FEINBERG, Circuit Judge.

Rafael Duverge Perez (hereafter Duverge) appeals from a judgment of conviction and sentence of 10 years imprisonment entered in the United States District Court for the Southern District of New York (Preska, J.), following his plea of guilty to a narcotics offense, in violation of 21 U.S.C. §§ 841(a)(1), 841(b)(1)(A) and 18 U.S.C. § 2. Judgment was originally imposed by Judge Preska in April 2000, and Duverge appealed to this court. In December 2000 we remanded for reasons set forth below. In April 2001 the judge adhered to her prior sentence, and Duverge again appealed. We now affirm.

I. Background

In November 1998, Duverge was arrested when he delivered 110.9 grams of cocaine base (crack) to an undercover officer. In December 1998, the Government filed a one-count indictment, charging Duverge with distributing and possessing with intent to distribute 50 grams and more of mixtures containing a detectable amount of crack. Shortly thereafter, Duverge pleaded guilty to this charge. Title 21, United States Code, Section 841(b)(1)(A) provides a 10-year mandatory minimum sentence for distributing crack.

In September 1999, the district court held a sentencing evidentiary hearing pursuant to United States v. Fatico, 579 F.2d 707 (2d Cir. 1978), to consider Duverge's argument that the doctrines of sentencing manipulation and sentencing entrapment entitled him to a downward departure. Two witnesses, Duverge and Luis Perez (hereafter Perez), the Government's cooperating witness, testified at the hearing. Duverge claimed that he originally had agreed to sell only powder cocaine to Perez, while Perez claimed that Duverge had agreed to sell him crack.

Duverge testified that on November 20, 1998, he received a phone call from a friend, Armando Duran. Duverge had previously worked for Duran as a dealer in powder cocaine. Duran asked Duverge if he could "get him 125" grams of powder cocaine for a customer of Duran's friend, Perez. Duverge obtained the cocaine from another dealer. According to Duverge, on the morning of November 23, 1998, Duran and Perez came to Duverge's apartment with a paper bag containing a coffee pot, received the powder cocaine from Duverge, and "cooked" the cocaine into crack in Duverge's kitchen. Duverge claimed he stated at that time that he wanted nothing to do with the "cooking" and wanted only to be paid. According to Duverge, Duran and Perez then told him that to be paid, he had to deliver the crack, which Duverge agreed to do upon a promise of additional payment. When Duverge delivered the crack to Perez and an undercover officer later that day, Duverge was arrested.

Perez testified that he had telephoned Duverge on November 20, 1998 and had requested 125 grams "of crack." He stated that he arrived at Duverge's apartment alone on November 23, 1998 to inspect the crack, saw the crack sitting in a white plate, and told Duverge to keep the crack and deliver it for him later in the day. He denied that he or Duran cooked the cocaine powder into crack.

The testimony of these two witnesses at the Fatico hearing thus differed primarily as to what drug Duverge was originally asked to get for Perez's customer and what occurred on the morning of November 23, 1998 in Duverge's apartment.

Before the hearing, Duverge had moved to call an experienced polygrapher, Glenn Almas, to testify on the results of two extensive polygraph examinations of Duverge in April 1999 regarding the events in question. Specifically, Almas would have testified that the examinations showed that Duverge was truthful when he answered "no" to a number of questions as to whether crack was mentioned in Duverge's conversations with Duran or Perez and whether Duverge had ever sold crack.

Almas also asked Duverge two questions during the polygraph examination about whether Duran and Perez had brought a coffee pot with them when they came to Duverge's apartment, but Almas apparently could draw no conclusion with respect to Duverge's responses to those questions.

The district court refused to accept the polygraph evidence, stating:

[A]ssuming that I could hear the testimony of the polygraph expert, I think under [Fed.R.Evid.] 403 I elect not to. Given that we all seem to agree the issue is strictly one of credibility, it seems to me that given the varying views on reliability of polygraph testimony and, therefore, the somewhat limited probative value, that the time it will take to put it in really outweighs its probative value, whereas the attitude and demeanor of the participants is what has all of the probative value. So, for those reasons, I decline to hear the polygraph testimony.

