Download Invincible - Knowledge Based Systems Group

Captain
Miner
Total
0.6
0.1
1.0
0.2
0.4
1.4
0.2
0.5
1.0
1.0
1.0
This naturally invalidates the probabilities of piece 2 and those of the marshal on the
board so the matrix effects of this change must be distributed over the whole matrix.
First, the probabilities of piece 2 are normalized. Each element in the column belonging
to piece 2 is divided by the total volume of that column (1.4)
Marshal
Captain
Miner
Total
Piece 1
0.3
0.6
0.1
1.0
Piece 2
0.57
0.14
0.29
1.0
Piece 3
0.3
0.2
0.5
1.0
Total
1.17
0.94
0.89
This looks even worse, now all the horizontal totals are wrong. All totals have an error
less than 40% though, which was the worst error in the previous table. The next step is to
normalize the rows. Each element is divided by the total of the row it is in.
Marshal
Captain
Miner
Total
Piece 1
0.26
0.64
0.11
1.01
Piece 2
0.48
0.15
0.33
0.96
Piece 3
0.26
0.21
0.56
1.03
Total
1.0
1.0
1.0
The totals of the columns are now not equal to 1, but again the errors are smaller than
after the previous step. One more normalization over the rows results in
Marshal
Captain
Miner
Total
Piece 1
0.26
0.63
0.11
1.0
Piece 2
0.5
0.16
0.34
1.0
Piece 3
0.25
0.20
0.55
1.0
Total
1.01
0.99
1.0
Now all errors are less than 1% which is a nice target accuracy for a manual calculation.
Comparing this table with the starting situation:
old
Piece 1
Piece 2
Piece 3
Total
Marshal
0.3
0.4
0.3
1.0
Captain
0.6
0.2
0.2
1.0
Miner
0.1
0.4
0.5
1.0
Total
1.0
1.0
1.0
new
Marshal
Piece 1
0.26
Piece 2
0.5
Piece 3
0.25
Total
1.01
72