EASTERBROOK, Circuit Judge.
We reheard this case en banc to address two questions on which we solicited supplemental briefs from the parties:
whether, as a matter of law, evidence regarding arbitration and a company’s collective bargaining agreement is inadmissible in a Title VII suit to show: 1) that an employer’s response to sexual harassment was reasonable for the purpose of determining employer liability; and 2) that the employer did not act with the state of mind necessary for the imposition of punitive damages.
The court now holds that this evidence is not relevant to liability and therefore is inadmissible under Fed.R.Evid. 402 with respect to that subject, but that the evidence is relevant to the propriety and amount of punitive damages, see Fed. R.Evid. 401, and therefore is admissible on that issue unless its probative value is “substantially outweighed” by other factors. Fed.R.Evid. 403. Because the district judge thought the evidence irrelevant across the board, we remand for an appropriate exercise of discretion under Rule 403—and, if necessary, a new trial limited to punitive damages.
I
Gary Amos worked for many years in Ameritech’s coin center and its small business unit. Most of the other employees in both places were women, and Amos behaved offensively among them. As long ago as 1975 Barbara Huckeba complained to her supervisor that Amos had exposed himself to her on three occasions. Ameri-tech’s response was to fire Huckeba, “explaining” that Huckeba was more likely to find satisfactory employment elsewhere than was Amos. Two other co-workers also complained in 1975 about sexually offensive conduct; unlike Huckeba, they kept their jobs. So did Amos, who was not disciplined. The record establishes other misconduct in 1988, 1989, 1990, 1991, 1992, 1993, and 1994, including Amos’s telling female co-workers that he was in love with them, flashing them, sending notes with sexual messages or propositions, grabbing them and rubbing their hair or buttocks (some times with his hands, sometimes with his erect penis), and allowing himself to be seen masturbating at his desk.
In 1989 a flashing incident led to a warning but no other discipline. In 1990 six women complained; five of the six informed Ameritech that Amos had pressed his erect penis against them. Amos was suspended for two weeks by a supervisor who neglected to read Amos’s personnel file and thus was unaware of his history of similar conduct. Complaints lodged by women in 1991 and 1992 did not lead to discipline. Although supervisors told [520]*520Amos to desist, the misconduct continued. But on December 18, 1992, Ameritech’s equal employment opportunity coordinator recommended that Amos be fired. That recommendation had to be approved by Ameritech’s labor relations manager, who was on vacation. By the time the labor relations manager returned and reviewed the file, more than 30 days had elapsed since Amos’s most recent documented misconduct — and the collective bargaining agreement between Ameriteeh and its union required discipline to occur within 30 days. So once again Ameriteeh did nothing concrete, though it did tell Amos that any further misconduct could result in suspension or discharge. “Further misconduct” was not long in coming; fresh complaints in February 1993 led to another investigation, at the end of which Ameri-tech did — nothing, yet again. All through 1993 Amos directed a stream of sexually charged comments and notes to his female co-workers, brushed against them deliberately (sometimes with an erection), showed them pictures of partially undressed women, commented on their clothing, and so on. Their complaints within Ameritech’s hierarchy were ineffectual. Another public-masturbation incident in March 1994 at last produced Amos’s removal.
Amos’s co-workers did not turn outside the company until April 1993, when Lori Everts filed with the eeoC a charge of discrimination. The eeoc was sufficiently disturbed by Ameritech’s pattern of neglect that it has litigated as the champion of Everts and the other women. Conduct more than 300 days before that charge cannot be reached under Title VII of the Civil Rights Act of 1964, though earlier events may be considered to place the more recent ones in context. Another significant date is November 21, 1991, when the Civil Rights Act of 1991 took effect. The 1991 Act authorizes compensatory and punitive damages for sex discrimination. 42 U.S.C. § 1981a. Because it is not retroactive, see Landgraf v. USI Film Products, 511 U.S. 244, 114 S.Ct. 1483, 128 L.Ed.2d 229 (1994), only Ameritech’s conduct (and injuries female workers sustained) after that date can lead to compensatory or punitive damages.
