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

[Forest Simmons]

On Sun, 3 Aug 2003, [iso-8859-1] Kevin Venzke wrote:

>  --- 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.

True, it's not exactly Joe's strategy, but has some of max likelihood
spirit.

> 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.

The recursive rule is more in this spirit than the other rules.

The recursive rule takes more of a global look at the distribution of
viabilities with an eye towards this semi-final position.

In fact, in the case of n candidates, three slots, a voter can
use his apriori viabilities and his full ordering of the candidates to
decide where to put the two cutoffs.

Suppose that your apriori viabilities were

1,2,8,6,5,4,7,3 .

The recursive rule says first look at 6,5,4,7,3 and then 6,5,4, and then
merge (5,4).

Now we have 1,2,8,5,7,3 where the 5 represents (5,4).

[note we are unable to update the viabilities, because we're still in the
process of filling out the ballot]

The recursive rule leads to the merger of (7,3).

Now we have 1,2,8,5,7 where the 7 represents (7,3).

Then 1 merges with 2, leaving

2,8,5,7

So 5 merges with 7 leaving

2,8,7 .

Expanding the abbreviations we see that we have

(1,2),8, ((5,4),(7,3)) .

So our best strategy (according to this method) is to put the greatest
viability candidate alone in the middle slot.

This is always the case in three slot MVP as long as the greatest
viability candidate is neither favorite nor most detested.

In fact, this method says that the greatest viability candidate should
always be alone in one of the three slots.

If the greatest viability candidate is at one of the extremes, then the
other cutoff is determined recursively:  The second greatest candidate is
also alone if at one of the extremes of its sublist L, otherwise the
weaker side merges with it:

Case I:   9 | 8 | 1,2,7,3,5,4
Case II:  9 | 2,3,6,7,4,5 | 8
Case III: 9 | 2,3,5,8 | 4,6,7
case IV:  9 | 2,3,7,4 | 8,6,5

It just came to me that in the n-slot case, one could actually use these
three slot (or even two slot) recommendations to get the updated
viabilities for the merger from n to n-1 slots.  That should take care of
the clone problem once and for all!

Forest