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.


1 Answer 1


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 $\sigma_i$, $$u_i(s_i^*,\sigma_{-i})\ge\sum_{s_i\in S_i}\sigma_i(s_i)u_i(s_i,\sigma_{-i}) \tag*{$\blacksquare$}$$

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.