Before trial Ameriteeh informed the district judge that it planned to defend in part on the theory that the collective bargaining agreement tied its hands and thus excused its failure to discharge Amos until 1994. Ameriteeh wanted to argue not only that the delay at the end of 1992 foreclosed discipline but also that, if it had fired Amos any earlier than March 1994, an arbitrator might well have concluded that Ameriteeh lacked “just cause” and reinstated Amos. According to Ameriteeh, one arbitrator recently had done just that with another employee discharged for making women’s working fives miserable, and it feared a repetition. The judge, however, issued a pretrial order excluding this proposed theory of defense and its associated evidence.
The jury concluded that, by tolerating Amos’s misconduct, Ameriteeh had engaged in sex discrimination, in violation of Title VII. It awarded three of Amos’s coworkers a total of $15,000 in compensatory damages and $1,050,000 in punitive damages. The district judge reduced the punitive damages to a total of $635,000, and the eeoc has not challenged this reduction on appeal. A panel of this court unanimously concluded that Ameritech’s supine posture despite ample knowledge made it answerable for Amos’s misconduct. 214 F.3d 813, 820-22 (7th Cir.2000). See also Burlington Industries, Inc. v. Ellerth, 524 U.S. 742, 759, 118 S.Ct. 2257, 141 L.Ed.2d 633 (1998) (explaining the circumstances under which an employer’s feckless responses to intentional misconduct by an employee means that the misconduct is attributed to [521]*521the firm). The panel also held that Ameri-tech’s decision not to protect its female workers from Amos’s abusive treatment adversely affected their conditions of employment on account of sex, because Amos’s misconduct was sufficiently severe and frequent, indeed that the evidence supported punitive damages as well. Ibid. See also Clark County School District v. Breeden, 532 U.S. 268, 121 S.Ct. 1508, 1509-10, 149 L.Ed.2d 509 (2001); Oncale v. Sundowner Offshore Services, Inc., 523 U.S. 75, 78, 81-82, 118 S.Ct. 998, 140 L.Ed.2d 201 (1998); Harris v. Forklift Systems, Inc., 510 U.S. 17, 21, 114 S.Ct. 367, 126 L.Ed.2d 295 (1993); Meritor Savings Bank, fsb v. Vinson, 477 U.S. 57, 72, 106 S.Ct. 2399, 91 L.Ed.2d 49 (1986). But a majority of the panel held that the district judge erred in foreclosing Ameri-tech’s proposed defense, 214 F.3d at 822-25, and it remanded for a new trial. We have reheard the case en banc to consider that subject. The remainder of the panel’s decision, vacated on the grant of rehearing en banc, is now reinstated and stands as the opinion of the full court on all issues other than those discussed below.
II
Ameritech contends that the 30-day, just-cause, and arbitration provisions of the collective bargaining agreement demonstrate that its actions were reasonable, and thus that Amos’s misconduct should not be attributed to the employer. Its argument has two principal strands: first, Amos’s reinstatement (if an arbitrator should conclude that he should not have been fired) could be costly to the company in terms of back pay; second, reinstatement could be costly to Amos’s co-workers because it would make Amos that much harder to get rid of in the future. The second line of argument also implies an adverse effect for women who worked elsewhere in the company. An arbitrator’s award reinstating Amos might impede efforts to remove harassers throughout the firm, and might correspondingly embolden these men for as long as the current (or a similarly worded) collective bargaining agreement lasts. These are serious concerns, and given the posture of the case we must assume that Ameritech could establish them to a jury’s satisfaction. Nonetheless, neither of these reasons is sufficient to defeat liability under Title VII. Three principal considerations lead to this conclusion.
