[EM] (3) ?goal of a better election method?

[Kevin Venzke]

Hi Steve,
Well, it seems clear to me that page 15 and slides 158-163 are talking about the same concept of being "partially strategy-proof-in-ranking." In both places the explanation offered by the authors is purely talking about how many candidates (out of a pair of two) a single voter might be able to adjust (in terms of their final grade). I think Kristofer gave the same interpretation on Jan 4 (although at the time I didn't understand the point).
That is to say, when you describe page 15 as suggesting that a voter is unlikely to be able to change a result, I feel this is not relevant to this page. It's mathematically not possible at all to contrive a scenario where a single voter who rated A>B can change his ballot in a way to adjust the final grades of both A and B when (in the original outcome) B had a higher final grade than A. They're saying it may be possible for a voter to affect the grade of one, but he won't ever be able to affect the grade of both.
On page 15 as you quote, they seem to conclude from this discussion that MJ "cuts in half the probability of manipulation." I find it confusing that they would call this "cut in half" and that they would call this "manipulation" without some additional explanation, but I don't see how page 15 offers any other interpretation of what they're referring to.
Regarding pages 197-198, I suspect these are related to slides 166-171. They both seem to discuss a naive estimate of the possibility that a voter could affect the outcome. That would seem to be a different issue from what's on page 15, yet both are called either "manipulability" or "probability of manipulation." I would note that they say the lowest possible probability is n + 1/2n, but they don't say what is the highest possible probability. We may guess 100%, but would any method actually correspond to that?
I don't really grasp what they are showing with the math: If n is a count of voters, why would probability be expressed in terms of n? Is this really the percentage of voters that can change the result? And if so would that really be just over 50% in the best case? (I wonder if Kristofer has already explained all this and I just didn't read it...) 
To clarify one item:>>K: This means that if the relative order of two candidates' final grades is not to the liking of some voter, that voter may be >>able to adjust the grade of one of the candidates, but not both. "One and not two" seems promising as something that has >>been cut by half.
>S:  Does your last phrase indicate that you now accept the validity of B&L’s claim that MJ “cuts” manipulation in “half”?

It indicates that I think "one and not two" is where they are getting "half." If so, I accept that this has been "cut in half," but I'm skeptical that this concept is a good representation of the "manipulability" of an SGF.
Kevin



      De : steve bosworth <stevebosworth at hotmail.com>
 À : "stepjak at yahoo.fr" <stepjak at yahoo.fr>; "election-methods at lists.electorama.com" <election-methods at lists.electorama.com> 
 Envoyé le : Jeudi 23 février 2017 8h53
 Objet : Re: [EM] (3) ?goal of a better election method?
   
 <!--#yiv8039895279 P {margin-top:0;margin-bottom:0;}-->HI Kevin,
Message: 2
Date: Sun, 19 Feb 2017 19:20:34 +0000 (UTC)
From: Kevin Venzke <stepjak at yahoo.fr>
To: steve bosworth <stevebosworth at hotmail.com>,
        "election-methods at lists.electorama.com"
        <election-methods at lists.electorama.com>
Subject: Re: [EM] goal of a better election method
Message-ID: <1682602380.1138980.1487532034974 at mail.yahoo.com>
Content-Type: text/plain; charset="utf-8"

