[EM] Proportional Lottery Application

[Forest Simmons]

If my intuition is correct, then the appropriate "first place margins"
defeat strength for A>B would be the following difference of sums ...

Sum f(X) over X not defeated by B
Minus
Sum f(Y) over Y defeating A

Does that make sense?

-Forest



On Sun, Oct 23, 2022, 1:46 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:

> Thanks, Jobst. I worked carefully through the Random Ballot Favorite Chain
> Building lottery example, but in the leap from RBFCB to the Ranked Pairs
> interpretation I carelessly forgot to accommodate clones.
>
> Here's what I think it should be:
>
> The defeat strength for the pairwise defeat A>B should be the sum of f(X)
> over all X that are not defeated by B ... not just fA).
>
> So if A is the CW, the defeat strength of A>B would be 100 percent, no
> matter the candidate B or the clone B' of B.
>
> Does that seem right?
>
> Also, if A is not the CW, and A' covers B, then the defeat strength of
> A>B, would be no greater than the defeat strength fo A'>B.
>
> So it seems to be the defeat strength we want, whether or not our leap
> from the RBFCB lottery to Ranked Pairs was justified logically.
>
> After Kevin and Kristofer test it experimentally we'll know it it's any
> good. The proof is in the pudding!
>
> -Forest
>
> On Sun, Oct 23, 2022, 12:11 AM Jobst Heitzig <heitzig at pik-potsdam.de>
> wrote:
>
>> Hi all, just shortly due to lack of time: When you say "benchmark
>> lottery" you mean random ballot, right? But then we measure defeat strength
>> by number of first-place votes, right? So how do we achieve
>> clone-independence then? Shouldn't we measure defeat strength by approval
>> score instead to make it clone-independent? Or is some decloning involved
>> to construct your notion of benchmark lottery that I missed? All the best,
>> Jobst
>>
>> Am 23. Oktober 2022 00:36:41 MESZ schrieb Forest Simmons <
>> forest.simmons21 at gmail.com>:
>>>
>>> A nice convergence of ideas has brought us to a practical proposal.
>>>
>>> The first idea is Jobst's Proportioal Lottery (democratic fairness)
>>> notion. If 30 percent of the voters bullet vote for X, then a fair lottery
>>> ought to select X at least 30 percent of the time.
>>>
>>> The second idea, also due to Jobst, is that pairwise defeat strength
>>> doesn't have to be overtly pairwise. Jobst was first to point out that DMC
>>> can be construed as Condorcet with winning approval as the gauge of defeat
>>> strength.
>>>
>>> The third idea introduced to the EM list by Jobst was Rivest's Condorcet
>>> Lottery ... a non-proprtional, but Dutta efficient, lottery lacking only in
>>> Monotonicity short of an ideal Condorcet method.
>>>
>>> The  fourth idea, also Jobst's, was Chain Climbing as a Banks efficient
>>> selection procedure.
>>>
>>> The fifth idea central to the evolution of our practical proposal was
>>> the importance of first place votes for de-incentivizing burial and Chicken
>>> strategy, while simultaneously conferring Landau efficiency ... the result
>>> of doggedly pursuing the Holy Grail of fpA-fpC when everybody else
>>> despaired.
>>>
>>> Kristofer's important "Friendly" classification of candidates arose in
>>> that context.
>>>
>>> Kristofer, Kevin, and Chris Benham (among many others) also continued
>>> championing the kernel of good worth emulating in IRV ...  hence Benham's
>>> method as the original genesis of DMC, and our widespread use of first
>>> place counts for emulating IRV's clone independence, and avoiding burial,
>>> Chicken dilemma, etc.
>>>
>>> The idea of Kristofer, Benham, Bristow-Johnson, and many others that we
>>> should keep the strategic burden off the backs of the voters by striving to
>>> bring into the Universal Domain the successes first achieved outside that
>>> domain ... the ones that benefitted greatly from the help of Approval
>>> cutoffs, Voting Published Rankings, candidate asset/proxy options, etc.
>>>
>>> For a long time the only monotone, clone independent, Landau efficient
>>> methods we could devise (like score based chain climbing) depended on
>>> approval or other (non-UD) score orders. Eventually, [with the advent of a
>>> de-cloned Kemeny-Young version we called Swap Cost] we got a clonefree,
>>> monotone, UD finish order that we could chain climb to get a Banks
>>> efficient method entirely within the UD domain.
>>>
>>> De-cloning Kendall-tau in order to de-clone K-Y) relied heavily on the
>>> benchmark lottery probabilities, as well as on the anti-favorite lottery
>>> probabilities.
>>>
>>> [The cost of swapping (reversing) an adjacent pair order in a
>>> permutation of the candidates ... is the product f(D)f'(A) where f is the
>>> first place probability of the descending candidate D, and f'(A) is the
>>> anti-benchmark lottery probability of the ascending candidate A; the cost
>>> of lowering a candidate is proportional to its percentage of the top votes,
>>> while the cost of raising it is proportional to its percentage of the
>>> bottom votes.]
>>>
>>> Finally, as Jobst pointed out recently, the prominent rôle of first
>>> place votes in Friendly Voting helps prevent weak centrists, from taking
>>> undue advantage of Condorcet's reliance on relative, as opposed to absolute
>>> preferences.
>>>
>>> Which brings us, without further ado, to our basic proposal:
>>>
>>> Condorcet (first place vote strength).
>>>
>>> That's it ... Condorcet, whether RP, CSSD, or River (it makes no
>>> difference) ... with defeat strength gauged by the defeater's benchmark
>>> lottery probability.
>>>
>>> That's all there is to it!
>>>
>>> At risk of detracting from the simplicity of the proposal, I cannot
>>> resist the temptation to point out some advantages that the average voter
>>> would not have any interest in, but are of great interest to every voting
>>> methods connoisseur:
>>>
>>> 1. Without trying to be, this method turns out to be Landau efficient;
>>> no covered candidate can win! This sets a new standard of immunity... there
>>> is no good excuse any more for electing covered candidates ...  immunity
>>> against various complaints must now include this one ... "But our candidate
>>> not only defeated the alleged winner (head-head), she also defeated every
>>> candidate the winner defeated ...  andno candidate that defeated the winner
>>> defeated our candidate. So how do you explain that?"
>>>
>>> 2. The method, like IRV, is Chicken proof, burial proof, bullet proof,
>>> etc.
>>>
>>> 3. The method is decisive; ties are as rare as first place ties under
>>> FPP Plurality. Natural tie breakers for defeat strength can simply
>>> substitute some other proportional lottery for the benchmark lottery.
>>>
>>> Since tie breaking is an important detail of every election law, humor
>>> me to elaborate on this detail. The benchmark lottery probability that we
>>> are using to gauge defeat strength for candidate X, is simply the
>>> percentage of the ballots on which X is ranked highest. Here's a related
>>> proportional lottery that could be used as a tie breaker: From a randomly
>>> drawn ballot B, elect the highest ranked [tied] candidate that has a
>>> beatpath to every other candidate ranked on B.
>>>
>>> Enough said for now!
>>>
>>> -Forest
>>>
>>
>> --
>> Diese Nachricht wurde von meinem Android-Gerät mit K-9 Mail gesendet.
>>
>
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