A
Ameritech supposes that both the collective bargaining agreement and the system of arbitration were imposed on it by forces outside its control, and that it therefore cannot be liable for what arbitrators do (or for what Ameritech itself does in anticipation of what arbitrators may do). Not so.
A collective bargaining agreement is just an employment contract, with organized rather than individual workers. No collective bargaining agreement is imposed on an employer; it is an agreement, and the firm has consented to every part of the arrangement. Employers cannot by their consent (in or out of a collective bargaining agreement) avoid duties under federal law. This would be clear enough with an individual contract of employment. Suppose Ameritech hired a new manager under a contract prohibiting the manager’s discharge for a term of years unless the division became unprofitable. If the manager took to harassing his female subordinates, or giving them fewer promotions or lower raises than otherwise identical men, Ameritech could not escape responsibility by saying that the manager’s contract precluded his removal. Likewise a university could not escape liability for the [522]*522misconduct of a tenured professor, or a football coach with a long-term contract. The just-cause standard in the collective bargaining agreement offered Amos less protection than any of the employees in these hypotheticals. If employers are liable for the acts of employees they have promised by contract not to discharge, likewise they are liable for the acts of employees who enjoy weaker “just cause” contractual protection.
Arbitration itself is a contractual arrangement. Ameritech treats arbitration as increasing the amount of misconduct that an employee may commit before becoming eligible for discharge. If, as we have just explained, a contractual prohibition on discharge does not avoid liability under Title VII, then a procedural device pointing in that direction cannot do so.
Some arbitrators doubtless are tolerant of employees who engage in sexual harassment—indeed, more tolerant than the law allows. This problem was well known to anyone who read the case reports before 1992, when the specter of arbitration first affected Ameritech’s handling of Amos. See, e.g., Newsday, Inc. v. Long Island Typographical Union, 915 F.2d 840 (2d Cir.1990); Communication Workers v. Southeastern Electric Cooperative, 882 F.2d 467 (10th Cir.1989). Ameri-tech could have negotiated for a provision defining sex harassment (or other discrimination) as “just cause,” or limiting arbitrators’ authority to reinstate persons discharged for events that violate federal law. But it did not do this, and it is in no position to insist that its female employees suffer the consequences of its negotiating strategy. Ameritech must itself bear the consequences of its labor-relations decisions. Cf. W.R. Grace & Co. v. Rubber Workers, 461 U.S. 757, 766-70, 103 S.Ct. 2177, 76 L.Ed.2d 298 (1983). Seniority provisions in collective bargaining provisions receive special treatment under Title VII. 42 U.S.C. §§ 2000e-2(h), 2000e-5(e). This strongly implies that other terms of labor agreements do not exculpate conduct that would violate Title VII if prescribed unilaterally by employers. Nor does it matter that the collective bargaining agreement is bilateral rather than unilateral; a union cannot surrender employees’ rights under Title VII. See Wright v. Universal Maritime Service Corp., 525 U.S. 70, 75-76, 119 S.Ct. 391, 142 L.Ed.2d 361 (1998); Alexander v. Gardner-Denver Co., 415 U.S. 36, 51, 94 S.Ct. 1011, 39 L.Ed.2d 147 (1974); Pryner v. Tractor Supply Co., 109 F.3d 354 (7th Cir.1997).
