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For discrete time replicator dynamics, fitness is defined only up to a multiplicative factor. I.e. if you pick some number $k$ then $(f_1, ..., f_n)$ defines the same fitness function as $\hat{f} = (f_1/k, ..., f_n/k)$. In particular, this mean for any $p$ you can always pick a $k$ such that $\langle \hat{f} \rangle_p$ = 1. So using your final equation, ...

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