In recognition of this phenomenon defendants will want some type of instruction to the jury to carefully consider the motives of the cooperating and immunized witnesses. Here is the one that Judge Pauley gave in the Daugerdas retrial where cooperating witnesses testified as to their own culpability and the defendants':.
Testimony of Cooperating Witnesses
You have heard from witnesses who testified that they were actually involved in planning and carrying out certain of the crimes charged in the Indictment. The law allows the use of cooperating witness testimony. Indeed, it is the law in federal courts that such testimony may be enough in itself for conviction, if the jury believes that the testimony establishes guilt beyond a reasonable doubt.
However, because of the interest a cooperating witness may have in testifying, his or her testimony should be scrutinized with special care and caution. The fact that a witness is a cooperator can be considered by you as bearing upon his or her credibility. Like the testimony of any other witness, cooperating witness testimony should be given such weight as it deserves in light of the facts and circumstances before you. If you find that a witness testified falsely in one part, you still may accept his or her testimony in other parts, or may disregard all of it.
You should ask yourselves whether a cooperating witness would benefit more by lying, or by telling the truth. Did the witness believe that his or her interests would be best served by testifying falsely or by testifying truthfully? Did this motivation color his or her testimony? If you find that the testimony was false, you should reject it. However, if you are satisfied that the witness told the truth, you should accept it as credible and act upon it accordingly.
Finally, you have heard testimony from cooperating witnesses who pleaded guilty to charges arising in part out of the same facts that are at issue in this case. You are instructed that you are to draw no conclusions or inferences of any kind about the guilt of the defendants on trial merely from the fact that a prosecution witness pleaded guilty to similar charges. That witness’ decision to plead guilty was a personal decision about his own guilt. It in no way changes the presumption of innocence that applies to each defendant who is on trial here, a presumption that remains with them throughout the trial and into your jury deliberations unless and until you conclude that the Government has proven a particular defendant’s guilt by competent evidence beyond a reasonable doubt.
You have heard testimony from two other types of witnesses. First, from witnesses who testified pursuant to a non-prosecution agreement, which means that the Government promised each of these witnesses that, in exchange for testifying truthfully, he or she would not be prosecuted for any crimes admitted either in Court or in interviews with prosecutors. And second, from witnesses who have testified under a grant of immunity, which means the witness’s testimony may not be used against him or her in any criminal case, except a prosecution for perjury or otherwise failing to comply with the immunity order.
Now, it is perfectly acceptable for the Government to enter into non-prosecution agreements and to request orders of immunity, and it is entitled to call witnesses who are subject to these agreements or orders. However, like cooperating witnesses, the testimony of a witness who has been promised that he or she will not be prosecuted or who has been granted an order of immunity should be examined with greater care than the testimony of an ordinary witness. You should scrutinize it closely to determine whether or not it is colored in such a way as to place guilt upon the defendants in order to further the witness’s own interest; for such a witness, confronted with the realization that he or she can win his or her own freedom by helping to convict another, has motive to falsify his or her testimony. On the other hand, if you believe it to be true, and determine to accept the testimony, you may give it such weight, if any, as you believe it deserves.The full set of jury instructions is linked here.