What is more, Ameritech is wrong to suppose that an arbitrator is some outside force even ex post its agreement to a given arbitration clause. As Eastern Associated Coal Corp. v. United Mine Workers, 531 U.S. 57, 121 S.Ct. 462, 148 L.Ed.2d 354 (2000), observes, an arbitrator exercises powers delegated from the parties who have agreed to arbitrate. The arbitrator acts on their (joint) behalf, with whatever authority the contract bestows. The resulting award is effectively the parties’ joint decision. “[W]e must treat the arbitrator’s award as if it represented an agreement between” the parties themselves. 121 S.Ct. at 467. An employer is free to delegate, to arbitrators or anyone else, but the holder of the delegated power still speaks for the employer. This would be clear if Ameritech (unilaterally or through a collective bargaining agreement) had delegated final authority, not to an arbitrator, but to its vice president for human resources. If the VP had announced that “mere harassment” is not just cause warranting discharge, Ameri-tech would be responsible for that decision. The firm’s board of directors or CEO could not plead helplessness as a result of the delegation and ask a jury to send the [523]*523plaintiff packing; the VP would speak for the firm. Ellerth establishes that much. See also Faragher v. Boca Raton, 524 U.S. 775, 118 S.Ct. 2275, 141 L.Ed.2d 662 (1998). If the delegation of final authority is to a person called an arbitrator rather than a person whose title is vice president, the situation is the same: the employer has by contract endowed a particular person with the authority to make a final decision on its behalf.
Suppose that, instead of being a coin-counter who harassed co-workers, Amos had been a truck driver who ran down a pedestrian while on company business. In the ensuing tort litigation, could Ameritech defend by saying that, although it knew Amos to be a poor driver, it had not fired him because it feared that an arbitrator would just reinstate him? Surely that would not be a defense. Indeed, it would not be a defense to liability even if Ameri-tech had fired the driver, and an arbitrator had reinstated him over the company’s vigorous opposition. It would remain vicariously liable for its employee’s misdeeds. Likewise under Title VII: Once the employer has been put on notice (as Ameritech was in 1975 and repeatedly since), it must act to protect other employees and bear the consequences of failure.
B
Ameritech proposed to argue not only that the collective bargaining agreement limited its options but also that, had it fired Amos and seen the decision reversed by an arbitrator, both the firm and women in its employ would have been worse off. We tackle the financial argument here and postpone the other branch to Part II.C below.
That failed efforts to discharge an employee may lead to costly back pay awards is undeniable. Likewise, successful efforts to discharge employees for sexual harassment have ended in liability—though appellate courts have been less sympathetic than juries to the harassers. See Mackenzie v. Miller Brewing Co., 241 Wis.2d 700, 623 N.W.2d 739 (2001) (reversing a jury’s award of $24 million in favor of an executive discharged for harassing women in the workplace); Stuck in the Middle: Employers Can Face Lawsuits from Accused as Well as Accusers, 69 U.S.L.W. 2539 (Mar. 13, 2001). So we grant Ameri-tech’s premise that keeping Amos around may have reduced its expected costs of doing business.
Nonetheless, arguments that expense justifies discriminatory conduct met their Waterloo in Los Angeles Department of Water & Power v. Manhart, 435 U.S. 702, 98 S.Ct. 1370, 55 L.Ed.2d 657 (1978). Women challenged a requirement that they make larger monthly contributions to its pension fund than their male counterparts. The city conceded that it was treating men and women differently but argued that it had not violated Title VII because women five longer, so it costs more to provide a given level of retirement benefits to women. The Court held that the cost differential was no defense, remarking:
[The city’s] argument might prevail if Title VII contained a cost-justification defense comparable to the affirmative defense available in a price discrimination suit. But neither Congress nor the courts have recognized such a defense under Title VII.... [W]e conclude that the ... practice violated Title VII.
435 U.S. at 716-17, 98 S.Ct. 1370. Similarly, the employer in Automobile Workers v. Johnson Controls, Inc., 499 U.S. 187, 111 S.Ct. 1196, 113 L.Ed.2d 158 (1991), argued that it could exclude fertile women from plants where lead was in the air, because lead might injure the employees’ offspring and lead to costly liability in tort. But the [524]*524Supreme Court held that neither health nor cost considerations could support the employer’s policy; Title VII entitles women to work without regard to the potentially higher costs that employers might experience. See also Orzel v. Wauwatosa Fire Department, 697 F.2d 743, 755 (7th Cir. 1983) (concluding that after Manhart a “cost-justification defense [is] not available in Title VII action”).
