Plaintiff filed this lawsuit individually and as administrator of the estate of his deceased 19-year-old son, Ronald Sherrod. The third amended complaint was in two Counts. The first Count was for wrongful death and alleged that defendant Willie Berry, a Joliet, Illinois, policeman, violated 42 U.S.C. § 1983 when he shot and killed Ronald. Berry supposedly violated “the customary standard police procedures for making felony stops of motor vehicles and investigating the occupants therein, thereby setting in motion a chain of events which resulted in the death of Ronald Sherrod on December 8, 1979.” Berry was also charged with violating national standards of police procedure as to the proper use of a gun and with using unreasonable force against Ronald.
Plaintiff additionally alleged in Count I that the other defendants, Joliet Chief of Police Frederick Breen and the City of Joliet, violated the same statutory provision, inter alia, through conduct “that resulted in a policy of the use of excessive force,” and through failure to train and supervise Joliet police officers with respect to the proper procedures for stopping vehicles and the use of force. Plaintiff sought $13,-000. 000 for loss of support in Count I.
In Count II, based upon the “right to raise a family,” plaintiff complained that defendants deprived him of his liberty without due process of law and his “right to the enjoyment of the result of rearing a child, his right to raise a child, [and] his right to raise a family as he sees fit.” Count II sought $1,000,000 compensatory damages.
Under Count I the jury awarded Ronald’s estate $850,000 for the value of his life, $300,000 for pecuniary loss to the estate and $1,700 for funeral expenses. Under Count II plaintiff individually was awarded $450,000 for the loss of parental association with Ronald. The total award was $1,601,700. In ruling on defendants’ motion for remittitur, the district judge decided that the verdicts of the jury were not excessive and added that were it in his power he would have granted an additur instead of ordering a remittitur (Defendants’ App. 52).
The evidence showed that on Saturday afternoon, December 8, 1979, Gary Duck-worth stole $50 to $80 from a cash register at Ziggy’s Plant and Gift Shop in Joliet. The owner, Janet Youngquist, went to an adjacent shop and was there directed toward Duckworth as he approached his parked car. He ran away, discarding the money which Ms. Youngquist retrieved. The police were then called and informed that there had been a robbery at the shop. Soon thereafter, the fleeing Duckworth entered Sherrod’s auto repair shop and asked for a jump start for his car, which he had left near Ziggy’s shop. He and Ronald left the auto repair shop in Ronald’s black 1969 Cadillac, with Ronald driving.
After defendant Berry heard a radio dispatch about the robbery, he told fellow officer Richard Klepfer that the description of the suspect fit Gary Duckworth, who had been involved in petty crimes such as purse snatchings in the past, but who, according to Berry, had never committed any acts of physical violence (Tr. 1144). The dispatch did not suggest that a weapon had been used in the robbery and did not describe the suspect as being armed. Indeed a subsequent dispatch reported that a “sneak thief” had taken some money and that an employee had managed to retrieve it after giving chase.1
*199Berry, accompanied by Klepfer, drove a Scout to investigate the robbery and saw Duckworth and Ronald, both blacks, in the 1969 Cadillac in a parking lot behind a bank near Ziggy’s. Berry assertedly recognized Duckworth from 100 feet away. The Cadillac began to exit the bank parking lot at 5 miles per hour. Berry signaled the Cadillac to stop a few feet to the left of the driver’s side of the police car. Berry dangerously chose to confront the car head-on instead of following recognized procedures for a proper felony stop, whereby he would have positioned his car behind the Cadillac, allowing him to observe the occupants and radio for assistance from other officers if necessary. The Cadillac obeyed all of Berry’s commands, but the officers drew their guns on a hunch that Duckworth was the Ziggy store robber. Berry pointed his gun at the Cadillac and ordered Ronald and Duckworth to raise their hands, which they did.2 As Berry approached the Cadillac on foot, Ronald, who was the driver, moved his right hand, which was clenched in a tight fist, toward the center of his chest. Berry testified that Ronald placed his right hand into the left inside pocket of his coat (Tr. 1132-1133). At that point, Berry fired his gun directly at Ronald’s left temple, killing him instantly. Berry later claimed that he thought Ronald was pulling a gun when Ronald, who was unarmed, put his hand into his jacket.3 Instead he was apparently reaching for a driver’s license. Berry purportedly believed that Ronald might have been an armed robber because he was in Duckworth’s company.
The following October Berry resigned from the Joliet Police Department, although he had not been disciplined with
Whether this was a reasonable inference to draw was a question of fact for the jury. respect to the Sherrod killing. Both plaintiff and his wife requested Chief Breen to delete a reference on Ronald’s death certificate which implied that he was a suspected robber and to discipline Berry. After several visits from Mrs. Sherrod, Breen threatened her with arrest if she tried to see him again. Being unsuccessful with Breen or the City, plaintiff filed this suit.
