[EM] Elections methods performance criterion

[Adam Tarr]

Ken Johnson wrote:

>Borda performed surprising well. I think this is because Borda, in effect, 
>tries to reconstruct a CR profile from preference rankings, and thus tends 
>to perform somewhat like CR. However, the simulations are only for 
>"sincere" Borda, and in practice voting strategy may degrade performance 
>in the same way that CR performance is degraded when people cast polarized 
>ballots (effectively equivalent to StAV). However, there is a catch: With 
>Borda, you can't adopt the same strategy without incurring a loss of 
>voting power. For example, suppose candidates are rated on a scale of 
>0...5. There are 4 candidates, C1 ... C4 and your corresponding sincere CRs are
>    C1(0), C2(1), C3(2), C4(3), C5(4), C6(5)
>Your optimum CR strategy is polarize your ratings,
>    C1(0), C2(0), C3(0), C4(5), C5(5), C6(5)
>With Borda, the sincere ranking translates to a Borda count identical to 
>the above sincere CR rating,
>    C1 < C2 < C3 < C4 < C5 < C6  -->  C1(0), C2(1), C3(2), C4(3), C5(4), C6(5)
>But if you try to use rank equality to mimic CR strategy, it doesn't work 
>quite the same,
>    C1 = C2 = C3 < C4 = C5 = C6  -->  C1(1), C2(1), C3(1), C4(4), C5(4), C6(4)

I believe that insofar as Borda allows equal rankings, you would score that 
ballot as
C1(0), C2(0), C3(0), C4(1), C5(1), C6(1).

>The strategy reduces your rating range from 0-5 to 1-4.

Borda would actually reduce your rating range from 0-5 to 0-1 in that 
case.  That said, Borda is resistant to such strategy only in elections 
with zero information.  If you know the probabilities of candidates winning 
the election, then you have incentive to pack as many candidates between 
your favorite and his/her closest competitors as possible.  Say, for 
example, that your sincere CR are:

A 100
B 75
C 65
D 50
E 49
F 30
G 20
H 0

And the expected probability of each candidate winning the election was

A 3%
B 35%
C 34%
D 2%
E 20%
F 3%
G 3%
H 3%

Your optimal Borda ballot then looks something like:

B>A>D>F>G>E>C.

In other words, you put as much space as possible between B (the 
front-runner you prefer) and his/her closest competitors (C and E).  This 
way, you get to cast six votes in the most important contest (between B and 
C).  You place C in last place, even though C is your third favorite 
candidate!  The strategy here is extreme and somewhat bizarre, but it would 
be common in Borda elections.

>Does this mean Borda is somewhat less susceptible to strategy than CR?

I think I just demonstrated why it is not.  CR strategy never becomes any 
more perverse than "who should I give 100 and who should I give zero?"  In 
Borda, you could have incentive to put your second favorite candidate alone 
in last place.

>(Or is there a possible way to make CR more strategy-resistant by 
>effectively diminishing the voting power of polarized ballots?)

I don't think so - this is why approval is popular around here.

-Adam