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May 21

Reduction by dominance?

I just saw this on my textbook and I'm really confused! For example, if the matrix is:

5 6 1 5 3 4

7 3 9 6 8 7

how can I reduce the columns by the principle of dominance. I can cancel out the first column, but what about the rest? Convex linear combination? Can someone explain what it is? I know I sound very stupid, sorry for that! Ryoga (talk) 06:26, 21 May 2014 (UTC)[reply]

A convex linear combination of columns refers to mixed strategies. It means choosing a vector

[p_{1},p_{2},p_{3},p_{4},p_{5},p_{6}]

with

\sum p_{i}=1

and multiplying column

i

by

p_{i}

, then adding them up (i.e. doing matrix multiplication). If the resulting column dominates (is always less than or equal to) one of the columns of the matrix, and your vector had a 0 in that column's location, then you can remove that column.

For example, consider

[0,.5,.5,0,0,0]

. This mixed strategy gets us the column

0[5,7]^{T}+.5[6,3]^{T}+.5[1,9]^{T}+0[5,6]^{T}+0[3,8]^{T}+0[4,7]^{T}=[3.5,6]^{T}

. This strongly dominates the first, fourth and sixth columns, and it has 0s in the first, fourth and sixth entries, so you can remove those three columns.--80.109.80.78 (talk) 15:06, 21 May 2014 (UTC)[reply]

Wow, thanks so much for the explanation! Ryoga (talk) 10:58, 22 May 2014 (UTC)[reply]

A convex combination of vectors

v_{i}

is

\sum _{i}p_{i}v_{i}

where

\sum _{i}p_{i}=1

and also

p_{i}\geq 0

. Bo Jacoby (talk) 18:10, 23 May 2014 (UTC).[reply]

Some additional relevant links: game theory, mixed strategy, strategic dominance, payoff matrix. -- ToE 22:55, 24 May 2014 (UTC)[reply]

Double covering cube nets of dodecominos?

There are 11 hexomino nets for the cube (ignoring rotations and reflections). Does anyone have any suggestions for figuring out how many of the 58937 dodecominos without holes (63600-4663) would provide a double covering for the cube? This number is definitely more than the number of possible dodeconimos made from two copies of the 11 hexonimo nets, for example

MMOOOOOO
MMMMMMMM
OOOOOOMM

is a valid double covering but not severable into two hexonimo coverings. (and there has *got* to be a better way to show a dodeconimo than that)Naraht (talk) 17:31, 21 May 2014 (UTC)[reply]