At the end of the Fatico hearing, the judge ruled that she credited the testimony of Perez over that of Duverge. The judge stated:

I do not credit [Duverge's] testimony that the folks came into his home with the coffee pot in a paper bag, opened the refrigerator and took out the baking soda and, before his very eyes, cooked the crack. I find that not to be credible testimony. I find more credible the thought that the crack was [already] cooked as suggested by [Perez].

The district court also rejected Duverge's motion for a downward departure on the ground of sentencing manipulation or sentencing entrapment, stating that (1) Duverge had not testified truthfully about who cooked the crack; and (2) she did "not find [the necessary] level of improper conduct here [to meet the definitions of sentencing entrapment or manipulation] even if one credits the defendant's testimony."

In April 2000, as already indicated, the district court sentenced Duverge to the statutory mandatory minimum term of incarceration for 10 years. The district court found that Duverge's false testimony made him ineligible for a downward departure under the "safety valve" provisions of U.S.S.G. §§ 5C1.2, 2D1.1(b)(6) and 18 U.S.C. § 3553(f), which allow a court to depart below a statutory minimum sentence if, among other requirements, "the defendant has truthfully provided to the Government all information and evidence . . . concerning the offense. . . ." As a result of Duverge's false testimony, the district court also applied an offense level enhancement for obstruction of justice, but allowed an offense level reduction for acceptance of responsibility, based on Duverge's adherence to his story "in the face of known dire consequences."

Thereafter, Duverge appealed his sentence. In October 2000, this panel heard oral argument, and reserved decision. Shortly after oral argument, the Government learned that cooperating witness Perez had been arrested in Maine on federal narcotics charges unrelated to the charges against Duverge. The Government notified Duverge's attorney about this newly-discovered evidence. With the consent of the Government and at the request of Duverge, we remanded the case to the district court in December 2000 for consideration of that evidence.

In April 2001, the district judge re-opened the Fatico hearing. At the beginning of the hearing, Perez's attorney advised the district court that Perez would invoke his Fifth Amendment right to refuse to answer questions that might incriminate him regarding his newly-discovered crimes, but that Perez would answer questions about Duverge's case, and about the testimony he gave during the September 1999 Fatico hearing. Duverge called Perez to the witness stand for further cross-examination. Perez asserted his Fifth Amendment right not to answer the questions he believed related to his newly-discovered crimes. Following the hearing, Duverge moved to strike Perez's testimony "in its entirety," or in the alternative that the district court infer that had Perez not invoked his Fifth Amendment privilege, his answers would have contradicted his testimony given in the September 1999 hearing. The Government opposed this request.

In October 2001, the district court issued an opinion reaffirming its earlier decision to credit the testimony of Perez over that of Duverge. The court denied Duverge's motion to strike Perez's testimony, stating:

The witness's direct testimony need not be stricken, . . . "[i]f a witness asserts the privilege on cross-examination regarding `collateral' matters." Brooks, 82 F.3d at 54. A "witness's testimony about other unrelated crimes may be collateral." Id.; see Dunbar, 612 F.2d at 694. . . .

The court ruled that the questions Perez refused to answer concerned collateral matters unrelated to the drug deal at issue in Duverge's case.

The court also denied Duverge's request that the court infer that Perez's answers to the unanswered questions during the April 2001 hearing would have contradicted his testimony during the September 1999 hearing. The district court held that it

continue[s] to credit Perez's testimony over [Duverge's testimony] . . . for two reasons.

