
626 F.3d 313 (2010)
UNITED STATES of America, Plaintiff-Appellant/Cross-Appellee,
v.
UNITED TECHNOLOGIES CORPORATION, Defendant-Appellee/Cross-Appellant.
Nos. 08-4256, 08-4257.
United States Court of Appeals, Sixth Circuit.
Argued: August 3, 2010.
Decided and Filed: November 18, 2010.
*315 ARGUED: Alan S. Gale, United States Department of Justice, Washington, D.C., for Appellant. Jeffrey A. Hall, Bartlit Beck Herman Palenchar & Scott LLP, Chicago, Illinois, for Appellee. ON *316 BRIEF: Alan S. Gale, United States Department of Justice, Washington, D.C., for Appellant. Jeffrey A. Hall, Bartlit Beck Herman Palenchar & Scott LLP, Chicago, Illinois, David C. Greer, Bieser, Greer & Landis LLP, Dayton, Ohio, David Z. Bodenheimer, Crowell & Moring LLP, Washington, D.C., for Appellee.
Before: SILER and SUTTON, Circuit Judges; CLELAND, District Judge.[*]

OPINION
SUTTON, Circuit Judge.
Fighter jet engines propel planes faster than the speed of sound, a sight that may be as exhilarating to watch as it must be to experience. Not so the procurement process for awarding contracts to make jet engines.
In the early 1980s, the Air Force set up a bidding process to make fighter jet engines. Attempting to win as many engine contracts as possible, United Technologies manipulated its cost estimates to convince the Air Force to award the company a large share of the work. After the Air Force discovered that United Technologies had played with its numbers, the United States filed a lawsuit under the False Claims Act and common law, seeking reimbursement for the fraud. The district court (1) ruled for the government on liability under the False Claims Act but held that it had suffered no damages and (2) ruled that the government's administrative proceedings precluded it from raising its common law claims. We affirm the liability determination but reverse the district court's damages determination and its claim-preclusion ruling.

I.
Pratt and Whitney is a division of United Technologies Corporation. For many years, the United States Air Force relied exclusively on Pratt to provide it with F-15 and F-16 fighter jet engines. In 1982, fearful of the risks of relying on just one company to produce the engines, the Air Force conducted a fighter engine competition (FEC) for future fighter jet engine contracts. Only Pratt and General Electric had the capacity to manufacture the Air Force's desired fighter jet engines and thus only the two companies competed for the contract.
In May 1983, the Air Force issued a request for proposal, seeking offers for the contract. Each offer, the Air Force explained, should include three options: (1) a price for one purchase order covering three years and three "not-to-exceed price options" for the Air Force to make annual purchases for an additional three years; (2) a price for a five-year purchase order with one not-to-exceed price option for the last year; and (3) a price option for a one-year purchase order, firm fixed-price options for annual purchases the second and third years and not-to-exceed price options for the last three years. Appeals of United Techs. Corp., 04-1 BCA P 32556, 2004 WL 483216 (A.S.B.C.A. Feb.27, 2004), finding #3 (Initial ASBCA Decision).
Faced with the prospect of losing some of its fighter jet engine contracts to GE, Pratt prepared an offer that made it difficult for the Air Force to enter into dual-source engine contracts with Pratt and GE. Pratt submitted its initial proposal in August 1983. The Air Force reviewed the proposal, prepared some objections and met with Pratt to go over its concerns. After the meeting, the Air Force gave *317 Pratt a list of questions about cost estimates for Pratt to answer in its final proposal. Pratt submitted its best and final offer in December 1983.
