We have consolidated the appeals in two employment discrimination cases that raise the same two issues: Are arbitration clauses in collective bargaining agreements other than in the maritime, railroad, and other transportation industries subject to the Federal Arbitration Act, 9 U.S.C. §§ 1 et seq.1 And can a collective bargaining agreement compel an employee to arbitrate a claim that he may have under one of the federal statutes, such as Title VII of the Civil Rights Act of 1964, the Age Discrimination in Employment Act, or the Americans with Disabilities Act, that confer litigable rights on employees? The first issue is critical to our jurisdiction of these appeals; the second is the issue on the merits — which, of course, we can reach only if we satisfy ourselves that we have jurisdiction.
Each of the two plaintiffs is a former employee of one of the defendants, and each *356was discharged by his employer in alleged violation of federal law. Pryner is black and complains that he was discharged in violation of Title VII and of 42 U.S.C. § 1981 because of his race and also because he had complained about Tractor Supply’s previous racial discrimination against him before he was fired; he also has a claim under the Americans with Disabilities Act. Tractor Supply has counterclaimed against Pryner, but the counterclaim is not involved in this appeal. Sobierajski complains of having been discharged because of his age (58 at the time of the discharge), in violation of the Age Discrimination in Employment Act, and because of a disability, in violation of the ADA.
Both plaintiffs were employed under collective bargaining agreements. The agreement between defendant Thoesen and Sobierajski’s union, the automobile mechanics union, provided that “in accordance with applicable Federal and State law, neither the company nor the Union will discriminate against employees covered by this collective bargaining agreement in regard to any terms or conditions of employment on the basis of race, creed, religion, national origin, sex or age.” The agreement between Tractor Supply and Pryner’s union, the teamsters, contains a similar provision but does not refer to federal or state law and adds to the prohibition against discrimination in regard to any “terms or conditions of employment” a prohibition against discrimination “with respect to hiring [or] compensation.” Both agreements authorize the employer to impose discipline on an employee, up to and including discharge, for “just cause.” And both contain a clause that creates a grievance procedure, making the union the employee’s griever and culminating in arbitration if the matter is not resolved in the earlier stages of the procedure, for disputes involving “interpretation or application” of the agreement.
Both Sobierajski and Pryner invoked the grievance procedure in their respective collective bargaining agreements. Pryner’s union struck out at the earlier stages of the grievance procedure, and has demanded arbitration. Sobierajski’s grievance, however, was abandoned, though whether by his own actions or those of the union is unclear. Pryner and Sobierajski then filed these discrimination suits, seeking damages (including punitive damages in the case of Pryner), attorneys’ fees, and reinstatement. The defendants moved to stay the suits pending arbitration of the plaintiffs’ claims. The motions were denied (Pryner’s in Pryner v. Tractor Supply Co., 927 F.Supp. 1140 (S.D.Ind.1996)), and the defendants have appealed from these denials. Sobierajski may be barred from seeking arbitration, having failed to exhaust his remedies. In that event, the stay would be pointless. But failure to exhaust contractual remedies would presumably bar his suit if, as the employers argue, the arbitration clauses in these collective bargaining agreements preclude suits to enforce statutory rights; otherwise the worker could circumvent the clause by disabling himself from seeking arbitration. So Thoesen also filed a motion in the district court to dismiss Sobierajski’s suit, and it was also denied. The question whether failure to exhaust the prearbitration grievance procedure would be a bar to suing is independent of the effect of the arbitration clause and was not raised in Thoesen’s motion. We express no view on it.
The issue of our appellate jurisdiction is whether these two appeals are authorized by the Federal Arbitration Act. The denial of a stay of proceedings before the court asked to grant the stay is not a final decision within the meaning of 28 U.S.C. § 1291, but when it is the denial of a stay of those proceedings pending arbitration it nevertheless is appealable immediately, at least (and probably at most, see Central States, Southeast & Southwest Areas Pension Fund v. Central Cartage Co., 84 F.3d 988, 990-92 (7th Cir.1996)) if the motion is filed under the arbitration act, which expressly authorizes such appeals. 9 U.S.C. § 16(a)(1)(A). See IDS Life Ins. Co. v. SunAmerica, Inc., 103 F.3d 524, 526 (7th Cir.1996). The act covers maritime transactions and transactions involving interstate commerce. 9 U.S.C. § 2. But “contracts of employment of seamen, railroad employees, or any other class of workers engaged in foreign or interstate commerce” are specifically excluded by section 1.
