POSNER, Circuit Judge.

A case can be complex without being difficult and difficult without being complex—and it can also be both complex and difficult, yet difficult for reasons unrelated to its complexity. This case is complex because of multiple parties, four separate proceedings (a mixture of arbitrations and adjudications), a multitude of complex written instruments, and sprawling submissions—the parties and amici curiae have filed a total of seven briefs, running to hundreds of pages. But these complexities—many of which we'll ignore—have little to do with whether to affirm or reverse.

Roughneck, the appellant, is a Chicago construction company engaged in cutting and drilling concrete. Some of this cutting and drilling is for a building's plumbing, some for its electrical system, some for other parts of the building. It employs plumbers for concrete work related to the building's plumbing, electricians for concrete work related to its electrical system, and laborers for the rest, and has collective bargaining agreements with the three local unions representing plumbers, electricians, and laborers respectively.

Its agreement with the plumbers ordained the creation of several pension and benefits funds, but for simplicity we'll pretend there was only one fund. An audit conducted by the fund determined that Roughneck had not employed plumbers to do work that the collective bargaining agreement entitled them to do, and therefore that Roughneck owed the fund the contributions that it would have had to make had it given the work to plumbers— some $2.2 million.

The collective bargaining agreement established a Joint Arbitration Board to resolve disputes arising out of it. The fund kicked off the arbitration process by filing a grievance with the Board. Roughneck filed its own grievance, arguing that the work that the fund contended had been plumbers' work had actually been within the jurisdiction of the electricians' and laborers' unions. (We don't know why Roughneck filed its own grievance rather than just opposing the fund's.) The laborers' union supported Roughneck, which had made contributions to that union's benefits funds for the work that the plumbers' fund contends should have been given to plumbers. Maybe the laborers' union worried that its benefits funds might be ordered to return the money that Roughneck should have given the plumbers' fund.

A hearing before the Joint Arbitration Board on the warring grievances was scheduled for the morning of July 22, 2008. Six days earlier, Roughneck had written the administrator of the Plan for the Settlement of Jurisdictional Disputes in the Construction Industry—the "National Plan," as it is called—charging that the grievances that Roughneck and the plumbers' fund had submitted to the Joint Arbitration Board were "impediments to job progress." The National Plan is an agreement between construction companies and the international unions that are the parents of local unions such as the plumbers' and electricians' and laborers' local unions in this case. The National Plan is admitted to bind the plumbers' local union and Roughneck, but the plumbers' fund is not a party, which may be significant (though we think not). The National Plan defines "impediment to job progress" to include "filing a grievance under a collective bargaining agreement" if the issue presented by the grievance "involv[es] a jurisdictional dispute." Roughneck's letter pointed out that both Roughneck and the plumbers' local are subject to another local arbitral entity (besides the Joint Arbitration Board), called the Joint Conference Board, which was created to resolve jurisdictional disputes, and that the issue between the parties was which craft of workers had "jurisdiction" over the work that the plumbers' fund believed should have been done by plumbers.

Either the plumbers' local union or Roughneck could have filed a grievance with the Joint Conference Board, but hadn't done so. They seem to have thought that that board could hear only jurisdictional disputes arising out of current employment, and not ones over completed work, as in this case, although the distinction is not found in the agreement between the employers and the local unions that had created the Board; the agreement doesn't define "jurisdictional dispute."

The administrator of the National Plan notified the international plumbers' union of Roughneck's letter and scheduled a hearing before an arbitrator named Paul Greenberg for the afternoon of July 21, 2008, the day before the scheduled hearing of the Joint Arbitration Board. Shortly before Greenberg's hearing, the fund's attorney wrote the international union's president saying that the Joint Arbitration Board was the appropriate forum to resolve the dispute.

