CVD wants Alt.V to be fairer but it isn't: misleadingwebsite

[Bart Ingles]

Craig Carey wrote:
> 
> The Approval Vote has the problem that people can grab at power by
> adding preferences. That sounds like fun.
> 
> Once they add the k-th preference, then they stab themselves down
> and the important parties voted for with earlier preferences miss
> out. Which party would win depends a lot on the nuances of the hints
> on the voting papers that say how many marks ought be written onto
> the paper.

Yes, with approval voting it is to the voter's advantage to vote for
neither too few nor too many choices.  Congratulations.

With IRV it is often stated that your lower choices can never help
defeat your higher choices.  While only true if you ignore the
possibility of voter "bluff" strategy, there is a worse problem: your
higher choices can act as spoiler, and help defeat a more front-running
lesser-evil candidate who is really your best hope.  In this case, you
need to know enough to vote the lesser-evil ahead of your true favorite.

But to do this you need to be able to identify the front-running
lesser-evil.  This is the precisely the same information you need in
order to use best strategy in Approval.  But Approval strategy doesn't
require you to place your favorite below your lesser-evil choice, so
Approval is actually easier and more forgiving.

---------------------------------------------------------------

On a separate note, I have been looking at worst-case situations with
Approval, assuming there is a complete lack of polling info, and that
each voter uses the best zero-info strategy of voting only for
candidates with above-mean utility.  

As a possible measure of worst-case performance, use the ratio of the
winning candidate's utility to that of the utility maximizer, expressed
as a fraction.  It should be easy to see that the worst-case ratio in
the three-candidate case is 1/3 for FPTP, and 1/4 for Runoff or IRV. 
Pairwise and Borda have no worst-case guarantees.

For Approval, the worst case seems to be 1/2 with three candidates
(unless someone can come up with an example with a lower ratio:

            Utility
Votes   1.0                  0.5                   0
        ---------------------------------------------
  34     A                    B                    C
  33     B                    C                    A
  33     C                    B                    A

Zero-info Approval strategy in this case is for everyone to
bullet-vote.  The ratio of winning utility/maximizing utility is:

u(A) / u(B)
 = (34 * 1.0) / [(34 * 0.5) + (33 * 1.0) + (33 * 0.5)]
 = 34 / (33 + 33.5)
 = 0.51

The A and C voters' middle choices are all exactly at the mean, so if
any of these voters had even a slightly higher opinion of B, they would
cast an additional approval vote for B, making B the winner.

Generally, I suspect that the minimum ratios for elections with n
candidates, where n >= 3, is 1/(n - 1).  If true, this would make
Approval the only method (at least of those being discussed on this
list) with a stronger worst-case guarantee than FPTP.

With two candidates, the minimum ratio for all methods discussed here is
1/2.