Condorcet's Advantages (fwd)

[Mike Ossipoff]

Mike Ossipoff writes:


[Sorry about the extra header--I don't have a way to delete
lines or blocs of text. Or, if I do, I haven't pursued finding
it yet. My editor that I use goes all haywire if I try to
delete lines]

> From dfb Mon Jul  8 17:00:18 1996
> Subject: Condorcet's Advantages
> Date: Mon, 8 Jul 96 17:00:15 PDT
> From: Mike Ossipoff <dfb at bbs.cruzio.com>
> Cc: dfb at bbs.cruzio.com
> X-Mailer: ELM [version 2.3 PL0]
> Message-ID:  <9607081700.aa02949 at cruzio.com>
> 
> 
> 
> As I was saying, Condorcet is the best in regards to the 
> lesser-of-2-evils problem. I'll put that in precise terms
> in this letter, using precisely-stated yes/no tests referred
> to as criteria. A criterion is a benchmark for measuring
> compliance with a standard.
> 
> I talked in my previous letter about why the lesser-of-2-evils
> (LO2E) problem is important. It's the reason why practically
> no progressive here is willing to vote honestly, and why
> they all feel they must "hold their nose", and vote for
> a lesser evil, instead of voting for their favorite. They
> actually vote for people they dislike, to keep someone worse
> from winning. The problem is that if Ralph Nader has a win,
> then we'll never know it, because of LO2E voting.
> 
> Condorcet's method counts "votes-against". That means that if,
> in a Dole, Clinton, Nader election, if you vote
> 
> 1. Nader
> 2. Clinton
> 
> ...this counts as votes against Dole.  What that means is that
> the mere fact that you've ranked Nader & Clinton over Dole
> means that your vote against Dole is reliably counted at full-
> strength.  So right there it's evident how Condorcet gets rid
> of the lesser-of-2-evils problem: If you've merely voted the
> lesser-evil somewhere over the greater evil, then you're
> helping to deny victory to that greater evil. 
> 
> This, of course, isn't the case with the Instant-Runoff, where
> Clinton could be eliminated in the 1st count, and if his voters
> have ranked Dole 2nd, then Dole could win because you ranked
> Clinton 2nd instead of insincerely ranking him 1st.
> 
> So Condorcet gets rid of the LO2E problem in a really transparent
> way, based on its choice rule that I stated in my previous
> letter.
> 
> In particular, if a majority of all the voters have voted Clinton
> over Dole, there's no way that Dole can win, based on Condorcet's
> rule, unless _every_ candidate similarly has a majority against him.
> If you're one of the voters who ranked Clinton 2nd, over Dole, 
> then you're counted as part of that majority that deny victory
> to Dole.
> 
> Some criteria:
> 
> The Generalized Majority Criterion:
> 
> A choice rule meets the Generalized Majority Criterion if &
> only if an alternative over which a full majority of all the
> voters have ranked another alternative can't win unless every
> alternative in the set from which that choice rule is to choose
> has an alternative ranked over it by a full majority of all the
> voters.
> 
> 
> Of the single-winner (sw) methods proposed, Condorcet is the only
> one that meets the Generalized Majority Criterion (GMC).
> 
> 
> The Lesser-of-2-Evils Criteria:
> 
> If all the voters in a set, M, of voters, consisting of a majority
> of all the voters, have ranked all the alternatives in some set
> S1 over all of the alternatives in some other set, S2, then
> a method meets Lesser-of-2-Evils Criterion #1 (LO2E-1) if &
> only if paragraph a) is true for that method. The method meets
> Lesser-of-2-Evils Criterion #2 (LO2E-2) if & only if paragraph
> b) is true for that method:
> 
> a) No alternative in set S2 can win unless either order-reversal
> (defined below) has taken place on a scale sufficient to change
> the election result, or every alternative in the set of alternatives
> from which that method is to choose has another alternative preferred
> to it by a full majority of all the voters.
> 
> [Order-reversal is the pracatice of ranking a less-liked alternative
> over a more-liked one]

[Actually, I'd rather substitute the words "preference falsification"
for "order-reversal", since "preference falsification" includes
voting a preference between alternatives between which one doesn't
have a preference].

