Resolution of Godambe's paradox

Two versions of a statistical paradox involving an ancillarity principle have recently been proposed by Godambe (1979, 1982). In this note, we point out that the two versions of his paradox are actually quite different, and we resolve both of them by investigating their relationship to randomization. The first version of Godambe's paradox was criticized earlier by Good (1980), but we believe his argument to be flawed.