[EM] STAR
Colin Champion
colin.champion at routemaster.app
Fri Aug 18 13:58:48 PDT 2023
Forest - I read fpdk's post as an implicit argument for cardinal voting
(which was why it was relevant to STAR). Each friend states the utility
of each topping to himself or herself: which topping do they choose
collectively? And the answer is the one whose sum of utilities is
greatest. I don't think there's a better answer.
CJC
On 18/08/2023 21:50, Forest Simmons wrote:
> He posed a pizza choice among friends pronlem.. a problem of consensus
> as opposed to "tyranny of the majority" ... how to find the best
> consensus decision when a simple majority first place preference would
> not be ideal.
>
> On Fri, Aug 18, 2023, 10:41 AM Colin Champion
> <colin.champion at routemaster.app
> <mailto:colin.champion at routemaster.app>> wrote:
>
> Forest – I may be being slow, but... what problem are you trying
> to solve? The problem which fpdk (quite plausibly, to my mind)
> said was optimally solved by cardinal voting? Or the problem which
> I claimed was optimally solved by decision theory? Or something to
> do with tactical voting?
> CJC
>
> On 18/08/2023 18:32, Forest Simmons wrote:
>> It's been a while since I thought about this but here's something
>> that somebody with some number crunching resources should
>> experiment with ... a lottery method that I used to call "the
>> ultimate lottery" back before Jobst invented MaxParC, which
>> arguably has at least an equal claim to ultimateness:
>>
>> Ballots are positive homogeneous functions of the candidate
>> probability variables. The homogeneity degree doesn't matter as
>> long as all of the ballots are of the same degree.
>>
>> The candidate probabilities are chosen to maximize the product of
>> the ballots.
>>
>> This candidate probability distribution can be realized as a
>> spinner. The spinner is spun to determine the winner.
>>
>> How would this work for our pizza example?
>>
>> For example, each voter's ballot could be her pizza desirability
>> [score] expectation as a function of the lottery probabilities.
>>
>> Then each A faction voter would submit the same ballot ... namely
>> the function given by the expression
>> 100pA+80pC, while each B faction voter would submit the expression
>> 100pB+80pC.
>>
>> When these ballots are multiplied together, we get the product
>> (100pA+80pC)^60×(100pB+80pC)^40.
>>
>> The p values that maximize this product (subject to the
>> constraint that they are non-negative and sum to 100 percent) are
>> pA=pB=0, and pC=100%.
>>
>> The lottery that maximizes the expectation product is called the
>> Nash lottery after John Nash who first used this idea for
>> efficient allocation of limited resources.
>>
>> Since expectations are linear combinations of the probabilities,
>> they are homogeneous of degree one ... one person, on vote. Their
>> product is homogeneous of degree n ... so n people, n votes.
>>
>> Instead of using voter expectations for their ballots, the voters
>> could have used other homogeneous expressions ... for example, by
>> simply replacing each sum of products by a max of the same products.
>>
>> The product of these modified ballots would be ...
>>
>> [max(100pA,80pC)]^60
>> ×[max(100pB,80pC)]^40.
>>
>> Maximization of this product with the same constraints as before,
>> yields the same consensus distribution ... pC=100%.
>>
>> This information is new in the sense that it has never been
>> submitted for official publication ... it's an exclusive bonus of
>> Rob Lanphier's EM list archive... first posted to this list back
>> in 2011 after Jobst and I published our 2010 paper on the use of
>> mixed strategies for achieving consensus.
>>
>> Anyway, it turns out that using the Max operator in place of the
>> Sum operator yields a distribution with less entropy whenever the
>> two distributions are not identical.
>>
>> Less entropy means less randomness, which means less chance,
>> which in this context, means more consensus.
>>
>> In our example, the candidate distribution turned out to be 100
>> percent candidate C ... zero randomness ... zero entropy ... 100
>> percent consensus.
>>
>> Now you can see why I mentioned the need for number crunching
>> capability ... experimenting with these ballot product
>> maximizations requires some serious number crunching.
>>
>> The field is wide open. Is the Ultimate Lottery Method strongly
>> monotonic? For that matter, how about even the Nash Lottery?
>>
>> Can MaxParC be formulated in terms of the Ultimate Lottery?
>>
>> Somebody with some grad students should get them going on this!
>>
>> fws
>>
>> On Thu, Aug 17, 2023, 11:10 AM Forest Simmons
>> <forest.simmons21 at gmail.com <mailto:forest.simmons21 at gmail.com>>
>> wrote:
>>
>> Suppose voter utilities for three kinds of pizza are
>>
>> 60 A[100]>C[80]>>B[0]
>> 40 B[100]>C[80]>>A[0]
>>
>> Suppose the voters must choose by majority choice between
>> pizza C and the favorite pizza of a voter to be determined by
>> randomly drawing a voter name from a hat.
>>
>> The random drawing method would give voter utility
>> expectations of
>>
>> 60%100+40%0 for each A groupie, and
>> 40%100+60%0 for each B groupie.
>>
>> The max utility expectation would be 60.