As part of this final offer, Pratt submitted a document (Exhibit 3.8.1) that explained the company's cost estimates and responded to the Air Force's questions. Exhibit 3.8.1 contained three false statements, all designed to drive up the prices for split-award contracts and to discourage the Air Force from splitting up the work. One: Pratt said that it had used the most recent inflation forecasts to calculate the not-to-exceed prices when it had used a different inflation index. Two: in preparing its cost estimates, Pratt said that it had applied a "decrement factor"an estimate of how much the engine prices would decrease based on Pratt's prior negotiations with engine-parts supplierseven though Pratt had not factored decrement data into the final offer. Three: Pratt said that it had used the most current engine configurations to calculate its prices, even though it had relied on an earlier engine model without accounting for engine components that did not go into the updated engine. The method to the scheme was to drive up the prices for split-award contracts, then provide a variation-in-quantity formula to reduce prices if the Air Force awarded a greater percentage of the engines to Pratt. Pratt does not dispute any of these facts.
Based on Pratt's and GE's proposals, and despite Pratt's efforts to discourage the Air Force from splitting the work, the Air Force chose to exercise a one-year, split-award option for the first year of the FEC and executed the contract with Pratt and GE. The contract allowed the Air Force to exercise 1-year, 3-year or 5-year options, the 1-year option allowing the Air Force to determine its engine quantities for one year, the 3-year option allowing the Air Force to determine its engine quantities for three years, and so on. The contract breaks each year into FY and FEC terms, with FY85 for example corresponding with FEC I and FY86 corresponding with FEC II. The contract obligated the Air Force to purchase at least 25 engines from Pratt in FEC I, and at least 100 engines from Pratt for each of the remaining five years.
In February 1983, the Air Force briefed Congress on its decision to award engine contracts to Pratt and GE on a competitive basis. After the hearings, Congress passed the Defense Department Authorization Act for FY85, which prohibited the Air Force from making a contract to purchase engines if the warranty pricewhich is to say the costs the engine provider would have to pay the government if the engine failedexceeded 10 percent of the total contract price. See Pub.L. No. 98-525, 98 Stat. 2591 (1984).
Because Pratt's warranties in the final offer exceeded the statutory cap, the Air Force sent a letter to Pratt indicating that the Air Force would "review and consider any update or revision to [Pratt's] warranty offer." Appx. at 618. The Air Force also suggested that Pratt could use this opportunity to revise other terms and conditions of the contract. Pratt responded with an offer to reduce its warranty prices in exchange for a $17.22 million cap on its previously unlimited liability. Pratt also offered several other improvements to the contract. The Air Force accepted the warranty modification and incorporated it into the contract. The Air Force also accepted several other modifications proposed by Pratt, including one that reduced the Air Force's minimum engine purchases. The Air Force exercised its option under these revised terms and awarded 40 engines to Pratt and 120 engines to GE for FEC I.
*318 To update and potentially improve the terms of the contract, as required under federal law, see A.S.P.R. § 1-1505(c)(iii), (d)(2), (f)(2), the Air Force began a "call for improvement" process, Initial ASBCA Decision, finding # 33. Before the Air Force exercised its option for FEC II, it sent a letter to Pratt asking if Pratt had any updates or improvements to its option prices. The Air Force repeated this process for FEC III, IV, V and VI. Before exercising its options for each year, the Air Force sent a letter to Pratt offering the company an opportunity to "update and improve the terms and conditions" of "options under [Contract F33657-84-C-2014]." Appx. at 680 (FEC III), 692 (FEC IV), 708 (FEC V), 734 (FEC VI). The Air Force indicated that it would consider each improvement individually, requiring Pratt to offer each revision on a "stand alone basis." E.g., Appx. at 682 (FEC III), 694 (FEC IV), 713 (FEC V), 736 (FEC VI). The Air Force also explained that it did not "intend to and will not conduct an additional formal competition," disclaiming any suggestion that Pratt should treat the letter as "a request for proposal." E.g., Appx. at 681 (FEC III), 694 (FEC IV), 711-12 (FEC V), 735-36 (FEC VI).
Pratt responded with updated options for each FEC cycle. Pratt offered improvements on its engine prices, its warranty prices and other contract provisions. In each year, the Air Force accepted Pratt's improvements, modified the contract and exercised its options under the revised contract terms. Due to these improvements, the Air Force awarded Pratt an increasing percentage of the engines after FEC I.