The hearing before Greenberg was conducted as scheduled on July 21; present were representatives of Roughneck and of the three parent international unions. That evening Greenberg ruled that the effort of the plumbers' fund to obtain benefits based on hours worked by employees represented by other local unions was beyond the authority of the Joint Arbitration Board, and he ordered that the upcoming hearing before the Board be cancelled and that the Board dismiss the grievances. He explained that "this effort to collect fringe benefit payments is patently jurisdictional in nature, and thus prohibited. If UA Local 130 [the plumbers' local union] believes it has a legitimate jurisdictional claim to work being performed by other crafts employed by Roughneck, the collective bargaining agreement has a clear mechanism available for UA Local 130 to vindicate its jurisdictional rights and the rights of the workers it represents. But the claim now scheduled for hearing on July 22 is not the appropriate mechanism, but instead is barred" (emphasis in original). The authority of the National Plan's arbitrator to make a binding interpretation of the collective bargaining agreement is not questioned.

Neither the plumbers' fund nor anyone else notified the Joint Arbitration Board of Greenberg's order, so the Board went ahead with its hearing the next morning. Roughneck did not attend. The Board ruled that the plumbers' fund was authorized to invoke the Board's jurisdiction, and, on the merits, that Roughneck had violated its collective bargaining agreement with the plumbers' union by having employees not represented by that union do plumbing work. The Board ordered Roughneck to pay the fund more than $3.3 million in delinquent contributions, plus interest and penalties.

Which brings us at last to the litigation out of which the appeals (functionally one appeal) that we are asked to decide arise. Roughneck filed two suits in the district court, which we've consolidated in this court: one to vacate the Board's order and the other to enforce Greenberg's order. Both suits were brought under section 301(a) of the Labor Management Relations Act (Taft-Hartley), 29 U.S.C. § 185(a), because they charge the defendants with violating a labor contract—the National Plan as interpreted by arbitrator Greenberg. And both seek the same relief—invalidation of the Joint Arbitration Board's $3.3 million award to the plumbers' fund.

Roughneck lost both cases, and appeals both judgments. The district court ruled that by submitting its grievance to the Joint Arbitration Board, and later by failing to appear at the hearing before the Board, Roughneck had waived any objection to the Board's jurisdiction and was therefore bound by the Board's order and could not avail itself of arbitrator Greenberg's order in its favor.

The applicable limitations period for a suit to set aside Greenberg's order was 90 days, as that is the deadline in Illinois law for a challenge to an arbitration award. 710 ILCS 5/12(b). For want of a statute of limitations in the Labor Management Relations Act, courts look to local law for a limitations period, see United Auto. Workers v. Hoosier Cardinal Corp., 383 U.S. 696, 704-06, 86 S.Ct. 1107, 16 L.Ed.2d 192 (1966); Sullivan v. Gilchrist, 87 F.3d 867, 870 (7th Cir.1996); Sullivan v. Lemoncello, 36 F.3d 676, 681 (7th Cir.1994), and the applicable local law in this case, all agree, is that of Illinois. The fund did not file a suit within 90 days; we pause to consider whether that was a fatal mistake, or whether the fund protected itself by opposing Roughneck's suit to enforce Greenberg's order.

Roughneck had filed its suit to set aside the award by the Joint Arbitration Board within the 90-day deadline for challenging an arbitration award. In defending against that suit, the fund challenged the validity of Greenberg's order, on which Roughneck relied for its argument that the Joint Arbitration Board's award had exceeded the Board's authority. Roughneck's suit to enforce Greenberg's order came a month later, but was timely because Illinois law provides a five-year statute of limitations for suits on (that is, suits to enforce) awards of arbitration, 735 ILCS 5/13-205, as distinct from suits challenging them. The difference in limitations periods is an example of the law's bias in favor of arbitration, a bias that seems largely motivated by a desire to limit judicial workloads. Why otherwise prefer nonjudicial to judicial dispute-resolution processes? Cf. Gotham Holdings, LP v. Health Grades, Inc., 580 F.3d 664, 666 (7th Cir.2009) ("the Federal Arbitration Act eliminates hostility to private dispute resolution; it does not create a preference for that process. . . . People do not `violate' or `undermine' any federal policy if they litigate rather than arbitrate. Federal policy favors arbitration only in the sense that it favors contracts in general").

