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
.

1999-04-01