[EM] How to convert a set of CR ballots to a set of Approval ballots

[Kevin Venzke]

Forest,

First, I have some thoughts on the below:

 --- Forest Simmons <fsimmons at pcc.edu> a écrit : 
> Yes, Joe Weinstein's strategy is what I had in mind.

I'm starting to think that the method is not so much about what strategy is
used, as about the high quality of information that each voter effectively
ends up with.

For example, consider the "weak centrist" ABC scenario.  I've argued in the
past that, using "BTE" strategy, the two factions won't approve B if his
utility is below midrange.  (I don't believe Weinstein's strategy gives a
decisive recommendation in this case, because the perceived odds of A and C are
equal.)

But in MPCR:
48: A>B>C
3: B>_>AC         
49: C>B>A
viabilities: A 96, B 103, C 98

new ballots:
48: AB>C
3: B>AC
49: C>BA
final score: A 48, B 51, C 49

Effectively what has happened is that the first faction has learned that they 
do not have the votes.  With this knowledge, no strategy would suggest bullet-
voting for A.

> 
> There are two reasons for this.
>[...]
> (2) Optimizing expected outcome pays off in the long run, but in the short
> run (one election) you really want to maximize the chance that your vote
> will make a positive difference.
> 
> To see this suppose that you must choose between a 50% percent chance of
> winning a thousand dollars and a 1% chance of winning sixty thousand
> dollars.  If you had the opportunity to play this game an hundred times,
> you would choose the higher expectation choice every time.  But if it were
> a one time opportunity, and you desperately needed the thousand dollars to
> make ends meet, you would probably take the 50% chance.

I do want to say that, as far as Approval itself, I think this analogy is a bit
off.  If we desperately need to make ends meet, then the utilities of those
prizes should be fairly close, and "BTE" strategy would also recommend the
50% game.  So the analogy only translates to elections if the voters care more
about affecting the result (=="getting some money") than getting the best result.

Let me give an example.  Say there are three candidates:
A: 40% odds, utility 10 for me
B: 10% odds, utility 2
C: 50% odds, utility 0

BTE strategy: Expectation is 4.2, so only approve A.
Weinstein's: You're more likely to be pivotal if you approve A and B.  But it
isn't worth it.  You may get yourself an outcome of utility 2 instead of 0,
but you're almost as likely to keep yourself from getting utility 10.

In MPCR you'll probably end up approving B, but (I'd say) it's because you've
learned that A's odds are probably 0%, not because approving B was the best
strategy to begin with.


Ok, enough of that.  Programming refinement #3 was very easy because of its
similarity to #1 (merge pair that minimizes sum of viability).  I'm afraid that
refinement #1 still looks to be the best in terms of picking the CW.  Here are
my stats for a standard 9-candidate simulation I'm using:

Refinement 3 (pair minimizing sum from pairs minimizing greater viability):
matches        8560
sdindec        246           2.87
mpcr<>cw       116           1.35    <--
mpcr=cw        5262          61.47
unanimous      286           3.34
mpcr=borda     138           1.61
mpcr=sd        520           6.07
borda=sd       1209          14.12
no matches     783           9.14

Refinement 1 (pair minimizing sum of ranks' viability):
matches        9085
sdindec        294           3.23
mpcr<>cw       88            .96     <--
mpcr=cw        5621          61.87
unanimous      502           5.52
mpcr=borda     207           2.27
mpcr=sd        718           7.9
borda=sd       1014          11.16
no matches     641           7.05

Refinement 2 (farther pair from frontrunner which includes min viability rank)
matches        3541
sdindec        101           2.85
mpcr<>cw       63            1.77    <--
mpcr=cw        2155          60.85
unanimous      312           8.81
mpcr=borda     65            1.83
mpcr=sd        318           8.98
borda=sd       316           8.92
no matches     211           5.95

"(method)<>cw" for some other methods with this setup:
Random Ballot: 39.38%
Random Candidate: 56.82%
place cutoff at Borda winner, nudged towards center: 11.85%

Maybe tomorrow I'll post an unusual outcome or two.  And about clones, I hope.

Kevin Venzke
stepjak at yahoo.fr


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