*357If “engaged in foreign or interstate commerce” is given its usual modern legal meaning, virtually all employment contracts are within the exclusion, including the two collective bargaining agreements upon which the defendants based their motions for a stay. Some courts so read it. Pritzker v. Merrill Lynch, Pierce, Fenner & Smith, Inc., 7 F.3d 1110, 1120 (3d Cir.1993); Willis v. Dean Witter Reynolds, Inc., 948 F.2d 305, 310-12 (6th Cir.1991); Herring v. Delta Air Lines, Inc., 894 F.2d 1020, 1023 (9th Cir.1989); United Food & Commercial Workers v. Safeway Stores, Inc., 889 F.2d 940, 943-44 (10th Cir. 1989). Others confine the exclusion to collective bargaining agreements, as in United Electrical, Radio & Machine Workers v. Miller Metal Products, Inc., 215 F.2d 221, 224 (4th Cir.1954); see also Domino Sugar Corp. v. Sugar Workers Local Union 392, 10 F.3d 1064, 1067 (4th Cir.1993). Others deem the arbitration act superseded with regard to such agreements by section 301 of the Taft-Hartley Act, 29 U.S.C. § 185 — as did we in Martin v. Youngstown Sheet & Tube Co., 911 F.2d 1239, 1244 (7th Cir.1990), and the cases cited there. But in International Union of Operating Engineers v. Murphy Co., 82 F.3d 185, 188-89 (7th Cir.1996), we assumed the opposite (that the arbitration act is not superseded by section 301), without citing Martin. And in Chicago Typographical Union v. Chicago Sun-Times, 935 F.2d 1501, 1504 (7th Cir.1991), decided between Martin and International Union of Operating Engineers, we had described the issue of supersession as an open one.
A footnote in United Paperworkers Int’l Union v. Misco, Inc., 484 U.S. 29, 40 n. 9, 108 S.Ct. 364, 372 n. 9, 98 L.Ed.2d 286 (1987), says that the arbitration act does not apply to labor arbitration. But the footnote gives no reason; is inconsistent with the Supreme Court’s subsequent decision in Gilmer v. Interstate/Johnson Lane Corp., 500 U.S. 20, 25 n. 2, 111 S.Ct. 1647, 1651, 114 L.Ed.2d 26 (1991), which expressly left the question open; and goes on to cite with apparent approval our decision in Pietro Scalzitti Co. v. International Union of Operating Engineers, 351 F.2d 576, 579-80 (7th Cir.1965), which, limiting the exclusion of employment contracts in section 1 to workers engaged in the physical movement of goods in interstate or foreign commerce, holds that the arbitration act applies to labor arbitration in all industries except transportation. See also Briggs & Stratton Corp. v. Local 232, 36 F.3d 712, 714-15 (7th Cir.1994); Miller Brewing Co. v. Brewery Workers Local Union No. 9, 739 F.2d 1159, 1162 (7th Cir.1984); Cole v. Burns Int’l Security Services, 105 F.3d 1465, 1469 (D.C.Cir.1997); Asplundh Tree Expert Co. v. Bates, 71 F.3d 592, 596-602 (6th Cir. 1995); Erving v. Virginia Squires Basketball Club, 468 F.2d 1064, 1069 (2d Cir.1972). This is acknowledged to be the “prevailing view” of the scope of the exclusion by an opponent. Matthew W. Finkin, “ ‘Workers’ Contracts’ Under the United States Arbitration Act: An Essay in Historical Clarification,” 17 Berkeley J. Employment & Labor L. 282, 290 (1996).
But aren’t we on both sides of the issue? We have cases like Martin that hold that section 301 supersedes the Federal Arbitration Act, a case that assumes it does not (International Union of Operating Engineers), and cases such as Briggs & Stratton that imply not by confining the exclusion in section 1 to transportation workers and saying nothing about supersession by section 301. This internal conflict is, fortunately, illusory. The cases that say that section 301 supersedes the arbitration act for cases within section 301’s domain hold (all but Martin itself and Cleveland v. Porca Co., 38 F.3d 289, 296 n. 5 (7th Cir.1994)) only that particular provisions of the act, such as its limitations periods (three months for moving to vacate an award, one year for enforcing it), are superseded, and do not mention the provision allowing interlocutory appeals. We shall see that these holdings are consistent with the act’s being generally applicable to collective bargaining agreements provided that the workers covered by the agreement are not engaged in transportation and federal common law created under the aegis of section 301 is used to determine any substantive issues that arise in proceedings to vacate or enforce the arbitration award. See, e.g., Glass, Molders, Pottery, Plastics & Allied Workers Int’l Union v. Excelsior Foundry Co., 56 F.3d 844, 848 (7th Cir.1995); Miller *358Brewing Co. v. Brewery Workers Local Union No. 9, supra, 739 F.2d at 1162.