First, . . . Perez had little or no incentive to lie about the drug deal with Duverge. At the time of Duverge's hearing, Perez had already assisted the government in the arrest of at least 30 people and thus did not need to lie to obtain the substantial assistance motion pursuant to U.S.S.G. § 5K1.1 that he hoped to receive from his cooperation with the government. Indeed, Perez had a disincentive to lie because if it were determined that Perez lied about the drug deal with Duverge, he could lose the benefits of his cooperation agreement. Thus, the potential downside from lying far outweighed any potential upside. On the other hand, [Duverge] had a strong motivation to lie during the Fatico hearing — his sentence could be reduced if he could demonstrate that he had been manipulated into selling crack instead of cocaine. Second, I continue to find that Duverge's story that Perez and another individual came into Duverge's "home with the coffee pot in a paper bag, opened the refrigerator and took out the baking soda and, before his very eyes, cooked the crack" is simply not a credible one.

(Footnote omitted).

In October 2001, Duverge's appeal was reinstated and referred to this panel in accordance with our prior order. We requested and received supplemental briefs addressing matters raised by the district court's October 2001 opinion, and took the appeal on submission.

II. Duverge's arguments to this court

Since Duverge's original appeal has been reinstated, we have before us the arguments raised both in the original appeal and in the supplemental briefs addressing Judge Preska's decision after remand.

A. Exclusion of polygraph evidence

In his original appeal, Duverge argued that the district court erred in excluding the polygraph evidence. He asserted that the standard for admitting scientific evidence at sentencing is more relaxed than at trial, that a credibility determination was at the heart of the sentencing, and that the reliability of polygraph evidence has improved. He therefore argued that the district court abused its discretion in refusing to accept it.

The Government responded that it is not clear in this circuit whether a sentencing court may admit polygraph results, as this court has expressly reserved judgment on that issue. United States v. Messina, 131 F.3d 36, 42 (2d Cir. 1997). Moreover, the Government asserted, it was well within the district court's discretion to deny admission of the evidence, as the Supreme Court has observed that there is no consensus that polygraph evidence is reliable. United States v. Scheffer, 523 U.S. 303, 309-12, 118 S.Ct. 1261, 140 L.Ed.2d 413 (1998). The Government also argued that regardless of the general reliability of polygraph evidence, it did not bear directly in this case on whether Perez and Duran had cooked the crack because the relevant questions during the lie detector test centered on whether Duverge had discussed crack prior to the day of the sale and whether he had ever sold crack. The Government also argued that the evidence would have been cumulative, and that admission of the evidence would have been time-consuming. Additionally, the Government argued that the rules of evidence do not apply to sentencing proceedings, citing United States v. Brown, 52 F.3d 415, 425 (2d Cir. 1995) and Fatico, 579 F.2d at 711.

We essentially agree with the Government. A district judge has broad discretion "as to what types of procedure are needed" at a sentencing proceeding for determination of relevant disputed facts. United States v. Slevin, 106 F.3d 1086, 1091 (2d Cir. 1996). The discretion of a sentencing court is similarly broad "either as to the kind of information [it] may consider, or the source from which it may come." United States v. Carmona, 873 F.2d 569, 574 (2d Cir. 1989). We review such determinations for an abuse of discretion. Slevin, 106 F.3d at 1091.

When evaluating whether to adopt a particular procedure to resolve sentencing disputes, a district court must analyze, among other things, the probative value and burdens of the proposed procedure. See United States v. Prescott, 920 F.2d 139, 144 (2d Cir. 1990). The crucial issue before the court at the sentencing hearings both before and after the remand was, as Judge Preska stated, "strictly one of credibility" at which she would have a chance to see for herself "the attitude and demeanor" of both Duverge and Perez. In addition, polygraph evidence, even if credited by the judge, would have had little bearing on the district court's decision that Duverge's "story" that Perez and Duran had cooked the crack in his apartment using a coffee pot they had brought with them "was simply not a credible one." Considering the record of the proceeding as a whole, the limited probative value of the polygraph evidence and, as the judge stated, "the time it will take to put it in," we hold that the district court did not abuse its discretion in refusing to hear this testimony.

