[EM] [CES #15268] Re: PAR: nearly-equivalent rules
Jameson Quinn
jameson.quinn at gmail.com
Mon Nov 14 12:12:14 PST 2016
Ted is right. This latest version of PAR is in fact O(N²) summable as a
whole. The tallies you need:
P(x): the number of "prefers" for X
a(x): the number of explicit "accepts" for X
B(x): the number of blanks for X on ballots that have at least one explicit
"reject"
A(x,y):the number of "accepts" for X on ballots that do not prefer Y
Here are the steps expressed using those quantities:
1. Define A(x) to be a(x) for x such that P(x)<25%, and a(x)+B(x) for all
other x. Define R(x) to be N-P(x)-A(x).
2. If there are any candidates such that R(x)<=50%, find the y that
maximizes P(y)*I(R(y)<=50%) (where I(...) is 1 if ... is true and 0
otherwise).
3. Define T(x) as P(x)+A(x,y).
4. If y maximizes T(y), then y wins. Otherwise, the winner is the candidate
z who maximizes N-R(z) (or equivalently, P(z)+A(z).)
2016-11-14 14:13 GMT-05:00 Ted Stern <dodecatheon at gmail.com>:
> The first round of Step 3 could be made summable with the addition of an
> NxN array where row i accumulates, from a ballot that does not Prefer the
> i-th candidate, Accept points cast for any candidates besides the i-th
> candidate.
>
> I'm not sure if that same array could be used for step 4, however. In
> fact, step 4 is not entirely clear to me.
>
> Ted
>
> On Mon, Nov 14, 2016 at 10:14 AM, Jameson Quinn <jameson.quinn at gmail.com>
> wrote:
>
>> I'm still working on getting PAR expressed in the best format. Here's the
>> latest:
>>
>>
>> 1. *Voters Prefer, Accept, or Reject each candidate.* On ballots
>> which don't explicitly use "Reject", or for candidates with less than 25%
>> "Prefer", blanks count as "Reject"; otherwise, blanks count as "Accept".
>> 2. *Tally 1 point for each "Prefer"* for each candidate.
>> 3. Out of the candidates (if any) with no more than 50% "Reject",
>> find the one with the most points. *For every ballot which doesn't
>> "Prefer" this frontrunner, add 1 point for each "Accept".*
>> 4. If the frontrunner changed, add 1 for each "Accept" that's not yet
>> tallied (the ones which "Prefer" the original frontrunner). *Most
>> points wins.*
>>
>>
>> I now believe that the system as expressed above meets the voted majority
>> Condorcet criterion. Which means that FBPPAR (almost) meets both FBC and
>> voted majority Condorcet! Impossible, you say? No; approval meets both of
>> these criteria (because it reduces "voted Condorcet" to a triviality), and
>> FBPPAR reduces to approval when voters strategize expecting a Condorcet
>> cycle.
>>
>> What do I mean by "(almost)" above? Well, FBPPAR allows voters to rate a
>> candidate as "prefer, but do not make them frontrunner because of my vote".
>> This is useful for cyclical and near-cyclical situations (that is, for A>B
>> voters who know that most B votes are B>C and that C beats A). But if this
>> option is used in cases where it is not actually strategic, it breaks the
>> voted majority Condorcet property.
>>
>> So FBPPAR is a system that meets FBC; deals with center squeeze via a
>> fairly strong Condorcet-like property; deals with chicken dilemma without a
>> slippery slope; can be counted in at most 3 rounds, each of which is O(N)
>> summable; meets majority and mutual majority; meets a weakened version of
>> later-no-help; outputs tallies that are summable with other approval
>> tallies; and is basically all-around awesome. Personally, I feel that
>> FBPPAR essentially dominates other 3-rank systems in terms of strategic
>> properties; and that PAR is, in the real world, a very close approximation
>> of FBPPAR that's justified by its greater simplicity.
>>
>> 2016-11-13 18:04 GMT-05:00 Jameson Quinn <jameson.quinn at gmail.com>:
>>
>>> Yet another revision of the rules. This one fixes it so that, if no
>>> candidates pass both thresholds, "accept" votes count.
>>>
>>>
>>> Prefer Accept Reject (PAR) voting works as follows:
>>>
>>> 1. *Voters Prefer, Accept, or Reject each candidate.* On ballots
>>> which don't explicitly use "Reject", blanks count as "Reject"; otherwise,
>>> blanks count as "Accept".
