[EM] Tideman Weber example

[MIKE OSSIPOFF]

Tideman suggested estimating the proportion relationship of the tie
probabilities by saying that the probability of i being in a tie
for 1st place is proportional to the square root of his probability
of winning. He gave a justification for that rough estimate. I
described his geometrical justification, and suggested another
justification too, at http://www.barnsdle.demon.co.uk/vote/sing.html
, in the Approval & Plurality Strategy article listed in the main
menu.

It's just an estimate, for when there's nothing else to go on.

Unlike tie probabilities, win probabilities are something that
people have a natural feel for. If someone declares candidacy,
we have an impression of how realistic their candidacy is.

Of course we don't need the actual tie probabilities, just a set
of numbers that are in the same proportion relation to eachother as
the tie probabilities are.

That's what the square roots of the win probabilities can be used to
estimate.

Of course one might rate merit or winnability by giving the best
or most winnable candidate 1, 10, or 100, and then rating the others
as fractions of that. Or by giving the worst or least winnable one
1, and rating the others as multiples of that. When writing the
winnability ratings below, I forgot to do it that way, but I'm
using the example as-is, instead of starting over.

4 candidates: A, B, C, & D.

Rate how good they are:

A 10, B 8, C 6, D 4


Rate how realistic their candidacy seems:

A 3, B 5, C 7, D 9


Since the square root of win probability is estimated to be
the probability of being in a tie, then we can multiply those tie
probabilities of i & j to get an estimate of the probability that i
& j will be in a tie for 1st place. Only an estimate.
It seemed more convenient to do the square root of the product
instead of square-rooting before multiplying.

Strategic values of B & C:

Sb = sqr(3X5)(-2) + sqr(5X7)(2) + sqr(5X9)(4) = 30.9

Positive strategic value. Vote for B.

Sc = sqr(7X3)(-4) + sqr(7X5)(-2) + (7X9)(2) = -14.3

Negative strategic value. Don't vote for C.

So you vote for A & B.

For Plurality strategy, you need the strategic value of A:

Sa = sqr(3X5)(2) + sqr(3X7)(4) + sqr(3X9)(6) = 57.25

A has by far the largest strategic value, so we vote for A in Plurality.
Based on the winnability estimates and our utilities, our best strategy
in this example is to vote for the least winnable candidate, in
Plurality.

p.s.

There was a typo in a recent letter from me about my wager probability
definition. I'd said "The probabilities of the wagered-for outcomes of
the Wi are numbers such that..."

I meant: "The probabilities of the wagered-for outcomes of the Wi are
Pi values such that..."

As I said, if "if the Pi are all equal" is added, then the definition
specifies a certain particular set of Pi, and the definition is
a complete definition, whereas if the Pi needn't be equal then it only
describes a necessary condition for the Pi to be the probabilities.

For one isolated unrepeatable event, I'd say that its probability is
what it would be if lots of other events were added to it, to make
the series in the wager definition or the frequency definition.

No doubt the probability of an unrepeatable event has already been
defined, and could be looked up somewhere. I posted my definitions
not because I believe that no other definition is available, but to
show that I haven't been using the word "probability" without
a meaning all this time. My wager definition & frequency definition
are efforts to write what probability of an unrepeatable event has
always meant to me when I've used the word.

Mike Ossipoff




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