Professor Finkin argues that the prevailing view, which limits the exclusion in section 1 to employment contracts in transportation, is wrong. His review of the legislative history (the arbitration act was passed in 1925) has persuaded him that Congress’s intention was to exclude all employment contracts. Yet, as he acknowledges, the impetus for the exclusion came entirely from the seafarers union, concerned that arbitrators would be less favorably inclined toward seamen’s claims than judges were. Judges favored such claims, the union thought, in part because of a tradition that seamen were “wards in admiralty,” in part because of peculiarities of maritime law that would make it easy to slip an arbitration clause into a maritime employment contract without the seaman’s noticing it, and in part because the maritime employment relation was already heavily regulated by federal law. It was soon noticed that the railroad industry’s labor relations were also heavily regulated — by a statute (the Railway Labor Act) that included provisions for compulsory arbitration of many disputes. Motor carriers were not yet comprehensively regulated, but it may have seemed (and was) only a matter of time before they would be: hence the expansion of the exclusion from seamen to railroad to other transportation workers. It seems to us, as it did to the Third Circuit in Tenney Engineering, Inc. v. United Electrical, Radio & Machine Workers, 207 F.2d 450, 452-53 (3d Cir.1953), that this history supports rather than undermines limiting “engaged in foreign or interstate commerce” to transportation. To impress the modern meaning on the quoted term, moreover, would both make the reference to seamen and railroad workers superfluous and give the exclusion a breathtaking scope. It would mean that an arbitration clause in an employment contract between a giant multinational corporation and its chief executive officer would, though plausibly “involving” interstate commerce within the meaning of section 2, not be enforceable in federal court. Such a result would not answer to any concern expressed to or by Congress in the debates leading up to the passage of the arbitration act.
Even if as we believe the collective bargaining agreements in this ease are not excluded from the act’s coverage by section l’s reference to employment agreements, they might be excluded, as some courts believe, by section 301 of the Taft-Hartley Act. Section 301(a), which makes collective bargaining agreements enforceable in the federal courts, was interpreted in Textile Workers Union v. Lincoln Mills, 353 U.S. 448, 77 S.Ct. 912, 1 L.Ed.2d 972 (1957), to require that principles of federal common law be developed and used to interpret such agreements. There is no reference in section 301 to arbitration, which becomes an issue in a section 301 case only when a collective bargaining agreement happens to contain (as most such agreements do) an arbitration clause. The question is then whether, despite its silence about arbitration, section 301 repealed the Federal Arbitration Act with respect to collective bargaining agreements that would otherwise be within the act’s scope. This seems hardly likely, quite apart from the venerable but contested principle that repeals by implication are disfavored and so are to be avoided unless there is no way to reconcile the two statutes. TVA v. Hill, 437 U.S. 153, 189-90, 98 S.Ct. 2279, 2299-2300, 57 L.Ed.2d 117 (1978); Vaughn v. Sullivan, 83 F.3d 907, 913 (7th Cir.1996); Reich v. Great Lakes Indian Fish & Wildlife Comm’n, 4 F.3d 490, 493 (7th Cir.1993); Blanco v. United States, 775 F.2d 53, 61 (2d Cir.1985) (Friendly, J.).
The cases that decline to apply the arbitration act’s three-month statute of limitations to suits under section 301 to set arbitration awards aside (e.g., Sullivan v. Gilchrist, 87 F.3d 867, 870 (7th Cir.1996); Chauffeurs, Teamsters, Warehousemen & Helpers Local Union v. Jefferson Trucking Co., 628 F.2d 1023, 1025-26 (7th Cir.1980); Harry Hoffman Printing, Inc. v. Graphic Communications, Int’l Union, 912 F.2d 608, 610-13 (2d Cir.1990); Champion Int’l Corp. v. United Paperworkers Int’l Union, 779 F.2d 328, 331 (6th Cir.1985); cf. United Auto. Workers v. Hoosier Cardinal Corp., 383 U.S. 696, 704-05, 86 S.Ct. 1107, 1112-13, 16 L.Ed.2d 192 (1966)) stand for nothing more than that while the existence of two statutes both conferring federal jurisdiction over the same *359claim in the same case is not such an anomaly as would justify holding that one or the other was exclusive, if there are inconsistencies the courts are going to have to make a choice. See Lander Co. v. MMP Investments, Inc., 107 F.3d 476, 481 (7th Cir. 1997). Jurisdictional redundancies are commonplace: a claim under federal law against a citizen of another state is within the jurisdiction of the federal courts under both 28 U.S.C. § 1331 and 28 U.S.C. § 1332, provided the claim is for more than $75,000. It is true, as mentioned in Lander and emphasized in Harry Hoffman Printing, Inc. v. Graphic Communications, Int’l Union, supra, 912 F.2d at 611-12, that the arbitration act reaches only disputes that are within the jurisdiction of the federal courts under some other statute, such as the diversity statute or, as here, section 301 of the Taft-Hartley Act. But all that this means, so far as the jurisdictional issue in the present cases is concerned, is that whenever the arbitration act is properly invoked there are two jurisdictional statutes in play. That is not an argument for refusing to apply one of them. And if the arbitration act is better regarded not as a source of jurisdiction at all but merely as a prescription of procedures for a class of eases otherwise within federal jurisdiction, this would actually strengthen the argument against deeming the act superseded by the presence of a “truly” jurisdictional statute such as section 301.
