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In the lecture, my professor said "when we eliminate weakly dominated strategies, you might end up with different equilibria depending on the order of deletion!"

Can anyone give me an example?

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Consider the game below.

$$ \begin{array}{|r|c|c|} \hline &L&R\\\hline U&5,1&4,0\\\hline M&6,0&3,1\\\hline D&6,4&4,4\\\hline \end{array} $$

Both $U$ and $M$ are weakly dominated by $D$.

If we start by deleting $U$. This would lead to the removal of $L$ in the next step. Then $M$ would be deleted. The solution would be $(D, R)$.

However, if we start by removing $M$, then $R$ would be removed next, and then $U$. The solution from deleting in this order is $(D, L)$.

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