[Election-Methods] [english 94%] Re: method design challenge + new method AMP

[Juho]

Here's an example on how the proposed method might work.

I'll use your set of votes but only the rankings.
51: A>C>B
49: B>C>A

Let's then reverse the votes to see who the voters don't like.
51: B>C>A
49: A>C>B

Then we'll use STV (or some other proportional method) to select 2  
(=3-1) candidates. STV would elect B and A. B and A are thus the  
worst candidates (proportionally determined) that will be eliminated.  
Only C remains and is the winner.

- I used only rankings => also worse than "52 point" compromise  
candidates would be elected
- I didn't use any lotteries => C will be elected with certainty

Juho



On May 2, 2008, at 22:29 , Jobst Heitzig wrote:

> Dear Juho,
>
> I'm not sure what you mean by
>> How about using STV or some other proportional method to select  
>> the  n-1 worst candidates and then elect the remaining one?
>
> Could you give an example or show how this would work out in the  
> situation under consideration?
>
> Yours, Jobst
>
>> Juho
>> On Apr 28, 2008, at 20:58 , Jobst Heitzig wrote:
>>> Hello folks,
>>>
>>> over the last months I have again and again tried to find a   
>>> solution to
>>> a seemingly simple problem:
>>>
>>> The Goal
>>> ---------
>>> Find a group decision method which will elect C with near  
>>> certainty in
>>> the following situation:
>>> - There are three options A,B,C
>>> - There are 51 voters who prefer A to B, and 49 who prefer B to A.
>>> - All voters prefer C to a lottery in which their favourite has 51%
>>> probability and the other faction's favourite has 49% probability.
>>> - Both factions are strategic and may coordinate their voting   
>>> behaviour.
>>>
>>>
>>> Those of you who like cardinal utilities may assume the following:
>>> 51: A 100 > C 52 > B 0
>>> 49: B 100 > C 52 > A 0
>>>
>>> Note that Range Voting would meet the goal if the voters would be
>>> assumed to vote honestly instead of strategically. With strategic
>>> voters, however, Range Voting will elect A.
>>>
>>> As of now, I know of only one method that will solve the problem  
>>> (and
>>> unfortunately that method is not monotonic): it is called AMP and is
>>> defined below.
>>>
>>>
>>> *** So, I ask everyone to design some ***
>>> *** method that meets the above goal! ***
>>>
>>>
>>> Have fun,
>>> Jobst
>>>
>>>
>>> Method AMP (approval-seeded maximal pairings)
>>> ---------------------------------------------
>>>
>>> Ballot:
>>>
>>> a) Each voter marks one option as her "favourite" option and may  
>>> name
>>> any number of "offers". An "offer" is an (ordered) pair of options
>>> (y,z). by "offering" (y,z) the voter expresses that she is  
>>> willing to
>>> transfer "her" share of the winning probability from her  
>>> favourite  x to
>>> the compromise z if a second voter transfers his share of the  
>>> winning
>>> probability from his favourite y to this compromise z.
>>>     (Usually, a voter would agree to this if she prefers z to   
>>> tossing a
>>> coin between her favourite and y).
>>>
>>> b) Alternatively, a voter may specify cardinal ratings for all   
>>> options.
>>> Then the highest-rated option x is considered the voter's  
>>> "favourite",
>>> and each option-pair (y,z) for with z is higher rated that the mean
>>> rating of x and y is considered an "offer" by this voter.
>>>
>>> c) As another, simpler alternative, a voter may name only a   
>>> "favourite"
>>> option x and any number of "also approved" options. Then each
>>> option-pair (y,z) for which z but not y is "also approved" is   
>>> considered
>>> an "offer" by this voter.
>>>
>>>
>>> Tally:
>>>
>>> 1. For each option z, the "approval score" of z is the number of   
>>> voters
>>> who offered (y,z) with any y.
>>>
>>> 2. Start with an empty urn and by considering all voters "free for
>>> cooperation".
>>>
>>> 3. For each option z, in order of descending approval score, do the
>>> following:
>>>
>>> 3.1. Find the largest set of voters that can be divvied up into   
>>> disjoint
>>> voter-pairs {v,w} such that v and w are still free for  
>>> cooperation, v
>>> offered (y,z), and w offered (x,z), where x is v's favourite and  
>>> y is
>>> w's favourite.
>>>
>>> 3.2. For each voter v in this largest set, put a ball labelled  
>>> with  the
>>> compromise option z in the urn and consider v no longer free for
>>> cooperation.
>>>
>>> 4. For each voter who still remains free for cooperation after  
>>> this  was
>>> done for all options, put a ball labelled with the favourite  
>>> option of
>>> that voter in the urn.
>>>
>>> 5. Finally, the winning option is determined by drawing a ball  
>>> from  the
>>> urn.
>>>
>>> (In rare cases, some tiebreaker may be needed in step 3 or 3.1.)
>>>
>>>
>>> Why this meets the goal: In the described situation, the only   
>>> strategic
>>> equilibrium is when all B-voters offer (A,C) and at least 49 of the
>>> A-voters "offer" (B,C). As a result, AMP will elect C with 98%
>>> probability, and A with 2% probability.
>>>
>>>
>>>
>>> ----
>>> Election-Methods mailing list - see http://electorama.com/em for   
>>> list info
>> 	
>> 	
>> 		
>> ___________________________________________________________ All  
>> new Yahoo! Mail "The new Interface is stunning in its simplicity  
>> and ease of use." - PC Magazine http://uk.docs.yahoo.com/ 
>> nowyoucan.html
>> ----
>> Election-Methods mailing list - see http://electorama.com/em for  
>> list info
>


		
___________________________________________________________ 
The all-new Yahoo! Mail goes wherever you go - free your email address from your Internet provider. http://uk.docs.yahoo.com/nowyoucan.html