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I saw the lemma:

"In extensive form games with complete and perfect information, any mixed strategy for player i will result in a lower or equal utility for player i compared to some pure strategy available to player i."

I understand it intuitively, that a mixed strategy decision, would be sub-par to the best-response I can take, but I don't know how to prove it.

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I assume that implicit in the lemma is the condition "holding other players' strategies fixed". I'll also assume that the strategy space is finite.

Given a profile of the other players' strategies $\sigma_{-i}$, player $i$'s utility from playing a mixed strategy $\sigma_i$ is given by: \begin{equation} u_i(\sigma_i,\sigma_{-i})=\sum_{s_i\in S_i}\sigma_i(s_i)u_i(s_i,\sigma_{-i}) \end{equation} where $\sigma_i(s_i)$ is the probability of playing pure strategy $s_i$ according to $\sigma_i$. Now let $$s_i^*\in\mathop{\arg\max}_{s_i\in S_i}\;u_i(s_i,\sigma_{-i}).$$ Such a maximum must exist because $S_i$ is finite. It follows that, for all $\sigma_i$, $$u_i(s_i^*,\sigma_{-i})\ge\sum_{s_i\in S_i}\sigma_i(s_i)u_i(s_i,\sigma_{-i}) \tag*{$\blacksquare$}$$

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  • $\begingroup$ Thank you very much! $\endgroup$ Jul 29, 2022 at 9:56

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