[EM] Re: Multi-Winner Approval Strategy

[Gervase Lam]

For 0-info, I am almost convinced that "Vote for the above-mean
candidates" is the strategy to use.  I'll start from the beginning.  (Note 
that what I'll be doing won't be exactly how Weber presents his 
calculations inpublications.)

In order to work out what a voter should put on an Approval ballot, Weber 
used a technique where he pitted each candidate head-to-head with the 
other candidates.  In order to work out the result of each head-to-head, 
he used the following equation, which works out the "Weber score" (my 
terminology, not his) of how one candidate does against another.

Wij = Pi * Pj * (Ui - Uj)

Wij is Candidate i's Weber score against candidate j
Pi is the probability/chance of candidate Pi winning the Approval vote
Pj is the probability/chance of candidate Pj winning the Approval vote
Ui is the voter's utility of candidate i
Uj is the voter's utility of candidate j

Since this is a 0-info race, the field is wide open.  Nobody knows who is 
going to win.  This means that the probabilities (Pi and Pj) are the same 
for all candidates.  As no division or multiplication will be done with 
this formula, the Pi * Pj can be "divided away" from all the 
head-to-heads.  This leaves behind:

Wij = Ui - Uj

Next I'll bring in six candidates (A to F) who I'll mark out of 10.  10 is 
given to the best candidate, 0 to the worst.

A = 10
B = 5
C = 4
D = 3
E = 2
F = 0

OK.  So those are the utilities of the candidates sorted it out.

Now I can pit each candidate against each other in order to work out the 
Weber scores.  This is shown below in the matrix below (which needs a 
fixed width font to see properly):

    i = A        i = B       i = C       i = D       i = E        i = F
A[ 0 = 10-10] [-5 = 5-10] [-4 = 4-10] [-3 = 3-10] [-2 = 2-10] [-10 = 0-10]
B[ 5 = 10- 5] [ 0 = 5- 5] [-1 = 4- 5] [-2 = 3- 5] [-3 = 2- 5] [ -5 = 0- 5]
C[ 6 = 10- 4] [ 1 = 5- 4] [ 0 = 4- 4] [-1 = 3- 4] [-2 = 2- 4] [ -4 = 0- 4]
D[ 7 = 10- 3] [ 2 = 5- 3] [ 1 = 4- 3] [ 0 = 3- 3] [-1 = 2- 3] [ -3 = 0- 3]
E[ 8 = 10- 2] [ 3 = 5- 2] [ 2 = 4- 2] [ 1 = 3- 2] [ 0 = 2- 2] [ -2 = 0- 2]
F[10 = 10- 0] [ 5 = 5- 0] [ 4 = 4- 0] [ 3 = 3- 0] [ 2 = 2- 0] [  0 = 0- 0]

The above is basically a pairwise matrix.  Each element in the matrix 
contains the equation Wij = Ui - Uj.

(I suppose the Weber equation and pairwise matrix could be used to analyze 
Condorcet stuff, like the Condorcet Criteria.  I suppose an IRV Weber 
equation could even be used to analyze IRV.  But I expect using Weber 
would be fiddly and I get the feeling that the structure produced won't be 
a matrix.  Anyway, back to Approval...)

Weber then adds up each column to find out how good each candidate is in 
comparison with one another.  This results in the following Weber vector 
(again my terminology, not Weber's)...

     A            B           C           D            E           F
[36 = 60-24] [ 6 =30-24] [ 0 =24-24] [-6 =18-24] [-12 = 12-24] [-24 =0-24]

Weber represents an Approval ballot using the following "vector":

A    B    C    D    E    F
[vA] [vB] [vC] [vD] [vE] [vF]

Each element of the vector is either a 1 (a Yes vote) or 0 (No vote).  
Weber then merges the above two vectors together by multiplying together 
the columns of each vector.  This results in the following...

A         B         C         D         E          F
[36 * vA] [ 6 * vB] [ 0 * vC] [-6 * vD] [-12 * vF] [-24 * vF]

Values vA to vF are then selected so that when all of the elements of the 
merged vector are added up, the total is as high as possible (i.e. is a 
maximum).

As positive values (i.e. values greater than 0) have been given to 
candidates A and B in the Weber vector, vA and vB should be 1 in order for 
the values to contribute as much as each value can to the total.

As negative values (i.e. values less than 0) have been given to candidates 
E and F in the Weber vector, vE and vF should be 0 in order for the 
negative values to contribute nothing to the total.

Candidate C is a special case.  Candidate C's value is 0 in the Weber 
vector.  This means that any number multiplied by this value 0 is going to 
be 0.  Therefore, vC can be either 0 or 1.  It does not matter.

Therefore, the best ballot "vectors" are...

A   B   C   D   E   F
[1] [1] [0] [0] [0] [0]

...or...

A   B   C   D   E   F
[1] [1] [1] [0] [0] [0]

And these define how it is best to vote in the 0-info case on an Approval 
ballot.

The reason why candidate C is a special because C's utility is exactly the 
same as the average utility of the candidates is (10 + 5 + 4 + 3 + 2 + 0) 
/ 6 = 4.  Candidates A and B have above average utilities, and the best 
ballot vectors show that I should vote for them.  Candidates D, E and F 
have below average utilities, and the best ballot vectors show that I 
should not vote for them.

Notice that all the calculations that have been done only use utilities. 
 Utilities remain the same for all the candidates, regardless of whether 
top 2, 3, 4, etc... are elected because they are the "marks out of 10" I 
personally gave to the candidates.

Also note that in the Approval ballot, voting is for each candidate 
individually.  It is not for a slate of candidates.  For each candidate, 
Weber attempts to make the best independent use of its score total.