# Show that the order of deletion doesn’t matter for the set of strategies surviving iterated elimination of strictly dominated strategies

Question from M.W.G-8B4

Show that the order of deletion doesn’t matter for the set of strategies surviving iterated elimination of strictly dominated strategies.

My proof attempt

I assume $$G_n$$ the game generated in the n^th step of IDDs.

Suppose $$s_i^D \in \Sigma_i^n$$ dominates $$s_i\in \Sigma_i^n$$ for given $$\Sigma_{-i}^n$$

But let’s assume it is not deleted in this round.

Then at round $$n+1$$, $$s_i^D \in \Sigma_i^{n+1}$$ still dominates $$s_i\in \Sigma_i^{n+1}$$ for given $$\Sigma_{-i}^{n+1}$$

So, the order of deletion doesn’t matter.

— what do you think about my proof attempt? Is this enough to show this statement? Or how can improve this proof?

Any help will be appreciated. Thanks!