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Math Help - Finding recessive rows and columns in a matrix

  1. #1
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    Finding recessive rows and columns in a matrix

    Ok, I have been looking for 2 days on the web before posting and I think the web has only confused me even more as I have got into text books like linear algebra for dummies and such.

    Ok I will explain what I am trying to do and hopefully someone can tell me how to do this. So in Linear Algebra I am on the section of Game programming, determining saddle points and such. Well the next chapter was what to do if there was not specific saddle point and how to determine the value of the game and optimal strategy in a 2X2 matrix for both R and C. Which thats easy there are simple formula to determine it. Plug the numbers in and go

    Ok so here is where the problem comes in, suppose you have a 3X3 matrix or even higher. So the book tells us to find the recessive rows and just go ahead and eliminate them. While looking all over the web I don't find them called recessive rows other people seem to call them dominant rows. However no one seem to be able to give me a really good way to determine what rows to eliminate. To me it just seems like everyone is guessing but there has to be some logic to it which I am not able to get this logic anywhere.

    So here is an example:
    I have the matrix

    2 0 4
    1 2 3
    4 1 2

    Ok just looking at that matrix

    the 3rd column is dominant or basically has really large numbers, yes this is what I am gleaming from doing my homework, 3 hours of class and 2 days searching on the web, pick the big numbers.

    Which gives you the new matrix

    2 0
    1 2
    4 1

    So my next guess going with the pick really big numbers is the last row. because the 4 is higher than everything else. However, I am wrong. According the what I am reading the first column is dominant? Recessive? What ever, sorry I am just so frustrated at this point not being able to find clear answers to how to reduce matrices it really seems like guess work to me.

    Anyway your left with
    1 2
    4 1

    Which once I am at that point then piece of cake. I can determine the optimal strategy for both players and expected value. I just cant figure out how to reduce Matrices. So any help even a link to something I am missing on how to reduce the rows and columns.

    Another example:

    0 4 6
    5 7 4
    9 6 3

    According to the book reduces to
    0 6
    9 3

    Which I just stare at and can not see how they got this. I mean I understand they eliminated the middle rows and columns but why those? Unfortunately my book has 3 paragraphs explaining this concept yet every single problem has reduction in it and every single problem I have got wrong because I chose the wrong rows or columns to eliminate.
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  2. #2
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    in the matrix:

    [0 4 6]
    [5 7 4]
    [9 6 3]

    i don't see any purely dominant columns, or recessive rows. so i don't see how they got the reduced matrix.

    in your original matrix:

    [2 0 4]
    [1 2 3]
    [4 1 2]

    column 3 dominates column 2, so C will never choose that strategy over column 2, as it is worse in every case. that leaves us with:

    [2 0]
    [1 2]
    [4 1]

    clearly row 1 is recessive to row 3, so R will always prefer that strategy (row 3), resulting in:

    [1 2]
    [4 1].
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  3. #3
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    Quote Originally Posted by Deveno View Post
    clearly row 1 is recessive to row 3, so R will always prefer that strategy (row 3), resulting in:

    [1 2]
    [4 1].
    LOL, no clearly it is not, thats what I am trying to understand, how is it clearly recessive. How do I know it is recessive.

    I see 2 and 0 then a 1 and 2 and then 4 and 1. I see no patterns or anything. Why is it recessive to 4 and 1 and not 1 and 2? How are you able to look at them and say that it is clearly recessive. Thats what I keep running into. Everyone says that a certain row is recessive, but why?

    I am just looking for a rule or something to tell me why it is recessive because to me it is not clear. Like I said I understand the dominant thing, it seems to just be pick the biggest numbers. is recessive just pick the lowest numbers?

    So like add them up?
    2+0 = 2
    1+2 = 3
    4+1 = 5

    So just throw away the ones that add up the lowest?
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  4. #4
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    a row (or column) a dominates another b if ai ≥ bi for all i, and ai > bi for at least one i. player C ignores dominant strategies because they have the highest payoff for R.

    player R, on the other hand ignores recessive strategies, because they have the lowest payoff for R.

    it's not just the "higher numbers", every number has to be the same or bigger, and and least ONE has to be bigger. but you do it different for columns than rows.

    for columns, you discard the dominant columns, for rows you discard the recessive rows.

    however, i can make no sense of why the 2nd matrix redices as you say it does.
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