K: Ok, I read the entire .pdf.  I don't think the "half the manipulability" claim is related to slide 150. S: Sorry, I did not mean that this claim is explicit on slide 150. Slide 150 only again refers to the more limited scope for “manipulation” offered by MJ‘s social grading function (SGF) in contrast to any method using “point-summing” which allows “all voters [to] manipulate: “If the mechanism is a point-summing method (the mean with respect to some parametrization), for almost all profiles, all voters can manipulate.”Before slide 150, slide 142 promised that: “We are going to prove that majority judgement is strategy-prooffor a large class of utility functions. When it is not, it is shown that itcombats manipulationsin many well defined senses”.However, on page 15 in Chapter 1 of B&L’s book (Majority Judgment, 2011, MIT), they verbally illustrate why the above claims are true. I will quote this passage shortly but please correct me if I am mistaken in seeing it as making the following simple point: Using MJ for the French election, a voter honestly would give Royal a grade of at least “good” and Bayrou less than “good”. If she tries to manipulate the result, she has practically no chance of increasing the probability of Royal being elected simply by giving Royal a “very good” or an ”excellent” instead, and giving Bayrou a “poor” or “reject”. Whether she votes honestly or manipulatively, each candidate’s “median-grade” will almost certainly remain the same. She would only have the best chance of changing the result if she instead were to give Royal a grade lower than “good” and Bayrou “higher” than “good”, i.e. voted in a way diametrically opposed to her own honest evaluations of the candidates, presumably not her desire:Page 15: “How could a voter who graded Royal higher than Bayrou manipulate? By changingthe grades assigned to try to lower Bayrou’s majority-gauge and to raiseRoyal’s majority-gauge. But the majority judgment ispartially strategy-proof in-ranking: those voters who can lower Bayrou’s majority-gauge cannot raiseRoyal’s, and those who can raise Royal’s majority-gauge cannot lower Bayrou’s.For suppose a voter can lower Bayrou’s. Then she must have given BayrouaGoodor better; but having preferred Royal to Bayrou, the voter gave a gradeof better thanGoodto Royal, so she cannot raise Royal’s majority-gauge (cannotraise herp). Symmetrically, a voter who can raise Royal’s majority-gaugemust have given her aGoodor worse and thus to Bayrou a worse thanGood;so the voter cannot lower Bayrou’s majority-gauge (cannot increase hisq).Compared with other mechanisms, the majority judgment cuts in half thepossibility of manipulation, however bizarre a voter’s motivations or whateverher utility function. The majority judgment resists manipulation in still otherways that other methods do not, but to see how requires information foundin voters’ individual ballots ….”S: With regard to B&L’s claim to offer an exact mathematic “proof” that MJ reduces by about “half” the scope for “manipulation” as that offered by the “point-summing” methods, can you please explain your own view of this claim. More exactly, do the following quoted passages merely restate the above explanation by using only pure mathematic notations, or do they mathematically provide us will the exact relevant number close to one half? Either way, I would very much appreciate it if you would help me to fully understand the case B&L make in their book in the pages leading up to pp.197-8 which contain the following formulaes:“For, a priori, a judge who manipulates wishes to increase the final grade with probability ½ and wishes to decrease it with probability ½. And, a priori, any one of the n judges may cheat. Thus, when the input of grades is r, the probability for a judge to manipulate successfully when using an SGF with aggregation functionf is(1/2)(μ− (f (r)) + (1/2) (μ+ (f (r)) =  (μ (f (r) )                     n                                        n                       2np.198: “One would like this probability to be as small as possible in the worst case; this amounts to finding anf that solves min  max_  μ (f (r)) = minμ (f ).  f    r= (r1,...,rn)                         f               that is, minimizes the manipulability. Thus the minimum probability is slightly over one-half, n+1/2n, and is only achieved whenf is an order function.”
  S: Also, how exactly does the above relate to Laraki’s slide 170:  "Given an aggregation functionfand inputr= (r1, . . . , rn), letμ−(f,r) =nbre of judges who can decrease the final grade,μ+(f,r) =nbre of judges who can increase the final grade, Letl=probability a judge wishes to increase the final grade. Theprobability of effective-manipulabilityoffis EM(f) =max       max_    l μ+(f,r) + (1−l)μ−(f,r)            r= (r1,...,rn)     0≤l≤1                         n S: I can send you the above formulaes as an attachment if they have been scrambled by your inbox.
K: Leading up to slide 150, [Laraki] argues that *if* your goal is simply for candidate X to get a grade of 7, then under order functions like MJ, your best strategy is to rate that candidate a 7. Under Range this is not true, because if that candidate is sitting at a 1, rating him a 10 would drag him towards a 7 rating faster than just rating him a 7 would. There is no "half" to be found here; it doesn't seem that any SGF could outperform MJ on this measure.
On slide 158 they describe the concept of being "partially strategy-proof-in-ranking." This means that if the relative order of two candidates' final grades is not to the liking of some voter, that voter may be able to adjust the grade of one of the candidates, but not both. "One and not two" seems promising as something that has been cut by half. S: Does your last phrase indicate that you now accept the validity of B&L’s claim that MJ “cuts” manipulation in “half”?
K: Then on 163 they say that order functions are the only SGFs that have this property.
What do you think, is there a better suggestion for something that has been halved? I would sort of hope so because if the answer is always either "one candidate" or "both candidates" that means there are only two outcomes for a test we're taking as a representation of manipulability under SGFs. S: I hope to understand the point you are making here after receiving your help and comments on all the above. I look forward to your feedback.Steve

      De?: steve bosworth <stevebosworth at hotmail.com>
 ??: "stepjak at yahoo.fr" <stepjak at yahoo.fr>; "election-methods at lists.electorama.com" <election-methods at lists.electorama.com>
 Envoy? le : Dimanche 19 f?vrier 2017 3h03
 Objet?: [EM] goal of a better election method

Message: 2
Date: Sun, 12 Feb 2017 17:31:37 +0000 (UTC)
From: Kevin Venzke <stepjak at yahoo.fr>
To: steve bosworth <stevebosworth at hotmail.com>,? EM list
??????? <election-methods at electorama.com>
Subject: Re: [EM] ?goal of a better election method?
Message-ID: <1346455147.4984684.1486920697303 at mail.yahoo.com>
Content-Type: text/plain; charset="utf-8"

Hi Kevin:
> S: Secondly, I believe that B&L ? prove? (pp. 15, 19, 186-198) that MJ provides only about ? half? the incentives? >or opportunities for anti-democratic ?strategic? voting to be successful. If you disagree, please explain the flaw in their argument.