C
As for the possibility that discharge followed by reinstatement would have emboldened Amos (and other similar men) to continue their misconduct, and thus harmed women: why should we assume that discharge was Ameritech’s only or even its principal option? Ameritech could have taken steps short of discharge, such as transferring Amos to a job where there would have been fewer women around. An arbitrator may have been empowered to reinstate Amos, but he was not empowered to control who Amos’s co-workers would be. Ameritech would have been entitled to transfer Amos to an empty room and give him make-work tasks, if that were necessary to protect its female employees. If, by giving an arbitrator the power to keep Amos on the payroll, Ameri-teeh saddled itself with a useless employee, that would not be the first time. Featherbedding ensues from some collective bargaining agreements, and the lateral arabesque solves many a personnel problem. See Laurence J. Peter & Raymond Hull, The Peter Principle ch. 3 (1969). Because by changing Amos’s assignments Ameritech could have prevented his reinstatement from injuring the women in its employ, this line of argument collapses to the cost-justification defense that we have already rejected.
We have so far indulged Ameri-tech’s belief that an arbitrator would have been entitled to reinstate Amos. Even this is questionable, however. Although there is no free-roving “public policy exception” to arbitral awards, see Eastern Associated Coal, W.R. Grace, and Paperworkers v. Misco, Inc., 484 U.S. 29, 43-44, 108 S.Ct. 364, 98 L.Ed.2d 286 (1987), no arbitrator is entitled to direct a violation of positive law. An award that requires conduct “contrary to an explicit, well-defined, and dominant public policy, as ascertained by reference to positive law” must be set aside. Eastern Associated Coal, 121 S.Ct. at 467. Title VII explicitly forbids sex discrimination in the workplace. No award that required an employer to tolerate an ongoing violation of this statute could be enforced. So we have no fear that female employees could be made worse off by ruling out the defenses Ameritech proposed to make.
D
Ameritech recognizes that it cannot contract away its obligations under Title VII. Still, it insists, the costs and difficulties created by its collective bargaining agreements are relevant to the question whether its response to Amos’s misconduct was “reasonable,” and thus whether his misconduct is attributed to the firm. The syllogism goes like this: employers are liable under Title VII for misconduct by subordinate employees only when the employers deal negligently (or unreasonably) with that misconduct, see Faragher, 524 U.S. at 799-808, 118 S.Ct. 2275; Hostetler v. Quality Dining, Inc., 218 F.3d 798, 809-11 (7th Cir.2000); costs are an important ingredient in the negligence calculus; employers therefore are entitled to explain to juries each of the costs and difficulties that an employer encounters. We have no quarrel with the proposition that costs matter to reasonableness. Yet this proposition does not assist Ameritech, for two reasons: first, [525]*525the evidence Ameritech wants to present shows not the kind of costs all employers face, but additional costs that Ameritech has elected to bear voluntarily; second, the evidence does not explain the duration of Ameritech’s conduct.
Ameritech fails to distinguish between costs that all employers face (because of the way the world works) and costs that an employer brings on itself. Suppose Amer-itech had pledged to donate $100,000 to the United Way every time one of its employees committed sexual misconduct in the workplace. That promise would make it costly for Ameritech to discipline or discharge someone like Amos, for it would first have to find that misconduct occurred and pay the $100,000. Yet an employer could not argue to the jury that this cost, incurred voluntarily, made inaction in the face of women’s complaints reasonable. If every firm in the industry made the same promise — if cost-increasing conduct were customary — the higher costs still would be voluntary. Likewise with ordinary torts. A manufacturer of cigarette lighters must take cost-justified precautions to prevent injuries to children. See Todd v. Societe BIC, S.A., 9 F.3d 1216 (7th Cir.1993) (en banc), 21 F.3d 1402 (1994) (en banc). But the manufacturer could not include the inefficiency of its own plants among the costs that it could properly take into account. Industry-wide practices and competitive pressures that raised the expense of hiring engineers to design better cigarette lighters would create costs beyond the control of any one employer, and these could be included in the negligence calculus. But the costs to which Ameritech points are not external in this way. A firm may consider those vicissitudes that the human condition presents to any thoughtful employer. But it may not justify its actions as reasonable in the light of avoidable costs created or increased by its own decisions.
