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For a two person game, player one has strategies A,B,C. Inspection reveals strategy B is strictly dominated by some mixture between A and C.

How does one calculate the probabilities assigned to A and C in the dominant strategy?

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2 Answers 2

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The way we did it in my Combinatorics and Game Theory class was to let $p$ be the probability that we chose strategy A and $1-p$ be the probability that we chose strategy C. Then we drew a graph that has two vertical "axes" and a horizontal axis connecting the two in the middle. The horizontal axis is the value of $p$, ranging from 0 to 1. On the left "axis", put a dot corresponding to the value of strategy A when $p=0$ and another dot corresponding to the value of strategy C when $p=0$. Do the same for the right "axis" but when $p=1$. Thus, you will have two lines (that may or may not intersect) that show the values of strategies A and C depending on the choice of $p$. My guess is that you want the value of $p$ such that VA($p$) and VC($p$) are equal, which corresponds to the intersection of the lines you constructed on the graph.

I hope that's what you're asking for...

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In general, finding an optimal strategy in a finite two-person zero-sum game reduces to a linear programming problem.

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