OPINION
BOYCE F. MARTIN, JR., Chief Judge.The stakes are often high in gambling, and millions of dollars are riding on the interpretation of a consent judgment in this Indian gaming case. At issue is whether several Michigan Indian tribes are obligated to continue making payments to the State of Michigan under the terms of a consent judgment entered several years ago. According to the consent judgment, the obligation to make payments is contingent upon the Tribes holding the “exclusive right to operate” electronic games of chance in Michigan. Sault Ste. Marie Tribe of Chippewa Indians v. Engler, No. 1:90-CV-611 (W.D.Mich. Aug. 20, 1993). The district court determined that the Tribes still hold the exclusive right to operate electronic games of chance, and in reaching that conclusion it excluded extrinsic evidence regarding the interpretation of the consent judgment. Sault Ste. Marie Tribe of Chippewa Indians v. Engler, No. 1:90-CV-611 (W.D.Mich. March 17, 1997). We agree with the district court’s interpretation of the consent judgment and with its decision to exclude extrinsic evidence. We also hold that the district court ruled correctly in denying the Lac Vieux Desert Band of Lake Superior Chippewa Indians’ motion to alter or amend the judgment under Fed.R.Civ.P. 59(e). Sault Ste. Marie Tribe of Chippewa Indians v. Engler, No. 1.-90-CV-611 (W.D.Mich. May 20, 1997). We therefore affirm the district court’s decisions.
I.
This case grew out of a dispute between several Indian tribes and the State of Michigan over the issuance of gaming permits. The Sault Ste. Marie Tribe of Chippewa Indians, Grand Traverse Band of Ottawa and Chippewa Indians, Keweenaw Bay Indian Community, Hannahville Indian Community, Bay Mills Indian Community, and Lac Vieux Desert Band of Lake Superior Chippewa Indians filed suit against the State of Michigan in the United States District Court for the Western District of Michigan on July 10, 1990. The Tribes, which are all federally acknowledged Indian tribes, alleged that the State was not negotiating in good faith to conclude a Tribal/State gaming contract under the Indian Gaming Regulatory Act, 25 U.S.C. §§ 2701-2721. On March 26, 1992, the district court dismissed the case after determining that the Eleventh Amendment barred the suit against the State. Sault Ste. Marie Tribe of Chippewa Indians v. Michigan, 800 F.Supp. 1484 (W.D.Mich.1992). The Sixth Circuit found that it did not have jurisdiction over the appeal. Sault Ste. Marie Tribe of Chippewa Indians v. Michigan, 5 F.3d 147 (6th Cir.1993). While the appeal was pending, the Tribes filed an amended complaint on May 5, 1992, naming Michigan Governor John Engler, rather than the State of Michigan, as defendant.
The parties stipulated for the entry of a consent judgment, and the district court entered the judgment on August 20,1993. Under the consent judgment, the State and the Tribes were directed to enter a gaming compact under which the Tribes would operate electronic games of chance. The consent judgment called for the Tribes to make semiannual payments to the Michigan Strategic Fund of eight percent of the net win at each casino from electronic games of chance. The *371Tribes were obligated to continue making the payments “only so long as the tribes collectively- enjoy the exclusive right to operate electronic games of chance in Michigan... The Saginaw Chippewa Indian Tribe of Michigan was also added as a party for the purpose of the consent judgment. The court dismissed the ease with prejudice but retained jurisdiction to enforce the consent judgment. The parties operated under the terms of the judgment for about three years.
