[EM] Defeat Strength

[Kristofer Munsterhjelm]

On 9/9/22 07:12, Kevin Venzke wrote:
> Hi Kristofer,
> 
>>> I'm afraid that defeat strength is such a tangible concept that it inclines us to view it as
>>> a finished product that we can now simply customize as we please. Or worse: That we are
>>> duty-bound to measure it "accurately." We maximize some desired property, readily identified
>>> within a single pairwise contest, and then apply the minmax algorithm to this ranking, and
>>> expect it to result in a method whose properties have some kind of relationship to the
>>> property we were maximizing, or properties that will at least be as good as the defeat
>>> strength metric sounded.
> 
>> Anyway. I guess there are two ways to answer this kind of question:
>> first, we could say "I want to maximize some combination of strategic
>> resistance and VSE" and find out where our indifference curve hits the
>> Pareto frontier.  Or we could say "this concept is essential to
>> democracy as I imagine it and must be present no matter what" (e.g. that
>> majority rule must imply Condorcet).
> 
> I like to keep as few as possible of the second type of opinion.
> 
>> I'd say that for defeat strength, I can't see any obvious second-type
>> argument that say, margins or wv naturally generalize democracy and so
>> must be present.
> 
> Though, I think such an opinion could be possible. If you take specific method properties as
> axiomatic you could significantly constrain the available choices.
> 
>> Statistical models would disagree, too: if
>> bottom-ranking is considered to be the voter saying "I don't know where
>> to rank these candidates", that's a different thing to the voter
>> implicitly intending to bottom-rank every nonranked candidate.
> 
> From the standpoint of minimizing incentives, there is a similarity between these two: No
> voter obtains a complaint or an incentive over preferences they didn't have. (This is where
> symmetric completion hurts the performance, as we may yield to claims that are
> hypothetical.) I'm not sure how to see a difference in implication between "I meant them to
> be equal" or "I didn't know the relative order." Maybe an estimate of social utility
> would vary based on how voted indifference is interpreted?

For truncation in particular, I think equal last means that unranked 
candidates provides information to the method that the voter desired 
those candidates to be considered worse than everybody else, while "I 
don't know" indicates that the voting method shouldn't care at all.

As an extreme example of the distinction, I once visited a web site 
called "kitten duel" or something (it's no longer up), where the site 
would show you two kitten pictures side by side and ask you which is 
cutest. Suppose that there was such a kitten duel site with persistent 
identities. Then if I visit it while logged in and choose the left 
picture, that means I prefer the picture on the left to the right, but 
it doesn't mean that I consider every other kitten picture to be 
equal-ranked for worst. I just haven't been asked, so the method doesn't 
know. It's the difference between my contribution to the Condorcet 
matrix having 0 everywhere not explicitly 1, and having "?" or "N/A" 
everywhere but in two cells.

>> I tend to personally favor wv, but that's partly due to convention,
>> partly due to strategy resistance.
>>   
>> (Now that I think about it, I guess there's a third category: that
>> algorithm feature X should be present because it seems natural to the
>> voters, or that Y shouldn't because it seems to arrive at a conclusion,
>> even if it's a correct one, as by magic, and so won't be considered
>> legitimate by the voters.
>>   
>> Again there would be two perspectives: a relative one where this much
>> strategy resistance or VSE can make up for an incomprehensible
>> algorithm, and an absolute one where there's some level of complexity
>> the voters simply won't accept.)
> 
> "Marketability" I certainly acknowledge as a thing. Hopefully it doesn't force too many
> decisions, and hopefully marketability isn't mistaken for more fundamental merit.
> 
> I'm not completely sure what is the accepted way to determine VSE. Is this the same as the
> principle that a voter's zero info strategy is to submit a complete sincere ranking? I feel
> like that principle lies more on the marketability side, since it shouldn't be a realistic
> scenario.

VSE is roughly speaking expected normalized honest performance under 
some model. You create a number of synthetic voters whose utilities you 
know, have them vote honestly, and check the utility of the winner 
relative to random ballot (on the one hand) and the highest utility 
candidate (on the other).

There's a trade-off between honest performance and strategic resistance 
that creates a Pareto frontier - more of one leads to less of the other 
if your methods are perfectly efficient.

For instance, Random Ballot is entirely strategy-proof but has a nonzero 
probability of electing a candidate that everybody but a single voter 
ranks last, and thus has pretty bad honest performance. On the other 
hand, Borda tends to do well under honesty but has serious strategy 
problems.

-km