Before the incident involved in this lawsuit, Berry had been involved in other incidents in which he resorted to violence. On one occasion, Berry beat a suspect over the head with a flashlight and on another occasion he shot at a suspected burglar. Defendant police chief Breen told Berry after the second incident that he should not have used deadly force on the suspect. On still another occasion, Berry pulled his revolver while engaged in an altercation with his future brother-in-law. No serious sanctions were ever imposed on Berry as a result of these other incidents, although Chief Breen testified that Berry was suspended for 10 days following the incident with his brother-in-law.
The defendant police chief testified in detail about the training given to police officers, and the disciplinary procedures that were in place to deal with misconduct by officers. As to the incident involving Ronald, Breen testified that it might have been appropriate for a police officer to shoot a robbery suspect if the suspect moved his hand into his pocket, as Ronald allegedly did, and if the officer feared for his life or that of another. He concluded that Berry had been justified in shooting Ronald.
*200Expert testimony was received in evidence to show that Ronald’s death was caused because Breen and Joliet did not curb the use of excessive force by its police officers and did not maintain proper procedures for vehicle stops. According to the testimony, Berry should have radioed the Police Department before acting on his own. He also should have consulted fellow officer Klepfer and should not have arranged a head-on stop with the Cadillac.
Economist Stanley Smith testified that Lucien Sherrod’s opportunity loss due to Ronald’s death was $513,000 because he was unable to take his planned retirement. He also testified that the economic loss to Ronald’s estate was $598,000 and that the “hedonic [pleasurable] value” of Ronald’s shortened life caused his estate a $1.5 million to $15 million social loss. As already noted, the jury awarded much lesser amounts.
Judge Leighton filed an opinion allowing Ronald’s estate to recover for the value of his life, relying on various authorities. Sherrod v. Berry, 629 F.Supp. 159 (N.D.Ill. 1985). The opinion upheld economist Stanley Smith’s testimony concerning the hedonic or pleasurable value of a person’s life.
I. Jury Instructions
Plaintiff asserts that defendants have waived their right to object to various jury instructions by failing to comply with the specific procedures of Rule 51 of the Federal Rules of Civil Procedure. However, the defendants did file their proposed instructions with the court and the trial judge told them that it would be satisfactory if they merely named those whose refusal they protested instead of cataloguing their objections (Tr. 1704-1705). Subsequently he filed an extensive memorandum opinion disposing of the refusal to give certain defense instructions and the giving of certain plaintiff’s instructions (Defendants’ App. 1-53). Plaintiff has fully responded to the alleged errors raised by the defendants (Br. 27-39). Under these circumstances defendants should not be penalized by having their objections deemed waived. Bowley v. Stotler & Co., 751 F.2d 641, 646-647 (3d Cir.1985); Stewart v. Ford Motor Co., 553 F.2d 130, 140 (D.C.Cir. 1977); Irvin Jacobs & Co. v. Fidelity & Deposit Co. of Maryland, 202 F.2d 794, 801 (7th Cir.1953). We therefore turn to the merits of the claims of error with respect to the instructions.
Defendants contend that the trial court committed reversible error in refusing to give four instructions. In pertinent part their tendered instruction No. 26 would have told the jurors that if the defendants were merely negligent, the jury must find in their favor. The district court refused to accept that argument by stating that the evidence showed Berry deliberately drew his pistol and intentionally shot Ronald, so that a negligence instruction would have been improper as to Berry. The judge added that the other defendants had made no attempt to prove that they had acted only negligently, instead of intentionally and deliberately, so that the instruction would also have been improper as to them. Moreover, the trial judge gave over objection defendants’ instruction No. 8 which provided that municipal and supervisory liability could be based only on a finding that the City established and maintained a policy or custom evidencing deliberate indifference or gross negligence in training, supervising, and disciplining its police officers, and that such policy or custom was the proximate cause of Ronald’s death. Defendants have not been able to convince us that their case was defended on a negligence theory. Plaintiff’s case was tried on a theory of intentional, wanton or reckless conduct. Since the record shows that negligence was not put in issue, instruction No. 26 was entirely inappropriate.
The dissent argues at great length, based on the dissenting opinion in City of Springfield v. Kibbe, — U.S. -, 107 S.Ct. 1114, 1116, 94 L.Ed.2d 293, that defendants’ instruction No. 8, which was given verbatim, misled the jury concerning the degree of liability required to hold a municipality liable under § 1983. The dissent fails to mention, however, that this instruc*201tion was proposed by the defendants; The majority of the Supreme Court in Kibbe refused to review a jury instruction to the effect that a municipality could be held liable under § 1983 if it was grossly negligent in training its police officers because the defendant City did not object to the instruction and indeed proposed its own instruction to the same effect. Id. at 1115. The Court concluded that “there would be considerable prudential objection to reversing a judgment because of instructions that the petitioner accepted, and indeed itself requested.” Id. at 1116. In this Circuit, it “is well-settled law that a party cannot complain of errors which it has committed, invited, induced the court to make, or to which it consented.” International Travelers Cheque Co. v. Bankamerica Corp., 660 F.2d 215, 224 (7th Cir.1981). Accordingly, a party who requests an instruction .cannot complain about error if the instruction is subsequently given. See Williamson v. Handy Button Machine Co., 817 F.2d 1290, 1295-1296 (7th Cir.1987); Deland v. Old Republic Life Ins. Co., 758 F.2d 1331, 1337 (9th Cir.1985); 9 Wright & Miller, Federal Practice & Procedure § 2258, at 675-676 (noting, however, that a party who later recognizes an error in a requested instruction may seek to rectify that error by objecting when the instruction is given). The dissent’s argument about any error in defendants’ instruction No. 8 is thus improper.
