcome might be

A 42
B 40
C 10

In this example, A wins.  C voters who prefer B to A might have got a better result from their point of view by voting for B instead.  This would be strategic voting.

What's the difference between Insincere, Strategic, and Sophisticated Voting?

An insincere vote is simply any one other than an expression of true preference (the sincere vote).  For example if a voter prefers candidate X to Y, but refuses to rank either, this is insincere.  The voter may do this on a matter of principle.  If the voter intends that the insincere vote will improve the election outcome, this would be a strategic vote.  If this is based on good strategy, considering the information available about how others are voting, then it is a sophisticated vote.

Are any methods immune to strategic voting?

In other words, are there any methods in which a voters sincere vote is always equal to  the best possible sophisticated vote.  There are.  For example Random Ballot.  In this method, one ballot is drawn at random, and the voters first preference decides the election.  This method is obviously highly random.  Of course, a vote that didn't actually decide the result would also not give any reason for strategic voting.

But I just invented a voting method that has no random component but which is also strategy free.

Sorry, but you didn't.  It isn't just that no one has thought of such a method; it has been proven mathematically to be impossible.

What different kinds of strategic voting are there?

Compromising-  Voting for a less favoured candidate over a more favoured in the hopes of preventing an even worse candidate from being elected.  All non-random methods are effected by this to some extent.  The vote following the dash shows how a group of voters can improve the result by strategy.

For IRV
45 A B C
20 B A C
35 C B A -- B C A

For those Condorcet methods which are equivalent to Minmax for three candidates
45 A B C
20 B C A -- C B A
35 C A B

For Average Ratings

       A     B    C
40   100     0   60 -- 100   0   100
60     0   100   70

Burying-  In this strategy you lower a candidate in the hopes of defeating it without electing someone worse.  For example,

For those Condorcet methods which are equivalent to Minmax for three candidates
45 A B C -- A C B
20 B A C
35 C B A

For Approval
    A      B      C
40  1      0      1
60  1      1      0  -- 0  1  0 (assuming this groups preference is B>A>C)

Push-over- Here, you raise a weak candidate in the hopes that this will help defeat a stronger enemy without actually winning.

From IRV
45 A B C  -- 39 A B C,  6 B A C
25 B C A
30 C A B
In this example, it is very important to A that C is eliminated before B.  After all, C's votes will be transferred to A, but B's would be transferred to C.

What is strategic nominating?

Often the election can be greatly influenced by what candidates are in the running.  Many methods are affected by Vote-splitting.

For example, in plurality assuming the sincere preferences are
25 A B C
30 B A C
45 C A B
Then, if voters vote sincerely, the result is
25 A
30 B
45 C
C wins.  But if A and B could organize ahead of time that only one would run, for example A, the result would be
55 A
45 C
based on the same preference.

Teaming- Some methods have the opposite effect, which I call teaming.  Here's an example from Borda.
45 A B
55 B A
A is the winner.  But if the group represented by B ran two candidates, B1 and B2, the results could be like this

45 A B1 B2
55 B1 B2 A
A 90 B1 155 B2 55
So, B1 won.  Obviously, it makes sense to run more candidates.

Crowding-  In this, the additional candidates neither helps nor hurts the group running them, but affects someone else.  If these candidates are actually proposals before a legislative body, then the outcome may be altered by cleverly suggesting similar proposals.

From Young,
40 A B C
35 B C A
25 C A B
A wins

But, if there are two C candidates
40 A B C1 C2
35 B C1 C2 A
25 C1 C2 A B
B wins.  This is because A loses to all members of the C group.  So the more members there are, the worse A is scored.

How do different methods compare with regard to resistance to strategy?

This is a subject of tremendous ongoing debate.  Since all the methods people advocate have some strategy, this debate involves what kind of strategies people are most likely to use, and which are more harmful.  Some methods, like approval voting, are often advocated partly on the basis that strategy will actually be beneficial.

If the vote has been conducted using ranked ballots, you might ask the question, "does a majority prefer candidate X to Y, or vice versa?"

To answer this question, you would go through the ballots counting the total number who rank X above Y, and the total who rank Y above X.  These tests are called pairwise, because they look at candidates two at a time.

If one candidate gets a pairwise victory against every other candidate, this is the Condorcet winner.
For example
45 A B C
20 B C A
35 C B A

45 rank A higher than B.  55 rank B higher than A.  So, B has a pairwise victory (or majority) against A.
65 rank B higher than C.  35 rank C higher than B.  So, B has a pairwise victory (or majority) against C.
Since B has a pairwise victory against every other candidate, he is the Condorcet Winner.  All the methods listed as meeting the Condorcet Criterion will always pick the Condorcet Winner if one exists.

