# pure-poa is bigger that mixed-pos?

generally it's true that Pure price of anarchy is >= mixed price of stability. but I just find cases where they are equal and I can't think of a case of strict inequality.

can you help me please?

thanks (: benny.

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## 1 Answer

Consider the following two-player game $$\begin{matrix}& L & R \\ T & 1,1 & 0,0\\ B & 0,0 & n,n\end{matrix}.$$ The game has two Nash equilibria: $(T,L)$ and $(B,R)$; the price of stability is 1 (as $(B,R)$ is a socially optimal strategy), whereas the price of anarchy is $\frac{1}{n}$.

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