As the six-year jet engine contract drew to a close, the Air Force began an audit of the FEC program, which uncovered handwritten notes by a Pratt executive detailing the company's plans to play with the numbers to win a sole-source contract. The Air Force looked into the matter but eventually closed the investigation.
In 1997, a Department of Justice auditor investigated the engine contracts and noticed Pratt's false statements in its best and final offer. In response, the United States launched two proceedings against Pratt: an administrative action and a federal-court lawsuit. In 2001, it sought a contract price reduction before the Armed Services Board of Contract Appeals, arguing that Pratt had furnished defective cost or pricing dataPratt's final offer and other preparatory workin violation of the Truth in Negotiations Act. 10 U.S.C. § 2306(f). The Board held that Pratt's final offer was not cost or pricing data within the meaning of the Truth in Negotiations Act, that Pratt's preparatory work constituted "cost or pricing data" under the Act and that Pratt's failure to disclose its preparatory work violated the Act. The Board, however, did not reduce the contract price because it determined that Pratt's understatements in its cost estimates offset its overstatements.
Pratt moved for reconsideration. The Board changed its mind in part, concluding that the Air Force had not relied on Pratt's cost or pricing data to determine the reasonableness of Pratt's offers, an affirmative defense under the Truth in Negotiations Act. ASBCA Reconsideration, 05-1 BCA P 32860, 2005 WL 147601 (A.S.B.C.A. Jan.19, 2005). The Air Force appealed, and the Federal Circuit affirmed. Wynne v. United Techs. Corp., 463 F.3d 1261 (Fed.Cir.2006).
In March 1999, the government filed a lawsuit against Pratt in federal district court for violations of the False Claims Act and for relief under three common law theories: breach of contract, payment by mistake and unjust enrichment. Because the False Claims Act has a 10-year statute *319 of limitations, 31 U.S.C. § 3731(b)(2), the United States sought to recover only for False Claims Act violations after March 1989, which included the last three years of the contract.
For reasons that the record does not fully explain, the bench trial did not begin until October 2004. The trial lasted through December 2004, and the court heard closing arguments in April 2005. In August 2008, the court issued its decision, finding Pratt liable for 709 violations of the False Claims Act, a number that corresponds to the number of invoices that Pratt submitted under the contract after March 3, 1989. The court concluded that, despite the calls for improvements process, "the Government had the legal right to award the entire multi-year contract to Pratt and to have the contract performed based upon Pratt's [final offer]." R.397 at 18. At the same time, however, the court found that the government suffered no damages because Pratt's price reductions offset any cost overstatements in its final offer. It separately held that claim preclusion barred the government from bringing its common law contract claims because it litigated or could have litigated those claims before the Board of Contract Appeals. Both parties filed appealsPratt with respect to the liability ruling and the government with respect to the damages and claim-preclusion rulings.

II.
As to liability, the district court concluded that Pratt violated two provisions of the False Claims Act: §§ 3729(a)(1) and 3729(a)(2) of Title 31. (In 2009, Congress amended 31 U.S.C. § 3729, Pub.L. No. 111-21, § 4. In this opinion, all references to § 3729 are to the earlier version.) Section 3729(a)(1) prohibits an individual from "knowingly present[ing] or caus[ing] to be presented, to an officer or employee of the United States Government ... a false or fraudulent claim for payment or approval." 31 U.S.C. § 3729(a)(1). Section 3729(a)(2) prohibits "knowingly mak[ing], us[ing], or caus[ing] to be made or used, a false record or statement to get a false or fraudulent claim paid...." 31 U.S.C. § 3729(a)(2) (2006). The Act defines a "claim" as "any request or demand, whether under a contract or otherwise, for money or property...." 31 U.S.C. § 3729(c) (2006).