Why choose the Condorcet winner?

Condorcet advocates generally view this as a natural extension of majority rule.  After all, if we do not choose the Condorcet winner, we must act against the majority opinion on one of these pairwise comparisons.  It is also often argued that if a majority prefers one candidate to another, the winning candidate is more likely the better candidate.  It follows that if one candidate is preferred to all others, this candidate is the best.

Furthermore, if the alternatives available form a spectrum (for exactly from left to right) then the Condorcet winner represents the median of voting opinion.  This is not necessarily the average position, but is the alternative with a majority saying they want this or an alternative to the left, and a different majority saying they want this or an alternative to the right.  So, by one definition, the Condorcet winner is in the centre of political opinion, when a centre exists.

How can the Condorcet winner not exist?

40 A B C
35 B C A
25 C A B

The pairwise results are:
A over B, 65-35
B over C, 75-25
C over A, 60-40

Since every candidate has a pairwise defeat, there is no Condorcet winner.

What do Condorcet advocates recommend when there is no Condorcet winner?

There is little consensus on this subject.  There are a number of methods, what are referred to as Condorcet criterion or Condorcet completion methods, that choose the Condorcet Winner if one exists, but also describe who to choose if one does not.  Check the Criteria page for methods that meet the Condorcet Criterion.

What is the Smith set / criterion / method?

Sometimes, even though it is not possible to find any undefeated candidate, it is possible to say that some of the candidates are only defeated by each other.  The smallest possible such set of candidates is the Smith set.  Note that a "set" of candidates is essentially just a "group" of candidates, except that mathematicians use the term "group" to mean something else.

For example, if there are candidates A, B, C, that have defeats like
A>B
B>C
C>A
then there is no way we can carve off one or two of these candidates and not have any defeats against them by the third.  So, the Smith set contains all three candidates.  If, however, we had more candidates
D>E
E>F
F>G
and any of A, B, C pairwise beats any of D, E, F, then the Smith set is {A, B, C}, and does not include all the candidates.

If there is a Condorcet winner, then then the Smith set consists of this candidate.

The Smith Criterion says that the winner should be from the Smith set.  This, of course, implies the Condorcet Criterion.  The Smith Method (also called Schwartz Method) eliminates all candidates not it the Smith Set.  This is usually combined with some other method to then choose between them.

What is LIIAC?

The Local Independence from Irrelevant Alternatives Criterion.  It says that not only must a member of the Smith set win, but the presence or absence of candidates outside the Smith set should not affect the result.  It is an attempt to provide as much independence from irrelevant alternatives as is possible in a ranked method.

How is the strength of a pairwise victory measured?

Some methods assign pairwise victories a strength so that it can be determined which prevail when there is a conflict.  There have been a number of ways suggested, but mainly either margins or winning-votes are used.  Consider a contest of 15 to 5

Margins-- use the margin of victories, in this case 15-5=10
Winning-Votes-- use only the votes on the winning side, 15
Losing-Votes-- use only the votes on losing side, 5

For Margins and Winning-votes, a higher score means a victory of greater strength.  For losing-votes, a higher score means a weaker victory.

These measures are clearly equivalent as long as only full rankings are given.  That is, they only make a difference if some people rank candidates as equal.

How can I make a pairwise table?

Every candidate is given a row and a column.  In each cell is written the number of voters who ranked the candidate corresponding to the row over the candidate corresponding to the table.  Some note of the number of voters should also be made.  Down the diagonal from top left to bottom right, the row candidate equals the column candidate.  Here, you could right 0 to show than no candidate was ranked above itself, but it is clearer just to mark an X.
 
 

voters=100	Bill Smith	Mary Robinson	Drew Peterson
Bill Smith	X	35	35
Mary Robinson	40	X	20
Drew Peterson	40	40	X

This shows that 40 people voted Mary over Bill.  35 Voted Bill over Mary.  This leaves 25 who ranked them the same.

Most methods will not use this table directly, although all other tables may be derived from it.  One possibility is to add 1/2 to each cell for every voter who ranks the candidates equally.  The total of the each row would then be that candidates Borda count.  Nanson (original) or Borda-Elimination could be implemented by eliminating a candidates row and column when it is eliminated, and recalculating the new Borda count among the remaining candidates.

If you want to use winning-votes as mentioned above, you would want to replace the losing side with 0, and vice versa.  To make a marginal pairwise matrix, each cell should show the margin of victory for the row's candidate over the column's candidate, or in the case where there is no victory, a 0.

Is every pairwise table possible?

No.  For example, it is possible to build a pairwise table showing a cycle of unanimous decisions, but this cannot happen based on the ballots.

What is the connection between the Condorcet winner and strategy?