Roughneck argues that the fund's failure to file a timely suit to set aside Greenberg's order bars the fund from challenging that order in Roughneck's suit to enforce it. We disagree. It's true that because Greenberg's order purported to extinguish the award to the plumbers' fund by the Joint Arbitration Board, and thus was equivalent to denying the fund's grievance against Roughneck, the fund could have sued to set aside his order. Association of Contracting Plumbers of the City of New York, Inc. v. Local Union No. 2 United Ass'n of Journeymen, 841 F.2d 461, 466-67 (2d Cir.1988). "[S]omeone against whom a judgment is entered is entitled to the rights of a party." Lefkovitz v. Wagner, 395 F.3d 773, 778 (7th Cir.2005). But in the circumstances of this case, the fund's failure to bring an action to vacate the Greenberg award should not bar it from presenting defenses to that award. Roughneck had already sued to enjoin the Joint Arbitration Board's order, and the fund—the defendant in that suit— was entitled to defend by challenging the validity of Greenberg's order. It would have been redundant to make the fund file its own suit challenging Greenberg's order rather than permitting it to raise that challenge as a defense to Roughneck's suit to vacate the Joint Arbitration Board's order.

It would be different had Roughneck not sued to enjoin the Board's order, but instead had merely filed suit to enforce Greenberg's order. Since the latter suit was filed more than 90 days after Greenberg's order, it would have been too late for the fund to challenge the validity of the order by interposing a defense against Roughneck's suit. The 90-day deadline governs defenses to suits to enforce arbitration awards, since those defenses would have been grounds for a timely challenge to the awards. International Union of Operating Engineers, Local 841 v. Murphy Co., 82 F.3d 185, 188 (7th Cir.1996); International Union of Operating Engineers, Local 150 v. Centor Contractors, Inc., 831 F.2d 1309, 1311 (7th Cir.1987); Local 2322, Int'l Brotherhood of Electrical Workers v. Verizon New England, Inc., 464 F.3d 93, 98-99 (1st Cir.2006); Local 802, Associated Musicians of Greater New York v. Parker Meridien Hotel, 145 F.3d 85, 89 (2d Cir.1998). But the fund's defense was timely, because it had first been raised in Roughneck's mirror-image suit to set aside the order of the Joint Arbitration Board, and that suit had been filed within 90 days of that order.

With the procedural issue out of the way, we turn to the merits of Roughneck's appeal. Ordinarily, just as two parties to a dispute can agree to settle it, thereby surrendering the procedural rights they would have had if they had litigated to judgment, they can agree to arbitration even if by agreeing they give up procedural rights they would otherwise enjoy. For that matter they could agree to resolve their dispute by the flip of a coin, or by using a Ouija board to obtain a paranormal resolution.

This is provided that the agreement does not violate legally protected rights of third parties; "third parties' rights may be affected only with their consent." Gotham Holdings, LP v. Health Grades, Inc., supra, 580 F.3d at 666. There are third parties here—the international unions—and they may have a legally protected interest in enforcement of the National Plan that the agreement between Roughneck and the fund of a local union to arbitrate a jurisdictional dispute before the Joint Arbitration Board violated. But let's assume the agreement didn't step on the internationals' toes; the assumption simplifies our decision, while rejecting the assumption would not change it.

So if Roughneck had stuck to its agreement to arbitrate before the Board (which it did not do) and had lost (as it did), it could not challenge the Board's order on the ground that the Board had no jurisdiction, because Roughneck would by its action have consented to the Board's resolving its dispute with the fund. Environmental Barrier Co. v. Slurry Systems, Inc., 540 F.3d 598, 606 (7th Cir.2008); Slaney v. International Amateur Athletic Federation, 244 F.3d 580, 591 (7th Cir. 2001); Howard University v. Metropolitan Campus Police Officer's Union, 512 F.3d 716, 720-21 (D.C.Cir.2008); Schoenduve Corp. v. Lucent Technologies, Inc., 442 F.3d 727, 734-35 (9th Cir.2006). It could have consented to have Judge Judy resolve the dispute, and would have been bound even though the collective bargaining agreement did not authorize her to resolve disputes between Roughneck and the local union or the union fund. As we explained in Jones Dairy Farm v. Local No. P-1236, United Food & Commercial Workers Int'l Union, AFL-CIO, 760 F.2d 173, 175 (7th Cir.1985), "if a party voluntarily and unreservedly submits an issue to arbitration, he cannot later argue that the arbitrator had no authority to resolve it."