Under the Act, a "false or fraudulent claim" includes a false statement that is material to the government's decision to pay a claim. See United States ex rel. A + Homecare, Inc. v. Medshares Mgmt. Grp. Inc., 400 F.3d 428, 442-43 (6th Cir.2005). The parties do not dispute that Pratt's best and final offer may provide the basis for liability under the Act. They do not dispute that Pratt's final offer contained three false statements. They do not dispute that those false statements were material to the government's decision to enter into a contract with Pratt. And they do not dispute that, in the abstract, an invoice submitted in connection with a contract premised on a false statement amounts to a "claim" under the Acta "request ... under a contract ... for money." 31 U.S.C. § 3729(c) (2006).
That leaves the possibility that liability does not exist when the relevant invoices are sent in connection with a multi-year contract like this one, as opposed to a single-year or one-time contract. That distinction, however, does not change things. See United States ex rel. Marcus v. Hess, 317 U.S. 537, 542-44, 63 S.Ct. 379, 87 L.Ed. 443 (1943); Harrison v. Westinghouse Savannah River Co., 176 F.3d 776, 785-88 (4th Cir.1999). Hess establishes that an invoice, which itself does not contain a falsity, may supply the premise for a false claim if submitted in connection with a fraudulently obtained contract. 317 U.S. *320 at 542-44, 63 S.Ct. 379. Why? The invoices "caused the government to pay claims ... under contracts found to have been executed as the result of ... fraudulent bidding." Id. at 543, 63 S.Ct. 379. One variety of fraud is a knowingly false statement that has the "natural tendency to influence, or is capable of influencing" a government decision maker to pay the related invoice. A + Homecare, 400 F.3d at 445. False statements underlying multi-year contracts generate a stream of related invoices and cause the government to pay all of the invoices related to the contract.
That is just what happened here. In its request for proposal, the Air Force asked that all proposals include cost estimates with fully supported cost data. After Pratt submitted its initial proposal, the Air Force asked Pratt why it failed to apply decrements. The Air Force met with Pratt before the company submitted its final offer and asked it to explain the cost estimates. Exhibit 3.8.1 addressed the Air Force's questions. On this record, Pratt's false statements had the potential to influence the government's decision to sign the contract and thus to pay the invoices under it. See A+ Homecare, 400 F.3d at 445.
None of this matters, Pratt submits, because it made the best and final offer well before March 3, 1989, when the statute-of-limitations window closed on its initial interactions with the government and above all well before several non-fraudulent modifications to the contract. All that leaves, Pratt urges, are invoices submitted after March 1989, which did not themselves contain falsities and which were connected to six separate one-year fighter jet engine contracts with the Air Force after it modified the original contract through the calls for improvements process.
But it is not that easy to separate the original contract from the modifications to it. The original contract set price windows for six years and obligated the Air Force to purchase a minimum number of engines in each of these six years. Not surprisingly, as a result, all of the documentation through FEC VI refers to the same contract F33657-84-C-2014the one arising from the false statements in the best and final offer. In soliciting revisions, moreover, the Air Force indicated it would consider each revision on a "stand alone basis," not that it would reconsider the entire contract. E.g., Appx. at 682 (FEC III), 694 (FEC IV), 713 (FEC V), 736 (FEC VI). The government recorded the new terms as contract modifications, not as new freestanding contracts.
Pratt demurs, pointing to language in the government's letters about "canceling" and "superseding" the original contract. But when read in context, this language cancels contract clauses or attachments, not the entire contract. See, e.g., Appx. at 1131-32 (FEC IV), 1213-14 (FEC VI). Confirming the point, the language appears in a section outlining "enhancement[s]" to be "incorporat[ed]" into the fighter-jet engine contract, Appx. at 1123 (FEC IV), or is listed as one of several "changes" to the original contract, Appx. at 1212 (FEC VI).
