[EM] corrected error in my posting

[MIKE OSSIPOFF]

In my posting yesterday, about election-utility strategy for Approval,
I replaced Ua(Pb+Pc+Pc) with Ua(1-Pa), because when I wrote that
I must have already started interpreting the Pi as win-probabilities,
though they're actually just tie-or-near-tie probabilities.

Of course, as written, the replacement isn't valid. But if
kWi replaces Pi at that point in the demonstration, it works.
Wi is the estimated probability that candidate i will win, and
k is some constant that's the same for all the kWi.

So Ua(Pb+Pc+Pd) becomes Ua(kWb+kWc+kWd), or kUa(Wb+Wc+Wd)

= kUa(1-Wa)

So it's:

kUa(1-Wa) - [kWbUb+kWcUc+kWdUd] > 0

Ua > WaUa+WbUb+WcUc+WdUd

So, when Weber's many-voter assumptions, and the assumption that
Pi is proportional to Wi, are made, the utility-maximizing strategy
votes for the candidates who are better than the voter's expectation
for the election.

One possible remaining objection is that I've assumed that
the win probabilities add up to 1, even though there could be
ties. But when you guess the candidates' win probabilities, can
you guess them so accurately that you can say that they're decisive
win probabilities, not just probabilities of winning ultimately
(maybe after a tiebreaker)? So it isn't so unreasonable for the
Wi to add up to 1.

Even when the voter isn't explicitly using election-utility strategy,
it can still be said that if the voter is maximizing his utility
expection, then, by these assumptions, he's voting for the
candidates whose utility is better than that of the election for
that voter.

So, as Richard was discussing, Approval, when people
vote to maximize their utility expectation, maximizes the number of
voters for whom the outcome is better than they expected, or better
than the incumbant, if that's the utility that they expected,
or better than average, if it's zero-info.

Mike Ossipoff




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