SENTELLE, Circuit Judge:

Appellants the District of Columbia, the District's Mayor, and the Director of the District's Department of Human Services (collectively, the "District") appeal two separate awards of attorneys' fees and expenses for work performed from 2010 to 2012 on this 42 U.S.C. § 1983 Medicaid class action. In this consolidated appeal, the District raises three grounds for its position that the district court's decisions amounted to an abuse of discretion. First, the District argues that the district court abused its discretion by making targeted reductions in Plaintiff-Appellees' ("Plaintiffs") fee requests, as opposed to the District's requested across-the-board reductions. Second, the District contends that the district court abused its discretion in awarding hourly rates to Plaintiffs based on the Legal Services Index ("LSI") update to the Laffey Matrix. See Laffey v. Nw. Airlines, Inc. (Laffey I), 572 F.Supp. 354, 371 (D.D.C.1983), aff'd in part, rev'd in part on other grounds, Laffey v. Nw. Airlines, Inc. (Laffey II), 746 F.2d 4 (D.C.Cir.1984), overruled in part on other grounds en banc by Save Our Cumberland Mountains, Inc. v. Hodel (SOCM), 857 F.2d 1516 (D.C.Cir.1988). Finally, the District states that the district court improperly ordered the District to pay for the time Plaintiffs' counsel spent on a third-party appeal. We disagree. The district court thoroughly analyzed both parties' positions in awarding some, but not all, of the requested fees. For the reasons stated below, we affirm.

BACKGROUND

In 1993, Plaintiffs filed a class action against the District challenging the District's provision of medical assistance, including certain services for all enrolled children, under the District's Medicaid program. In 1996, the district court concluded that the District had violated 42 U.S.C. § 1983 by depriving Plaintiffs of their statutory and constitutional rights. Salazar v. District of Columbia, 954 F.Supp. 278, 334 (D.D.C.1996). In 1999, while the judgment was pending on appeal, the parties entered into a comprehensive settlement agreement (the "Settlement Order").

Under the terms of the Settlement Order, Plaintiffs' counsel is entitled to compensation for monitoring the District's compliance with the provisions of the Settlement Order, for representing individual class members to enforce their rights under federal Medicaid law and the Settlement Order, and for work not designated as monitoring work, such as enforcement, attorneys' fees, and appeals work. Paragraphs 64 and 65 of the settlement establish certain rates, which are adjusted annually for inflation, for Plaintiffs' counsel to monitor and enforce the District's compliance with the Settlement Order. For instance, if a purported member of the class seeks Plaintiffs' counsel's assistance, the reasonable time and expenses of Plaintiffs' counsel in determining whether the individual is a member of the class and in providing legal assistance "shall be deemed compensable monitoring of [the Settlement] Order under 42 U.S.C. § 1988 [at a rate of]" $75/hr. In addition, for rates set under Paragraphs 64 and 65, the Settlement Order specifies that those "hourly rates shall be adjusted annually, beginning on January 1, 1999, based on the U.S. Department of Commerce Consumer Price Index for Legal Services." Under Paragraph 66, the parties left open the rates for work not specified as monitoring work under Paragraphs 64 and 65 of the Settlement Order.

The Settlement Order provides no further guidance for determining an appropriate fee award where the rate is unspecified. However, this Court has developed a three-part analysis to assess appropriate fee awards under fee-shifting statutes in cases involving complex federal litigation. See Eley v. District of Columbia, 793 F.3d 97, 100 (D.C.Cir.2015) (citing SOCM, 857 F.2d at 1517). A court must: (1) determine the "number of hours reasonably expended in litigation"; (2) set the "reasonable hourly rate"; and (3) use multipliers as "warranted." Id. In addition, the "fee applicant bears the burden of establishing entitlement to an award, documenting the appropriate hours, and justifying the reasonableness of the rates," with the opposing party remaining "free to rebut [the] fee claim." Covington v. District of Columbia, 57 F.3d 1101, 1107-08 (D.C.Cir.1995), cert. denied, 516 U.S. 1115, 116 S.Ct. 916, 133 L.Ed.2d 847 (1996).

Here, the District challenges the district court's analysis of the hours Plaintiffs' counsel spent litigating the multiple issues arising from the monitoring and enforcement of the Settlement Order. Broadly, the District argues that the district court abused its discretion by failing to rein in Plaintiffs' counsel's fees further than it did.

