[EM] Dyadic approval implemented as CR
Richard Moore
rmoore4 at home.com
Thu Aug 23 22:22:50 PDT 2001
Roy wrote:
> CR is a great system in theory, but strategic voting
makes it
> degenerate into Approval.
I think I mentioned that (according to my new way of
thinking, anyway) this is true if you have either zero
information or "strong" information (whatever that is), but
in the more likely case where you only a little information
CR starts to need a little fuzzy logic.
Think about the ZI case: You would vote 100% for every
candidate you rate above mean utility, and 0% for every
other candidate. The cutoff point (mean) is well defined.
Say your utilities are 100, 90, 60, and 0; the cutoff point
for the second candidate is 160/3 = 53 and for the third
candidate the cutoff is 190/3 = 63 (you don't include the
candidate you are evaluating in the mean calculation). So,
in CR you would vote 100 for your top two candidates, and 0
for the others.
Now consider a case where you have strong information. Say
you know with 100% certainty that the winner will be either
the second or fourth candidate. Also, say but both are
equally likely. Then you only need to include those two
candidates in the mean calculation, so you would vote for
every candidate you rate 45 or higher (i.e., the top three
candidates would get 100).
In both those cases the cutoff point is crisp. You can plot
your optimum CR vote vs. your utility for a given candidate
(with your other utilities held constant) and get a step
function. Then CR does indeed collapse into Approval.
Now consider a case like Craig's earlier example:
> 9 A>B>C>D=E=F : 100>90>1>0=0=0 : ZI AB : St AB
> 38 B>D>A>C=E=F : 100>52>51>0=0=0 : ZI BDA : St B
> 40 C>B>A>D=E=F : 100>85>70>0=0=0 : ZI CBA : St C
> 9 D>C>B>A=E=F : 100>10>9>0=0=0 : ZI DC : St DC
> 4 E=F>A>B>C>D : 100=100>90>12>10>0 : ZI EFA : St EFA
In this case, we have a lot of information, but it turns out
that we would have a hard time figuring out how to get an
optimum strategy out of the information. The problem is that
we don't know how other voters are going to figure their
strategy. It turned out that B was always a front runner,
but the other front runner was either A or C, depending on
what assumptions we made about the other voters' strategies.
E and F can be ignored entirely (unless you're one of the 4
who actually like those two). So, let's say you conclude
that B has a "high" chance of winning and that A and C each
have a "medium" chance. We'll also conclude that D is low on
the probability scale. We don't know any of these
probabilities to any degree of precision.
If we are in the first group, we know we can vote 100 for A,
and 0 for D, E, and F. We could also rate C at 0, since we
wouldn't want to contribute to a possible victory of C over
either A or B. But should we rate B at 100 or 0? If we were
certain that C was more likely to win than A, and therefore
the most likely tie would be between B and C, we would vote
B=100 since this would hurt C's chances more than it would
hurt A's chances. But if we think the most likely tie is
between A and B, we would vote B=0. (Actually, since our
utility for B is high, the ratio between the two tie
probabilities would have to exceed the ratio between the
utility gaps -- 9:1 -- before we would switch, but the
cutoff would still be sharp.) Unfortunately, in many cases
we just don't have enough confidence in our estimate of the
tie probabilities to make that call. The clean step function
we saw before becomes a curve. The more uncertain our
probabilities are the broader the transition region. In that
case, CR gives us a way out. We can vote a rating for B that
matches our rough guess of the shape of that curve and where
B falls along the transition region. You can think of this
as a way of hedging your bets.
If this thinking is correct, then CR might have an
application to political elections after all.
Richard
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