What is the test for perjury concerning bargains made between witness and prosecution?

We agree, of course, that a witness who has committed a crime known to a prosecutor has a motive to cooperate with the prosecution. It is this fact which makes perjury concerning bargains made between witness and prosecution so vital. E. g., Giglio v. United States, supra. But once the witness has admitted his vulnerability to prosecution which could result in substantial imprisonment and his awareness that he would not be prosecuted, the jury has before it the elements of the defense claim that the witness is committing perjury to save his own skin.