The District also challenges the "reasonable hourly rate" determination. That determination requires showing at least three elements: (1) "the attorneys' billing practices"; (2) "the attorneys' skills, experience, and reputation"; and (3) "the prevailing market rates in the relevant community." Covington, 57 F.3d at 1107. As to these three sub-elements, the District focuses its challenge on the district court's determination of the "prevailing market rates in the relevant community."

A court calculating a prevailing market rate allows fee applicants to submit attorneys' fee matrices as one type of evidence. Covington, 57 F.3d at 1109. As we have previously noted in Eley, 793 F.3d at 100, "[t]he most commonly used fee matrix is the `Laffey Matrix' — the schedule of prevailing rates compiled in [Laffey I]."

The Laffey Matrix sets out a general guideline for awarding attorneys' fees based on experience. See, e.g., Eley, 793 F.3d at 101 (discussing Laffey Matrix rates). For instance, the Laffey Matrix sets the rate for "experienced federal court litigators in their 11th through 19th years after law school graduation" at $150/hr. Id. (citing Laffey II, 746 F.2d at 8 n. 14). But these 30-year-old rates must also be adjusted for inflation. See Eley, 793 F.3d at 101 (citing SOCM, 857 F.2d at 1525). For this reason, updated Laffey Matrices developed, including the two at issue here — (i) the Laffey Matrix as updated by the Legal Services Index ("LSI") of the Nationwide Consumer Price Index ("CPI") (the "LSI Laffey Matrix"), and (ii) the All-Items CPI for the Washington, D.C. area (also known as the "USAO Laffey Matrix"). See also Eley, 793 F.3d at 101-02 (elaborating on the differences between the LSI and the USAO Laffey Matrices).

In 2013, Plaintiffs filed two fee applications for work done from 2010 through 2012. The district court granted in part and denied in part those fee applications, which decisions the District appeals now. In both decisions, and in spite of the District's arguments otherwise, the district court ruled that the LSI Laffey Matrix provided the appropriate billing rates for work lacking a specified rate in the Settlement Order. See Salazar v. District of Columbia (Salazar III), 991 F.Supp.2d 39, 47 (D.D.C.2014); see also Salazar v. District of Columbia (Salazar IV), 30 F.Supp.3d 47, 51-52 (D.D.C.2014) (adopting analysis of Salazar III). Citing to previous resolutions of other fee applications in this on-going litigation, the district court reiterated that "the [LSI-adjusted][ [sic] Laffey index has the distinct advantage of capturing the more relevant data because it is based on the legal services component of the Consumer Price Index rather than the general CPI on which the U.S. Attorney's Office matrix is based." Salazar III, 991 F.Supp.2d at 47 (quoting Salazar v. District of Columbia (Salazar I), 123 F.Supp.2d 8, 14-15 (D.D.C.2000)) (adopting and citing Salazar v. District of Columbia (Salazar II), 750 F.Supp.2d 70, 72-74 (D.D.C.2011)); see also Salazar IV, 30 F.Supp.3d at 5152. After concluding that the LSI Laffey Matrix was the appropriate rate index, the district court addressed the District's many arguments related to the appropriateness of Plaintiffs' counsel's billing. See Salazar III, 991 F.Supp.2d at 49-64; Salazar IV, 30 F.Supp.3d at 52-65.

The district court awarded some but not all of the fees and costs Plaintiffs sought; and where the district court found that Plaintiffs' counsel's fees were not reasonable, the court either reduced the hours sought, or denied the fee request. See, e.g., Salazar III, 991 F.Supp.2d at 53 (reducing billing for some individual claims by 15% as the court found the hours spent "excessive"); Salazar IV, 30 F.Supp.2d at 61 (denying request for fees for categories of work that were "unrelated" to the case). In the end, the district court awarded Plaintiffs $655,587.98 for fees and expenses related to the 2011 fee application, and also awarded Plaintiffs $522,990.63 for fees and expenses related to the 2012 fee application. J.A. at 234445, 2390-91:

DISCUSSION

We review the district court's fee awards for abuse of discretion, "and will not upset its hourly rate determination absent clear misapplication of legal principles, arbitrary fact finding, or unprincipled disregard for the record evidence." Eley, 793 F.3d at 103 (citation and quotation marks omitted). "This limited standard of review is appropriate in view of the district court's superior understanding of the litigation and the desirability of avoiding frequent appellate review of what essentially are factual matters." Id. at 104 (quoting Covington, 57 F.3d at 1110). We do, however, "examine de novo whether the district court applied the correct legal standard." Id. (citation and quotation marks omitted).

