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I followed this lecture : https://www3.nd.edu/~apilking/Math10120/Lectures/Topic%2029.pdf

It gives a way to reduce a n x m matrix to a 2x2 matrix where we know how to solve a problem of mixed game strategy. We are searching for an optimal mixed row and column strategies and the value of the zero sum game defined by the given matrix.

However, if no lines nor columns are dominated, we can't reduce the matrix.

I would like to solve (for example this matrix, I ve simply taken an example of the pdf and changed values in order to make incomparable any lines / columns )

\begin{array}{c|rrrr} & A & C & D \\\hline X & 6 & 7 & 3 \\ Y & 7 & 5 & 2 \\ Z & 8 & 3 & 9 \end{array}

How should I solve this problem ? I'm searching for a general method in order to then solve harder problems.

Marine Galantin
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1 Answers1

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If there is a 2x2 matrix, you usually find a probability $p$ for player 1 to make player 2 indifferent between his two actions. Then you find probability $q$ for player 2 to make player 1 indifferent between his two actions. This is then a mixed Nash equilibrium, because both are indifferent between any of the two available actions, so they can play both with positive probability.

The problem is the same with a 3x3 matrix, but you will have to solve for 4 rather than 2 probabilities. First, you need to find probabilities $p_1$ and $p_2$ with which player 1 plays his first and second action, respectively (and he will play the third with probability $1-p_1-p_2$), so that player 2 is indifferent between his available actions. Then you need to find probabilities $q_1$ and $q_2$ with which player 2 plays his first two actions (and the third with probability $1-q_1-q_2$), so that player 1 is indifferent between all of his actions.

How to set up the appropriate equations is explained here in an example.

Nameless
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  • Thanks for the answer! In the link you gave me, the question is about matrices with 2 possipibilities within each cases. What s the link with my problem? It is a really interesting problem, but I don't see any link with what you said? – Marine Galantin Apr 09 '19 at 09:02
  • Are you considering the "edit" part? – Marine Galantin Apr 09 '19 at 09:28
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    Yes the EDIT part is about computing the mixed strategies. This answer is for a 3x3 matrix, i.e., a game where both players have 3 possible actions. Hence, it is exactly the case you are considering. You may be confused because the matrix gives the payoff for each of the two players in every 3x3 cell. – Nameless Apr 09 '19 at 12:13