The nature of the revisions points in the same direction. They arose from the calls for improvements process, a federally mandated opportunity to facilitate further price competition between Pratt and GE. The Air Force designed the process to benefit itselfto seek "improvements" to the contract from the government's perspective, an opportunity to sweeten a deal that happened only because the Air Force entered into the contract based on fraud in the first place. The Air Force never issued a new request for proposal; it sought only improvements from parties to the existing jet engine contracts.
*321 Also unavailing is Pratt's reliance on Allison Engine Co. v. United States ex rel. Sanders, 553 U.S. 662, 128 S.Ct. 2123, 170 L.Ed.2d 1030 (2008). Construing a prior version of § 3729(a)(2), Allison Engine held that a plaintiff must prove that a defendant intended to use a false record to get the government to pay a claim. Id. at 665, 128 S.Ct. 2123. In response, Congress amended the False Claims Act, retroactive to claims pending in June 2008, providing that a claimant no longer has to prove that a defendant intended to get a false claim paid. See Fraud and Enforcement Recovery Act of 2009, Pub.L. No. 111-21, 123 Stat. 1617 (2009). Even if the Allison Engine standard for liability under § 3729(a)(2) applies to Pratt, a point we need not decide, the government satisfied it. Pratt intended the government to pay the fraudulent invoices that it submitted directly to the government, while Allison Engine had no such intent when it submitted invoices to a contractor, not the government. Allison Engine, 553 U.S. at 666, 671-72, 128 S.Ct. 2123. Even if that were not the case, moreover, Pratt still violated § 3729(a)(1), a provision not at issue in Allison Engine, by submitting fraudulent invoices to the government.
Pratt separately invokes a procurement regulation obligating the Air Force to solicit cost or pricing data if the Air Force modifies a contract by over $500,000. A.S.P.R. § 3-807.3. The regulation makes an exception if the price is "based on adequate price competition." Id. Because the Air Force never solicited cost or pricing data during the calls for improvements process, Pratt claims that the Air Force must have based the contract prices on adequate price competition rather than on Pratt's final offer or cost or pricing data. Yet the link missing in this chain of reasoningwhy all of this releases Pratt from liabilityis never explained, and we think it is inexplicable. The False Claims Act makes an individual liable for presenting a fraudulent claim to the government regardless of whether the government decides to pay the claim based on price competition. 31 U.S.C. § 3729(a)(1); see A+ Homecare, 400 F.3d at 443-44. And under § 3729(a)(2), a false statement must be "material" to the government's decision to pay, meaning that it must have the objective, natural tendency to influence a government decision maker. Liability does not depend on whether the government relied in fact on the false statement to pay out the claim. A + Homecare, 400 F.3d at 445. The False Claims Act, unlike the Truth in Negotiations Act, does not contain an affirmative defense if the government does not rely on a false statement. Cf. 10 U.S.C. § 2306a(e)(2). The regulation thus does not alter the material reality under the False Claims Actthat Pratt's false statements had the natural tendency to influence the Air Force to enter into this six-year contractand at most goes to the government's damages, not Pratt's liability.

III.
The government says that the district court erred in concluding that Pratt's conduct caused no damages. Under the False Claims Act, the government may recover "actual damages," the difference between what it paid and what it should have paid for the goods. United States v. Bornstein, 423 U.S. 303, 317 n. 13, 96 S.Ct. 523, 46 L.Ed.2d 514 (1976); United States ex rel. Roby v. Boeing Co., 302 F.3d 637, 646 (6th Cir.2002); United States ex rel. Compton v. Midwest Specialties, Inc., 142 F.3d 296, 304 (6th Cir.1998). In a case covering as many years and involving as many complicated procurement regulations as this one, it should not come as a surprise that the district court encountered several difficult damages calculations. *322 And, as a result, it perhaps also should not come as a surprise that hindsight allows us to identify a few mistakes, some of which require a new calculation by the trial court.
