MESKILL, Circuit Judge:

This appeal arises out of a trademark dispute involving the use of the name "Orient Express" in connection with the sale of various articles of retail merchandise. The plaintiffs-appellants appeal from a judgment and orders entered in the United States District Court for the Southern District of New York, Motley, J., cancelling nineteen of their registered trademarks and dismissing their claims of trademark and service mark infringement and unfair competition. They also challenge Judge Motley's award of attorneys' fees to the defendants-appellees.

Following oral argument, we remanded this case to the district court for clarification as to the standard of proof employed in the appellees' counterclaim seeking cancellation of appellants' registered marks. In response to that remand, Judge Motley amended her opinion and an accompanying order. For the following reasons, we now affirm.

BACKGROUND

Appellant Robert M. Lee is the sole owner of appellant Hunting World, Inc., a merchandising business that operates through a New York City retail outlet and an annual mail order catalog. Based on Lee's self-styled reputation as a hunter and outdoorsman, the company specializes in clothing, luggage and giftware related to sporting and safaris. Lee also is majority owner of appellant Orient Express Trading Co., which was set up to import goods from China for sale in this country.

In 1975, Lee applied to the United States Patent and Trademark Office (PTO) for registration of a trademark using the name "Orient Express." The words were in an Oriental-style script and appeared with a picture of a train. Lee said in his application that he had first used the mark on December 31, 1974, and that he would continue to use it in connection with the sale of five classes of articles — cosmetics, jewelry, luggage, giftware and clothing. The so-called "Chinese train logo" was registered by the PTO in 1976. During this same time period, Lee also sought registration of a service mark that read "Canton Fair at Orient Express," for use at the New York Hunting World store. The PTO registered the mark in 1978.

In 1977, defendant-appellee SeaCo, Inc. purchased two of the original cars from the famous Orient Express train that traveled between Paris and Istanbul from 1883 until 1977. SeaCo planned to use the cars as part of a commercial re-creation of the train that would travel between London and Venice. SeaCo also planned to market a travel-related line of products called "Collection Orient Express." The company conducted a trademark search and discovered the American registration of Lee's Chinese train logo. SeaCo officials subsequently wrote to Lee to see if he would either sell any rights he might have in the name Orient Express or enter into a joint venture with SeaCo. Thereafter, the parties held three meetings in New York in late 1981.

After those meetings, and after two visits to the New York Hunting World store, SeaCo officials became skeptical about Lee's financial resources and about the scope of his use of the Orient Express marks. They became convinced that Lee had used the Chinese train logo infrequently and that it only had been used in connection with a narrow range of imported Chinese items and not in connection with any goods comparable to those that SeaCo planned to sell. SeaCo thereafter notified Lee that it was no longer interested in a joint business relationship. SeaCo also changed the name of its product line to "Collection Venice Simplon-Orient-Express." The name of the train, which had its inaugural run in May 1982, was similarly changed, to reflect the fact that it would travel through the Simplon tunnel in Switzerland.

Meanwhile, on the day before the parties' third 1981 meeting, as it apparently was becoming clear that a joint venture was unlikely, Lee filed seventeen PTO applications for the registration of trademarks using the name Orient Express in connection with the sale of a wide variety of goods. He again claimed to have first used the name in connection with those goods on December 31, 1974. The PTO registered those marks at various times between 1983 and 1985. Furthermore, in 1982, Lee filed affidavits with the PTO pursuant to sections 8 and 15 of the Lanham Act, 15 U.S.C. §§ 1058, 1065 (1982), claiming that the original Chinese train logo had been in continuous use for five years since the date of its registration, in connection with all the goods listed in his 1975 application. In 1984, he filed similar affidavits concerning the Canton Fair service mark. The effect of these affidavits was to establish a twenty year period of duration on the marks, see 15 U.S.C. § 1058, and to establish their incontestability, see 15 U.S.C. § 1065.

Subsequently, Lee and his two companies instituted this action against SeaCo and its marketing affiliates. Appellants claimed: (1) infringement of registered trademarks, in violation of 15 U.S.C. § 1114 (1982); (2) unfair competition and false designations of origin, in violation of section 43(a) of the Lanham Act, 15 U.S.C. § 1125(a) (1982); (3) common law unfair competition and trademark infringement; and (4) unfair competition in violation of N.Y.Gen.Bus. Law § 368-d (McKinney 1984). The appellees counterclaimed, seeking a declaratory judgment of non-infringement and cancellation of all of appellants' registered trademarks, pursuant to 15 U.S.C. §§ 1064(c), 1115 (1982 Supp. IV 1986). After a nine day bench trial, Judge Motley entered an order cancelling nineteen of appellants' registered trademarks for fraudulent statements made to the PTO. Thus, appellants' claims for infringement of the registered marks were dismissed. As to all other claims, Judge Motley entered judgment for the appellees. Finally, she awarded attorneys' fees to the appellees pursuant to 15 U.S.C. § 1117 (1982 Supp. IV 1986).

