Are Uniformly Most Powerful Unbiased Tests Really Best?

In teaching the development of uniformly most powerful unbiased (UMPU) tests, one rarely discusses the performance of alternative biased tests.It is shown, through the comparison of two independent Bernoulli proportions, that a biased test (the Z test) can be more powerful than the UMPU test (Fisher's exact test.randomized) in a large region of the alternative parameter space.A more general example is also given.