First, the district court subtracted the full amount of Pratt's warranty price reductions from the court's estimate of the government's damages without calculating the value of the new warranties. In its best and final offer, Pratt offered to give the government engine warranties without a cap on Pratt's liability to the government. When Pratt later reduced the warranty prices, as part of the calls for improvement process, it capped its liability at 300% of the warranty price or $17.22 million. While the government paid less for the engine warranties, it also got something less in returnfar less insurance for engines gone awry. Nothing in the court's opinion shows that it accounted for this reality. To calculate the difference between what the government paid and what it should have paid, see Roby, 302 F.3d at 646, the court must account for the diminished value of the new warranties.
None of this is necessary, Pratt counters, because the government gave up nothing by agreeing to a liability capthat the value to the government of an unlimited warranty was no different from a warranty capped at $17.22 million. That is hardly self-evident, and indeed nothing in the record supports the notion that $17.22 million encompasses all damages that a faulty fighter-jet engine might cause or all repairs that such an engine might need.
Second, the district court incorrectly subtracted the government's total savings from the final offer prices to calculate damages. The district court reduced the government's damages for FEC III, IV and V by the amount Pratt reduced its prices over the course of the entire contract, meaning that it compared FEC VI prices to the original contract prices. But in FEC III, IV and V, the government did not pay FEC VI prices because those revisions had not yet occurred. The district court should calculate the difference between what the government paid each yearFEC III, IV and Vand what it should have paid each year. It should not subtract the price reductions the government won for FEC VI when it determines whether the government suffered any damages during FEC III, IV and V.
Third, we are not confident that the district court applied the correct damages standard for the years under review. The district court calculated the difference between the 1984 contract prices and what the government paid for FEC III-VI. But, as noted, the district court should have calculated the difference between what the government paid and what it should have paid for what it received. Neither party maintains that the 1984 contract prices represent the fair market value of the goods the government received or that the government should have paid those prices. The court, to the contrary, found that the FEC I prices turned on false claims. On remand, the district court should calculate what the government eventually paid each year during FEC III, IV, V and VI, what it should have paid each year based on what the government received, then take the difference between the two. If the court concludes that Pratt's prices during FEC III, IV, V and VI represent the fair market value of the fighter jet engine contractor were below fair market valuethe government is not entitled to damages.

IV.
After a final judgment on the merits, a party cannot relitigate claims it brought in the first proceeding, and it *323 cannot litigate any claim it could have raised in the first proceeding. Mitchell v. Chapman, 343 F.3d 811, 819 (6th Cir. 2003). The district court concluded that claim preclusion barred the government from pursuing its common law claims in this case. We disagree.
The district court first concluded that the government litigated Count III the contract claimbefore the Board of Contract Appeals and therefore cannot have a second bite at the apple. Count III alleges that Pratt breached its contractual obligation "to provide accurate, complete and current cost or pricing data ... and failed to adjust its [Alternate Fighter Engine] Contract prices to reflect the impact and effect on those prices." R.334 at 25. The Board decided, however, only whether Pratt had violated the Truth in Negotiations Act, not whether Pratt had breached its contractual obligations or overstated its prices. The decision left open the possibility that Pratt's false statements were actionable on other grounds, including breach of contract. See Initial ASBCA Decision, Conclusion of Law III. The factual premises of this case and the Board's case also differ. Before the Board the government argued that Pratt's failure to provide its preparatory work was a violation of the Truth in Negotiations Act, but we are not concerned with Pratt's preparatory work here, only its final offer. The government thus did not litigate Count III before the Board, leaving us to decide whether Count III, like Counts IV and V, could have been litigated before the Board, not whether it was.
The district court concluded that the administrative litigation barred the government's common law claims because the government could have litigated those claims before the Board. But prior litigation precludes a claim only if the initial tribunal had jurisdiction over the claim. Rawe v. Liberty Mut. Fire Ins. Co., 462 F.3d 521, 529-30 (6th Cir.2006). According to all relevant precedent at the time the government filed its administrative claims, the Board did not have jurisdiction over these common law claims. Because these precedents directed the government not to bring these claims before the Board, the government is not precluded from litigating the claims before the district court.
