[EM] A Proportionally Fair Consensus Lottery for which Sincere Range Ballots are Optimal

[Juho]

On Nov 21, 2009, at 11:39 AM, Jobst Heitzig wrote:

> if you want it to be as deterministic as possible


Some short (high level) comments on randomness vs. deterministic  
results.

First I'd like to say that there are different needs and different  
methods. Sometimes randomness is a good thing, sometimes not,  
sometimes neutral.

Random ballot is very random. It offers one type of proportionality.  
But in many single winner elections what one wants is majority  
decisions and decisions that pick the same best possible consensus  
candidate every time with certainty. Decisions on whom to elect as the  
next dictator for life for a country and which pizza to take or who  
will use the only ticket to movies are different.

Some level of randomness is often acceptable even if one wants the  
election to be deterministic in principle. For example in the case  
when there is no Condorcet winner one could well accept some level of  
randomness when picking one of the leading candidates. This can be  
done e.g. to eliminate some strategy related problems (or why not, to  
collect sincere opinions in the ballots).

Electing one of the non-leading candidates is often not acceptable.  
Practically all methods contain some randomness in any case, due to  
districting, sequential elimination in STV, need to solve ties etc.,  
so small additional doses of randomness does not change anything. But  
election of Hitlers and alike should in many cases (not all though)  
have strictly zero probability.

So, sometimes full (maybe proportional) randomness, sometimes small  
amount is ok, sometimes as little as possible. Random and  
deterministic methods may both be good, often just for different  
needs. Pick a method based on what you need. No need to try to find a  
single method or class of methods that would be "generally best for  
all purposes".

Juho







On Nov 21, 2009, at 11:39 AM, Jobst Heitzig wrote:

> Folks,
>
> you probably overlooked that I have already described a variant which
> works *completely* without Random Ballot and will definitely elect one
> of the top-3 range options (as determined from the 'strategic'  
> ballot):
>
>> Method "Range top-3 runoff" (RT3R)
>> ===================================
>>
>> 1. Each voter separately supplies
>>   a "nomination" range ballot and a "runoff" range ballot.
>>
>> 2. From all "nomination" ballots, determine
>>   the options A,B,C with the top-3 total scores a>b>c.
>>
>> 3. Let L be the lottery in which B wins with probability
>>   p = max(0,(2b-a-c)/(b-c))  and C wins with probability 1-p.
>>
>> 4. Let q be the proportion of "nomination" ballots
>>   on which the lottery L has an expected rating
>>   below the rating of A on that ballot.
>>
>> 5. Option A wins if,
>>   on at least the same proportion q of all "runoff" (!) ballots,
>>   the lottery L has an expected rating
>>   below the rating of A on that ballot.
>>   Otherwise B wins with probability p and C wins with probability 1- 
>> p.
>>
>
> So if you want it to be as deterministic as possible, you can do it  
> like
> this or similarly. If you modify it further and set q=1/2, you even  
> get
> a majoritarian version if you want that.
>
> Yours, Jobst
>
>
>
> Warren Smith schrieb:
>> --yep.  Only reason I did what I did was simplicity (kind of a pain  
>> if
>> voters have to submit
>> both a range-type and a condorcet or approval-type ballot).
>> But your way is better in that it tends to yield a better winner  
>> than my way.
>>
>> Also, note -- which is even more obnoxious -- we could have each  
>> voter
>> submit TWO ratings-style ballots,
>> the "honest range ballot" and the "dishonest range ballot"; then the
>> HRB is used to
>> decide between DHR and random ballot...
>>
>>
>> On 11/20/09, Raph Frank <raphfrk at gmail.com> wrote:
>>> This is effectively performing random ballot and then giving the
>>> voters the option to roll the dice a second time.
>>>
>>> Any single seat method could be used to select the first candidate.
>>> If you used a good single seat method to pick the compromise winner,
>>> then the random ballot would rarely if ever be activated.
>>>
>>> For example.
>>>
>>> 1) Voters submit ratings ballot and also a ranked or an approval  
>>> ballot
>>> 2) Determine the winner using condorcet or approval (or other  
>>> method)
>>> 3) Determine the random ballot odds for each candidate
>>> 4) If a majority prefer the winner in 2 to the expectation in 3),  
>>> then
>>> the winner from 2 wins.
>>> 5) Otherwise, use the random method
>>>
>>> Ofc, using a majority instead of a unanimous decision breaks some of
>>> the properties of the pure consensus method.
>>>
>>
>>
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