>>> 2. *Of the candidates that are both "viable"* (at least 25% Prefer) *and
>>> "acceptable"* (no more than 50% reject), *the one with most
>>> "Prefer"s is called the leader.*
>>> 3. Each "Prefer" is worth 1 point. For viable candidates, each
>>> "Accept" on a ballot which doesn't prefer the leader is also worth 1 point. *Most
>>> points wins.*
>>>
>>>
>>> If I could do it without it sounding too complicated, I'd make it so
>>> that explicit "accept" votes can count for non-viable candidates. But I
>>> don't think it's worth even the minor extra complication it would take to
>>> explain that.
>>>
>>> Note that changing the above system so that it passes FBC requires only
>>> giving a "prefer for the purposes of step 3, but count step 2 as if I
>>> rejected" option.
>>>
>>> I think this system is pretty great in outcomes; and that, while it's
>>> certainly not the simplest method, it's not too complex.
>>>
>>> 2016-11-12 23:33 GMT-05:00 Jameson Quinn <jameson.quinn at gmail.com>:
>>>
>>>>
>>>>
>>>> 2016-11-12 17:45 GMT-05:00 Neal McBurnett <nealmcb at gmail.com>:
>>>>
>>>>> I haven't been paying that much attention. But this definition
>>>>> doesn't define what this means "a rival is explicitly rejected". Is
>>>>> "rejected" the opposite of "viable"? Or does it mean that no other voter
>>>>> cast any "reject" vote for any other candidate? Or ???
>>>>>
>>>>
>>>> If candidates are ABCD, a ballot that prefers A and leaves the rest
>>>> blank counts as rejecting BCD, but one that prefers A and rejects B and
>>>> leaves the rest blank counts as accepting CD.
>>>>
>>>>
>>>>
>>>>> So if I have an "accept" on a ballot that doesn't "prefer" the leader,
>>>>> that means 0 points?
>>>>> Since votes depend on how other votes are interpreted and who the
>>>>> leader is, is the tally guaranteed to be attracted to a single stable
>>>>> equilibrium tally?
>>>>>
>>>>> Thanks,
>>>>>
>>>>> -Neal
>>>>>
>>>>> On Sat, Nov 12, 2016 at 05:27:53PM -0500, Jameson Quinn wrote:
>>>>> > I think this is the best phrasing I've come up with so far for PAR.
>>>>> I think it's not exactly equivalent to the definitions I've
>>>>> > given before, but it is close enough to meet all the same properties
>>>>> and give the same result in any basic scenario type.
>>>>> >
>>>>> > Prefer Accept Reject (PAR) voting works as follows:
>>>>> >
>>>>> > 1. Voters can Prefer, Accept, or Reject each candidate. Blanks
>>>>> count as "Reject" if no rival is explicitly rejected; otherwise,
>>>>> > blank is "Accept".
>>>>> > 2. Candidates with at least 25% Prefer, and no more than 50%
>>>>> reject, are "viable". The most-preferred viable candidate (if any) is
>>>>> > the leader.
>>>>> > 3. Each "prefer" is worth 1 point. For viable candidates, each
>>>>> "accept" on a ballot which doesn't prefer the leader is also worth
>>>>> > 1 point. Most points wins.
>>>>> >
>>>>> >
>>>>> > 2016-11-12 13:47 GMT-05:00 Jameson Quinn <jameson.quinn at gmail.com>:
>>>>> >
>>>>> > Thinking about PAR and the electoral college, I realized that
>>>>> there is a different way to state the PAR rules:
>>>>> >
>>>>> > 1. Voters can Prefer, Accept, or Reject each candidate. Default
>>>>> is "Reject" for voters who do not explicitly reject any
>>>>> > candidates, and "Accept" otherwise.
>>>>> > 2. A candidate is "viable" if they are rejected by under 50%.
>>>>> > 3. Each ballot gives 1 point to each candidate it prefers.
>>>>> Ballots which prefer no viable candidates also give 1 point to each
>>>>> > candidate they accept, so long as that candidate is
>>>>> preferred by at least 25%.
>>>>> > 4. Now find the viable candidate with the most points, if any,
>>>>> and redo step 3 from scratch as if only that candidate were
>>>>> > viable.
>>>>> > 5. The winner is the candidate with the most points.
>>>>> >
>>>>> > This could potentially differ from PAR in that it waits slightly
>>>>> longer to "reveal" the preferences of candidates with under
>>>>> > 25% preferences. In practice, I doubt this would typically make
>>>>> any difference.
>>>>> >
>>>>> > The procedure above is more complicated than PAR's, but the
>>>>> advantage is that it produces counts which include the disqualified
>>>>> > candidates, and thus is suitable for combining with totals from
>>>>> non-PAR systems such as approval, plurality, or
>>>>> > "pre-elimination totals" from IRV.
>>>>> >
>>>>> >
>>>>> > --
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>>>>>
>>>>
>>>>
>>>
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