DISCUSSION

First, we find no error in Judge Motley's decision to cancel appellants' nineteen registered marks. Although Lee had obtained incontestable status for two of the marks, such marks are still subject to the defense of fraud. See 15 U.S.C. § 1115(b)(1). Moreover, any other registered mark is subject to "any legal or equitable defense or defect which might have been asserted if such mark had not been registered." 15 U.S.C. § 1115(a). A party seeking cancellation of a registered trademark on grounds of fraud must demonstrate the alleged fraud by "clear and convincing evidence." Beer Nuts, Inc. v. Clover Club Foods Co., 711 F.2d 934, 942 (10th Cir. 1983); Money Store v. Harriscorp Finance, Inc., 689 F.2d 666, 670 (7th Cir. 1982); Knorr-Nahrmittel Aktiengesellschaft v. Havland International, Inc., 206 U.S.P.Q. 827, 834 (T.T.A.B. 1980). Cf. Jaskiewicz v. Mossinghoff, 822 F.2d 1053, 1058 (Fed. Cir. 1987) (applying same standard to attorney fraud before PTO). The allegedly fraudulent statements may not be the product of mere error or inadvertence, but must indicate a "deliberate attempt to mislead the [PTO]." Money Store, 689 F.2d at 670. Moreover, the knowing misstatement must have been with respect to a material fact — one that would have affected the PTO's action on the applications. See Citibank, N.A. v. Citibanc Group, Inc., 724 F.2d 1540, 1544 (11th Cir. 1984); Rick v. Buchansky, 609 F.Supp. 1522, 1536-37 (S.D.N.Y.), appeal dismissed, 770 F.2d 157 (2d Cir. 1985).

Based on the evidence before her, Judge Motley concluded that appellants' dates of first use on the trademark applications and the subsequent affidavits filed pursuant to sections 8 and 15 of the Lanham Act were "greatly exaggerated." She also found that Lee had been "disingenuous" in filing the seventeen 1981 applications and the subsequent affidavits, intentionally attempting to capitalize on SeaCo's anticipated business profits and preparing for "vexatious" litigation. Finally, she concluded that appellants' claimed scope of use of the registered marks had been exaggerated and that the continuous use claimed in the section 8 and 15 affidavits had, in fact, been "sporadic, at best." In Judge Motley's view, appellants' statements to the PTO had not reflected the "uncompromising candor" that is required of applicants before that agency. See T.A.D. Avanti, Inc. v. Phone-Mate, Inc., 199 U.S. P.Q. 648, 655-56 (C.D.Cal. 1978).

In the conclusion of Judge Motley's original opinion, dated January 28, 1987, she said that SeaCo and its affiliates had shown appellants' fraud "by a preponderance of the credible evidence." However, much language in the body of her opinion suggested that she had, in fact, applied the correct "clear and convincing" standard. By order dated December 15, 1987, 838 F.2d 1203, we remanded this case for clarification. Judge Motley subsequently amended her opinion and order, holding that the appellees had shown fraud by clear and convincing evidence and that the appellants had thereafter failed to establish their remaining claims by a preponderance of the evidence. We find no clear error in the district court's factual findings and no error in its legal conclusions on this matter and we therefore affirm the cancellation of appellants' registered marks and the dismissal of the claims for infringement of those marks.

Even if appellants' registered marks are cancelled, however, the use of the name Orient Express could still be protected from unfair competition under section 43(a) of the Lanham Act, 15 U.S.C. § 1125(a). See Centaur Communications, Ltd. v. A/S/M Communications, Inc., 830 F.2d 1217, 1220 (2d Cir. 1987); Thompson Medical Co. v. Pfizer Inc., 753 F.2d 208, 212 n. 5 (2d Cir. 1985). Assuming that appellants would have a protectible interest under the trademark laws, see Centaur, 830 F.2d at 1220-25, they would not be entitled to relief under section 43(a) unless there was a likelihood of confusion among an appreciable number of consumers about the source of the products at issue. Id. at 1225. See also Charles of the Ritz Group Ltd. v. Quality King Distributors, Inc., 832 F.2d 1317, 1321 (2d Cir. 1987); Mushroom Makers, Inc. v. R.G. Barry Corp., 580 F.2d 44, 47 (2d Cir. 1978) (per curiam), cert. denied, 439 U.S. 1116, 99 S.Ct. 1022, 59 L.Ed.2d 75 (1979).