Now let us consider duration. Costs are important to the acquisition of knowledge; an employer is not required to place microphones and cameras in every corner of the workplace, so it may take a while to learn what is going on. The eeoc does not contend, however, that Am-eritech failed to implement reasonable strategies to learn about the conditions its female employees were encountering. Costs also are important to an employer’s initial response to accusations of misconduct. For example, an employer may defer acting while attempting to learn the truth of the matter, and it may begin with steps such as counseling if it seems likely that they will succeed. Again the eeoc does not fault Ameritech’s initial responses. (Although Ameritech’s decision to fire Huckeba rather than Amos is indefensible, that occurred outside the period of limitations.) It did not take Ameri-tech’s managers long to learn, however, that Amos is incorrigible. An employer knowing that working conditions are worse for women than for men may experiment reasonably with strategies to cure that problem, but it can not be reasonable to string that process out for a decade, as Ameritech did, no matter how high the costs of correction. Once managers knew not only of Amos’s misbehavior but also of his unwillingness (or inability) to change, it had to do something more effective. Doing nothing can never be the sort of “reasonable” step that prevents imputation to the firm of the inferi- or working conditions encountered by a protected class of employees. Yet apart from a single (and futile) suspension in 1990, a sanction whose modesty is attributable to a manager’s negligent failure to read Amos’s employment file, “do nothing” was the employer’s only strategy. Even if the firm acted reasonably in not discharging Amos, its do-nothing ap[526]*526proach cannot have been reasonable, which means that Ameritech is responsible for the conditions Amos created in the workplace.
Ill
Whether evidence inadmissible to establish a defense on the merits may have a role when assessing liability for (and the amount of) punitive damages is a distinct question. We proceed in three steps. First we inquire whether Ameritech has preserved its right to argue that the evidence is admissible with respect to punitive damages. Second we take up the relevance issue under Rule 402. Third we consider the possibility that any error was harmless.
Ameritech wanted to use the evidence with respect to both liability and punitive damages; the district judge ruled that the evidence could not be used at all; Ameritech did not respond with a request that use be limited to punitive damages (which likely would have entailed a request for a bifurcated trial). Did this omission doom Ameritech’s entitlement to argue in this court that, if the evidence is irrelevant to liability, it may still be relevant to punitive damages? We believe that an argument for such a distinction is open in this court, for two principal reasons.
First, the district judge made a definitive ruling excluding the evidence for all purposes. A party need not except to an evidentiary ruling; a claim is preserved for appeal if the party makes an offer of proof, as Ameritech did. Fed.R.Evid. 103(a)(2). In particular, “[o]nce the court makes a definitive ruling on the record admitting or excluding evidence, either at or before trial, a party need not renew an objection or offer of proof to preserve a claim of error for appeal.” Fed.R.Evid. 103(a). Although the language we have quoted did not become effective until December 1, 2000, Wilson v. Williams, 182 F.3d 562 (7th Cir.1999) (en banc), reached the same conclusion under prior law. Litigation could not be improved by requiring a litigant, told by a judge that particular evidence is inadmissible, to step through a series of counterproposals to use the evidence for each potential subsidiary issue in a case. That would take time, cost money, and try everyone’s patience, to little purpose. Ameritech therefore did not have to protest the district court’s ruling or ask the judge to reconsider. It is entitled to argue on appeal that the ruling was wrong either in whole or in part.