The arbitration act covers almost the whole field of arbitrable disputes, only a subset of which (and a small one in 1925) grow out of collective bargaining agreements. The act’s limitations periods were not designed with such agreements in mind, and may not suit them. When the question arose what the statute of limitations should be in suits under section 301 to challenge arbitration awards based on such agreements (the statute contains no limitations period), the courts cast about for analogous claims from which to “borrow” a limitations period. In so doing they did not hold that section 301 had repealed the arbitration act so far as collective bargaining agreements were concerned; and when in 1988, long after section 301 had been enacted, Congress amended the arbitration act to make denials of stays pending arbitration expressly appealable despite the absence of finality, it did not carve an exception for arbitration pursuant to collective bargaining agreements. Judicial Improvements and Access to Justice Act, Pub.L. No. 100-702, 102 Stat. 4671 (1988). That may have been an oversight, but we suspect not; for no reason is suggested why, if immediate appeals of denials of such stays are a good thing with respect to the arbitration of other types of dispute, they are a bad thing when the dispute is a labor dispute. If anything, the public interest in resolving labor disputes by arbitration, and therefore in allowing erroneous refusals to defer to arbitration to be corrected promptly, is greater than in an ordinary commercial arbitration.
The big thing that section 301 did (besides creating federal jurisdiction over suits to enforce collective bargaining agreements), so far as labor arbitration is concerned, was to ordain the creation of a body of federal common law to govern disputes arising out of such arbitration. Included in that body are limitations periods, which by analogy to the substance-procedure distinction worked out in the wake of the Erie decision fall on the substantive side of the divide. Guaranty Trust Co. v. York, 326 U.S. 99, 109-10, 65 S.Ct. 1464, 1469-70, 89 L.Ed. 2079 (1945). The question whether a particular type of interlocutory order is immediately appealable, however, is a quintessentially procedural question to which the Federal Arbitration Act provides an answer that creates no tension with anything in either section 301 or the common law of collective bargaining agreements that has evolved under it; that in fact effectuates the policy of section 301, which is to encourage the resolution of labor disputes by means other than industrial warfare — and arbitration is prominent among those means.
Our resolution of the jurisdictional issue can be criticized as creating an arbitrary difference between the appeal rights of parties to employment contracts, including collective bargaining agreements, in nontransportation industries and the corresponding appeal rights of, parties to collective bargaining agreements and other employment con*360tracts in transportation. Parties to employment contracts in other industries can appeal from orders denying stays pending arbitration. Parties to employment contracts in the transportation industries cannot because, excluded from the Federal Arbitration Act by section 1, they cannot rely on section 16; and section 301 of the Taft-Hartley Act has no provision, corresponding to that section, authorizing interlocutory appeals. We need not consider the likely success of efforts to construct a right to take such an appeal from other statutes, such as 28 U.S.C. § 1292(a)(1), or from judge-made doctrines of appealability such as the collateral-order doctrine; let us assume for the sake of argument that they would fail. The resulting anomaly (different treatment of labor arbitration in transportation versus nontransportation industries) would be the unavoidable consequence of an interpretation of section 1 of the arbitration act that differentiates between employment contracts in the transportation and nontransportation sectors, and that is a distinction to which this court is committed.
Another anomaly concerns the specific statutes of limitations that the courts have borrowed for use in section 301 arbitration cases. Always these are statutes of limitations applicable to arbitration awards rather than anything specially to do with labor. See, e.g., Sullivan v. Gilchrist, supra, 87 F.3d at 870. We have even borrowed from arbitration statutes that are inapplicable to collective bargaining contracts! Stevens v. Gateway Transportation Co., 696 F.2d 500, 502 (7th Cir.1982); Hemmings v. Barian, 822 F.2d 688, 690. (7th Cir.1987). Why should a state statute of limitations for arbitration cases be thought more apt to section 301 than the statute of limitations in the Federal Arbitration Act? We cannot find an answer in the cases and are given additional pause by the decisions that say, Hygrade Operators, Inc. v. Local 333, 945 F.2d 18, 22 (2d Cir.1991), or more commonly assume, Local 232 v. Briggs & Stratton Corp., 837 F.2d 782, 788-89 (7th Cir.1988); Evans v. Einhorn, 855 F.2d 1245, 1246 (7th Cir.1988) (per curiam), that disputes growing out of labor arbitration arise under section 301, without mentioning the arbitration act. But American law is rarely as clear as one might like; and for the reasons that we explained earlier, we think the better view is that we do have jurisdiction of these appeals under section 16 of the arbitration act. So we may turn at last to the issue on the merits, which is whether a collective bargaining agreement can compel the arbitration of a federal anti-discrimination claim.