In the November 5, 1996, general election, the people of Michigan voted on and adopted “Proposal E,” the Michigan Gaming Control and Revenue Act. (codified at Mich. Comp. Laws Ann. §§ 432.201-432.216 (West Supp. 1998)). The Michigan Gaming Control and Revenue Act became effective on December 5 and was amended by S.B. No. 669, 89th Leg. Reg. Sess. (Mich.1997) (codified as amended at Mich. Comp. Laws Ann. §§ 432.201-432.226 (West Supp.1998)). The Act established the Michigan Gaming Control Board and authorized the Board to grant up to three licenses for casino gaming in Detroit. Mich. Comp. Laws Ann. § 432.204, § 432.206 (West Supp.1998). Under the current law, before the applicant can apply to the Michigan Gaming Control Board, Detroit must pass an ordinance regulating gaming, the applicant must establish a development agreement with the city, and the applicant must have a plan for community investment or involvement. Mich.. Comp. Laws Ann. § 432.206(1) (West Supp.1998). When these criteria, among others, are met, the application is forwarded to the Michigan Gaming Control Board. Mich. Comp. Laws Ann. § 432.206(1) (West Supp.1998). Under Michigan law, a casino is “a building in which gaming is conducted,” and gaming is defined as operating “any gambling game or gambling operation.” Mich. Comp. Laws Ann. § 432.202(g), (x) (West Supp.1998). Therefore, a casino license necessarily includes the right to operate electronic games of chance.
In response to the passage of “Proposal E,” several of the Tribes notified the State that they would-no longer make their semiannual payments because their exclusive' rights to operate casino games had been terminated. On December 9,1996, Governor Engler filed a motion to compel compliance with the consent judgment. On March 14, 1997, the district court issued an opinion and order granting in part and denying in part Governor Engler’s motion to compel compliance with the consent judgment. On March 28, the Lac Vieux filed a motion under Rule 59(e) to alter or amend judgment. On May 20, the court denied the Lac Vieux’s Rule 59(e) motion. The Lac Vieux filed a timely motion of appeal of the March 17 and May 20 orders on June 13. The Lac Vieux tribe is the only tribe involved in the appeal.1
II.
A. Meaning of “Exclusive Right to Operate”
At issue in this case is the interpretation of the consent judgment. A district court’s interpretation of a consent decree or judgment is a matter of law subject to de novo review, and the underlying findings of fact are reviewed for clear error. Huguley v. General Motors Corp., 67 F.3d 129, 132 (6th Cir.1995). Where as here, though, we are reviewing the interpretation of a consent judgment by the district court that crafted the consent judgment, it is probably more accurate to describe our standard of review as “deferential de novo.” It is only sensible to give the court that wrote the consent judgment greater deference when it is parsing its own work. As this Court has noted: “[a]t first blush, giving substantial deference to the district court’s interpretation of the [consent] decree appears to be inconsistent with de novo review. Yet, in Brown v. Neeb, 644 F.2d 551, 558 n. 12 (6th Cir.1981), we explained that the district court’s reading of the decree was merely an additional tool for contract interpretation.” Huguley v. General Motors Corp., 52 F.3d 1364, 1369-70 (6th Cir.1995). As the Court noted in Brown, *372“[flew persons are in a better position to understand the meaning of a consent decree than the district judge who oversaw and approved it.” 644 F.2d at 558 n. 12. We agree, and we will review the district court’s decision accordingly.
Consent decrees and judgments are binding contracts. See id. at 557. The interpretation of a consent decree or judgment is a question of contractual interpretation. Huguley, 67 F.3d at 132. Because this contract was formed' in the State of Michigan, it is interpreted under Michigan law. Sawyer v. Arum, 690 F.2d 590, 593 (6th Cir.1982). Under Michigan law, “[t]he primary goal in the construction or interpretation of any contract is to honor the intent of the parties.” Rasheed v. Chrysler Corp., 445 Mich. 109, 517 N.W.2d 19, 29 n. 28 (1994). The Court “must look for the intent of the parties in the words used in the instrument.” Michigan Chandelier Co. v. Morse, 297 Mich. 41, 297 N.W. 64, 67 (1941).
This dispute centers on the words “exclusive right to operate,” and even more directly on the words “exclusive right.” What is an exclusive right? The Lac Vieux contend that when “Proposal E” was signed into law the Tribes lost their exclusive right and therefore no longer needed to make payments to the State. The district court held that “the TribesTose their ‘exclusive right to operate’ when the [Michigan Gaming Control Board] issues a license to operate a casino to a person or entity other than the Tribes.” Although the State originally contended that the Tribes’ obligation to make payments lasted until another casino operator actually began to operate a casino, it now agrees with the district court’s middle position that the obligation lasts until another operator receives a license.