But the arbitrator in Jones Dairy had made his award; it was only after losing that Jones Dairy Farm argued that he hadn't been authorized to arbitrate its dispute with the union. Had it prevailed in the arbitration it wouldn't have made the argument. It was playing heads I win, tails you lose. Jones Dairy Farm makes clear that we don't permit that. See also WellPoint, Inc. v. John Hancock Life Ins. Co., 576 F.3d 643, 647-48 (7th Cir.2009); Environmental Barrier Co. v. Slurry Systems, Inc., supra, 540 F.3d at 606; Opals on Ice Lingerie v. Bodylines Inc., 320 F.3d 362, 368 (2d Cir.2003); Nationwide Mutual Ins. Co. v. Home Ins. Co., 330 F.3d 843, 846 (6th Cir.2003); Nghiem v. NEC Electronic, Inc., 25 F.3d 1437, 1440 (9th Cir. 1994). Is that what happened here?

After agreeing to (indeed initiating) arbitration by the Joint Arbitration Board, Roughneck woke up to the fact that the Board didn't have jurisdiction over Roughneck's dispute with the plumbers' fund, and decided (correctly as matters turned out) that the dispute should be resolved under the National Plan, and obtained an arbitration award under the Plan the day before the Joint Arbitration Board's scheduled hearing. It's true that in Nghiem v. NEC Electronic, Inc., supra, as in this case, the party (Nghiem) seeking to set aside the arbitrator's award had challenged the arbitrator's authority before the arbitrator made his final ruling, yet the court held that Nghiem was nevertheless bound by that ruling. Our case is distinguishable. Nghiem's abandonment of arbitration came after he had attended hearings before the arbitrator, presented evidence, and filed a fifty-page closing brief. Although poky in challenging the authority of the Joint Arbitration Board, Roughneck did manage to withdraw its consent to arbitration before any substantive proceedings were conducted. Furthermore, Nghiem had no valid ground for questioning the arbitrator's authority; Roughneck had a compelling ground.

Relying on arbitrator Greenberg's order that the Board dismiss the Joint Arbitration Board's proceeding, Roughneck did not attend the Board's hearing. The fund did not mention Greenberg's order to the Board, and so the Board went ahead and— unsurprisingly, given the absence of the adverse party, Roughneck—made a generous award to the fund. Roughneck says the fund was notified of the order, and should have told the Board. The fund has never denied that it was notified, though neither has it admitted it—explicitly. Its lawyer was evasive when pressed at oral argument; to the following remark from the bench—"apparently Greenberg sent his decision to the funds, so he must have thought they were interested parties"—the lawyer responded: "That may be, your honor. That certainly doesn't bind the Joint Arbitration Board, nor does it provide that the funds knew of the decision at the time that the arbitration proceeded." The lawyer, who also represented the fund in both arbitration proceedings, must have known whether his client had received notice of Greenberg's order; and we interpret his evasive answer as acknowledging that the fund indeed was notified.

It wouldn't have mattered, however, if the fund hadn't been aware of Greenberg's order. It knew there was going to be a proceeding before the National Plan in which Roughneck would be asking for such relief—Roughneck had invoked the National Plan on July 16, five days before the hearing conducted by Greenberg and six days before the Joint Arbitration Board's hearing, and had notified the plumbers' fund, the local plumbers' union, the international union, and the secretary of the Joint Arbitration Board that it was going to ask a National Plan arbitrator for relief. Those actions unmistakably signaled Roughneck's withdrawal of consent to arbitration by the Board.

Roughneck was not faultless in this messy matter of overlapping arbitrations. It should not have dawdled in invoking the National Plan, it should have notified the Board of Greenberg's order, and it should have shown up at the Board's hearing to make sure that everyone was aware of the order. But were these errors in lawyering fatal, as the fund contends? Generally lawyers' errors are forgiven when no one is prejudiced by them, and if there is prejudice, still the court endeavors to fit the punishment to the crime, so that if the prejudice is slight the sanction imposed on the lawyers or their client is light, consistently with the principle of proportionality that governs court-devised sanctions. Montaño v. City of Chicago, 535 F.3d 558, 563 (7th Cir.2008); Allen v. Chicago Transit Authority, 317 F.3d 696, 703 (7th Cir. 2003); Smith v. Gold Dust Casino, 526 F.3d 402, 405 (8th Cir.2008); Malot v. Dorado Beach Cottages Associates, 478 F.3d 40, 45 (1st Cir.2007); Doyle v. Murray, 938 F.2d 33, 34 (4th Cir.1991).

