The following example can be regarded as a generalization of Pigeonhole Principle.

Example 1
If every person likes at least
of the books in a library, then the library has a book, which at least of people like.
Proof: Suppose there are
When dealing with problems solved by the pigeonhole principle, we know the total number of pigeons and pigeonholes. For problems solved by averaging argument, we usually don’t see the number of pigeonholes or the number of pigeons or both of them. In this case, we “imagine” the number of pigeons and pigeonholes and keep them in mind.
When talking about probability, the case is always the latter, because we don’t count the exact number of pigeons and their holes.
Let’s introduce two formal discriptions of averaging argument.
Averaging Argument
Let
and be sets, and be a prediction on . Let be a real number in the interval . If for each in , there exists at least in that satisfy . Then, there exists a in such that there exist at least elements in that satisfy .
Proof: We can construct a proof of the same style as the proof of Example 1.
Notice that averaging argument can be generalized to continuous sets
Averaging Argument
Let
be a function. If we have a circuit such that when is chosen uniformly at random and is chosen independently from some distribution over , we have holds with probability at least . Then there exists a single string such that .
Proof: Keeping the unseen pigeonhole number in mind, the proof might get simple. But we can also prove it in probabilistic style.
If there’s no such
Averaging Argument is a generalization of Averaging Argument, since it makes no limitation of
Averaging argument are used in the proofs of some interesting results of complexity. A simple example is the proof of
Theorem
.
Proof: For any
For each
The basic idea of Averaging Argument can be used elsewhere.
The next exam is quite important, which will be used in the proof of Goldreich-Levin Theorem.
Example 2
Let
be a function and defines as where . Let . Suppose there exists a
algorithm with running time and satisfy Then there exists a set of size at least such that for every it holds that
Proof: Still, we keep in mind the unseen number of pigeonholes. We have “1” pigeonholes. “1” means if we choose a pigeonhole uniformly at random from all the pigeonholes then we have probability “1” that we chosen from the set of all pigeonholes… This is not verbose, since there are cases that we choose a pigeonhole from a subset with less than “1” pigeonhole.
We assume we have probability