Proof
Let . For every input, we have . Therefore, . Using Fubini's theorem, since all terms are non-negative we can switch the order of summation, giving . By the Pigeonhole principle, there must exist an algorithm so that, concluding the proof.
As mentioned above, this theorem can also be seen as a very special case of the Minimax theorem.
Read more about this topic: Yao's Principle
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