In most methods it is possible, if not practical, for a majority of people to get a chosen candidate elected.  For methods meeting the Majority Criterion, this can be done if the majority ranks their chosen candidate first.  For methods like Average ratings, the majority can give their chosen candidate the highest rating, and all others the lowest.  Note that if a majority does this, it is not possible for the minority to change the result, even with strategic voting.

If their is a Condorcet winner for the voter's sincere preferences, and this candidate does not win, then by definition, a majority of the voters prefer this candidate to the winner.  As I have explained, this majority could have used strategy to get a result that would have been favourable to all its members.

We can therefore conclude that when the Condorcet winner loses in any of these methods, even those that do not meet the Condorcet criterion, it is because voters did not have enough information or were unwilling to use strategy.

If there is no sincere Condorcet winner, then there is no candidate whose victory is stable against strategy.  For example,
45 A>B>C
20 B>C>A
35 C>A>B

Whichever candidate wins, there will be a majority who prefer a different candidate.  This majority can get their way by ranking this candidate first.

Any way of judging the relative merits of electoral methods is a standard.

What is a Criterion?

A criterion is much like a standard except that

• It is worded as a pass/fail test
• It can be mathematically proven which criteria pass it
• This judgment can be made based only on the results of the method, without regard to the procedure used

Note that the word "criteria" is plural and the word "criterion" is singular.

What is a clone/twin?

It has been noted that some methods seem to hurt a party that runs multiple candidates, while some encourage it.  These are the vote-splitting, teaming, crowding issues mentioned above.  To try to define this issue in such a way that mathematical proofs would be possible, the idea of a was developed. 

It makes more sense to think of what constitututes a clone set then what is an individual clone.  A clone set is comprised of alternatives which are ranked the same relative to the other alternatives by all voters.  In a rated method, you would consider them to be alternatives which are rated the same by all voters.  For example

D A B C
B A C D
C B A D

In every ballot, A and B are ranked the same relative to candidate C and to candidate D.  In each ballot if A is above C, B is above C.  If A is bellow D, B is bellow D, and so on.  Another way of saying this is that A and B are always together on every ballot.  They never have any candidates outside the clone set between them.

The set of all alternatives is not considered to be a clone set, even though it otherwise would be one.

The idea of clones allows us to mathematically judge whether or not a method will be effected by the number of candidates representing a particular ideology.

What is an Irrelevant Alternative?

According to Arrow, an irrelevant alternative is any that does not win.  For example, in the following example,

40 A B C
25 B C A
35 C B A

What is monotonicity?

Consider

1. You have ballots which result in a loss for a particular candidate
2. You lower the candidates standing on some of the ballots
3. The candidate wins for these altered ballots

C is eliminated.  B wins.
But, if we change some, but not all, of the ballots from ACB to CBA, we can get

39 A C B
30 B A C
31 C B A

B is eliminated.  A wins.

The important point is that A won because of a change in the ballots that only demonstrated a reduction in A's support, or an increase of that of B and C.  Non-monotonicity (the property of having monotonicity violations) is common in methods that use elimination of candidates.  This is because in most methods, it helps a candidates standing to have more candidates he defeats, and hurts to have candidates who defeat him.  Often in these methods, if a candidate defeats another too badly, the result is that the candidate is eliminated, and the winner drops in standing.

How can I properly use Criteria?

Criteria are a useful tool for analyzing methods.  They can help to determine if a method meets your goals.  They should not be assumed to be goals in and of themselves, or to be criticized for not being so.

I suggest the following comment as an example of a common misunderstanding regarding criteria.

Sure, Borda fails Independence of Clones (ICC), but twins will never exist in a real election, so it doesn't matter.

In essence the speaker is criticizing ICC for being to easy to pass, and uses this as a justification for why Borda fails it.

Another problem occurs with a statement like the following:

Plurality, Minmax and Borda all fail ICC

Using the above definition of an Irrelevant Alternative, Arrow suggested the criterion that no Irrelevant Alternative should affect the result of an election.  This is called the Independence from Irrelevant Alternatives Criterion.
In short, Arrow proved that no method can satisfy all of a number of reasonable sounding criteria.  In particular no method can:

1.  Allow voters to rank candidates.
2.  Always choose the majority winner in a two-candidate race.
3.  Meet the Independence from Irrelevant Alternatives Criterion.

This can be easily proven from the following example

40 A B C
35 B C A
25 C A B

The pairwise results are:
A over B, 65-35
B over C, 75-25
C over A, 60-40

Are there any Methods that meet IIAC?
Yes.  This is done either by breaking criterion 1 or 2 above.  Here are some possibilities:
1.  Ignore the vote.  Allow a dictator to make the decision.  This violates 2.
2.  Only accept first-place choices.  This violates 1.
3.  Use Average Ratings.  Here, each voter gives a score (for example from 0 to 100)  to each candidate.  The candidate with the highest average score wins.  Although this is not a ranked ballot, ranks could be determined from this information, provided no two candidates are given the same rating.  The method clearly violates 2.

