[EM] Generalized Approval Strategy Example
Richard Moore
rmoore4 at cox.net
Wed Apr 17 23:39:00 PDT 2002
Richard Moore wrote:
> So my modified formula is
>
> delta-Pij = -sum_over_X(Fi(X-Ai)*Gj(X-Aj)*product_over_k(Bik(X-Ak)))
>
> The sum is over all integer values of X for which Fi(X-Ai) is non-zero;
> that could range from 0 to Xmax if Ai = 0, or from 1 to Xmax+1 if Ai = 1.
> For convenience we could just express this as the sum over all integers
> from 0 to Xmax+1. Note that j represents the candidate we are considering
> whether or not to vote for, and i represents the candidate whose win
> probability we want to determine the effect of our j vote upon. (Yeah, I
> know, a preposition is a terrible thing to end a sentence with.)
One more correction -- since we haven't yet decided whether to voter for j,
we calculate delta-Pij with Aj set to zero. So the final formula is:
delta-Pij = -sum_over_X(Fi(X-Ai)*Gj(X)*product_over_k(Bik(X-Ak)))
> I hope to have a small example worked out soon.
4 candidates, 12 voters (other than myself), zero information. That should
be large enough to make it interesting. My utilities are 100, 60, 50, and 0
for candidates 0, 1, 2, and 3, respectively.
For this zero-info case, I'll populate the Fi(X) cells with a binomial
distribution of the 12 voters' choices, assuming a 50% chance a voter
will approve a given candidate. There is another philosophical
question that can be raised at this point. By assuming a binomial
distribution, and 50% approval probabilities, I'm making assumptions
about how the other voters will vote. For one thing, we don't even
know if we will get a full turnout. Even if a full turnout is guaranteed,
the model predicts that each voter has a 1/16 chance of approving no
candidates, and a 1/16 chance of approving all candidates, which would
mean the voters don't know approval strategy. And even if the voters
know approval strategy, are they using zero-info strategy as we are?
Do they know about small-population voting strategies? Since I can't
answer these questions, I have assumed the simple case of independent
binomial distributions of votes for each candidate. There's certainly
an opportunity for someone (else) to develop a statistical model that
could take these other factors into account.
For the initial values of delta-Pij, j = 0, I'll use A0 = A1 = A2 = A3
= 0. So the score probabilities are:
Score probabilities (Fi)
X F0(X-A0) F1(X-A1) F2(X-A2) F3(X-A3)
0 0.000244 0.000244 0.000244 0.000244
1 0.00293 0.00293 0.00293 0.00293
2 0.016113 0.016113 0.016113 0.016113
3 0.053711 0.053711 0.053711 0.053711
4 0.12085 0.12085 0.12085 0.12085
5 0.193359 0.193359 0.193359 0.193359
6 0.225586 0.225586 0.225586 0.225586
7 0.193359 0.193359 0.193359 0.193359
8 0.12085 0.12085 0.12085 0.12085
9 0.053711 0.053711 0.053711 0.053711
10 0.016113 0.016113 0.016113 0.016113
11 0.00293 0.00293 0.00293 0.00293
12 0.000244 0.000244 0.000244 0.000244
13 0 0 0 0
The cumulative probabilites are:
Cumulative score probabilites (Ck)
X C0(X-A0) C1(X-A1) C2(X-A2) C3(A3)
0 0.000122 0.000122 0.000122 0.000122
1 0.001709 0.001709 0.001709 0.001709
2 0.01123 0.01123 0.01123 0.01123
3 0.046143 0.046143 0.046143 0.046143
4 0.133423 0.133423 0.133423 0.133423
5 0.290527 0.290527 0.290527 0.290527
6 0.5 0.5 0.5 0.5
7 0.709473 0.709473 0.709473 0.709473
8 0.866577 0.866577 0.866577 0.866577
9 0.953857 0.953857 0.953857 0.953857
10 0.98877 0.98877 0.98877 0.98877
11 0.998291 0.998291 0.998291 0.998291
12 0.999878 0.999878 0.999878 0.999878
13 1 1 1 1
The Gj values are:
Differential of cumulative probabilities (Gj)
X G0(X-A0) G1(X-A1) G2(X-A2) G3(X-A3)
0 0.000122 0.000122 0.000122 0.000122
1 0.001587 0.001587 0.001587 0.001587
2 0.009521 0.009521 0.009521 0.009521
3 0.034912 0.034912 0.034912 0.034912
4 0.08728 0.08728 0.08728 0.08728
5 0.157104 0.157104 0.157104 0.157104
6 0.209473 0.209473 0.209473 0.209473
7 0.209473 0.209473 0.209473 0.209473
8 0.157104 0.157104 0.157104 0.157104
9 0.08728 0.08728 0.08728 0.08728
10 0.034912 0.034912 0.034912 0.034912
11 0.009521 0.009521 0.009521 0.009521
12 0.001587 0.001587 0.001587 0.001587
13 0.000122 0.000122 0.000122 0.000122
Taking the sum of the products of the appropriate columns (see
the formula) gives these values for delta-Pi0:
0.162174 -0.05406 -0.05406 -0.05406
and the strategic value of candidate 0 (from the scalar product
of these values and my utilities) is 10.27101. This is positive,
so I set A0 = 1.
Setting A0 to 1 shifts the first column of the Fi(X-Ai) down one
position. C0 and G0 are similarly affected. Repeating the
calculations then gives delta-Pi1 (for the case of A0 = 1) as
-0.07 0.147163 -0.03858 -0.03858
and the strategic value of candidate 1 is -0.09919. Therefore
I should only approve candidate 0.
Increasing the utility of candidate 1 to 61 is enough to change
my vote for this candidate from "no" to "yes".
All this was done in a spreadsheet. For larger elections, a
PERL script or a C program would be easier to write (especially
if a different number of candidates is needed).
-- Richard
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