[EM] election-utility strategy

[MIKE OSSIPOFF]

I've been meaning to post this message for several days, but didn't
have the opportunity to spend enough time on the computer, till today.
Joe & Richard have just posted on that topic, and so it might seem
as if that's why I'm posting on it today. But I've been meaning to
post this for days. I mention that because their messages might make
this one seem a little on the elementary side, and so I don't want
it to seem as if this message is supposed to be some sort of
rebuttal or enhancement to their messages.

Most likely someone here as already checked this matter out, and
has maybe posted about it.

I like the election-utility strategy for Approval, which is why
I asked Richard about what's been demonstrated about it. It's
one of the useful, practical Approval strategies, and is especially
easy to use.

Though election-utility depends on an assumption that might be
more approximate than the assumptions that Weber-Tideman uses,
all the inputs for these methods are approximate anyway, and so
it's questionable whether any serious loss of accuracy results when
using approximate methods.

Say there are 4 candidates: a,b,c & d. Using Weber's many-voters
simplifying assumptions, and his method for maximizing one's
utility expectation, we should vote for candidate a if & only if:

Pab(Ua-Ub)+Pac(Ua-Uc)+Pad(Ua-Ud) > 0

One assumption of Tideman's estimating method is that we can
replace that with:

PaPb(Ua-Ub)+PaPc(Ua-Ub)+PaPd(Ua-Ud) > 0

, where Pa is the probability, if there's a tie or near-tie for
1st place, a will be in it.

That can be written:

Pa[Pb(Ua-Ub)+Pc(Ua-Uc)+Pd(Ua-Ud)] > 0

We can drop the Pa, and have: Pb(Ua-Ub)+Pc(Ua-Uc)+Pd(Ua-Ud) > 0

That can be written:

Ua(Pb+Pc+Pc) - [PbUb+PcUc+PdUd] > 0

Ua(1-Pa) - [PbUb+PcUc+PdUd] > 0

Ua > UaPa+UbPb+UcPc+UdPd

If Pa were the probability that a would win, then this would
be a statement of the election-utility strategy.

Under the assumptions of this discussion, the election-utility
strategy maximizes utility expectation if the probability that
candidate a will be in a tie or near tie for 1st place if there is
one is the same as, or proportional to, the probability that
candidate a will win.

Well, it's reasonable to assume that the candidate a's
tie-or-near-tie probability is proportional to his probability
of winning. If a is more likely to win than b, it's because a is
a stronger candidate, and s/he is also more likely to be in a
tie or near tie for 1st place.

Now, Tideman's approximate assumption that a candidate's
tie-or-near-tie probability is proportional to the square root of
his win-probabiliity is probably better.

But, as I said, the inputs of these methods are so approximate anyway,
that we needn't quibble about accepting an approximation, and
the election-utility strategy is a very useful one.

In fact, I wouldn't hesitate to use election-utility, Weber-Tideman,
top-2-contenders, etc, even in committee elections, for the reason
stated in the previous paragraph, and because though 3-way ties
aren't vanishingly unlikely in committees, they're still significantly
less likely than 2-way ties.

Mike Ossipoff


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