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

[Steve Eppley]

On 1/11/2020 2:42 PM, Kristofer Munsterhjelm wrote:
> On 11/01/2020 17.46, Steve Eppley wrote:
>> On 1/11/2020 6:48 AM, Kristofer Munsterhjelm wrote:
>>> On 10/01/2020 12.41, Steve Eppley wrote:
>>>> For the criterion that matters most to me, I don't have a
>>>> rigorous definition.  Here's a non-rigorous definition:  The
>>>> voting method should give candidates who want to win a strong
>>>> incentive to take positions that the voters themselves would
>>>> collectively choose given a well-functioning direct democracy...
>>>> even on issues that most voters don't care strongly about.
>>>> Here's how I relate that to voting methods like Maximize Affirmed
>>>> Majorities (MAM), which facilitate competition, count all
>>>> pairwise majorities, and pay attention to the sizes of the
>>>> majorities:  Suppose candidate Alice wants to win, and is considering
>>>> taking position p on some issue.  Although she knows a majority of the
>>>> voters prefer alternative q over p, her wealthy campaign donors favor p
>>>> and most voters care more about other issues.  Given a voting method
>>>> like MAM, the risk to Alice is that by advocating p, she would create an
>>>> opportunity for another candidate Bob to enter the race, take position q
>>>> and copy Alice's positions on all other issues.  The larger the
>>>> majority who prefer q over p, the larger the majority who would
>>>> tend to rank Bob over Alice.  Defeating Alice.  A deterrent
>>>> against taking unpopular positions to benefit donors.
>>> How about this? If you clone A into A1 (Bob) and A2 (Alice), and A1 is
>>> ranked above A2 on more ballots than A2 is ranked above A1, then if the
>>> original winner was A, the new winner should be A1.
>>>
>>> That most voters care about other issues than p vs q means that Alice
>>> and Bob should be near-clones, since "Alice but with q" is a slight
>>> improvement to "Alice with p", but not enough of an improvement that
>>> some other candidate is ranked between A1 and A2.
>>>
>>> If voters care more about q vs p, then A1 and A2 will no longer be
>>> near-clones, but hopefully the method should generalize robustly from
>>> the clone case so that it follows the spirit of the criterion.
>> If by "How about this?" you're suggesting satisfaction of that clone
>> criterion ("... the new winner should be A1") implies satisfaction of my
>> non-rigorous criterion ("create a strong incentive to take positions the
>> voters would choose"), I don't see why that would be so. 
>>
>> Instant Runoff would elect A1 in that clone criterion's scenario, yes?
> 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).

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.

Thanks for spending time on this.  I hope you can continue.

--Steve