Second, if by inaction after the district court’s ruling Ameritech did forfeit its entitlement to distinguish using the evidence in the liability phase from using it in the punitive-damages phase, the eeoc by its inaction has forfeited any entitlement to the benefit of Ameritech’s forfeiture. Claims of waiver may be waived in turn; claims of forfeiture may be forfeited (or waived). Burris v. Parke, 95 F.3d 465, 469-70 (7th Cir.1996) (en banc); United States v. Newman, 144 F.3d 531, 542 n. 11 (7th Cir.1998); United States v. Leichtnam, 948 F.2d 370, 375 (7th Cir.1991). The eeoo’s brief does not contend that Ameritech has forfeited its opportunity to seek reversal with respect to punitive damages. After our supplemental briefing order clearly separated the liability and punitive-damages issues, the eeoc had a second opportunity to argue that Ameri-tech had forfeited any opportunity to argue for different outcomes on these issues. Once again the eeoc did not argue that Ameritech has forfeited any legal position by failing to make appropriate requests or protests after the district judge’s ruling in limine. We could hardly penalize Ameri-tech for forfeiture while overlooking the [527]*527eeoc’s decision not to make forfeiture an issue (a decision that deprived Ameritech of any opportunity to respond to an argument that it has forfeited part of its position). Both issues presented by our supplemental briefing order therefore are open to decision.
To establish liability under Title VII, the eeoC had to prove that Ameritech as the employer (and not just Amos personally) had engaged in sex discrimination, which is a form of intentional wrongdoing. It did this by showing that, with actual knowledge of Amos’s behavior, Ameritech’s managers permitted Amos to continue wreaking havoc among his female co-workers. To establish an entitlement to punitive damages, the eeoc had to show more than intentional discrimination.
A complaining party may recover punitive damages under this section against a respondent (other than a government, government agency or political subdivision) if the complaining party demonstrates that the respondent engaged in a discriminatory practice or discriminatory practices with malice or with reckless indifference to the federally protected rights of an aggrieved individual.
42 U.S.C. § 1981a(b)(l). “Malice” toward women and “reckless indifference to the federally protected rights” of female workers are considerably more blameworthy than the carelessness (such as allowing a 30-day time limit to lapse during a manager’s holiday) and general thoughtlessness that characterize much of Ameritech’s conduct.
In Kolstad v. American Dental Ass’n, 527 U.S. 526, 535, 119 S.Ct. 2118, 144 L.Ed.2d 494 (1999), the Supreme Court concluded that § 1981a(b)(l) conditions punitive damages on “the employer’s knowledge that it may be acting in violation of federal law, not its awareness that it is engaging in discrimination.” The minimum requirement for a punitive award is that the employer “discriminate in the face of a perceived risk that its actions will violate federal law”. Id. at 536, 119 S.Ct. 2118. See also Gentry v. Export Packaging Co., 238 F.3d 842, 851-52 (7th Cir.2001). But this minimum may not be sufficient, for at least two reasons. First, an employer aware of a risk nonetheless may believe that its conduct is lawful, making a punitive award inappropriate. See 527 U.S. at 537,119 S.Ct. 2118 (discussing situations “in which the employer discriminates with the distinct belief that its discrimination is lawful”). Second, § 1981a(b) authorizes but does not compel an award of punitive damages. Circumstances that fail to knock out punitive damages as a matter of law still may persuade a jury that the employer’s conduct was not so malicious or so reckless that damages should be added to the compensatory award.
Moreover, because the size of a punitive award reflects the jury’s assessment of the defendant’s blameworthiness, many circumstances that bear on this issue become relevant, even though they could not defeat liability. When in tort law a plaintiff requests punitive damages, it is common for both sides to introduce a great deal of evidence that could not be deemed relevant to liability. See, e.g., BMW of North America, Inc. v. Gore, 517 U.S. 559, 565-66, 116 S.Ct. 1589, 134 L.Ed.2d 809 (1996); Kemezy v. Peters, 79 F.3d 33 (7th Cir. 1996); Zazú Designs v. L’Oréal, S.A., 979 F.2d 499 (7th Cir.1992). Consider again the truck driver example from Part II.A. An arbitrator’s award reinstating a poor driver, who later injured a pedestrian, would not eliminate the employer’s tort liability for the pedestrian’s actual loss, but it might persuade a jury not to award [528]*528punitive damages or to reduce the size of any punitive award.