Moreover, we are also troubled by the dissent’s legal argument that municipal liability under § 1983 may not be based on gross negligence. Although the four dissenters in Kibbe did indeed hold that a municipality’s liability under § 1983 for inadequate training must be premised on a finding of “deliberate indifference or reckless disregard” and not “negligence ... alone,” 107 S.Ct. at 1121, that view has yet to be adopted by a majority of the Court. In Daniels v. Williams, 474 U.S. 327, 106 S.Ct. 662, 88 L.Ed.2d 662, also cited by the dissent, the Supreme Court expressly held that “this case affords us no occasion to consider whether something less than intentional conduct, such as recklessness or ‘gross negligence,’ is enough to trigger the protections of the Due Process Clause.” Id. at 667 n. 3. Furthermore, this Circuit, while recognizing that something more than “mere negligence” is required to hold a municipality liable under § 1983, has yet to reject “gross negligence” as a basis for municipal liability. See Jones v. City of Chicago, 787 F.2d 200, 203-206 (7th Cir. 1986). We therefore think that the dissent is on rather dubious ground in arguing that reversal is required based on the position taken by the dissenters in Kibbe.
Defendants’ instruction No. 30 was tendered to tell the jury that police officer Berry was entitled “to use any force which he reasonably believes to be necessary to effect any arrest and any force which he reasonably believes to be necessary to defend himself or another from bodily harm while making an arrest.” The remainder of the instruction thrice refers to arrest. Here there is no claim that Berry was attempting to arrest Ronald or Duckworth, making the instruction inapt. Furthermore, the instruction was not an accurate statement of the law since the Supreme Court in Tennessee v. Garner, 471 U.S. 1, 105 S.Ct. 1694, 85 L.Ed.2d 1, held unconstitutional a Tennessee statute which provided that an officer may use all necessary means to effect an arrest. Instead, Judge Leighton gave defendants’ instruction No. 2 which correctly guided the jury as to a police officer’s use of deadly force. That instruction stated:
If you find that at the time in question Willie Berry shot and killed Ronald Sherrod that he reasonably believed that the use of such force was necessary to prevent death or great bodily harm to himself, then you must find in favor of all of the defendants, regardless of your finding concerning Frederick Breen and the City of Joliet.
The refusal to give defendants’ instruction No. 30 was not erroneous. See, e.g., Richardson v. City of Indianapolis, 658 F.2d 494, 502 n. 2 (7th Cir.1981), certiorari denied, 455 U.S. 945, 102 S.Ct. 1442, 71 L.Ed.2d 657; Brandes v. Burbank, 613 F.2d 658, 669 (7th Cir.1980).
*202Defendants next claim that Judge Leighton should have told the jury through their proposed instruction No. 31 that the police chief and City of Joliet’s failure to discipline Berry for killing Ronald was insufficient by itself to show that they maintained a policy of excessive force. This instruction was refused on the ground that it was argumentative because it omitted to mention their liability for failing to discipline Berry as well as other Joliet police officers in other instances of excessive force. See Spesco, Inc. v. General Electric Co., 719 F.2d 233, 239 (7th Cir.1983). As the trial judge pointed out, Berry’s shooting of Ronald did not stand alone as a basis for determining the liability of the City and the police chief. The plaintiff had proved a series of incidents involving excessive force by Joliet police officers, all of which were known to the police chief and the City and which resulted in little or no disciplinary action. Thus the defendants’ instruction lacked an evidentiary basis. Moreover, the tendered instruction is an erroneous statement of the law, for a single incident can under appropriate circumstances give rise to an inference of an actionable municipal policy under § 1983. Pembaur v. City of Cincinnati, 475 U.S. 469, 106 S.Ct. 1292, 1298-1300, 89 L.Ed.2d 452. Finally, defendants’ instruction No. 8 was an adequate substitute because it told the jury in part, in accordance with the defendants’ theory, that
[sjhowing that individual officers violated a person’s constitutional rights on an isolated occasion is not enough by itself to show that adequate supervisory procedures were not provided.
Defendants’ final claim of instruction error was the refusal to give their instruction No. 10 and the giving of plaintiff’s instruction No. 28.1. Number 10 provided:
Evidence of the alleged prior misconduct of Defendant Willie Berry can be used only to show the prior knowledge of Defendants Frederick Breen and the City of Joliet and is not to be considered in determining the liability of Willie Berry for the shooting of Ronald Sherrod.