> 
> b) The voters in set M can always ensure that the winner won't
> come from set S2, and they can achieve that without ranking
> a less-liked alternative equal to or over a more liked alternative.
> 
> ***
> 
> These 3 criteria, GMC, LO2E-1, & LO2E-2, are the things that set
> Condorcet's method apart from all other methods.
> 
> ***
> 
> Condorcet's method, as I've defined it, strictly meets GMC
> & LO2E-1. With a simple rule added to it, Condorcet strictly
> meets LO2E-2. But even without that rule, Condorcet meets LO2E-2
> for all paractical purposes, which is more than can be said for
> any other method. 
> 
> First I'll state the added rule, called the "subcycle rule", and
> then I'll talk about why Condorcet meets LO2E-2 even without
> the subcycle rule.
> 
> In fact, after stating the subcycle rule, I'll demonstrate
> Condorcet's compliance with all 3 criteria.
> 
> ***
> 
> The subcycle rule:
> 
> Definitions:
> 
> A "cycle" is a set of alternatives that beat eachother in circular
> fashion: A beats B beats C beats A, for example.
> 
> Cycle A is a "subcycle" of cycle B if every alternative in cycle
> A beats everything in cycle B that is beaten by something in cycle
> A, and if every alternative in cycle A is beaten by everything that
> beats something in cycle A.
> 
> What it amounts to is that cycle A is an element of cycle B. A
> cycle that's an element of another cycle.
> 
> 1. Before choosing a winner, solve any cycles that exist.
> 
> 2. Before solving a cycle, first solve any subcycle of that cycle.
> 
> 3. To solve a cycle, apply Condorcet's choice rule to the alternatives
>    in that cycle, choosing a winner among those alternatives according
>    to the Condorcet choice rule. Eliminate from the election every
>    member of that cycle except for the winner. If that cycle, A, is a
>    subcycle of another cycle, B, then the winner in A replaces cycle
>    A in cycle B, occupying the same position in cycle B that cycle
>    A occupied. This last sentence isn't strictly necessary as part
>    of the rule, since it follows from the definition of a subcycle
>    that the winner in A beats what the other members of a beat
>    and is beaten by what beats them.
> 
> ***
> 
> As I said, with the use of the subcycle rule, Condorcet's method
> strictly meets LO2E-2, as well as GMC & LO2E-1.
> 
> ***
> 
> As for why Condorcet meets those criteria:
> 
> GMC:
> 
> This follows directly from the statement of Condorcet's rule.
> If alternative X has a majority ranking something over it, and
> no other alternative has that many people ranking something
> over it, then  X can't possibly win, since Condorcet chooses
> its winner by counting votes-against, by counting how many
> voters have ranked over an alternative the alternative which,
> among those that beat it, is ranked over it by the most voters.
> 
> 
> LO2E-1:
> 
> This follows from GMC. The alternatives in set S2 all have
> 1 or more alternatives ranked over them by a majority. If
> not every alternative has another alternative preferred to it
> by a majority, and if no one falsifies a preference,
> then it's impossible for every alternative to have another
> alternative ranked over it by a majority.
> 
> So LO2E-1 is just a re-statement of GMC in different terms.
> 
> 
> [Actually, instead of "order-reversal", I should say "order-
> falsification", the voting of a preference ordering that isn't
> one's true preference]
> 
> 
> LO2E-2:
> 
> Again, every alternative in set S2 has at least 1 alternative
> ranked over it by a majority. Since the voters in M constitute
> a majority, they have the power to ensure that any particular
> alternative _doesn't_ have a majority ranking anything over it.
> 
> By so doing, they can ensure that the winner won't come from
> S2.
> 
> In particular, if they don't include any S2 alternatives in
> their ranking, then no non-S2 alternative can possibly have
> an S2 alternative ranked over it by a majority. The only
> way that every non-S2 alternative can have another alternative
> ranked over it by a majority is if every non-S2 alternative
> has another non-S2 alternative ranked over it by a majority.
> 
> That's a cycle among the non-S2 alternatives. The subcycle
> rule gets rid of that cycle before choosing a winner, eliminating
> everything but the winner of that cycle. What that is done,
> the remaining winner in the cycle no longer has anything ranked
> over it by a majority, and so none of the S2 alternatives can
> possibly win, since all the S2 alternatives have at least
> 1 alternative ranked over them by a majority. Since Condorcet
> complies with GMC, no S2 alternative can possibly win.
> 
> This was accomplished by the M voters merely by refusing to
> include any S2 alternatives in their rankings. They didn't
> have to rank any less-liked alternative equal to or over
> any more-liked alternative.
> 
> So Condorcet strictly complies with LO2E-2, as well as LO2E-1
> & GMC, when the subcycle rule is used.
> 
> ***
> 
> As for why Condorcet complies with LO2E-2 for all practical
> purposes, even without the subcycle rule:
> 
> If the S2 voters hope to use order-reversal to create a
> circular tie (A beats B beats C beats A), and to win
> that circular tie, by Condorcet's scoring rule, even though
> no M voter has included any S2 alternative in his/her ranking,
> then they need a cycle to exist among the non-S2 alternatives.
> Either they have to be lucky enough for such a cycle to exist
> spontaneously, or they have to be lucky enough to have an
> opportunity to make such a cycle, and they have to have
> sufficient predictive knowledge to be able to take advantage
> of that opportunity. This is so improbable that the S2 voters
> certainly can't safely use order-reversal if the voters in
> M have refused to rank any S2 alternatives.
> 
> In fact, successful order-reversal by S2 votes is even
> more improbable & impossible to predict when another
> refinement, called the "Smith set" is used. The Smith
> set has other advantages too, and I'll describe it in a
> subsequent message.

I should add here that the reason why the Smith set makes it
even more highly improbable for LO2E-2 to be successfully violated,
even when the subcycle rule isn't used, is that to do so, the
order-reversers would have to contrive to create a cycle, and
also a subcycle on the cycle. That's rather a lot to expect of
them. They'd have to do that, because the circular tie would have
to include S2 alternatives, in order to qualify them to win. And
there'd have to also be a cycle among non-S2 alternatives.

> 
> 
> 
> 
> 
> Mike Ossipoff
> 
> 
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> 
> -- 
> .-
> 


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