[EM] whipping-boy Borda

[Richard Moore]

David Gamble wrote:

 > Year 1
 >
 > An election is held using Borda for a single seat. There are two
 > candidates A and B. The voters give the following rankings:
 >
 > 12 A
 > 40 A>B
 > 30 B>A
 > 18 B
 >
 > A obtains 52 points (52x1 + 30x0), B obtains 48 points (48x1 +
 > 40x0). A wins.
 >
 > Year 2
 >
 > This time there are 3 candidates A,B and C. The voters give
 > identical rankings as in the previous election. Nobody expresses a
 > preference for C making candidate C the most irrelevant of
 > irrelevant alternatives.
 >
 > 12 A
 > 40 A>B
 > 30 B>A
 > 18 B
 >
 > This time the points are A= 52x2 + 30x1 and B= 48x2 + 40x1. A
 > obtains 134 points and B wins with 136 points yet identical ballots
 > have been cast in both elections.

Yes, a problem might be indicated here, but the real problem is not 
the IIAC violation. In fact, it's doing something we might expect it 
to do. (The following is not an endorsement of Borda, just a close 
look at what's really going on with Borda's IIAC violations.)

Suppose the only information you were given about the second election 
is that there are three candidates, that A was strictly preferred to C 
by 82% of the voters, and that B was strictly preferred to C by 88% of 
the voters. Given only that information you might be inclined to think 
B is a better fit. Perhaps the presence of C is revealing something 
about the true levels of demand for A and for B. In other words, C is 
adding information to the race.

Let's expand this example by adding another candidate D, with the 
following ballots cast:

Year 1 (with A, B, and D the only candidates):

12 A>B>D
40 D>A>B
30 B>D>A
18 B>A>D

Scores: A=82, B=108, D=110. D wins.

Year 2 (C also ran):

12 A>B=C>D
40 D>A>B>C
30 B>D>A>C
18 B>A=C>D

Scores: A=164, B=196, C=30, D=180. B wins.

Now there is no Condorcet winner in either year. C is still the 
Condorcet loser in the second year. The fact that 88% strictly prefer 
B to C, while only 70% strictly prefer D to C, has tilted the balance 
away from D and towards B in the second election.

The real problem with Borda is not that it processes the information 
afforded by C's presence in the race, but rather the indiscriminate 
way it processes that information. If both C and C's clone, C1, are in 
the race, then the effect of the information contributed by C's 
presence is doubled. The distortion that results when the candidate 
distribution in the issue space does not match the distribution of the 
voters in that space is the culprit (this is a generalization of the 
clone problem), rather than IIAC violations per se.

If clones are added from a region of issue space that is also heavily 
populated with voters, the effect isn't as bad, since the amount of 
boost provided by a set of clones to a candidate X depends on how many 
voters rank X above that clone set. Lots of candidates in Borda would 
be a good thing, *provided* they are distributed over the issue space 
in a representative way. Also, a single candidate from a sparsely 
populated and remote region of issue space can be as distorting as a 
pair of clones from a moderately populated region. Nothing is said in 
the example about how far C is from the more  populated regions; and 
as my 2nd year example suggests, in the larger candidate field of that 
example, C might be an acceptable candidate to 30% of voters.

The other real problem with Borda, of course, is the strong incentive 
it gives for voters to reverse their preferences.

  -- Richard