How can contradictory majority preferences be explained?

It is a commonly held opinion that the majority should rule.  However, in examples like the following:

40 A B C
35 B C A
25 C A B

The pairwise results are:
A is preferred by a majority over B, 65-35
B is preferred by a majority over C, 75-25
C is preferred by a majority over A, 60-40

Actually, there is a much simpler case that may be more familiar to you.  Consider a plurality election with results

40 A
35 B
25 C

Two points should be made here.  First, although there are conflicts in majority preferences, this doesn't mean that the individual voters feel a conflict.  Remember that when we say that "the majority" is saying contradictory things, the majorities that are in conflict are made up of different people.

Several ways have been suggested for resolving this paradox:

• Majority rule should be abandoned.
• Majorities that can conflict must not be true majorities.  pairwise majorities should be abandoned as a source of information.
• Although the majority may be more likely to be right, majorities can be wrong.  It is not surprising then that they can come into conflict.

Partly as a result of the problems with applying majoritarianism, and partly on its own merit, some people have tried to apply the common principle of "greatest good to greatest number" to elections.

Average Ratings- One way this can be done is with a method called Average Ratings.  Each person gives each candidate a score from 0 to 100.

          A            B        C
5         0           10      100
40      100            0        0
55       90          100        0

The average scores are:
A 89.5
B 55.5
C 5.0

The conclusion is that A has the greatest average (and total) support and will provide the greatest utility.  This is despite the fact that B is the first choice of a majority.

It should be pointed out that in Average Ratings there is no benefit to limiting the spread between best and worst rating.  That is, voters should rate their favourite candidate at 100 and least at 0.  Since the actual utility to different people may not vary by a common amount, this cannot be thought of as measuring utility, but of normalized utility.  I will refer to this as "satisfaction".

Approval- is equivalent to average ratings when the scale is limited to 0 or 1.  Although this is a rather crude approximation, some people justify approval on the basis that it will maximize satisfaction.

Borda- Borda assigns a different number to each rank.  If the difference in satisfaction between each two consecutive ranks on the ballot is equal, then this will maximize satisfaction.  In any case, it is often justified as estimating this.

Other Positional Methods- Borda's point assignments are not the only possible.  Often people suggest using different point assignments, usually giving higher scores to the top of the ballot, as a way to more closely represent maximum satisfaction.  Plurality is one such method.  If you believe that people are only concerned about their first choice, and uninterested in their others, then plurality maximizes satisfaction.

Do these methods behave less strangely than other methods?

This, of course, depends on what is meant by "strangely".  It is true that in some ways these methods correspond to most peoples intuition.  For example, if for a particular elected method candidate X is the winner for one group, and candidate X is the winner for another group, then one might expect that candidate X would be the winner in a larger group combining all these voters.  This is the case in the utility methods, but not necessarily in others.

One can also ask whether it is possible that a sincere vote might cause an election outcome less favourable to the voter than if the ballot had not been cast.  This is not possible using the utility methods, but it is under the others.

How have people applied the principle of majoritarianism to electoral methods?

Lots of ways.  Here are four requirements that are often suggested:

• A candidate who wins a majority of first place votes should win the election.  This is the Majority Criterion.
• A candidate who loses a pairwise contest should not win unless all candidates lose some pairwise contest.  This is equivalent to the Condorcet Criterion.
• A candidate who in a pairwise contest has a majority of the total voters against him should not win unless all candidates do.  That is, a majority of the voters participating in a pairwise contest is not necessarily a majority of the voters as a whole.  This requirement ensures that majorities of all the voters take precedence.
• If there is a set of candidates such that all candidates in the set pairwise beat all candidates outside the set, then the winner must be from inside the set.  This is the Smith Criterion.
• If a majority of voters rank all the members of a set over all the members outside the set, then the winner must be chosen from inside the set.  This is the Mutual Majority Criterion.
• The winner must receive a majority approval vote.

Lots of ways.  In particular, Borda is often advocated as a probabilistic method.  For every pair of candidates X and Y, we can suspect that the better X is than Y, the more people will see this and in fact rank X over Y.  Borda, therefore attempts to sum these numbers to get a most likely winner.  Borda has, however, been criticized on the basis that it assumes that these pairwise decisions are made independently in order to sum them.

Condorcet methods are also argued for on this basis.  If one decides that a pairwise majority decision is more likely to be right than wrong, then a candidate that receives a majority over all others can be thought to be the most probable best candidate.

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