e^x
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#1
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We're given the matrix:

\begin{pmatrix} 0 & 8 & 5 \\8 & 4 & 6 \\12 & -4 & 3 \end{pmatrix}

and we have to simplify the game in order to use the graphical method so we have to remove a row or a column. I understand how to remove a column but i dont understand how I would decide which row to eliminate? I' m struggling to decide which linear combinations to take?
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ghostwalker
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(Original post by e^x)
We're given the matrix:

\begin{pmatrix} 0 & 8 & 5 \\8 & 4 & 6 \\12 & -4 & 3 \end{pmatrix}

and we have to simplify the game in order to use the graphical method so we have to remove a row or a column. I understand how to remove a column but i dont understand how I would decide which row to eliminate? I' m struggling to decide which linear combinations to take?
I can't see a combination of rows, but there is a combination of columns that would allow you to exclude column 3.
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e^x
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(Original post by ghostwalker)
I can't see a combination of rows, but there is a combination of columns that would allow you to exclude column 3.
Yes, you can use columns 1 and 2 to eliminate column 3. But my lecturer said you can also remove a row. I would just ask him to explain it but he is away from uni and doesn't have access to emails.
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ghostwalker
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(Original post by e^x)
Yes, you can use columns 1 and 2 to eliminate column 3. But my lecturer said you can also remove a row. I would just ask him to explain it but he is away from uni and doesn't have access to emails.
I can't see how any pair of rows can dominate the other one, since in all three cases, the other one will contain the maximum value for one of the columns.
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e^x
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(Original post by ghostwalker)
I can't see how any pair of rows can dominate the other one, since in all three cases, the other one will contain the maximum value for one of the columns.
I guess my lecturer made a mistake then because i cant see how you would remove a row too.
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RichE
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(Original post by e^x)
I guess my lecturer made a mistake then because i cant see how you would remove a row too.
Having excluded a column, can one then exclude one of the (truncated) rows? Just trying to help - game theory not my thing.
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ghostwalker
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(Original post by RichE)
Having excluded a column, can one then exclude one of the (truncated) rows? Just trying to help - game theory not my thing.
After removing column 3, none of the remaining truncated rows can be excluded, however it is solvable graphically.

For dominance of rows:
We can exclude a row, if there exists a linear combination of the other rows, such that the corresponding entries in the linear sum are greater than the entries in the row to be excluded. The multipliers for the linear sum are probabilites, and hence in [0,1] and sum to 1.

In effect it's saying we'd never play the excluded row, since there is a stratergy involving the other rows which is always better (has a greater payoff) - hence we discard it.

A similar argument holds for dominance of columns, except we discard columns where the linear sum is always less than the row to be discarded.
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