Two competing interests have to be considered. One is the interest in allowing unions and employers to establish a comprehensive regime for the adjustment of employment disputes; it argues for allowing the collective bargaining agreement to force all such disputes into the grievance and arbitration groove even when the dispute arises out of a claim that a worker’s statutory rights have been infringed. The other is the interest in the effective enforcement of rights designed for the protection of workers whom Congress has classified as belonging to vulnerable groups, generally and in these two cases minority groups — blacks, the disabled, and the aged (though whether persons 40 years old and older, the group protected by the age discrimination law, should be considered a vulnerable segment of the population may certainly be questioned as an original matter). That interest will be impaired if the right to bring suit in federal district court to enforce these rights is taken away from the workers.
Or will it? The employers in our two cases, supported by employer groups that have filed amicus curiae briefs, argue that the workers will actually-be better off if the employers’ position is adopted. Because the Equal Employment Opportunity Commission has an enormous backlog and limited resources for litigating, the vast majority of workers who have claims under any of the statutes that the Commission enforces have perforce to bring and finance their own lawsuits; they cannot rely on the Commission to do so for them. In contrast, a grievance is prosecuted by the worker’s collective bargaining representative (the union) at no cost to the worker. It is true that the plaintiffs in these cases do not wish to abandon their right to invoke the grievance machinery created by the collective bargaining agreements; *361they just want the option of pursuing judicial remedies on top of or in lieu of their arbitral remedies. But unless the arbitral remedy is exclusive, the company’s incentive to agree to negotiate an arbitration clause broad enough to encompass statutory violations will be reduced, as the only effect of such a clause will be to multiply the employee’s remedies.
The employers emphasize that these collective bargaining agreements do not take away any of the workers’ substantive statutory rights, but merely substitute an arbitral for a judicial proceeding as the means of vindicating the rights. Although the arbitrator’s award, whether or not confirmed, 4 Ian R. Macneil, Richard E. Speidel & Thomas J. Stipanowich, Federal Arbitration Law § 39.6 (1994); G. Richard Shell, “Res Judicata and Collateral Estoppel Effects of Commercial Arbitration,” 35 UCLA L.Rev. 623, 642 (1988), can — this is implicit in the notion of binding arbitration — be pleaded as res judicata in the worker’s federal district court suit, Rudell v. Comprehensive Accounting Corp., 802 F.2d 926, 930 (7th Cir.1986); Wolf v. Gruntal & Co., 45 F.3d 524, 528 (1st Cir.1995); Greenblatt v. Drexel Burnham Lambert, Inc., 763 F.2d 1352, 1360 (11th Cir.1985); American Law Institute, Restatement (Second) of Judgments § 84(1) (1982), subject to limitations unnecessary to dwell on here, id., §§ 84(2)-(4), this is true if the worker wins before the arbitrator as well as if he loses. So there is no curtailment of his substantive rights. The only difference is the forum; the defendants are right. The plaintiffs riposte that it isn’t clear that the collective bargaining agreements in our two cases are completely coextensive with the plaintiffs’ statutory rights; and it isn’t. But a court can properly stay a suit before it if any issue in the suit is arbitrable, even if some issues are not. The arbitration act says in fact that the court shall stay the suit, not just a piece of the suit, if the suit is “brought upon” an arbitrable issue, 9 U.S.C. § 3, though the cases, perhaps concerned lest the tail wag the dog, treat the question whether to stay the entire case as discretionary in eases involving both arbitrable and nonarbitrable issues. American Recovery Corp. v. Computerized Thermal Imaging, Inc., 96 F.3d 88, 96-97 (4th Cir.1996); McCarthy v. Azure, 22 F.3d 351, 361 (1st Cir.1994); Genesco, Inc. v. T. Kakiuchi & Co., 815 F.2d 840, 856 (2d Cir.1987); 2 Macneil, Speidel & Stipanowich, supra, § 23.3.1, pp. 23:12-:13. This interpretation was assumed in Dean Witter Reynolds Inc. v. Byrd, 470 U.S. 213, 218-19 and n. 5, 105 S.Ct. 1238, 1241-42 n. 5, 84 L.Ed.2d 158 (1985), though in the context of issues nonarbitrable as a matter of law, as distinct from those the parties may just not have agreed to arbitrate.
