We give a brief statement and proof of Yao's technique for lowering bounding the randomized complexity of a problem. Yao's technique is essentially an application of VonNeumann's mini-max principle.

Notation: Let A be a generic deterministic algorithm, or equivalently a pure strategy in game theoretic terms. Let be a generic randomized algorithm algorithm, or a equivalently a generic distribution over deterministic algorithms, or equivalently a mixed strategy in game theoretic terms. Let I be a generic input. Let and be a generic distribution over the inputs. We use E[A, I] to denote the time used by algorithm A on input I, to denoted the expected time used by algorithm on input I, to denoted the expected time used by algorithm on input distribution , and to denote the expected time of algorithm A on input distribution .

Statement:

That is, you can lower bounded the randomized time complexity by the average case time on any distribution.

Proof: Consider any input distribution . Since the optimal distribution puts all the probability on the best algorithm, it follows that:

Now

since the maximizer may choose to be . Then

since the optimal choice for the maximizer is to put all the probability on the input that maximizes the expected running time for

Example Application: Since the expected running time of every deterministic comparison-based algorithm for sorting is it follows by Yao's technique that the expected running time of every randomized comparison-based algorithm is .