Lawyers and their clients are, it is true, held strictly to waivers. But a waiver is the deliberate relinquishment of a known right, and that did not occur. Roughneck never said it wouldn't invoke the National Plan, and there was no necessary inconsistency in Roughneck's invoking arbitration by the Joint Arbitration Board as well, given possible uncertainty about which arbitral route was the proper one; after all, the fund continues to insist that the Board did have jurisdiction. Eventually Roughneck had to choose and did; it should have chosen earlier but what harm was done by its dawdling? None that we can see, given the fund's failure to complain about having to pay its lawyers more because a hearing was held before the Joint Arbitration Board even though Greenberg had in effect enjoined it. This is a case of no harm, no foul.

The plumbers' fund argues that because it is not a party to the National Plan, Greenberg's order exceeded his jurisdiction, and furthermore that a disagreement over the results of an audit cannot be a jurisdictional dispute as that term is used in the National Plan and the collective bargaining agreement, so that even if Greenberg had jurisdiction his order was erroneous. The fund and Roughneck each defend the arbitral order favorable to it by reference to the principle that judicial review of arbitration is exceedingly narrow, which it is, but the principle falls out when a court has to choose between two inconsistent arbitration awards.

True, multiple arbitral awards often are enforced even when they are based on inconsistent theories. "Since judicial review of an arbitration award is so limited as to be little better than a rubber stamp, arbitral awards based on diametrically opposed interpretations of the identical contract can all withstand judicial review." Consolidation Coal Co. v. United Mine Workers of America, 213 F.3d 404, 406 (7th Cir.2000). But not if it's impossible for the parties to comply with both awards. Id.; Local # 850, Int'l Ass'n of Machinists & Aerospace Workers v. T.I.M.E.-DC, Inc., 705 F.2d 1275, 1276-78 (10th Cir.1983). And that's the case here. Greenberg's order says that the "UA Local 130, the Plumbing Contractors Association and the Joint Arbitration Board are hereby ORDERED to dismiss the matter currently scheduled to be heard July 22, 2008." There is no way to give effect to this arbitral award and enforce the Joint Arbitration Board's award. To enforce the former award is to hold that the latter was ultra vires and therefore cannot be enforced.

In a case that addresses the situation, analogous to what we face here, in which an employer faces arbitration awards that assign the same work to different unions, we have ruled that neither award should be enforced and instead that a new arbitration should be conducted, in which the employer and both unions can present their arguments to a single arbitrator. Local 416, Sheet Metal Workers Int'l Ass'n v. Helgesteel Corp., 507 F.2d 1053, 1058 (7th Cir.1974); to the same effect, see Office & Professional Employees Int'l Union, AFL-CIO v. Sea-Land Service, Inc., 210 F.3d 117, 123 (2d Cir.2000); Retail, Wholesale & Dep't Store Union, Local 390 v. Kroger Co., 927 F.2d 275, 277-78, 281-82 (6th Cir.1991). That in essence is what the National Plan is empowered to order and did order in this case—that the bipartite Joint Arbitration Board proceeding be suspended so that a tripartite resolution, in which benefits would be divided between the plumbers and the laborers rather than Roughneck's being required to pay duplicate benefits, could be conducted by the Joint Conference Board.

And as between the two orders, that of the Joint Arbitration Board and Greenberg's, the latter takes precedence. The National Plan assigns arbitrators to resolve disputes governed by the plan, and that includes determining whether an impediment to job progress has occurred. With "impediment to job progress" defined to include filing grievances with the Joint Arbitration Board in jurisdictional disputes, Greenberg, the National Plan arbitrator, had to determine whether there was a jurisdictional dispute and his determination that there was established the invalidity of the Joint Arbitration Board's award. The judgment is therefore reversed with instructions to enforce Greenberg's order and vacate the order of the Joint Arbitration Board.

REVERSED AND REMANDED, WITH INSTRUCTIONS.