The defendants concede that to the extent that the rights conferred by the collective bargaining agreements, or the sanctions available to the arbitrators, fall short of fully vindicating the plaintiffs’ substantive and remedial statutory rights, the plaintiffs will be free to resume their suits after the arbitrators render their awards, having filed the suits within the statute of limitations. But the findings made' by the arbitrators might be entitled to collateral estoppel effect in the resumed suits. Benjamin v. Traffic Executive Ass’n Eastern Railroads, 869 F.2d 107, 110 (2d Cir.1989). We say “might” to emphasize the readiness of courts to deny that effect to arbitral awards when the arbitrator has failed to explain his findings adequately or to follow procedures likely to lead to reliable findings — “adequacy” and “reliability” being relative to the importance of the claim that the arbitrator’s findings are sought to be used to block. Dean Witter Reynolds Inc. v. Byrd, supra, 470 U.S. at 222-23, 105 S.Ct. at 1243-44; McDonald v. City of West Branch, 466 U.S. 284, 289, 104 S.Ct. 1799, 1802-03, 80 L.Ed.2d 302 (1984); Kulavic v. Chicago & Illinois Midland Ry., 1 F.3d 507, 513-15 (7th Cir.1993); Universal American Barge Corp. v. J-Chem, Inc., 946 F.2d 1131, 1137 (5th Cir.1991).
The honey-tongued assurances of the employers’ able counsel and their amici, unusually and perhaps opportunistically aligned with the unions’ interest in controlling their members’ access to remedies, do not persuade us that there is no genuine conflict between employer and employee interests in these cases. The plaintiffs’ rights under the collective bargaining agreements are not as extensive as their statutory rights, so that to *362obtain complete relief they may have to undergo two trials, one before the arbitrator and the other in the district court. And by being forced into binding arbitration they would be surrendering their right to trial by jury — a right that civil rights plaintiffs (or their lawyers) fought hard for and finally obtained in the 1991 amendments to Title VII, and that they also have under the age discrimination and disability acts.
Most important, the grievance and arbitration procedure can be invoked only by the union, and not by the worker. The worker has to pei'suade the union to prosecute his grievance and if it loses in the early stages of the grievance proceedings to submit the grievance to arbitration. Vaca v. Sipes, 386 U.S. 171, 190-91, 87 S.Ct. 903, 916-17, 17 L.Ed.2d 842 (1967); cf. Martin v. Youngstown Sheet & Tube Co., supra, 911 F.2d at 1244. The defendants point out that if the union arbitrarily refuses to prosecute a grievance, let alone refuses on racial or other invidious grounds to do so, the worker can bring a suit against the union for breach of its duty of fair representation of all members of the bargaining unit. DelCostello v. Teamsters, 462 U.S. 151, 164, 103 S.Ct. 2281, 2290-91, 76 L.Ed.2d 476 (1983); Garcia v. Zenith Electronics Corp., 58 F.3d 1171, 1176 (7th Cir.1995); Conn v. GATX Terminals Corp., 18 F.3d 417, 420 (7th Cir.1994); LaBuhn v. Bulkmatic Transport Co., 865 F.2d 119, 120 (7th Cir.1988); Trnka v. Local Union No. 688, 30 F.3d 60, 61 (7th Cir.1994). This raises the spectre of three suits to enforce a statutory right — the suit against the union to force it to grieve and if necessary arbitrate the grievance, the arbitration proceeding, and the resumed district court proceeding if the workers’ rights under the collective bargaining agreement are more limited than their statutory rights. In any event, the union has broad discretion as to whether or not to prosecute a grievance. It may take into account tactical and strategic factors such as its limited resources and consequent need to establish priorities, just as other “prosecutors” must do, as well as its desire to maintain harmonious relations among the workers and between them and the employer. Vaca v. Sipes, supra, 386 U.S. at 191-92, 87 S.Ct. at 917-18; Cole v. Burns Int'l Security Services, supra, 105 F.3d at 1479; Garcia v. Zenith Electronics Corp., supra, 58 F.3d at 1176; Kulavic v. Chicago & Illinois Midland Ry., supra, 1 F.3d at 515. Corresponding to this expansive and ill-defined discretion, the scope of judicial review of its exercise is deferential. See, e.g., Air Line Pilots Ass’n v. O’Neill, 499 U.S. 65, 78, 111 S.Ct. 1127, 1135-36, 113 L.Ed.2d 51 (1991); Trnka v. Local Union No. 688, supra, 30 F.3d at 61; Nida v. Plant Protection Ass’n National, 7 F.3d 522, 526 (6th Cir.1993). The result is that a worker who asks the union to grieve a statutory violation cannot have great confidence either that it will do so or that if it does not the courts will intervene and force it to do so. While the grievance machinery could in principle offer the worker a cheaper alternative to suing, it seems unlikely that the union would be any more willing to prosecute a marginal case than a lawyer asked to handle it on a contingent-fee basis. Indeed, the union might for strategic reasons decline to prosecute a claim that would have enough merit to enable the worker to retain a lawyer on a contingent-fee basis were the worker not bound to the union.