The Lac Vieux raise several points of contention with the district court opinion. They argue that, “[s]ince Proposal E became effective ... the operation of electronic games of chance is no longer prohibited in the State of Michigan.” They hang this argument on a definition of “prohibited” extracted from the Supreme Court opinion in California v. Cabazon Band of Mission Indians, 480 U.S. 202, 107 S.Ct. 1083, 94 L.Ed.2d 244 (1987). In Cabazon, the Court distinguished between “criminal/prohibitory” laws and “civil/regulatory” laws in determining that federal law did not permit California to regulate tribal bingo enterprises. Id. at 209-11, 107 S.Ct. 1083. The Lac Vieux contend that Caba-zon ’s definition of “prohibit” is relevant because the district court in this case determined that the Tribes have an exclusive right “as long as others are prohibited or shut out from the ‘right to operate’ such games” and that, following the passage of “Proposal E,” electronic games of chance are regulated but not prohibited in Michigan.
The Lac Vieux’s argument has multiple flaws. First, the Cabazonc&se centers on statutory interpretation and deals with the enforceability of state laws on Indian reservation land. This case does not deal with statutory interpretation and has nothing to do with the enforceability of state laws. The Lac Vieux do not argue that Michigan never had the ability to collect the payments from the Tribes; the Lac Vieux merely contend that the State no longer has that ability. Therefore, the Cabazon definition of “prohibit” is not necessarily relevant to the interpretation of a consent judgment.
. Even if the restricted definition of “prohibit” were germane, the Lae Vieux ignore the fact that the district court posits its qualification for exclusivity in the disjunctive. Exclusivity depends on others being “prohibited or shut out.” Even if we were to accept the Lac Vieux’s definition of “prohibit,” the Lac Vieux would not prevail because other operators are nonetheless “shut out” from operating electronic games of chance. Operation of electronic games of chance without a license is against the law in the State of Michigan. See Mich. Comp. Laws Ann. § 750.303 (West 1991 & Supp.1998).
The district court defined “exclusive” as “limited to possession, control or use by a single individual or group.” The Lae Vieux argue that the court misapplied its own definition because the court considered only “possession” and “use” and ignored the term “control.” The district court did not determine whether the Tribes controlled casino gambling in Michigan because, once again, *373the district court definition is in the disjunctive. It is “possession, control or use,” and there is, therefore, no need to show all three elements to prove exclusivity. As Governor Engler points out, it makes no sense to read the consent decree as conditioning the Tribes’ payments on their control of casino gambling because they never controlled it. If the Tribes controlled casino gambling, they would never surrender their exclusivity or pay money to the State — making much of the consent judgment surplusage. To the extent any entity controls casino gambling it is the state legislature.
At this time, no one but the Tribes has the right to operate electronic games of chance. The Tribes can point to no one who is infringing their right. They will continue to have an exclusive right until the Michigan Gaming Control Board grants casino licenses to non-Tribal groups. The district court was correct in holding that the Tribes maintain their exclusive right until another group receives a casino license.
B. Admissibility of Extrinsic Evidence
In finding the plain meaning of “exclusive right to operate,” the district court also determined that it did not need to consider extrinsic evidence. The determination of whether a contract is ambiguous, thereby making extrinsic evidence admissible for interpretive purposes, is a question of law and therefore subject to de novo review. Wulf v. Quantum Chem. Corp., 26 F.3d 1368, 1376 (6th Cir.1994). A court “does not have the right to make a different contract for the parties or to look to extrinsic testimony to determine their intent when the words used by them are clear and unambiguous and have a definite meaning." Michigan Chandelier, 297 N.W. at 67. “A contract is ambiguous if ‘its words may reasonably be understood in different ways.’” UAW-GM Human Resource Ctr. v. KSL Recreation Corp., 228 Mich.App. 486, 679 N.W.2d 411 (1998) (quotingRaska v. Farm Bureau Ins. Co., 412 Mich. 365, 314 N.W.2d 440 (1982)). The Lae Vieux argue that if the district court did not adopt their interpretation of the phrase “exclusive right to operate,” then the phrase must be ambiguous. The Lac Vieux contend that the variance between their definition and the district court’s definition creates sufficient ambiguity to merit the admission of extrinsic evidence. Simply claiming the contract is ambiguous does not, however, make it so. The district court was correct in determining that the terms of the consent judgment were unambiguous.
