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Problem 10

# Reduce the given payoff matrix by dominance. $$\left.\begin{array}{rrr} p & q & r \\ a & 2 & 0 & 10 \\ b & 15 & -4 & -5 \end{array}\right]$$

Expert verified
Since there are no strictly dominated strategies for either the row player or the column player, the dominance reduced payoff matrix remains the same as the original matrix: $$\left.\begin{array}{rrr} p & q & r \\ a & 2 & 0 & 10 \\ b & 15 & -4 & -5 \end{array}\right]$$
See the step by step solution

## Step 1: Identify the strictly dominated strategies for the row player.

To identify the strictly dominated strategies for the row player, we must compare each strategy's payoffs for all column player strategies. Let's compare the strategies a and b: - For columns p: a (2) < b (15) - For columns q: a (0) > b (-4) - For columns r: a (10) > b (-5) Since there isn't any strategy for the row player that is always better or equal and sometimes better, there is no strictly dominated strategy for the row player.

## Step 2: Identify the strictly dominated strategies for the column player.

Now let's compare the strategies of the column player (p, q, and r) by analyzing the payoffs from the row player's perspective: For column p VS column q - For row a: p (2) > q (0) - For row b: p (15) < q (-4) For column p VS column r - For row a: p (2) < r (10) - For row b: p (15) > r (-5) For column q VS column r - For row a: q (0) < r (10) - For row b: q (-4) > r (-5) Once again, as we can see from this comparison, there isn't any strictly dominated strategy for the column player.

## Step 3: Result of the dominance reduction.

Since no strictly dominated strategies exist for both the row and column players, we cannot further reduce the given payoff matrix through dominance. The dominance reduced payoff matrix remains the same as the original matrix: $$\left.\begin{array}{rrr} p & q & r \\ a & 2 & 0 & 10 \\ b & 15 & -4 & -5 \end{array}\right]$$

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