The Board derives its authority from the Contract Disputes Act of 1978, which sets up an administrative dispute resolution process for government contracts. 41 U.S.C. § 607. The Board may hear appeals only from a "decision" of a "contracting officer." Id. §§ 606, 607(d). Contracting officers may hear claims "relating to a contract." Id. § 605(a). Yet the Contract Disputes Act does not authorize "any agency head to settle, compromise, pay, or otherwise adjust any claim involving fraud." Id. (emphasis added).
Because agency heads do not have the authority to resolve "any claim involving fraud," fraud-related claims (such as the False Claims Act claims here) are excluded from the Contract Dispute Act's dispute resolution process and, by extension, from the Board's jurisdiction. The same, we think, goes for the government's common law claimsbreach of contract, payment by mistake and unjust enrichmentwhich in essence are alternative pleadings to its fraud claims under the False Claims Act.
The government's entire case, including its common law claims, "arises from Pratt's knowing false certifications, non-disclosures, and active concealments" during the negotiation and performance of the jet engine contract. R.334 ¶ 2. It may be the case, to be sure, that not every contract or unjust enrichment claim involves fraud. But to describe these allegations is to show that they involve fraud. The complaint *324 describes "Pratt's [alleged] fraud," id. ¶ 4, "Pratt's [alleged] fraudulent" behaviors, id. at 18, and Pratt's alleged knowingly false representations and active concealment of material facts, see, e.g., id. ¶¶ 2, 30, 30E, 35, 36, 37, 45, the last of which amount to fraud by another name, see A+ Homecare, 400 F.3d at 443-44; Restatement (Second) of Torts § 525. No less important than the labels we put on the government's claims is the reality that all of the claims are premised on the same set of facts. See R.334 ¶¶ 2, 59, 60, 63, 68. To resolve the one set of claims (the common law claims) requires the resolution of the other (the False Claims Act fraud claim), and the Contracts Disputes Act does not allow the agency to make that determination.
That, however, by itself does not prove the government is right. The more difficult question is whether the Contract Disputes Act allows contracting officers (and therefore the Board of Contract Appeals) to hear claims involving fraud. The Contract Disputes Act prohibits only an "agency head" from settling claims involving fraud, 41 U.S.C. § 605(a) (emphasis added), but there is no similar restriction on contracting officers' ability to do so. Congress defined "contracting officer" and "agency head" to refer to different individuals. 41 U.S.C. § 601(1), (3). The decision of the contracting officer, not the agency head, forms the basis for the Board's appellate jurisdiction, id. §§ 606, 607(d), which means that, if a contracting officer could issue a decision on a claim involving fraud, the Board could have heard the government's common law claims. (No carve out to the Board's jurisdiction exists. Id. §§ 606, 607(d).)
While we may have to resolve whether contracting officers can issue decisions on claims involving fraud at some point, this case does not require it. All available precedent at the time, it turns out, directed the government not to bring its common law claims before the Board of Contract Appeals. The initial litigation began before the Board, whose decisions the Federal Circuit directly reviews. By the time the government moved for a contract price reduction before the Board, the Federal Circuit had already ruled that "Congress did not intend fraud claims by the government to be included in the dispute process outlined by section 605(a)." Martin J. Simko Constr., Inc. v. United States, 852 F.2d 540, 545 (Fed.Cir.1988). The Board by then also had concluded that Congress excluded fraud-related claims from the jurisdiction of contracting officers and the Board. See Appeal of Comada Corp., 83-2 BCA P 16681, 1983 WL 13084 (A.S.B.C.A. June 20, 1983). But see Appeal of Dry Roof Corp., 88-3 BCA P 21096, 1988 WL 97213 (A.S.B.C.A. Aug.9, 1988) (suggesting the Board has not definitively answered the question). Indeed, every court that has considered the issue has determined that contracting officers and administrative boards do not have jurisdiction over claims involving fraud. See, e.g., United States v. Hardrives, Inc., No. 92-15224, 8 F.3d 31, 1993 WL 385498, at *7 n. 10 (9th Cir. Sept.30, 1993); United States ex rel. Rille v. Sun Microsystems, Inc., No. 4:04CV00986, 2008 WL 4756170, at *5 n. 66 (E.D.Ark. Oct. 28, 2008); United States v. United Techs. Corp., No. 5:92-CV-375, 1996 WL 653620, at *2 (D.Conn. Oct.11, 1996) (listing district court opinions). In addition to this precedent, a Federal Acquisition Regulation says that contracting officers cannot hear claims involving fraud. See 48 C.F.R. § 33.210.
