[EM] RE: FBC comparison: WV, margins, MMPO, DMC
simmonfo at up.edu
Mon Sep 12 15:50:38 PDT 2005
Dear Adam (and other interested parties),
The John and Jane dialogue must have given you the wrong impression that my other discussions of the possibilities were limited to the three candidate case. Please read them again with the idea in mind that they apply to any finite number of candidates.
I won't quibble about whether an exhaustive consideration of cases is more thorough than a course of simulations. They are both valuable and complement each other, because each can give insights to improve the next version of the other.
Before the dialogue, I considered two cases: (1) Compromise is sure to beat Favorite. (2) Compromise is not sure to beat Favorite.
So far these two cases exhaust the possibilities, but that won't stop us from considering some subcases later.
In case (1) DMC gives no incentive at all for Favorite Betrayal, because approval of Compromise already reinforces the Compromise>Favorite defeat to the max.
However, in the same case under Shulze, Ranked Pairs, or River (whether margins or wv) it sometimes helps to betray Favorite by ranking Compromise strictly ahead of Favorite. I'll give an example below, for those that have never seen this before.
Near the end of my previous message (in a part not quoted by Adam) I showed that (in the Bubble Sorted Approval formulaton of DMC) only in (what I called) case 2d would Favorite Betrayal payoff, and that case 2d is not only unlikely, but also very difficult to trust the polls on.
(I would appreciate a similarly thorough analysis of cases (1) and (2) from the wv folks.)
Here is case 2d.
Favorite has more approval than Compromise (so is seeded above Compromise), and Favorite gets defeated by someone seeded above Compromise before being challenged (but not defeated) by Compromise.
At this point we might regret that we helped Favorite block
Compromise from a chance of proceeding to the top.
However, given that Compromise had less approval than Favorite and that
Favorite beat Compromise, we shouldn't expect Compromise to have great
chances of beating all the guys ahead of Favorite, notwithstanding
pre-race polls to the contrary.
My further comment on 2d was this:
This last case (2d) would be a pretty flimsy excuse for voting Compromise
over Favorite. I suppose it could happen if your preference of Favorite
over Compromise were extremely weak, or if you had extremely precise,
detailed information in advance about voter preferences that gave Compromise a sure
chance of beating every candidate that could beat Favorite despite
Compromise having less approval than Favorite and being unable to beat
The John/Jane dialogue only served to dramatize the difficulty of convincing someone to rank Compomise over Favorite under case 2d (with three candidates for simplicity).
Now here's the promised example (in tournament form) of a case where wv voters would regret not having ranked Compromise ahead of Favorite, even though Compromise did end up beating Favorite, pairwise.
First the wv pairwise results without the Favorite betrayal. Here F stands for Favorite, and C stands for Compromise, and the defeats are expressed in the form A(65)D, meaning that A beat D by 65 to something:
C(55)F(70)A(65)D(75)F together with A(60)C(20)D
After weakest defeats in their cycles, i.e. the ones of strength 20, 55, and 65, respectively, we are left with a beatpath D(75)F(70)A(60)C . Candidate D wins under Beatpath, River, and Ranked Pairs.
Now suppose that 25 voters go back in a time machine and reverse their F>C marks to C>F.
Then the C(55)F defeat becomes C(80)F, and all of the other defeats retain their directions and wv strengths.
This time the weakest defeats in their respective cycles are the ones with strengths 20, 60, and 65.
Already by the time we remove the defeats of strength 20 and 60, candidate C is undefeated.
If we proceed as in Ranked Pairs the respective defeats of strengths 80, 75, 70, and 20 are set in place, with C as winner.
In River we set in place the defeats of strength 80, then 70, and then 65, which coincidentally yields the strongenst beatpath C(80)F(70)A(65)D through the candidates.
If I am not mistaken, River, Beatpath, and Ranked Pairs all agree on C as winner.
I hope that this earns some respect for DMC's Favorite Betrayal resistance :-)
From: Adam Tarr [mailto:ahtarr at gmail.com]
Sent: Fri 9/9/2005 7:18 PM
To: Simmons, Forest
Cc: election-methods-electorama.com at electorama.com; Condorcet at yahoogroups.com
Subject: Re: [EM] RE: FBC comparison: WV, margins, MMPO, DMC
On 9/9/05, Simmons, Forest <simmonfo at up.edu> wrote:
Adam and Kevin have argued that Winning Approval tends to encourage Favorite Betrayal more than Winning Votes does.
But a thorough examination of the possibilities does not seem to support that view.
A "thorough examination" would really involve some sort of simulation or aggregation of a whole host of reasonable scenarios; something neither you nor I have done in any form. All we have are our gut instincts about the likeliness of various scenarios, which disagree. Kevin has discussed running a simulation to test this, which seems like a great idea.
When winning votes is used as a measure of defeat strength, you know that if your compromise does beat your favorite, then you can still help your compromise by strengthening that defeat. In order to do this, you have to rank your compromise strictly ahead of your favorite.
Let's examine the case you're hypothesizing here.
A priori, assume our sincere preference is A>C>B. You said that compromise beats favorite, so C>A. Since compromise doesn't already win, we know B>C. Since the case is trivial (and hopeless) if B is the Condorcet winner, we assume the cycle A>B>C>A.
Since you state a need for favorite betrayal ("helping your compromise") here, we can further assume that the weakest defeat is A>B, causing B to be elected.
So strengthening C>A does nothing for our cause. What we actually need to do is WEAKEN B>C or C>A, so that A>B is no longer the weakest defeat.
But a priori, C>B and A>C are our sincere preferences. We are already doing everything in our power to weaken those defeats.
So order-reversal cannot help our case here. Winning votes is immune to this particular favortie beytrayal scenario. Your scenario does not make sense, as I understand it. Maybe I made a mistake in my analysis; if you believe so, then please present an example.
Summary: This conversation shows that it would take a lot of confidence in a set of unlikely (but possible) conditions to entice a rational voter into Favorite Betrayal under DMC even if condition (3) is thought to be likely. Furthermore, if condition (3) is known to be false, then there is no Favorite Betrayal incentive at all.
Does that help?
What would be more convincing to me would be to show a reasonable scenario where wv has favorite betrayal incentive, but DMC does not. I believe I have shown the opposite.
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