As we have frequently noted, any examination of a "likelihood of confusion" in such cases should be guided by the multifactor balancing test first set out by Judge Friendly in Polaroid Corp. v. Polarad Electronics Corp., 287 F.2d 492 (2d Cir.), cert. denied, 368 U.S. 820, 82 S.Ct. 36, 7 L.Ed.2d 25 (1961). The eight factors to be balanced are:

(1) the strength of the mark, (2) the degree of similarity between the two marks, (3) the proximity of the products, (4) the likelihood that the senior user will bridge the gap, (5) evidence of actual confusion, (6) the junior user's bad faith vel non in adopting the mark, (7) the quality of the junior user's product, and, finally, (8) the sophistication of the relevant consumer group.

Centaur, 830 F.2d at 1225 (citing Polaroid, 287 F.2d at 495). See also Charles of the Ritz, 832 F.2d at 1321. These factors, however, are not all-inclusive elements of every successful cause of action. Rather, they are "variables," see Polaroid, 287 F.2d at 495, that are to be taken into account in balancing the equities in any given case. "[T]he court may have to take still other variables into account." Id. Although we have noted the importance of employing the "right formulas" in trademark cases, see Centaur, 830 F.2d at 1219, we do not mean to suggest that district courts must slavishly recite the litany of all eight Polaroid factors in each and every case. A district court need only consider sufficient factors to reach the ultimate conclusion as to whether or not there is a likelihood of confusion. On appeal, we review that conclusion de novo. Charles of the Ritz, 832 F.2d at 1321; Plus Products v. Plus Discount Foods, Inc., 722 F.2d 999, 1004-05 (2d Cir. 1983).

Here, Judge Motley expressly considered four of the Polaroid factors. First, she found that there was no evidence of actual confusion between Lee's and SeaCo's products, a conclusion that may often weigh against a finding of a likelihood of confusion. See McGregor-Doniger, Inc. v. Drizzle Inc., 599 F.2d 1126, 1136 (2d Cir. 1979). Second, Judge Motley found that the purchasers of both product lines at issue here were "sophisticated and knowledgeable" consumers. Third, she concluded that appellants' marks were not particularly strong or distinctive. Rather, she found the name Orient Express to be "in common use." Fourth, Judge Motley concluded that SeaCo had adopted its marks and trade names in good faith. In fact, she concluded that it was Lee who had acted in bad faith, attempting to capitalize on SeaCo's use of the name. All these conclusions militate heavily against a finding of a likelihood of confusion to support a successful section 43(a) claim. Moreover, we believe that Judge Motley's factual findings would support an additional conclusion that the lack of any real proximity between the product lines at issue here would significantly reduce any consumer confusion. Judge Motley found that "until [SeaCo's] luxury train service was inaugurated . . ., the name Orient Express was used [by Lee] only in connection with merchandise imported from China." J.App. at 27. These were not the types of products that would have been confused with the travel-related merchandise offered with the "Collection Venice Simplon-Orient-Express." In sum, we agree with the district court's decision to dismiss appellants' section 43(a) claim. For the same reasons, we agree that appellants' common law and state law unfair competition claims also should have been dismissed. See Allied maintenance Corp. v. Allied Mechanical Trades, Inc., 42 N.Y.2d 538, 543-46, 399 N.Y.S.2d 628, 631-33, 369 N.E.2d 1162, 1165-67 (1977).

Finally, we decline to disturb Judge Motley's award of attorneys' fees pursuant to 15 U.S.C. § 1117. We have held that such an award is justified if the "losing party prosecuted or defended [the] claim in bad faith." Universal City Studios, Inc. v. Nintendo Co., 797 F.2d 70, 77 (2d Cir.), cert. denied, ___ U.S. ___, 107 S.Ct. 578, 93 L.Ed.2d 581 (1986). Given Judge Motley's conclusion that Lee filed the seventeen 1981 trademark applications and the section 8 and 15 affidavits with the intent to capitalize on SeaCo's profits and to instigate "vexatious" litigation, we believe that she was within her discretion in awarding fees.

CONCLUSION

For all of the foregoing reasons, the judgment and orders of the district court are affirmed.