Under these circumstances, we cannot say that the government should have brought its common law claims before the Board. See Browning v. Levy, 283 F.3d 761, 772-73 (6th Cir.2002). A plaintiff who relies on the initial tribunal's decisions on *325 jurisdiction conserves judicial resources, serving one of the goals of claim preclusion. See 18 Charles Alan Wright, Arthur R. Miller & Edward H. Cooper, Federal Practice & Procedure § 4415 (3d ed.2010). If we were to apply claim preclusion based on a disagreement with the Federal Circuit's view of the Board's jurisdiction, that would force litigants to raise their fraud claims before the contracting officer and seek rehearing en banc in the Federal Circuit or certiorari at the Supreme Court. Otherwise, the contracting officer and the Board would reject the claims based on one jurisdictional view, and we could reject them based on the other. While the government might have opted to urge the Federal Circuit to rethink its decision in Martin Simko, we will not punish it for choosing not to do so.
Pratt insists that the Contract Disputes Act covers claims "relating to a contract," 41 U.S.C. § 605(a), and because the government's claims relate to the jet engine contract they fall within the Board's jurisdiction. That may be true, but this ignores the carve out in § 605(a) for claims "involving fraud."
Pratt adds that the government's argument amounts to a "collateral attack" on the Board's jurisdiction. But that assumes the Board had jurisdiction over the government's common law claims, assumes that the Board asserted jurisdiction over those claims and assumes that the government wishes to attack collaterally an adverse ruling on those claims. Pratt's argument fails at the first assumption, and the rest of the assumptions collapse with it. The government did not raise these claims before the Board; otherwise we would be talking about issue, as opposed to claim, preclusion. The Board, it may be true, considered and rejected the government's argument that the information contained in Pratt's final offer constituted cost or pricing data under the Truth in Negotiations Act. But the Board made clear that it "express[ed] no opinion" on whether purported defects in the final offer could be remedied under "other contract clauses, procurement regulations or statutes." Initial ASBCA Decision, Conclusion of Law III. Neither the Board nor the Federal Circuit exercised jurisdiction over the government's common law claims, leaving nothing for the government to attack, collaterally or otherwise. See Charles Alan Wright et al., Federal Practice & Procedure § 4413 (3d ed.2010).
At the end of the day, we acknowledge, this claim-preclusion ruling may matter little to Pratt or the government. If the government recovers False Claims Act damages, there may be nothing left for it to recover on its common law claims. If the district court concludes that the government suffered no damages, it is doubtful that it could recover on its common law claims, which as shown amount to alternative ways of seeking relief on the same facts.
Finally, at places in its brief, Pratt raises an issue-preclusion defense with respect to some of these questions. Because the district court ruled that the litigation before the Board precluded the government from litigating its common law claims, however, it had no reason to consider the defense. The district court should address the defense in the first instance.

V.
We affirm the district court's liability determination, reverse its damages calculation, remand for a recalculation of damages, reverse its claim-preclusion ruling and remand for further proceedings.
NOTES
[*]  The Honorable Robert H. Cleland, United States District Judge for the Eastern District of Michigan, sitting by designation.
