A Manipulable Voting Example

[Bruce Anderson]

I came across the following interesting example while looking for something 
else.  (I eventually found the other example, but it's nowhere near as 
interesting as the one given here.)

Suppose that a community has 101 voters, that A, B, and C are the only 
candidates under consideration, that the polls are open all 24 hours on election 
day, that 99 of the 101 voters voted in the morning, that the other 2 have not 
voted yet, and that the community is having a party on election night.  Suppose 
that, for any of several reasons, the 99 voters who have already cast their 
ballots are very vocal about their voting.

Suppose that the 2 voters who have not yet voted completely agree with each 
other about the candidates and issues, and they begin to take notes on how the 
other 99 voters say they voted.

Suppose that the voting method being used in the election is the version of 
Condorcet's method as "officially" defined on this list (EM-Condorcet).  
Accordingly, ranked ballots are used, with ties and truncations allowed.

Suppose that, at the party, the 2 aforementioned voters compile the following 
data concerning the other 99 votes:

 27:  A,(B&C)
 16:  (A&B),C
  7:  B,A,C
 23:  (B&C),A
 26:  C,(A&B)
---
 99 = # of votes already cast.

Suppose that those 2 voters have a strong preference for B over C, and they also 
have a strong preference for C over A (so they have any very strong preference 
for B over A); i.e.,

  2:  B,C,A
---
101 = total # of voters.

Doing a quick hand calculation, those 2 voters see that, if they leave the party 
before midnight and vote honestly, then their second choice, C, will win.

Then they wonder, maybe they could just stay at the party and C would win 
anyway.  So they compute who would win if they did not vote.  They find that if 
they stay at the party, and the election is determined by the 99 votes above, 
then their first choice, B, will win.

But suppose they feel that they have to vote, and they wonder if they can 
manipulate the result so that B wins.  They consider truncation, i.e., voting:
  2:  B,(A&C)
instead of:
  2:  B,C,A,
even though they strongly prefer C over A.  Sure enough, B wins again.

Then they notice that, if they truncate, they would be the only voters who don't 
express a preference between A and C.  Since the aggregated tabulation of 
preferences will be publicly reported, they decide that they'd rather not 
truncate in this manner if they didn't have to (and yet still be able to 
manipulate the result so that B wins).  They don't want to try order-reversal, 
i.e., voting:
  2:  A,C,B
instead of:
  2:  B,C,A.
But what about pair-reversal, e.g., voting:
  2:  B,A,C
instead of:
  2:  B,C,A?
Sure enough, B wins in this case also.

In short, according to the EM-Condorcet voting method, if those 2 voters vote 
honestly, then their second choice, C, wins; but if they don't vote, they 
truncate, or they pair-reverse, then their first choice, B, will win.  For 
comparison (assuming I did the calculations correctly):

Voting                   Stay             Pair-
Method           Honest  Home  Truncate  Reverse 
---------------  ------  ----  --------  -------
EM-Condorcet:       C     [B]     [B]      [B]  
Condorcet(1\2):     C      C     [B&C]     [B]
Borda:              B      C       B        B
Hare:               C      A       A        A
Kemeny:             C      C     [B&C]     [B]
Nanson:             C      C       C      [B&C]
Plurality:          C      C       C        C
Pl-runoff:          C      A       A        A
Reg.-Champ.:        C      C       C        C

Candidates in brackets, i.e., [B] or [B&C], indicate places where strategic 
advantage is gained by manipulation.

This example certainly does NOT mean that the EM-Condorcet voting method is 
necessarily worse than any other particular voting method.  What does it mean 
(if anything)?  Perhaps that should be the subject of future messages.

Bruce