Number 28.1 provided:
Evidence as to any misconduct by Willie Berry prior to the killing of Ronald Sherrod can be considered only for the purpose of determining liability on the part of Fred Breen and/or the City of Joliet based on any notice or knowledge on their part as to any propensity of Willie Berry towards the use of excessive force. However, for the purpose of determining whether Willie Berry reasonably believed that the use of deadly force was necessary to prevent death or great bodily harm to himself or another under the circumstances of this case, you may consider any prior reprimands Willie Berry may have received with regards to his use of force before December 8, 1979.
The two instructions parallel each other. Thus refused No. 10 and the first part of given No. 28.1 are virtually the same, making No. 10 unnecessary as redundant. The second part of No. 28.1 suggested that prior reprimands would be germane to determining the reasonableness of Berry’s belief that deadly force was necessary at the time he shot Ronald. The defendants object that the standard for determining whether the use of force is justified is a purely objective one. Hence they argue that Berry’s state of mind at the time he shot Ronald is immaterial to whether a reasonable person would conclude that deadly force was necessary to prevent imminent death or great bodily harm to himself or another. Judge Leighton believed that it was proper to instruct the jury on the factors to be considered in determining whether Berry’s use of deadly force in shooting Ronald was necessary under the circumstances. He thought that the existence of prior reprimands would be relevant to whether a reasonable person in Berry’s position should have known that the kind of deadly force to which Berry resorted in the Sherrod incident was unwarranted.
Defendants are correct in arguing that the appropriate standard is an objective one. To the extent that the instruction attempted to interject a subjective element *203into the excessive force inquiry, it is indeed problematic. However, in reviewing the adequacy of jury instructions in a civil trial, this Court “must look to the instructions as a whole, in a common sense manner, avoiding fastidiousness, inquiring whether the correct message was conveyed to the jury reasonably well.” See Wilk v. American Medical Ass’n, 719 F.2d 207, 218 (7th Cir. 1983) (quoted in Ramsey v. American Air Filter Co., 772 F.2d 1303, 1312 (7th Cir. 1985)). Moreover, “[e]ven if we should discern error in one or more instructions, we will not reverse a judgment — especially after [a lengthy] trial — unless we are persuaded the jury’s understanding of the issue was seriously affected, to the prejudice of the [complaining party].” Wilk, 719 F.2d at 218-219; see Beard v. Mitchell, 604 F.2d 485, 498 (7th Cir.1979).
As set out above, the jury was instructed at the defendants’ request to return a verdict in favor of the defendants if it found “that at the time in question Willie Berry shot and killed Ronald Sherrod that he reasonably believed that the use of force was necessary to prevent death or great bodily harm to himself.” (Defendants’ instruction No. 2). They were also instructed not to consider evidence as to any prior misconduct by Berry in determining his individual liability. Although Judge Leigh-ton should not have instructed the jury to consider any prior reprimands which Berry may have received with regard to his use of force, the evidence presented at the trial indicated that Berry had been reprimanded only once prior to the Sherrod incident and, as just recounted, the jury was expressly told to disregard evidence of the misconduct leading to the reprimand. After viewing the instructions as a whole in light of the facts of this case and the evidence presented, we cannot conclude that this error was so prejudicial to the defendants as to constitute reversible error requiring a new trial. See Brandes v. Burbank, 613 F.2d 658, 668-669 (7th Cir.1980).
II. Evidentiary Rulings
Defendants have asked us to review three evidentiary rulings of the trial court. The plaintiff contends that the defendants have waived their right to challenge these rulings on appeal because they did not object to the admission of the evidence at trial. These matters, however, were raised in defendants’ motions in limine which sought to exclude: (1) the fact that Ronald was unarmed; (2) the lawsuits by Thompson and Bucciarelli against Joliet police officers, and (3) economist Stanley Smith’s testimony as to the value of life. All are treated in the district judge’s opinion overruling the new trial motion (Defendants’ App. 37-39, 35-37 and 24-25) and were argued on the first day of trial. With this background, it was unnecessary under Rule 46 of the Federal Rules of Civil Procedure for defendants to have renewed their objections at the time the evidence was admitted. Thronson v. Meisels, 800 F.2d 136, 142 (7th Cir.1986); Cook v. Hoppin, 783 F.2d 684, 691 n. 2 (7th Cir.1986); American Home Assur. Co. v. Sunshine Supermarket, Inc., 753 F.2d 321, 324-325 (3d Cir.1985); Sprynczynatyk v. General Motors Corp., 771 F.2d 1112, 1118-1119 (8th Cir.1985), certiorari denied, 475 U.S. 1046, 106 S.Ct. 1263, 89 L.Ed.2d 572; Sheehy v. Southern Pac. Trans. Co., 631 F.2d 649, 652-653 (9th Cir.1980); contra Collins v. Wayne Corp., 621 F.2d 777, 784 (5th Cir.1980). There was no waiver.
A. Ronald was not armed
Defendants sought to exclude evidence indicating that Ronald was unarmed at the time of the shooting. They claim that such evidence is not relevant to whether Berry reasonably believed that the use of deadly force was justified at the time of the shooting and that its admission was highly prejudicial to Berry.