>>
>> On the other hand, if voters decide to go with the sure deal
>> C, the assured utility fo every voter will be 80.
>>
>> Every rational voter faced with this choice will choose C.
>>
>> Here we have an ostensibly random method that is sure to
>> yield a consensus decision when voters vote ratkonally.
>>
>> More on this topic at
>>
>> https://www.researchgate.net/figure/Properties-of-common-group-decision-methods-Nash-Lottery-and-MaxParC-Solid-and-dashed_fig3_342120971
>> <https://www.researchgate.net/figure/Properties-of-common-group-decision-methods-Nash-Lottery-and-MaxParC-Solid-and-dashed_fig3_342120971>
>>
>> fws
>>
>> On Thu, Aug 17, 2023, 1:18 AM Forest Simmons
>> <forest.simmons21 at gmail.com
>> <mailto:forest.simmons21 at gmail.com>> wrote:
>>
>> The best methods that I know of for the friends context
>> are minimum entropy lottery methods characterized by max
>> possible consensus (min entropy) consistent with a
>> proportional lottery method with higher entropy fallback
>> to disincentivize gratuitous defection.
>>
>> Jobst's MaxParC (Max Partial Consensus) is the best example.
>>
>> Too late to elaborate tonight.
>>
>> fws
>>
>> I'll
>>
>> On Wed, Aug 16, 2023, 10:01 AM <fdpk69p6uq at snkmail.com
>> <mailto:fdpk69p6uq at snkmail.com>> wrote:
>>
>>
>> On Mon, Aug 14, 2023 at 12:09 AM C.Benham wrote:
>>
>> > I think this is an interesting point. We can
>> ask at a philosophical level what makes a good
>> voting method. Is it just one that ticks the most
>> boxes, or is it one that most reliably gets the
>> "best" result?
>>
>>
>> The one that most reliably gets the best result in
>> the real world. The difficulty with this approach is
>> accurately modeling human voting behavior and the
>> consequent utility experienced from the winner, but
>> it's still the better answer philosophically.
>>
>> (Note that VSE predates Jameson Quinn by decades, and
>> has had several different names:
>> https://en.wikipedia.org/wiki/Social_utility_efficiency
>> <https://en.wikipedia.org/wiki/Social_utility_efficiency>)
>>
>> > And that's partly because the premise of
>> Condorcet is essentially built on a logical
>> fallacy - basically that if A is preferred to B
>> on more ballots that vice versa then electing A must
>> > be a better result than electing B.
>>
>> I'd be interested in reading your explanation of
>> why you think that is a
>> "logical fallacy". What about if there are only
>> two candidates?
>>
>>
>> Ranked ballots can't capture strength of preference.
>> It's possible for a majority-preferred candidate to
>> be very polarizing (loved by 51% and hated by 49%),
>> while the minority-preferred candidate is
>> broadly-liked and has a much higher overall
>> approval/favorability rating. Which candidate is the
>> rightful winner?
>>
>> https://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html
>> <https://leastevil.blogspot.com/2012/03/tyranny-of-majority-weak-preferences.html>
>>
>> "Suppose you and a pair of friends are looking to
>> order a pizza. You, and one friend, really like
>> mushrooms, and prefer them over all other vegetable
>> options, but you both also really, /really/ like
>> pepperoni. Your other friend also really likes
>> mushrooms, and prefers them over all other options,
>> but they're also vegetarian. What one topping should
>> you get?
>>
>> Clearly the answer is mushrooms, and there is no
>> group of friends worth calling themselves such who
>> would conclude otherwise. It's so obvious that it
>> hardly seems worth calling attention to. So why is
>> it, that if we put this decision up to a vote, do so
>> many election methods, which are otherwise seen as
>> perfectly reasonable methods, fail? Plurality,
>> top-two runoffs
>> <http://en.wikipedia.org/wiki/Two-round_system>,
>> instant runoff voting
>> <http://en.wikipedia.org/wiki/Instant-runoff_voting>,
>> all variations of Condorcet's method
>> <http://en.wikipedia.org/wiki/Condorcet_method>, even
>> Bucklin voting
>> <http://en.wikipedia.org/wiki/Bucklin_voting>; all of
>> them, incorrectly, choose pepperoni."
>>
>> (And strength of preference is clearly a real thing
>> in our brains. If you prefer A > B > C, and are
>> given the choice between Box 1, which contains B, and
>> Box 2, which has a 50/50 chance of containing A or C,
>> which do you choose? What if the probability were 1
>> in a million of Box 2 containing C? By varying the
>> probability until it's impossible to decide, you can
>> measure the relative strength of preference for B > C
>> vs A > C.)
>> ----
>> Election-Methods mailing list - see
>> https://electorama.com/em <https://electorama.com/em>
>> for list info
>>
>>
>> ----
>> Election-Methods mailing list - seehttps://electorama.com/em <https://electorama.com/em> for list info
>
> ----
> Election-Methods mailing list - see https://electorama.com/em
> <https://electorama.com/em> for list info
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20230818/8026afbe/attachment-0001.htm>
More information about the Election-Methods
mailing list