A.

As noted, the District takes issue with the district court's calculation of the reasonableness of the attorney hours. Specifically, the District argues that the district court should have applied across-the-board reductions to Plaintiffs' counsel's fee applications, in light of purportedly excessive and vague billing. For example, the District would have preferred the district court to reduce the 2011 fee application by 20% and the 2012 fee application by 15% rather than the specific reductions in certain categories of the fee application that the district court made.

With respect to the district court's determinations about the reasonableness of the attorneys' hours in the fee applications, the court acted within its discretion. The district court conducted a comprehensive and careful analysis of the parties' positions, and its resolution of the parties' disputes concerning the amount of attorney hours was reasonable. See, e.g., Blum v. Stenson, 465 U.S. 886, 902 n. 19, 104 S.Ct. 1541, 79 L.Ed.2d 891 (1984) ("A district court is expressly empowered to exercise discretion in determining whether an award is to be made and if so its reasonableness."). Indeed, the district court addressed the adequacy of the time records at issue, purported overbilling or excessive billing, and the District's requested across-the-board reductions. See Salazar III, 991 F.Supp.2d at 50, 57, 64; see also Salazar IV, 30 F.Supp.3d at 52, 54, 61, 64-65. What is more, the district court has supervised this case for at least the last fifteen years, making it well-situated to determine the reasonableness of Plaintiffs' counsel's fees. Accordingly, we will not disturb the district court's decisions.

B.

The District also takes issue with the district court's determination that the LSI Laffey rates apply in this case. As a preliminary matter, we reject Plaintiffs' threshold contention that the District waived its challenge to the LSI Laffey rates. Plaintiffs maintain that the law-of-the-case doctrine bars the District from raising that challenge because the District failed to appeal the district court's prior fee decisions applying the LSI Laffey Matrix, as opposed to the USAO Laffey Matrix, in setting hourly rates under Paragraph 66 of the Settlement Order. See Salazar I, 123 F.Supp.2d at 11-15; Salazar II, 750 F.Supp.2d at 72-74. But the law-of-the-case doctrine does not apply here. Under that doctrine, "a legal decision made at one stage of litigation, unchallenged in a subsequent appeal when the opportunity to do so existed, becomes the law of the case for future stages of the same litigation, and the parties are deemed to have waived the right to challenge that decision at a later time." Kimberlin v. Quinlan, 199 F.3d 496, 500 (D.C.Cir.1999) (emphasis added) (citation and internal quotation marks omitted). While the District failed to appeal the district court's use of the LSI Laffey Matrix rates in Salazar I or Salazar II, the District is taking the opportunity now to challenge the application of the LSI Laffey Matrix rates to the 2010-2012 attorneys' fees, which were not at issue in those prior decisions. Therefore, we consider on its merits the District's challenge to the district court's market-rate determination for 2010-2012.

The district court did not abuse its discretion. The district court's selection of LSI Laffey rates is consistent with this Court's intervening decision in Eley. The district court appropriately required the Plaintiffs to demonstrate the propriety of the rates they sought (i.e., under the LSI Laffey Matrix). In Eley, we vacated a district court's fee award based on evidence submitted by the District tending to show that, in the particular context of IDEA claims, there is a submarket in which attorneys' hourly fees are generally lower than the rates in either of the Laffey Matrices. Eley, 793 F.3d at 105. In this case, by contrast, the District identifies no such submarket, instead acquiescing in the notion that the litigation at issue qualifies as complex federal litigation (as to which the Laffey Matrices apply). See, e.g., Appellant Br. at 48-49 (discussing that "in complex federal court litigation" the USAO update to the Laffey Matrix is a more appropriate method for determining the prevailing market rate).