The district court ruled that evidence as to whether Ronald was armed was both relevant and material to determining whether Berry acted reasonably when he shot and killed Ronald. Berry defended his shooting on the ground that Ronald made a “furtive” movement of his hand toward a pocket of his coat, leading him to infer that Ronald was armed and hence presented a danger to Berry. The district court was *204concerned that excluding the evidence which showed that Ronald was unarmed would unfairly prejudice the plaintiff by allowing the jury to labor under the false assumption that Ronald was in fact reaching for a gun with which to shoot Berry.
It is well settled in this Circuit that “[t]he district court has broad discretion to determine the admissibility of evidence, and thus [the Court of Appeals] will reverse the court’s evidentiary rulings only upon a clear showing of abuse of discretion.” United States v. Garver, 809 F.2d 1291, 1297 (7th Cir.1987); Davis v. Lane, 814 F.2d 397, 399 (7th Cir.1987). It is apparent that one of the parties would have suffered some prejudice regardless of what decision the district court reached regarding admissibility. Under these circumstances we are particularly hesitant to second-guess the district court’s balancing of prejudice and probative value. See, e.g., Davis, 814 F.2d at 399; Kier v. Commercial Union Insurance Cos., 808 F.2d 1254, 1258 (7th Cir. 1987); West v. Love, 776 F.2d 170, 174 (7th Cir.1985).
Defendants argue that any prejudice to the plaintiff resulting from exclusion of the evidence demonstrating that Ronald was unarmed could have been cured by a cautionary instruction to the effect that whether or not Ronald was armed was irrelevant to the determination of the reasonableness issue. However, defendants could just as easily have requested a similar instruction to minimize the prejudice to them of admitting the evidence. No such instruction was requested. Furthermore, as discussed in Part I supra, the jury was instructed, in accordance with Tennessee v. Garner, 471 U.S. 1, 3, 105 S.Ct. 1694, 1697, 85 L.Ed.2d 1, to find in favor of the defendants if “at the time in question Willie Berry shot Ronald Sherrod he reasonably believed that the use of such force was necessary to prevent death or great bodily harm to himself.” (Defendants’ instruction No. 2). Finally, we consider defendants’ entire line of argument to be somewhat disingenuous since if Ronald had been armed, we are confident that they would have insisted on introducing evidence proving that fact so that the jury would not unfairly speculate that Ronald was unarmed. See Davis v. Lane, 814 F.2d 397, 399 (7th Cir.1987) (fact that prison inmate was armed with a knife was properly admitted in § 1983 action against guard who shot him).
By allowing into evidence the fact that Ronald was unarmed, the district court reasonably resolved a difficult and close issue. We cannot say that its resolution was a clear abuse of discretion.4
B. The lawsuits against Berry and Joliet police officers
Evidence was admitted concerning the lawsuit by Michael Thompson, the suspect whom Berry had beaten over the head with a flashlight, against Berry, the Joliet Police Department, and the City of Joliet, and another lawsuit by a plaintiff named Bucciarelli against several other Joliet police officers, which was the result of an incident which occurred one month after the shooting of Ronald Sherrod. Defendants contend that this evidence was not *205probative of any issue in the case and was highly prejudicial. The district court admitted evidence of these matters for the limited purpose of showing that Breen and the City of Joliet knew that Berry and other police officers had been charged with the use of excessive force, belying the assertion of proper training and corrective action by Chief Breen and the City (Defendants’ App. 36-37). This Court has consistently recognized that the “trial court’s balancing of probative value and unfair prejudice is highly discretionary and its decision on admissibility will be accorded ‘great deference.’ ” West v. Love, 776 F.2d 170, 174 (7th Cir.1985) (quoting United States v. Medina, 755 F.2d 1269, 1274 (7th Cir.1985)); see United States v. Beasley, 809 F.2d 1273, 1278 (7th Cir.1987); United States v. Laughlin, 772 F.2d 1382, 1392 (7th Cir.1985). No error was involved in admitting this testimony.
The dissent objects to the admission of the fact of the Bucciarelli lawsuit on the ground that the incident giving rise to the suit occurred one month after Berry shot Ronald and hence was irrelevant to determining whether the City and the chief of police maintained an improper policy regarding the use of force at the time of Ronald’s death. Although the defendants on appeal complained about the admission of the fact of the Bucciarelli suit because it “had no probative value whatsoever regarding any issue in this lawsuit” (Def. Br. 37), they never specifically challenged its admission on the ground seized upon by the dissent. The dissent cannot urge reversal on a ground that has been waived. See Ohio Casualty Insurance Co. v. Bazzi Construction Co., 815 F.2d 1146, 1149 (7th Cir.1987); National Fidelity Life Insurance Co. v. Karaganis, 811 F.2d 357, 360 (7th Cir.1987).
