[EM] "Turkey" problem and MCA

[Kevin Venzke]

Gervase,

Interesting topic.

 --- Gervase Lam <gervase at group.force9.co.uk> a écrit : 
> > Date: Sun, 25 May 2003 10:10:12 +0200 (CEST)
> > From: =?iso-8859-1?q?Kevin=20Venzke?= <stepjak at yahoo.fr>
> > Subject: Re: [EM] The "Turkey" problem and limited ranks
> 
> > 48: A>B>C  (A worth 100, B worth 15, C worth 0)
> > 2: B (B worth 100, A and C worth 0)
> > 48: C>B>A  (C worth 100, B worth 15, A worth 0)
> 
> > In Approval, unless A or C look hopeless, only the 2 voters will
> > approve B.  The other 96 voters are better off trying to break a
> > 100-15 tie than a 15-0 one.  Thus the average utility of the winner
> > is improved by collecting information on preference priorities.
> 
> Interesting to see how things look with the "one step up" improved 
> Approval (i.e. MCA).
> 
> At first, I thought that if MCA were used, B would be the winner.  But 
> now, I'm not 100% sure.  It depends on how anxious 96 voters are at 
> showing on their ballots that B is better than their worst enemy by 
> putting B in the middle slot.  I think the voters most probably would.

Since B is only worth 15 to the 96 voters, I strongly suspect that
they wouldn't approve B at all.  They can easily cause Favorite to lose,
if Favorite would otherwise have had the most Favorite+Approved votes.
(Imagine that both A and C are shy of a majority.  It's not too late 
for Favorite to win.  But he WILL lose if his supporters approve B.)
There is not enough gain for the risk, to approve B.

> Thinking about this a little more, the 96 voters might even put B in the 
> top slot because their worst enemy is very close to the 50% mark.

I believe they would definitely do this if they believed Worst could get
a majority, but Favorite had no chance of a majority.  If Favorite cannot
win with "Favorite" votes, it's a useless gesture to delay approving B
until the "Approved" votes.

>  Would 
> the voters use Approval strategy if normal Approval were used?  If so, 
> they might do the same thing and vote for B as well as their favorite.  It 
> really depends on what type of "Approval" strategy the voters in MCA and 
> normal Approval use.

Hmm.  If their plain Approval strategy is to maximize their ability
to improve the result, then if:
1. A's perceived odds == C's perceived odds, and
2. B is the "middle" candidate for the voters we're talking about, and
3. abs(Favorite's worth - B's worth) > abs(Worst's worth - B's worth), then

The voters shouldn't approve B in plain Approval.  Pretty sure about that.
Their votes stand to gain them less if they do.

I bet it wouldn't be too hard to come up with the ideal strategies for 
MCA.  More odds would have to be estimated, though, such as "perceived 
odds of X getting a majority of Favorite rankings."

Actually, you go a long way if you can estimate "perceived odds of
NOBODY having a majority of Favorite rankings."  In that event
Favorite/Approved makes no difference.

I'm actually going to ponder this.  It seems like an easy assignment,
but I wonder if it might end up being a huge mess.

> May be Forrest is right in saying that the number of slots should be 
> square_root(Number of Candidates).

Hmm.  We were talking about Condorcet slots, I believe.  With
three-rank Condorcet, B would definitely go in the middle rank if he's
at all better than Worst.

How would more slots change MCA?  That's an interesting question.  It
seems to me that you would create a more obvious game of Chicken.  As
MCA already is, it's not "safe" to put candidates in the 2nd rank,
because your 1st rank candidate could still win.  You hope that other
voters put your 1st rank candidates in their own 2nd ranks.  In
other words, you want to withhold your concessions if you can.

I'd bet that if you had five-rank MCA, the 2nd and 3rd ranks would
often be unused.  I'll ponder that, too.

> 
> Thanks,
> Gervase.

Kevin Venzke
stepjak at yahoo.fr


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