[EM] Manipulability stats for more poll methods (fixed footnotes)

[Michael Ossipoff]

There’s no reason for the renormalization. Among A, B, C & D (in that order
of magnitude) if B is at the mean, then, with the A=0 & D=1
renormalization, B’s renormalized value is the mean of all of the
renormalized values.

The position of the mean among the candidates doesn’t change with
renormalization.



On Sat, May 4, 2024 at 15:25 Michael Ossipoff <email9648742 at gmail.com>
wrote:

>
>
> On Sat, May 4, 2024 at 14:45 Kristofer Munsterhjelm <km_elmet at t-online.de>
> wrote:
>
>>
>> Yes, that's right. But consider a voter with the following utilities:
>>
>> A: 0.57
>> B: 0.32
>> C: 0.23
>> D: 0.08
>>
>> Normalization to two steps fixes the highest value (0.57) to 1 and the
>> lowest value (0.08) to 0 and rounds off the intermediate values after
>> linearly scaling them.
>
>
> Yes. So far, so good. But…
>
> This in essence says that a value is rounded off
>> to 1 if it's greater than or equal to 0.325 (the midpoint between 0.08
>> and 0.57)
>
>
> What? You didn’t average the normalized values. You averaged two of the
> values before normalization. The midrange isn’t usually the same as the
> mean. You used the midrange as the mean.
>
> If you call the top value 1, & the bottom value 0,
> then a rating’s new value is the number that’s the same % of the way from
> 0 to 1 as the old number’s % from.08 to .57
>
> Average of those new values: .4475
>
> You still approve the best two.
>
>
>
> so the 0-1 normalized ballot is
>>
>> A: 1, B: 0, C: 0, D: 0
>>
>> On the other hand, the mean utility is 0.3. So the mean utility approval
>> ballot is
>>
>> A: 1, B: 1, C: 0, D: 0.
>>
>> > 4 "dimensions" sounds like a lot.  What are the "strategy attempts" ?
>> > How much and what information do the strategists have?  Are the
>> > strategists confined to just trying to get their favourites elected, or
>> > any candidate they prefer to the initial winner?
>>
>> The method works pretty much like this, for generating and testing a
>> single election. (I've simplified the exact order that strategies are
>> called upon, but this is in effect what happens.)
>>
>> ==== (Algorithm start) =====
>>
>> Draw candidate positions for each candidate (in this case, each is a
>> point on a 4D normal distribution with mean 0 and variance 1).
>> Draw voter positions for each voter, and create their honest ballots
>> based on the distances between the voter and candidates.
>> Pass the resulting ballots through the method to establish the honest
>> outcome.
>> If there's a tie, skip (because deciding what a strict improvement is
>> when there's a honest tie is ambiguous). Otherwise let the winner be W.
>>
>> For each candidate X who is not the winner W:
>>         For i = 1 to number of strategy attempts / number of candidates
>>                 Set the strategic ballots to the honest ballots.
>>
>>                 For every voter who prefers X to W:
>>                         Replace that voter's strategic ballot with a
>>                         ballot according to a strategy that depends on
>>                         i.
>>
>>                 Pass the modified strategic ballots through the method.
>>                 If X is now a winner, the method is manipulable in
>>                         this election. Return success.
>>
>> If we reach this point without any success, return failure; the method
>> is (probably) not manipulable in this election.
>>
>> ==== (Algorithm end) =====
>>
>> The indexed strategies are
>>         i=0: Compromising (raise X to unique top)
>>         i=1: Burial (lower W to unique bottom)
>>         i=2: Two-sided (do both at once)
>>         i>2: Coalitional strategy
>>
>> The compromising, burial, and two-sided strategies modify the voters'
>> otherwise honest ballots - for instance, compromising changes a
>> strategist's ballot so that X is at unique top and the rest of the
>> ballot is unchanged.
>>
>> The first time the coalitional strategy is called for a particular
>> election, candidate to strategize for, and value of i, it chooses a
>> random number of strategic ballots (between 1 and 3 inclusive). Each
>> strategic voter then picks one of these ballots at random. This
>> simulates strategies where every strategist ballot is equal, as well as
>> ones where there are a few groups each with their own ballot type, thus
>> covering more than JGA's simulations without becoming *too*
>> computationally expensive.
>>
>> So with the setup for the stats that I gave, the full setup for a single
>> method is like this:
>>
>> for j = 1 to 500k
>>         Run the algorithm detailed above.
>>         It returns one of three states: honest tie, success, or failure.
>>         Increment the corresponding counter, call it TIES, SUCCESSES or
>> FAILURES.
>>
>> manipulability = SUCCESSES/(500k - TIES)
>>
>>
>> So to answer your questions:
>>
>> The strategists don't adapt their strategy to the information available
>> to them, even though they strictly speaking have full information.
>> However, they get to try over and over again until they win. If there is
>> a full information strategy with not too many distinct ballots, then
>> this random sampling will eventually find it, given a high enough
>> strategy attempts value.
>>
>> For each non-winner X, everybody who prefers X to the current winner
>> gets to have a go. So not just their favorites: anybody they all prefer
>> to the current winner.
>>
>> >
>> >>
>> >> [2] The detailed stats suggest that pushover is a problem with
>> Smith//DAC
>> >
>> > You don't have enough candidates for a sub-cycle, and so the method
>> > can't fail mono-raise.  How can it have a Pushover problem?
>>
>> I did a bit more checking, and the full preference version doesn't have
>> this high an "other strategy" count. Since I think it's unlikely that
>> the version with truncation would have more pushover than the fully
>> ranked one, I'm going to retract this; most likely it's just an artifact
>> of the simulator's ballot reduction process that falsely attributes the
>> strategy to the "other" category for cardinal methods.
>>
>> >
>> >> - Margins-Sorted Approval, because I'm not sure how it works
>> >
>> > (I struggle to take this at face value.  Probably my promotion of MSA
>> > has convinced you that it is the best method and you were concerned
>> that
>> > your simulation wouldn't do it justice.
>>
>> I'd like to believe both that I have enough scientific integrity not to
>> do that, and that people know I have, too :-)
>>
>> Actually, I was planning on putting MSA at the same level as the other
>> "I don't know enough about these or their dynamics" methods (double
>> defeat Hare, MSMLV, and Max Strength Transitive Beatpath).
>>
>> > But our expert doing the
>> > simulation claiming he can't understand the method isn't a good look
>> for
>> > its proposability.)
>> >
>> > Why didn't you simply ask me to explain it to you?
>>
>> I think it's the sorting phase that does it. My vague idea of how it
>> works is that you essentially run a sorting algorithm on intermediate
>> values, and that seems a little too complex to me. But I might just have
>> got it wrong and then the initial impression of it as an intimidating
>> method stuck.
>>
>> Ted Stern pointed me at the Electowiki article for MSA, which in turn
>> led me to his Python implementation. I might port it if I have time, but
>> I feel a bit exhausted after gathering all these stats. We'll see :-)
>>
>> > What happened to separate entries for BTR,  Woodall and Benham?
>>
>> They're in the other post. I didn't want to add them all to the post
>> that was intended to focus on the new results. That's why I said "some
>> for comparison" - the others are here:
>>
>>
>> http://lists.electorama.com/pipermail/election-methods-electorama.com/2024-April/006029.html
>>
>> I could post all the stats - ordinal and cardinal methods' - in a
>> summary post if you or other EM members would like.
>>
>> -km
>> ----
>> Election-Methods mailing list - see https://electorama.com/em for list
>> info
>>
>
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