The essential conflict is between majority and minority rights. The collective bargaining agreement is the symbol and reality of a majoritarian conception of workers’ rights. An agreement negotiated by the union elected by a majority of the workers in the bargaining unit binds all the members of the unit, whether they are part of the majority or for that matter even members of the union entitled to vote for union leaders — they need not be. The statutory rights at issue in these two cases are rights given to members of minority groups because of concern about the mistreatment (of which there is a long history in the labor movement, see, e.g., Steele v. Louisville & Nashville R.R., 323 U.S. 192, 65 S.Ct. 226, 89 L.Ed. 173 (1944)) of minorities by majorities. We may assume that the union will not engage in actionable discrimination against minority workers. But we may not assume that it will be highly sensitive to their special interests, which are the interests protected by Title VII and the other discrimination statutes, and will seek to *363vindicate those interests with maximum vig- or. The employers’ position delivers the' enforcement of the rights of these minorities into the hands of the majority, and we do not think that this result is consistent with the policy of these statutes or justified by the abstract desirability of allowing unions and employers to cut their own deals. And we are given no reason to believe that the ability of unionized workers to enforce their statutory rights outside of the grievance machinery established by collective bargaining agreements is undermining labor relations.
The defendants cite provisions added in the early 1990s to Title VII, the Age Discrimination in Employment Act, and the Americans with Disabilities Act that “where appropriate and to the extent authorized by law, ... arbitration ... is encouraged to resolve disputes arising under” these laws. Pub.L. No. 102-166, § 118, 105 Stat. 1071, 1081 (1991); 42 U.S.C. § 12212. These provisions, a polite bow to the popularity of “alternative dispute resolution” and perhaps a mild sop to the judiciary, which has expressed alarm at Congress’s relentless expansion of the jurisdiction of the federal courts, encourage arbitration “where appropriate” — and if we are right it is not appropriate when it is not agreed to by the worker but instead is merely imposed by a collective bargaining agreement that he may have opposed. Nothing in the background of the amendments is inconsistent with this interpretation. It would be at least a mild paradox for Congress, having in another amendment that it made to Title VII in 1991 conferred a right to trial by jury for the first time, Pub.L. No. 102-166, § 102, § 1977A(c), 105 Stat. 1071, 1073 (1991), to have empowered unions, in those same amendments, to prevent workers from obtaining jury trials in these cases. We know that statutes, being products of compromise, frequently reflect inconsistent aims. But if that is the case here, we might expect' to find a hint of it in the statutory language or design, or in the legislative history, but we find none.
We are not holding that workers’ statutory rights are never arbitrable. They are arbitrable if the worker consents to have them arbitrated. If the worker brings suit, the employer suggests that their dispute be arbitrated, the worker agrees, and the collective bargaining agreement does not preclude such side agreements, there is nothing to prevent a binding arbitration. Austin v. Owens-Brockway Glass Container, Inc., 78 F.3d 875, 879 (4th Cir.1996); American Italian Pasta Co. v. Austin Co., 914 F.2d 1103, 1104 (8th Cir.1990); Gardner v. Shearson, Hammill & Co., 433 F.2d 367 (5th Cir.1970) (per curiam); Moran v. Paine, Webber, Jackson & Curtis, 389 F.2d 242, 246 (3d Cir.1968). All we are holding is that the union cannot consent for the employee by signing a collective bargaining agreement that consigns the enforcement of statutory rights to the union-controlled grievance and arbitration machinery created by the agreement.
Thus far we have analyzed the arbitrability of the disputes in these two cases without reference to the precedents that might be thought to govern the issue, even though the hundreds of pages of briefs are devoted to little else. That in itself is a sign that case law is unlikely to be dispositive of the issue. The' authority of a previously decided case is at its greatest when it is apparent that the case does not differ in any material particular from the case in which it is cited as authority; and to show the absence of a material difference usually is easy. When instead a lawyer tries to create a mosaic of case authority from a large number of previous eases all distinguishable from the present one, what purports to be an investigation of case law is really a quest for policies or principles that can be found in the previous cases and are thus legitimate factors to guide the decision of a new case. Those factors we have discussed. Both sides of these appeals also are able to cite a case or two that squarely support their position — cases that really do not differ in any material particular from the present one — but in doing so all they have done is identify a circuit split. Compare Austin v. Owens-Brockway Glass Container, Inc., supra, 78 F.3d at 885, with Varner v. National Super Markets, Inc., 94 F.3d 1209, 1213 (8th Cir.1996).
