I understand what weakly dominated strategies are, I understand action profiles, 3 player payoff matrixes and utilities, but I cannot seem to figure out what exactly is asked.
The utilities should be such that, by applying in two different orders the iterated elimination of weakly dominated strategies, two different and unique action profiles are returned.
Can someone clarify this for me?
This is getting at the fact that the result of the iterative removal of weakly dominated strategies can depend on the order of removal. Consider the game
\begin{array}{|c|c|} \hline & L & R \\ \hline A & (10, 4) & (5, 4) \\ \hline B & (0, 1) & (4, 6) \\ \hline \end{array}
Depending on the order of elimination, the set of strategies that remains after iterative removal of weakly dominated strategies can be $\{A, L\}$ or $\{A, R\}$. Note that both of these are Nash equilibrium's, but by using iterative removal of weakly dominated strategies we lose one of them.
For a 3x3 example this game has four Nash equilibrium $(A, L), (B, L), (C, L), (B, R)$. By using iterative removal of weakly dominated strategies we can get $(A, L), (B, L), (C, L)$ depending on the order of removal.
\begin{array}{|c|c|c|} \hline & L & M & R \\ \hline A & (0, 2) & (1, 0) & (0, 0) \\ \hline B & (0, 1) & (0, 0) & (1, 1) \\ \hline C & (0, 1) & (0, 1) & (0, 1) \\ \hline \end{array}