Furthermore, the case upon which the dissent relies, Magayanes v. Terrance, 739 F.2d 1131 (7th Cir.1983), did not involve municipal liability under § 1983. In Magayanes, this Court held that evidence of an incident occurring six months after the one at issue in the case was not relevant to whether the defendants at that earlier time had notice that the design of a police vehicle used to transport prisoners was defective. In contrast, in a case specifically involving municipal liability, the Fifth Circuit held that subsequent conduct by a municipal policymaker may be used to prove preexisting disposition and policy. Grandstaff v. City of Borger, 767 F.2d 161, 171 (5th Cir.1985), certiorari denied, — U.S.-, 107 S.Ct. 1369, 94 L.Ed.2d 6860. The court emphasized the difficulties that a § 1983 plaintiff is likely to encounter in attempting to prove the existence of an official municipal policy or custom at the time of the incident in which he was injured. Id.
C. Economist Smith’s testimony as to the value of Ronald’s life
The defendants also contend that the trial court erred in admitting the expert testimony of Stanley Smith, an economist, as to the economic value of a lost human life. In essence defendants object that the testimony was too speculative to be admissible. As will be discussed further below, it is well settled in this Circuit that § 1983 permits recovery on behalf of the victim’s estate for the loss of life. Bell v. City of Milwaukee, 746 F.2d 1205 (7th Cir.1984); Bass v. Wallenstein, 769 F.2d 1173 (7th Cir.1985). It is therefore axiomatic that plaintiffs seeking to recover the value of a decedent’s life must be entitled to submit expert testimony to help guide the jury in reaching an appropriate damages award.
Mr. Smith testified as to how a life is valued in the field of economics. Smith was well qualified to discuss this matter, and indeed defendants did not question his qualifications at trial. Sherrod v. Berry, 629 F.Supp. 159, 162 (N.D.Ill.1985). The defendants certainly could and should have submitted contrary expert testimony. They failed to do so and therefore cannot be heard now to complain that the plaintiff’s expert's testimony was too speculative or somehow unreliable.
The trial judge ably addressed defendants’ arguments contesting the admissibility of Mr. Smith’s testimony concerning the hedonic value of human life, 629 F.Supp. 159 (N.D.Ill.1985). Judge Leighton ex*206plained that the rule against recovery of “speculative damages” applies where it is uncertain whether the defendant caused the damages, or whether the damages flowed from his act. The fact that the measure or extent of the injury suffered may be uncertain does not bar recovery. Judge Leighton concluded, “The fact that the hedonic value of a human life is difficult to measure did not make either Smith’s testimony or the damages speculative.” 629 F.Supp. at 164.
The testimony of expert economist Stanley Smith was invaluable to the jury in enabling it to perform its function of determining the most accurate and probable estimate of the damages recoverable for the hedonic value of Ronald’s life. The trial court committed no error by admitting that testimony.
D. Berry’s prior police misconduct
Defendants contend that evidence of Berry’s prior police conduct involving his use of excessive force in three other incidents was wrongly admitted. We cannot agree. Municipal liability under § 1983 is premised on the plaintiff’s proving that his constitutional rights were violated as a result of an official municipal policy or custom. Monell v. Department of Social Services, 436 U.S. 658, 690-691, 694, 98 S.Ct. 2018, 2036, 2037, 56 L.Ed.2d 611. The Supreme Court has recently indicated that “[pjroof of a single incident of unconstitutional activity is not sufficient to impose liability under Monell, unless proof of the incident includes proof that it was caused by an existing, unconstitutional municipal policy, which policy can be attributed to a municipal policymaker.” Oklahoma City v. Tuttle, 471 U.S. 808, 105 S.Ct. 2427, 2436, 85 L.Ed.2d 791. Thus a plaintiff must prove a specific pattern of conduct or series of incidents violative of constitutional rights in order to establish the existence of a municipal policy or custom. See Hossman v. Blunk, 784 F.2d 793, 796-797 (7th Cir.1986); Grandstaff v. City of Borger, 767 F.2d 161, 171 (5th Cir.1985); Strauss v. City of Chicago, 760 F.2d 765, 767-769 (7th Cir.1985).
Defendants complain that several of the incidents admitted into evidence by the plaintiffs to establish a municipal policy involved Officer Berry. While this evidence was admittedly prejudicial with respect to Berry, it was highly probative as to the existence of a municipal policy of Joliet to maintain on its police force officers who consistently used excessive force in discharging their official duties, the touchstone of municipal and supervisory liability under § 1983. The trial court admitted the evidence of the prior incidents only against Chief Breen and the City to show that they had knowledge of Berry’s prior use of excessive force and failed to take adequate corrective action (Tr. 220-223, 463; Defendants’ App. 29-35). The incidents were never proved against Berry. As requested, a cautionary instruction was given at the time the evidence was admitted (Tr. 222-223, 463), and a final limiting instruction, plaintiff’s instruction No. 28.1 set out above in Part I, was given to the jury at the close of the trial. As explained above, the trial court’s balancing of the probative value and prejudicial effect of evidence is highly discretionary. West v. Love, 776 F.2d at 174. In this setting the trial court did not abuse its discretion in admitting the three prior incidents involving Berry into evidence.