Even if the district court had found the terms of the contract ambiguous, it is not clear how the admission of the Lae Vieux’s extrinsic evidence would have aided the process of contract interpretation. Among the extrinsic evidence the Lac Vieux propose to offer are an affidavit and testimony from an accountant, an excerpt from a brief filed by Governor Engler in a state lawsuit, and affidavits from two bankers. The accountant testified about the economic damage the Tribes would suffer from “Proposal E.” The Lac Vieux contend that the excerpt from the Engler brief shows that Governor Engler regarded the Tribes’ obligation to pay as terminated merely with the passage of legislation allowing gambling. The bankers testified to the difficulty the Tribes would experience in getting loans in light of “Proposal E.” None of the evidence relates to the formation of the contract. For extrinsic evidence to be relevant, it must relate to the formation of the contract. See, e.g., Sawyer, 690 F.2d at 593 (noting that under Michigan law extrinsic evidence may be admitted in absence of facial ambiguity in order to examine circumstances surrounding formation of contract); Gary Boat Club, Inc. v. Oselka, 31 Mich.App. 465, 188 N.W.2d 127, 130-31 (1971) (stating that with ambiguous contract “trial court should consider the language employed, the subject matter, and the surrounding circumstances under which the parties entered into the agreement; it should effectuate the intent of the parties when the agreement was consummated”); Genesee Merchants Bank & Trust Co. v. Sefa, 23 Mich.App. 423, 178 N.W.2d 826, 828-29 (1970) (“In determining the intent of the parties where the agreed language proves ambiguous, surrounding facts and circumstances may be considered for the purpose of interpretation, and statements of the parties contemporaneous with the making of the agreement are admissible as evidence.”). Post-hoc interpretations of this nature are *374irrelevant to the parties’ intent at the time of the agreement.
C. Motion to Alter or Amend Judgment Under Rule 59(e)
The Lac Vieux put forward several arguments in their motion to alter or amend. The district court denied the motion, determining that the arguments were either previously rejected or untimely raised. A denial of a motion to alter or amend a judgment under Rule 59(e) is typically reviewed for abuse of discretion. Keweenaw Bay Indian Community v. United States, 136 F.3d 469, 474 (6th Cir.1998). On appeal, the Lac Vieux are forwarding one argument — that “the District Court erred ... by failing to recognize that Proposal E transferred control over the exclusive right to operate electronic games of chance from the Tribes to the [Michigan Gaming Control Board] and the City of Detroit.” Putting aside the issue of the correctness of the argument, the Lac Vieux raised it tardily. The Lae Vieux acknowledge that they did not make the argument prior to filing their Rule 59(e) motion, but they argue that they did not raise the issue because they never thought the district court would determine otherwise. A motion under Rule 59(e) is not an opportunity to re-argue a case. FDIC v. World Univ. Inc., 978 F.2d 10, 16 (1st Cir.1992) (“Rule 59(e) motions are aimed at re consideration, not initial consideration. Thus, parties should not use them to raise arguments which could, and should, have been made before judgment issued. Motions under Rule 59(e) must either clearly establish a manifest error of law or must present newly discovered evidence.” (internal citations and quotation marks omitted) (emphasis in original)). The Lac Vieux did neither in their Rule 59(e) motion. Because the Lac Vieux could have, but did not, raise their argument before the district court ruled on the motion to compel compliance, the argument is barred.
III.
The district court’s decisions are affirmed.
. The Lac Vieux filed a separate case against the Michigan Gaming Control Board, and others, alleging that "Proposal E” was an impairment of contract in violation of the Michigan and United States Constitutions. The district court granted the defendants' motion for summary judgment. Lac Vieux Desert Band of Lake Superior Chippewa Indians v. Michigan Gaming Control Bd., No. 2:97-CV-67 (W.D.Mich. Oct. 31, 1997).