[EM] IRNR

[bql at bolson.org]

On Mon, 17 May 2004, MIKE OSSIPOFF wrote:

> Brian Olson--
>
> You probably have written a better method than IRV.
>
> You wrote:
>
> Instant Runoff Normalized Ratings
> (IRNR)
>
> Every voter casts a rating of each choice on a scale of -1.0 to 1.0 or
> some equivalent scale. Each voter's voting power is normalized, each
> rating is divided by the sum of the absolute values of the ratings
>
> I reply:
>
> The sum f the absolute values of his own ratings, or of all the ratings of
> him and the other voters?
>
> The rest of the definition seems clear.

Normalization happens within a voter's ballot, not between voters. If I
vote {1.0, 0.5, -0.5, -1.0} that normalizes to {2/6, 1/6, -1/6, -2/6}. If
C of A,B,C,D gets disqualified, my renormalized ballot is {0.4, 0.2, dq,
-0.4}.

> You continued:
>
> Condorcet: yes-ish.
>
> I believe that IRNR is more powerful than Condorcet
> because it also addresses the degree of preference and not just the order.
>
> I reply:
>
> That's how a method loses Condorcet's strategy advantages. If how far above
> Y you rank X is counted, then the only way you can fully vote X over Y is if
> you vote X in 1st place and Y in last place. That gives Borda its big
> strategy problems.
>
> People would like to be able to fully vote Compromise over Worst, while
> still fully voting Favorite over Compromise. That's impossible unless we
> count only the fact that you vote X over Y, disregarding how far over Y
> you've voted X.
>
> That's why CR doesn't have Condorcet's strategy advantages. But CR is still
> very good, because it has its own strategy advantages. Personally I like the
> advantages of Condorcet best, but I'd gladly settle for CR or Approval as
> the official public voting system.

The tradeoff is interesting. It seems that a ranking system such as
Condorcet treats rankings as absolute and total, each choice is
effectively infinitely better than the next. IRV makes this even more
explicit by its process of throwing a whole vote around. I guess I tend to
see the world in a more nuanced way.

I wanted the instant-runoff part in IRNR so that people could throw
whatever nuanced weight behind any choice they want and know that in the
end their whole vote would be counted. I single-pass CR without
normalization looses this nuance and becomes approval. I think a
single-pass CR with normalization could degenerate to a Single Vote
strategy because you don't want to waste part of your voting power on
someone who won't win. Thus, IRNR to make sure that after winnowing down
to 2 choices, your whole vote is being applied.

> "Strategy Free": maybe not. A 51% majority could rate candidate A at 0.02
> and B at 0.01, 49% could vote B 1.0 and A -1.0 . B would win. Does this
> violate SFC?
>
> I reply:
>
> A point system isn't likely to meet SFC. IRNR probably doesn't meet SFC.
> Your example that you give above, with the 51% and the 49% addresses a
> reason why IRNR is unlikely to meet CC or SFC.
>
> But it isn't a complete failure example. Can an example be written in which
> no one falsifies a preference, and a majority of all the voters prefer the
> CW to candidate B, and vote sincerely, but B wins anyway in IRNR? If so,
> then IRNR fails SFC.

[snip]

> B wins with IRNR in that 2-candidate example, doesn't it?

Yup.

plugged into the web toy:
*49 -1,1
*51 .02,.01

Condorcet choses A, IRNR choses B.

> You continued:
>
> Is it a just system anyway?
>
> I reply:
>
> Maybe. Failing SFC doesn't make a method unjust. Approval and CR fail SFC,
> and they're among the best methods. But I still like SFC compliance, which
> is why Condorcet wv is my favorite.

I think my optimizing for a global utilitarian satisfaction. Between the
previous and the next examples, the ratings show that a
most-loved/least-hated winner is not the Condorcet Winner. I suppose as in
many voting theoretic arguments, we're left to debate how likely the
various populations are.

> You continued:
>
> If IRNR were modified to expand
> votes out to a 1.0 to -1.0 scale before normalizing them the 51% vote
> would translate to A=1.0 and B=-1.0; A would win.
>
> I reply:
>
> If all the negative ratings becames -1, and all the posiive ratings became
> 1, wouldn't that change the method to Approval? Sure, then the majority
> would win in the 2-candidate example. That shows an advantage of Approval
> over CR: Approval doesn't give people a chance for sincere voters to be had
> by strategizers as can happen in CR.

No no, The lowest rating on a ballot is stretched to -1.0 and the highest
to 1.0, ratings in between scaled appropriately. A vote { .5, .25, 0 }
becomes { 1.0, 0.0, -1.0 }. This could be a potential strategy point for
rating votes, so maybe take it away from the poor voters who don't know
how to vote and build it into the system. Although, some people want a
voting system that rewards smart voters.