Judges retain, and should use, authority under Fed.R.Evid. 403 to keep the trial within sensible bounds. For example, Am-eritech’s desire to introduce the outcome of some other arbitration could divert the case into complex questions about whether that arbitrator’s decision was justified in light of what some other employee did to some other women; a judge may curtail such a presentation and perhaps could exclude it altogether if he concluded that it would be unjust to open the question at all without allowing the eeoc to respond with particulars. But district courts may not exclude as “irrelevant” under Rules 401 and 402 any evidence bearing on the defendant’s state of mind — and in particular may not categorically exclude evidence tending to show why the defendant acted as it did, for such evidence bears on “malice”, “reckless indifference to the federally protected rights of an aggrieved individual”, and the size of any appropriate punitive award. Accord, Palmetto State Medical Center, Inc. v. Operation Lifeline, 117 F.3d 142, 149-50 (4th Cir.1997); Global Petrotech, Inc. v. Engelhard Corp., 58 F.3d 198, 201-02 (5th Cir.1995). Cf. Beckwith v. Bean, 98 U.S. 266, 274-82, 25 L.Ed. 124 (1879).
Part II of this opinion concludes that the “just cause” clause of the collective bargaining agreement, and Ameritech’s concern about the potential decision of an arbitrator, are not relevant to liability. But Ameritech may be able to persuade a reasonable jury that its labor-relations decisions do not evince “malice” or “reckless indifference” to the federally protected rights of female employees. The eeoC has a strong argument that Ameritech’s inaction indeed demonstrates “reckless indifference”, but the debate should not be preempted by disabling Am-eriteeh from explaining itself. Moreover, as we have stressed, even “malice” and “reckless indifference” come in gradations. Keeping Amos around because of a concern with firm’s bottom line is bad, but not as bad as keeping him around because Ameritech’s supervisors agreed with his attitude toward, and treatment of, female workers. An employer is entitled to show that things were not as bad as they appeared, and thus to influence the jury’s assignment of punitive damages. The district court’s order enabled the eeoC to ask the jury rhetorically why any conscientious employer would have acted as Ameritech did unless it wanted harm to befall female workers, while disabling Ameritech from giving what may have been its best answer.
The eeoc suggests that the district court’s error was harmless because the jury instructions set a higher threshold for punitive damages than Kolstad later determined is appropriate. We do not believe, however, that an error in Ameriteeh’s favor on the standard of liability precisely offsets an error in the eeoü’s favor on how that standard is to be met. Whether or not the two errors canceled each other on the threshold decision to award some punitive damages, they could not have canceled each other on the quantum of damages. This is not to say that a punitive award aggregating $650,000 could not be sustained on a record containing Ameriteeh’s full explanation for its conduct. Our point, rather, is that only assurance that any reasonable jury must return an award of some $650,000 could demonstrate the error’s harmlessness. No such assurance is possible, given the great discretion juries possess when fixing the level of punitive damages.
[529]*529IV
The judgment of the district court with respect to compensatory damages is affirmed. The judgment awarding punitive damages is vacated, and the case is remanded for further proceedings consistent with Part III of this opinion.
Chief Judge Flaum and Judges Manion, Kanne, Rovner, Diane P. Wood, Evans, and Williams join Part I of this opinion. Judges Rovner, Diane P. Wood, Evans, and Williams join Part II. Chief Judge Flaum and Judges Manion, Kanne, and Evans join Part III. Every portion of the opinion therefore speaks for a majority.