[EM] An incentive to take positions a direct democracy would choose (was Re: Arrow's theorem and cardinal voting systems)

[Steve Eppley]

On 1/22/2020 6:50 PM, Kristofer Munsterhjelm wrote:
> On 12/01/2020 21.59, Steve Eppley wrote:
>> On 1/11/2020 2:42 PM, Kristofer Munsterhjelm wrote:
>>> I don't think it would in every such scenario. Consider this election pair:
>>>
>>> Before cloning:
>>>
>>> 110: A
>>> 100: X>A
>>> 100: Y>A
>>>
>>> X and Y are eliminated and then A wins.
>>>
>>> After cloning:
>>>
>>> 110: A2>A1
>>> 100: X>A1>A2
>>> 100: Y>A1>A2
>>>
>>> First A1 is eliminated, and then X and Y are eliminated, and then A2
>>> wins. But A1 is the CW and beats A2 pairwise 200-110.
>>>
>>> If the q-preferring majority ranks A1 and A2 low enough, then IRV may
>>> exclude A1 before it gets to determine who should win of A1 and A2. It's
>>> the usual center squeeze.
>>>
>>> Does that make the clone criterion more suited to your purposes, or
>>> would it have to be stronger? I suppose the clone criterion is a sort of
>>> local optimum criterion (if Alice exists, then Bob can copy all of
>>> Alice's positions except the one a majority dislikes, and overtake
>>> Alice), while your non-rigorous criterion is a global optimum criterion.
>>>
>>> (In passing, I think I see that LIAA + clone independence implies this
>>> clone criterion, as well.)
>> You're right that Instant Runoff fails "clone A1 should win."
>>
>> I don't know whether its satisfaction implies satisfaction of "the
>> incentive to take positions the voters would choose."  My election
>> method analysis skills are very rusty.
>>
>> I don't recall LIAA.  I assume you mean LIIA (Local Independence of
>> Irrelevant Alternatives, promoted by Peyton Young).
> Yes, that should have been LIIA.
>
>> There appears to be a flaw in that clone criterion.  Suppose 3
>> clones majority cycle: Bob > Alice > Charlie > Bob.  The premise of
>> the "clone A1 should win" criterion could hold: In the "original"
>> scenario where Bob doesn't run, Alice wins.  We don't have enough
>> information to show that Bob will win if Bob runs too.  Alice could
>> still win if the Bob>Alice majority is the smallest of the three
>> cyclic majorities. (When I described my thinking about the incentive
>> in MAM, I wrote: "The larger the majority who prefer q over p, the
>> larger the majority who would tend to rank Bob over Alice.")  But
>> that clone failure isn't necessarily a failure of the voting method
>> to create the strong incentive.  My hunch is that typically,
>> candidates like Alice won't be able to rely on a Bob>Alice majority
>> being the smallest in a cycle, when taking positions on issues.  The
>> chance that Bob>Alice won't be smallest in a cycle is a risk to be
>> avoided, all else being equal.
> I'm not entirely sure what you mean. Do you mean that even if "A1 should
> win" happens to be necessary, it isn't sufficient; or that even if it
> happens to be sufficient, it isn't necessary?
>
> I *think* you're saying it's not sufficient, because there could already
> be a clone of Alice, and then when Bob enters, he could have the
> smallest of the cyclic majorities and create a cycle, and he won't win.
>
> More generally, we can say that he creates a three-cycle and the cloning
> comes out so that, according to the cycle-resolution mechanism of the
> method in question, he doesn't win even though he's in the Smith set.
> But then it would seem that no matter what method you have, it's
> possible to construct the cloning so that the right clone loses.
>
> If that's right, then there has to be some kind of additional structure
> that makes it possible for the method to distinguish the right clone
> from the other clones. In the original example, that is that Bob copies
> all of Alice's positions except the disliked one, where he does better
> according to a majority. For the three clones to create a cycle, there
> has to be some set of properties so that a majority prefers Bob's to
> Alice's, Alice's to Charlie's, and Charlie's to Bob's. But then,
> wouldn't Bob have to differ from Alice by more than one property?

Where you ask whether I meant "isn't sufficient" or "isn't necessary," what I meant is that the clone criterion you proposed, "A1 must win," doesn't clearly distinguish between voting methods that create the desired incentive and voting methods that don't. (Neither does the "weaker" criterion "Alice must not win.")  Alice can still win even though clone Bob is ranked over Alice by a majority.  There may be voting methods that fail the clone criterion yet create the desired incentive anyway.  The clone criterion might not be necessary.

And perhaps some voting methods that satisfy the clone criterion may fail to create the desired incentive in more general, non-clone cases.  The clone criterion might not be sufficient.  A "clone recognition filter" could be tacked onto a bad voting method, contrived to recognize when the "A1 must win" premise exists in the votes and guarantee the defeat of A2 in the rare case when the premise holds, yet not create the desired incentive in other cases.  For example, the voting method "if clone A2 must lose then elect clone A1; else elect the Instant Runoff winner" would satisfy whatever clone criterion you like that implies A2 must lose, but fail to create the desired incentive.

When you say "there has to be some kind of additional structure that makes it possible for the method to distinguish the right clone" it looks like you're thinking specifically about clone criteria, and possibly not generally enough to cover non-clone cases too.  Although you may be right that any rigorous criterion that distinguishes voting methods that create the desired incentive must be some kind of clone criterion, I think that's just speculation.  The ability to contrive a "clone filter" in the previous paragraph suggests the speculation is wrong.

Regarding your question about whether Bob would need to differ from Alice by more than one property (policy) for there to be a "Bob>Alice>Charlie>Bob" majority cycle (in which the Bob>Alice majority might unfortunately be the smallest majority), the answer is No, Bob can differ from Alice on only one policy.  Suppose issue I1 is abortion and issue I2 is taxes.  Suppose that on abortion, a majority prefer policy aMaj over policy aMin.  Suppose that on taxes, a majority prefer tMaj over tMin.  Suppose Charlie takes positions aMaj and tMaj (consistent with my goal that the voting method should create an incentive to take majority-preferred positions).  Suppose Alice takes positions aMin and tMin, after cleverly calculating that the minority who prefer aMin and the minority who prefer tMin are "single issue voters" and thus will together comprise a majority coalition who prefer policy pair aMin&tMin over policy pair aMaj&tMaj.  Bob has two obvious options, either of which
produces the majority cycle: (1) Bob can take positions aMaj and tMin, or (2) Bob can takes positions aMin and tMaj.  In either option, Bob differs from Alice by only one property.  The option that's better for Bob depends on which issue majority is smaller (given a voting method that pays attention to the sizes of the pairwise majorities).

There may of course be other reasonably simple options that are even better for Bob, if I1 or I2 isn't a dichotomous issue.  For example, suppose policy aMaj is "no restrictions whatsoever on a woman's right to have an abortion" and aMin is "completely ban all abortions."  Bob could consider a range of abortion policies between those two extremes... for example "no restrictions during the first 6 months after conception, and after 6 months allow abortion only for medical need." (Note: issues are rarely one-dimensional.  Where I use one-dimensional terms like "between" I'm simplifying for the sake of discussion.)

--Steve