The dissent objects that evidence of one of the incidents involving Berry's prior use of force was improperly received against Berry himself. In this incident Berry shot at a suspected burglar named Waddell after an auto chase when Waddell emerged from his car in a crouched position with his hand in his coat. When evidence of this incident was introduced, Judge Leighton did not instruct the jury, as he had prior to the introduction of evidence of the other two incidents involving Berry, that the evidence was admissible only against the police chief and the City, and not against Berry individually. In addition, in a memorandum opinion disposing of the post-trial motions, Judge Leighton indicated that he believed that evidence of the Waddell incident could come in against Berry because it was so factually similar to the Sherrod incident that it showed a consistent pattern *207of conduct (Def.App. 35). However, at the conclusion of the trial, the jury was clearly instructed that:
Evidence as to any misconduct by Willie Berry prior to the killing of Ronald Sherrod can be considered only for the purpose of determining liability on the part of Fred Breen and/or the City of Joliet____
(Plaintiffs instruction No. 28.1). The instruction given made no exception for the Waddell incident. The jurors took a written copy of this final instruction into their deliberations and hence it was undoubtedly this instruction to which they referred for guidance in reaching their verdict. The importance of this final instruction is further evidenced by the fact that when Judge Leighton did give cautionary instructions prior to the introduction of evidence of the other two incidents, he told the jury that they would understand the effect of the limitation contained in the cautionary instructions only at the conclusion of the trial after he had instructed them on the law.
Consequently, we believe that this final limiting instruction sufficiently countered any confusion which might have existed at the time the evidence of the Waddell incident was introduced and that the jury did not consider that incident in determining whether Berry had a reasonable belief that deadly force was necessary when he shot Ronald. Evidence of the Waddell incident was without question admissible against the police chief and the City to establish an official municipal policy or custom, and any residual prejudice to Berry as a result of its admission was not so great as to warrant reversal.
III. Loss of life is compensable to the deceased’s estate
The district court permitted the jury to consider awarding Ronald’s estate the value of his life. As defendants admit (Br. 44), this ruling accords with our recent decisions in Bell v. City of Milwaukee, 746 F.2d 1205 (7th Cir.1984), and Bass v. Wallenstein, 769 F.2d 1173 (7th Cir.1985). Defendants have not persuaded us to overrule these authorities. See also R. Posner, Tort Law: Cases and Economic Analysis 121-122 (1982).
IV. Award to Luden for loss of parental association
Defendants concede that the parent-child relationship is a liberty interest protected by the Due Process Clause of the Fourteenth Amendment (Br. 44), see Bell, 746 F.2d at 1242-1245, but they assert its loss is not compensable absent a specific intent to sever it. This argument was waived because it was not raised in their post-trial motions nor in any proposed instruction requiring a showing of specific intent for a loss of parental association claim. In fact there was no objection to plaintiff’s instruction No. 19(a) which did not impose such a requirement.
Even if this argument were not waived, it is foreclosed by Bell and Bass, supra. In recognizing that the right to parental association was a liberty interest protected by the Due Process Clause, this Court did not impose any state of mind requirement necessary to a finding of a deprivation. Bell, 746 F.2d at 1243-1245. Defendants rely on Trujillo v. Board of County Comm’rs, 768 F.2d 1186 (10th Cir. 1985), in arguing that a specific intent to violate the right to parental association is required in order to state a claim under § 1983. Trujillo is distinguishable from Bell because the Tenth Circuit grounded the right to parental association in the First Amendment whereas we based the right on the Due Process Clause. Id. at 1189-1190. A First Amendment violation requires “proof that the state’s action was intended to repress an individual’s protected speech or association.” Id. at 1189. Consequently, the Tenth Circuit held that an allegation of specific intent to interfere with the parental relationship was required to establish a constitutional violation. Id. at 1190. The Fourteenth Amendment’s Due Process Clause, in contrast, contains no such requirement of specific intent. Id. (noting that Bell imposed no specific state of mind requirement to find deprivation of intimate associational rights). Despite the Tenth Circuit’s decision in Trujillo, we are *208bound by our decision in Bell and the plaintiff has sufficiently made out a violation under § 1983.
The defendants also argue that the Supreme Court’s recent decision in Daniels v. Williams, 474 U.S. 327, 106 S.Ct. 662, 88 L.Ed.2d 662, prohibits an award of damages to Lucien Sherrod for the loss of his right to parental association. In Daniels, the Court held that mere negligence cannot constitute a due process violation actionable under § 1983. Defendants argue that it is impossible to interfere intentionally with a parent-child relationship absent some knowledge of the existence of that relationship. However, the Supreme Court in Daniels acknowledged that liability under the Due Process Clause could be premised on recklessness or gross negligence. 106 S.Ct. at 667 n. 3. As discussed above in Part I, the record clearly indicates that the defendants acted recklessly and with gross negligence. Thus damages were properly awarded to Lucien Sherrod.
