[EM] Dyadic approval implemented as CR

[Richard Moore]

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