Accordingly, unlike in Eley, 793 F.3d at 103, the District does not argue for rates lower than both of the Laffey matrices, but instead argues that one Laffey Matrix should apply instead of the other. See Appellant Br. at 48-54 (arguing that the USAO Laffey Matrix is the preferable market-rate index). Moreover, our decision in Eley, 793 F.3d at 104-05, reaffirmed our prior decision in Covington, 57 F.3d at 1110, in which we held that the fee applicants "clearly" met their burden of justifying their requested rates and "were properly accorded a presumption of reasonableness." In Covington, the fee applicants' evidentiary submissions for the prevailing market rates in complex federal litigation included "the Laffey matrix, the U.S. Attorney's Office matrix, affidavits attesting to increases in the market rates since the original Laffey matrix, and memorandum opinions in district court cases which relied on these matrices." 57 F.3d at 1110. The District, in rebuttal, failed to cite any relevant cases supporting its requested rates. Id. at 1111. Therefore, we affirmed the district court's "determination that the relevant market is that of complex federal litigation." Id. at 1112.

As the District concedes that the relevant market is that of complex federal litigation, the only issue is whether Plaintiffs submitted sufficient evidence for the district court to conclude that the LSI Laffey Matrix applies. Like the fee applicants in Covington, 57 F.3d at 1110, Plaintiffs submitted "a great deal of evidence regarding prevailing market rates for complex federal litigation." Plaintiffs submitted evidence for their preferred Laffey Matrix update, including the affidavit of the economist that developed the LSI Laffey Matrix — Dr. Michael Kavanaugh. Dr. Kavanaugh's affidavit explained why the LSI is a better measure of the change in prices for legal services in Washington, D.C. than the USAO update to the Laffey Matrix. J.A. at 2038-65. The affidavit also noted other federal courts that have adopted the LSI Laffey Matrix. See J.A. at 2039-40 ¶ 6.

In addition to this evidence, Plaintiffs went further and submitted more evidence supporting the use of the rates approved by the district court than the submissions found adequate in Covington. For instance, Plaintiffs submitted billing rates tables demonstrating the difference between average national law firm rates and the LSI update to the Laffey Matrix, as well as the difference between average national law firm rates and the USAO update to the Laffey Matrix. J.A. at 2292. As an example, for lawyers with experience levels between eleven and nineteen years from the date of law school graduation, the average national law firm rate in 2012 to 2013 was $672. Id. For the same experience level in the same time frame, the LSI updated rate was closer to the average national law firm rate at $626, while the USAO updated rate was $445. Id. On average, the LSI Laffey Matrix rates were 14% lower than the average national law firm rates for all experience levels in this time period. See id. On the other hand, the USAO Laffey Matrix rates were 38% lower than the average national law firm rates. See id.

Furthermore, a 2012 National Law Journal Rates Survey showed that the rates for partners in Washington, D.C. on the high-end of the market far exceeded the rates in the LSI update. See J.A. at 2293. With these numbers and submissions in the record, the district court's point that "the LSI-adjusted matrix is probably a conservative estimate of the actual cost of legal services in this area," does not appear illogical. See Salazar III, 991 F.Supp.2d at 48 (citation and internal quotation marks omitted). The District, neither below nor on appeal, rebuts this logic with relevant arguments. The District makes much about the fact that this prolonged litigation is depleting public funds in a case in which Plaintiffs no longer need to pay LSI updated Laffey rates to "attract competent counsel." Appellant Br. at 49. But as the district court correctly noted, and as we have made clear, "fees should be neither lower, nor calculated differently, when the losing defendant is the government." Salazar III, 991 F.Supp.2d at 49 (quoting Copeland v. Marshall, 641 F.2d 880, 896 (D.C.Cir.1980) (en banc)).1

C.

As to the final matter on appeal, the District takes issue with the district court's order requiring the District to pay for the time Plaintiffs' counsel spent responding to an appeal involving an effort to obtain information used by one of the District's contractors. See Salazar III, 991 F.Supp.2d at 59-60 (awarding Plaintiffs' appeal fees). We affirm the award of fees for this work, despite the District not entering an appearance in the appeal. See id. at 60. In the particular circumstances of that appeal, where the information was necessary for Plaintiffs' counsel to litigate some of the claims underlying the Settlement Order, see id., the award of fees was not inappropriate.

CONCLUSION

For the foregoing reasons, and based on the particular facts of this case, the decisions of the district court are affirmed.

So ordered.