Each side has its favorite Supreme Court case that it has flogged mercilessly to yield *364the desired holding. Alexander v. Gardner-Denver Co., 415 U.S. 36, 94 S.Ct. 1011, 39 L.Ed.2d 147 (1974), arose out of a Title VII suit like Pryner’s brought by a black employee against his employer. The employee was a member of a bargaining unit represented by a union that had negotiated a collective bargaining agreement materially identical to that in Pryner’s case. His claim was pressed by the union in a grievance proceeding that eventuated in arbitration, and the arbitrator found that Alexander had been discharged for just cause, which in the employer’s view precluded a Title VII suit. The Supreme Court disagreed. It held that the arbitration of a contractual right not to be discriminated against does not preclude enforcement of a statutory right. This is not quite the same thing as holding that the worker does not have to go through the arbitration process. But as we mentioned earlier, the parties to our cases all assume that if Pryner and Sobierajski must go through arbitration they also must abide by its result, though one possible result is the arbitrators’ concluding that they lack authority under the collective bargaining agreements to render complete, or perhaps any, relief, in which event there would be room for further litigation.
By stressing, somewhat formalistically it might seem, the “distinctly separate nature” of contractual and statutory rights, 415 U.S. at 50, 94 S.Ct. at 1020-21, the Supreme Court sowed the possibility of distinguishing Sobierajski’s case. For his collective bargaining agreement mentions federal and state law and could therefore be deemed consent to the arbitrators’ enforcing statutory as well as contractual rights. The defendants note this, but put most of their weight on the Supreme Court’s subsequent decision in Gilmer v. Interstate/Johnson Lane Corp., supra. Gilmer, a stockbroker, signed a registration agreement with the New York Stock Exchange (that is, an agreement that permitted him to register as a broker with the exchange) in which he consented to the arbitration of any dispute arising out of his employment by a member of the exchange. Later he claimed that his employer had discriminated against him in violation of the Age Discrimination in Employment Act. The Court held that he had to submit his claim of age discrimination to arbitration. Although the arbitration clause was not in the employment contract (if it had been the Court would have had to decide the applicability of the Federal Arbitration Act to non-transportation employment contracts), but in the registration agreement to which the employer was not a party, the employer had required Gilmer to sign the registration agreement as a condition of his employment.
Although the Court did not overrule Alexander, it did say that the mistrust of the arbitral process that had permeated that opinion had been “undermined” by subsequent decisions evincing a positive attitude toward arbitration. 500 U.S. at 34 n. 5, 111 S.Ct. at 1656 n. 5. It also distinguished Alexander from Gilmer’s case on three grounds. The first was that the arbitration in Alexander had “occurred in the context of a collective-bargaining agreement,” creating a “tension between collective representation and individual statutory rights.” Id. at 35, 111 S.Ct. at 1657. An attenuated version of that tension was present in Gilmer itself, because Gilmer’s employer had forced him as a condition of his employment to sign a nonnegotiable form contract with the New York Stock Exchange that contained an arbitration clause. But Gilmer’s access to arbitration, if a dispute arose, was not controlled by a union, or any other entity or individual. At all events the Court treated his situation as being different from Alexander’s.
The second distinction drawn in Gilmer was that the issue in the earlier case had been the preclusive effect of an arbitration award concerning a contractual right on the litigation of a statutory right, rather than (as in Gilmer) the enforceability of the agreement to arbitrate the statutory claim-. The third distinction was that Alexander had not been decided under the Federal Arbitration Act. This was not to say that the act had been totally inapplicable, but only that, as we saw earlier, substantive issues in suits that are within the scope of both the arbitration act and section 301 of the Taft-Hartley Act are governed by section 301. So the third ground of distinction ties back to the first, and like the first puts our two cases on the Alexander side of the line — while the second *365distinction seems to be a distinction without a difference, since the Court assumed, consistent with what we said earlier about the preclusive effects of arbitration awards, that Gilmer could not bring an age discrimination suit if he lost the arbitration proceeding. 500 U.S. at 28, 111 S.Ct. at 1653. But for what the distinction is worth, it tugs our cases a little closer to Gilmer, since here as there the issue is the enforceability of the arbitration clause in a collective bargaining agreement rather than the effect of the award in an arbitration of a contractual right on the right to sue to enforce a statutory right.
On balance our case is closer to Alexander; but is enough left of Alexander to compel a decision in favor of the plaintiffs? Only the Supreme Court can answer that question; and we are timid about declaring decisions by the Supreme Court overruled when the Court has not said so. Khan v. State Oil Co., 93 F.3d 1358, 1364 (7th Cir.1996), cert. granted, — U.S. -, 117 S.Ct. 941, 136 L.Ed.2d 831 (1997). The conservative reading of Gilmer is that it just pruned some dicta from Alexander — and it certainly cannot be taken to hold that collective bargaining agreements can compel the arbitration of statutory rights. That issue was not before the Court or discussed by it. The Court may have so distinguished Alexander as to deprive it of any authoritative force; but that is the most it did and by doing so opened what till then had been a closed issue. It did not resolve the issue, which for the reasons stated earlier we believe should be resolved in favor of the plaintiffs’ right to sue. The district judges were therefore right to deny the defendants’ motions for stays of the judicial proceedings pending arbitration.
Affirmed.