B. The district court's credibility determination

Duverge launches various attacks on Judge Preska's determination that the testimony of Duverge was not as credible as the testimony of Perez. On his reinstated appeal, Duverge makes the additional argument that the district court erred in its treatment of Perez's refusal to answer questions on Fifth Amendment grounds in the April 2001 hearing. He claims that the district court should have (1) ruled that Perez waived his Fifth Amendment privilege against self-incrimination and directed him to answer the questions asked, or in the alternative, stricken Perez's testimony; (2) inferred that Perez's answers to the questions he refused to answer would have supported Duverge's positions; and (3) found that Perez's testimony was not reliable. Duverge also claims that Perez's refusal to answer certain questions about his truthfulness during his earlier sessions with the Government constituted a refusal to respond to questions on non-collateral matters, and required the district court to strike Perez's earlier testimony.

The Government argues that Judge Preska's finding that Perez was more credible than Duverge was not clearly erroneous. It also contends that the district court did not abuse its discretion by refusing to strike Perez's previous testimony, especially because Perez had asserted the Fifth Amendment privilege only on collateral matters. The Government also argues that Duverge abandoned his argument that Perez waived his Fifth Amendment privilege, and that even if Duverge had pressed this argument, it would fail.

We review the sentencing court's factual findings for clear error, United States v. Ortiz, 136 F.3d 882, 883 (2d Cir. 1997), and the decision refusing to strike testimony because a witness refused to answer a question on Fifth Amendment grounds for abuse of discretion, see United States v. Yip, 930 F.2d 142, 147 (2d Cir. 1991).

We find clear support in the record for the district court's decision to credit the testimony of Perez over that of Duverge. Judge Preska stated that her credibility findings were based on the demeanor of the witnesses, and her view that Duverge's version of the events was "simply not a credible one." Morever, as the Government points out, Judge Preska was aware of all of the facts that brought about the remand and second hearing, including the drug charge against Perez in Maine. We cannot say that on this record, the judge's finding that Perez's testimony regarding the drug transaction was more credible than Duverge's was clearly erroneous.

Moreover, the judge did not abuse her discretion in refusing to strike Perez's testimony after he refused to answer certain questions based on his assertion of his Fifth Amendment rights. The district court ruled that the questions Perez refused to answer concerned collateral matters not related to the drug deal at issue in this case, and that Duverge had been given a full and fair opportunity to cross-examine Perez about his direct testimony both at the September 1999 hearing and at the April 2001 hearing. On this record, we agree with both rulings.

C. Other arguments

Duverge's remaining arguments on his first appeal were that the district court had erred by (1) finding his testimony untruthful and thus denying him a downward departure under the "safety valve" provisions of U.S.S.G. §§ 5C1.2, 2D1.1(b)(6) and 18 U.S.C. § 3553(f); and (2) holding that the doctrines of sentencing entrapment or sentencing manipulation did not permit a downward departure.

These arguments are without merit. A district court's refusal to depart downward is generally unreviewable on appeal, see United States v. Tenzer, 213 F.3d 34, 42 (2d Cir. 2000), unless its action falls within a few narrow exceptions to the general rule. See, e.g., id. (there is a narrow exception to the general rule of non-appealability when the sentencing court mistakenly believes that it lacks the authority to depart). No exception applies here.

Finally, according to Duverge, "sentencing entrapment" occurs when government agents induce a defendant to commit an offense that the defendant was not otherwise predisposed to commit, and "sentencing manipulation" occurs "when the government engages in improper conduct that has the effect of increasing the defendant's sentence." United States v. Caban, 173 F.3d 89, 93 n. 1 (2d Cir. 1999). However, since Judge Preska credited Perez's version of events over Duverge's version, and this finding was not clearly erroneous, we will not consider Duverge's argument that the doctrines of sentencing entrapment or sentencing manipulation apply. Under Perez's version of events Duverge was knowingly involved in a crack deal the whole time, and therefore was not induced to commit an offense that he was not otherwise predisposed to commit.

We have considered all of Duverge's arguments and we find them to be without merit. Judgment affirmed.