[EM] STAR

Fri Aug 18 10:32:50 PDT 2023

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>
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
>
>
> fws
>
> On Thu, Aug 17, 2023, 1:18 AM Forest Simmons <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> 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
>>> )
>>>
>>> > 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
>>>
>>> "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 for list
>>> info
>>>
>>
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