K: What I understood from Kristofer's Jan 4 explanation of page 15 is that?this halving of the manipulability is not meant to be a comparison to?any other methods. It's a comparison to a (rather strange) hypothetical situation. If so, this claim on its own could be true but is of unclear value.
Possibly this argument is used to build up to a larger argument. But when it gets stated on its own it feels misleading to me, because there's no way for the reader to understand what this "half" is half of. S: No, B&L clearly explain why MJ provides only about ?half? of the incentives and opportunities to ?manipulate? the results as compared to all the methods that gain their results by ?summing? or ?averaging? all the votes.? In his reply to me, Kristofer also gave me a link to the following source in which B&L concisely state the same claim on slide 150.?  This is in the middle of their discussion of ?strategy?, i.e. between slides 143 & 185: Kristofer: See this slide set by B&L for more on that:
http://igm.univ-mlv.fr/AlgoB/algoperm2012/01Laraki.pdf . 
| Majority Judgement - Measuring, Ranking and Electing igm.univ-mlv.frTraditional Methods and results Incompatibility Between Electing and Ranking Majority Judgement: Two Applications Majority Judgement Measuring, Ranking and Electing |


I look forward to your feedback.
| Majority Judgement - Measuring, Ranking and Electingigm.univ-mlv.frTraditional Methods and results Incompatibility Between Electing and Ranking Majority Judgement: Two Applications Majority Judgement Measuring, Ranking and Electing |


SteveKevin
++++++++++++++++++++++
2. Re: ?goal of a better election method? (Kevin Venzke)


Message: 2
Date: Sun, 12 Feb 2017 17:31:37 +0000 (UTC)
From: Kevin Venzke <stepjak at yahoo.fr>
To: steve bosworth <stevebosworth at hotmail.com>,  EM list
        <election-methods at electorama.com>
Subject: Re: [EM] ?goal of a better election method?
Message-ID: <1346455147.4984684.1486920697303 at mail.yahoo.com>
Content-Type: text/plain; charset="utf-8"

Hi Kevin:
> S: Secondly, I believe that B&L ? prove? (pp. 15, 19, 186-198) that MJ provides only about ? half?  the incentives? >or opportunities for anti-democratic ?strategic? voting to be successful. If you disagree, please explain the flaw in their argument.

K: What I understood from Kristofer's Jan 4 explanation of page 15 is that? this halving of the manipulability is not meant to be a comparison to ?any other methods. It's a comparison to a (rather strange) hypothetical situation. If so, this claim on its own could be true but is of unclear value.
Possibly this argument is used to build up to a larger argument. But when it gets stated on its own it feels misleading to me, because there's no way for the reader to understand what this "half" is half of.

S: No, B&L clearly explain why MJ provides only about ?half? of the incentives and opportunities to ?manipulate? the results as compared to all the methods that gain their results by ?summing? or ?averaging? all the votes.  In his reply to me, Kristofer also gave me a link to the following source in which B&L concisely state the same claim on slide 150.  This is in the middle of their discussion of ?strategy?, i.e. between slides 143 & 185:

Kristofer: See this slide set by B&L for more on that:
http://igm.univ-mlv.fr/AlgoB/algoperm2012/01Laraki.pdf .
| Majority Judgement - Measuring, Ranking and Electingigm.univ-mlv.frTraditional Methods and results Incompatibility Between Electing and Ranking Majority Judgement: Two Applications Majority Judgement Measuring, Ranking and Electing |



| Majority Judgement - Measuring, Ranking and Electingigm.univ-mlv.frTraditional Methods and results Incompatibility Between Electing and Ranking Majority Judgement: Two Applications Majority Judgement Measuring, Ranking and Electing |




Majority Judgement - Measuring, Ranking and Electing<http://igm.univ-mlv.fr/AlgoB/algoperm2012/01Laraki.pdf>
igm.univ-mlv.fr
Traditional Methods and results Incompatibility Between Electing and Ranking Majority Judgement: Two Applications Majority Judgement Measuring, Ranking and Electing

I look forward to your feedback.

Steve

Kevin

 

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