> But that doesn't mean that the modified IRNR woiuld then meet SFC. It would
> no longer fail SFC in that particular example that I used before.  But with
> more candidates it would. Say there are 3 candidates, and the majority
> prefer B to C. Some of that majority have A as favorite. They give 1 to A
> and -1 to B & C. That's sincere, because they aren't falsifying a
> preference, voting a preference that they don't have. And they aren't
> failing to vote a preference that the balloting system in use would allow
> them to vote in addition to the preferences that they actually did vote.
> That's because if they gave 1 to B, they'd no longer be voting their
> preference for A over B. So they're voting sincerely when they give -1
> instead of 1 to B.
>
> The C voters, let's say, give 1 to C and -1 to B & A. Though they're a
> little short of a majority, the C voters have no trouble thereby making C
> get more points than B or A. Because, let's say, the B voters have given -1
> to A, which is sincere for the same reasons described above.
>
> So C wins even though a majority prefer B to C and vote sincerely. All that
> remains is to make B be the CW in the example. Sincere preferences:
>
> 40: CBA
> 25: BCA
> 35: ABC
>
> B is CW, and a majoritly prefer B to C, as SFC's premise requires. In my
> example, the voting is as follows:
>
> 40: 1C, -1B, -1A
> 25: 1B, -1A, -1C
> 35: 1A, -1B, -1C
>
> Those ballots are sincere for the reasons that I described, though all SFC
> requires is that the B & A voters vote sincerely.
>
> Point totals:
>
> A: -30
> B: -50
> C: -20

I submitted this to my web toy, http://bolson.org/voting/vote_form.html ,
as:

*40 -1,-.99,1
*25 -1,1,-.99
*35 1,-.99,-1

It does indeed extract the Condorcet winner as you say and IRNR chooses C.
If you look at the histograms it is clear that C is the most-loved and
least-hated choice.

> So no one is falsifying a preference, and a majority prefer the CW, B, to C,
> and vote sincerely, and C wins.
>
> C wins, even though the premise conditions of SFC are met. -1,1 CR, which is
> the really the same as Approval,  fails SFC.
[snip]
> You continued:
>
> Strong Defense Strategy: Yes. A majority casting votes can win without
> mis-ordering any votes.
>
> I reply:
>
> But SDSC requires more than that. A majority preferring X to Y has to be
> able to make Y lose without voting a less-liked candidate _equal to_ or over
> a more-liked one.
>
> (Where someone votes X equal to Y iff s/.he doesn't vote either over the
> other, but votes each over someone).
>
> Say there's a bare majority for X over Y. Say the Y voters give maximum to Y
> and minimum to everyone else. The majority must give nearly the maximum to X
> (and the minimum to Y) to make Y lose. Even if the X favorite voters give
> maximum to X, the members of the X>Y majority to whom X isn't favorite still
> have to give some nearly maximum amount to X, to make him beat Y. Sure,
> ideally they could give barely enough to X to put him over Y, and give
> maximum to their favorite. And, if they can give as finely-graded point
> ratings as they want to, they could still avoid voting less-liked candidates
> equal to more-liked candidates. So, it looks as if infinitely-finely-graded
> CR could meet SDSC.
>
> Your method isn't CR, and I don't know if it meets SDSC, but it might, if it
> allows infinitely-finely-graded ratings.

There's no reason why IRNR can't be implemented with sufficiently fine
resolution. Give everyone a thousand millivotes or a billion nanovotes. My
current implementations use 64 bit floating point numbers.

But it doesn't have to be too fine grained, here's how a bare majority for
X can beat a fanatical minority for Y, and then there's some Z.

*49 -1,1,-1
*51 1,-.1,-.2

The 51% could even give Z something between 1 and -0.1, it wouldn't
matter.

> You continued:
>
> Does SDS actually mean that the winner should be picked by the largest
> majority?
>
> I reply:
>
> No. SDSC requires that if a majority prefer X to Y, then they should have a
> way of voting that ensures that Y won't win, without voting a less-liked
> candidate equal to or over a more-liked one.
>
> (A voter votes A equal to B iff s/he doesn't vote either over the other, but
> votes each over someone).

OK, yeah. I think I've got that.

> Approval and CR meet FBC and WDSC. And, as I was saying, maybe CR can meet
> SDSC if it allows infinitely fine gradation of points ratings.
>
> So maybe your point system can meet those criteria too.
>
> The eliminations complicate the compliance determinations, and I can't say
> at first glance if IRNR meets those criteria.
>
> As someone already pointed out, the eliminataion process prevents you from
> saying that IRNR meets Participation based only on initial point scores.
>
> Mike Ossipoff

So, the failure mode for Participation is: When will additional ballots
which prefer X to Y cause an election to switch away from X? (right?)
(does it have to be to Y, or does it count if it's to Z - a compromise
candidate?) After running a few quick tests I'm inclined to think that
there isn't a way to trick IRNR in that way.

Brian Olson
http://bolson.org/