V. Evidence of business performance of Sherrod Auto Repair after Ronald’s death.
The district judge refused to admit evidence of the performance of Sherrod's auto repair shop after Ronald died. He did admit evidence showing the value of the auto repair business while Ronald worked there and contributed to its earnings, but held that events occurring there after Ronald’s death “were not probative of what the garage would have earned had he not been killed” (Defendants’ App. 43). There was no error in confining such evidence to Ronald’s lifetime because the evidence that was admitted provided a sufficient basis for calculating his projected contributions over his expected life span.
VI. Right to remittitur
As shown, Lucien Sherrod as administrator of Ronald’s estate recovered a total of $1,151,700. The jury awarded the estate $850,000 for the value of Ronald’s life, $300,000 for pecuniary loss to the estate, and $1,700 for funeral expenses. $450,000 was awarded Lucien individually for loss of parental association with Ronald under Count II. Defendants have made no real attempt to show, as required, whether the verdicts were monstrously excessive or so large as to shock the conscience of the court, thus justifying a remittitur. Mary Beth G. v. City of Chicago, 723 F.2d 1263, 1275 (7th Cir.1983). When asked at oral argument, the only suggestion of defendants’ counsel was to remit the $850,000 award, but they refused to provide the Court with any figures as to the approximate size of an appropriate award. The defendants’ brief attacked the entire Count I verdict of $1,151,700 and the Count II verdict of $450,000 (Br. 51), but again offered no guidance as to paring these amounts. The trial judge held that the verdicts were not excessive and that he would grant an additur if it were in his power (Defendants’ App. 52). In response, plaintiff has merely stated that any remittitur would be “offensive” (Br. 50).
In Memphis Community School Dist. v. Stachura, 477 U.S. 299, 106 S.Ct. 2537, 91 L.Ed.2d 249, a tenured public school seventh-grade teacher was suspended and later reinstated. He sued for damages under 42 U.S.C. § 1983. The jury found the defendant School District and others liable for a deprivation of procedural due process and for a violation of plaintiffs First Amendment rights. Plaintiff was awarded $266,758 in compensatory damages and $36,000 in punitive damages. Under erroneous instructions, the compensatory damages were based on plaintiff’s actual injury and on the abstract value of certain constitutional rights. In remanding for a new trial, the Supreme Court held that the appropriate level of damages for violations of constitutional rights should ordinarily be determined according to common law tort principles. It reaffirmed its ruling in Carey v. Piphus, 435 U.S. 247, 98 S.Ct. 1042, 55 L.Ed.2d 252, that no compensatory damages can be awarded for infringement of constitutional rights absent proof of actual injury. Damages must only compensate for actual harm.
In Stachura, the instructions “called on the jury to measure damages based on a *209subjective evaluation of the importance of particular constitutional values.” 106 S.Ct. at 2545. There were no such instructions in the present case. Likewise the damages here were awarded to compensate the plaintiff in both his capacities (as administrator and individually) for actual injuries caused by a constitutional violation. Therefore Stachura does not require a remand.
If we were the jurors, our award might have been substantially less than the $1,601,700 rendered, but we can take judicial notice that many higher awards have been sustained in loss of life cases. In this case there has been a complete absence of any suggestions at either the oral argument or in defendants’ principal brief as to the proper amount of a remittitur. The district court believed that none was proper. In their reply brief, defendants seem to say there should have been no allowance for the loss of Ronald’s life (for which the jury awarded $850,000) and no allowance for loss of parental association (for which the jury awarded $450,000) (Reply Br. 17-18). We cannot agree with such “all or nothing” arguments and therefore no remittitur will be directed.
VII. Sufficiency of evidence as to Police Chiefs and City’s liability
Defendants’ final argument is that for want of evidence, the judgments against Chief Breen and the City cannot stand. It is noteworthy that defendants’ reply brief makes no effort to support this argument.
In his memorandum opinion of November 15, 1985, Judge Leighton summarized the evidence as to Breen and the City as follows (Defendants’ App. 49-51):
In this case, the facts are entirely different [from those in People v. Lenard, 79 Ill.App.3d 1046 [35 Ill.Dec. 104, 398 N.E.2d 1054] (1st Dist.1979)]. There was evidence, much of it the testimony of Berry, the defendant Breen, and police officers of the City of Joliet, supported by records of the defendant City, which proved to the jury that Willie Berry was a violence-prone policeman who, on a number of occasions, had used excessive force, in fact, lethal violence on citizens with whom he came in contact; that this proneness for violence culminated in the killing of Ronald Sherrod by Berry on December 8, 1979. There was evidence that the chief of police, Breen, knew of Berry’s acts of violence, but he did not take steps to correct them. All of the facts concerning Berry’s conduct were recorded in documents in the possession of Breen and the defendant municipality. These facts were known to the corporation counsel of the city, who as the city’s lawyer, came in contact with police officers like Berry and learned of incidents involving the use of excessive force by Joliet police officers.
Finally, this case is to be distinguished from Lenard. The evidence in the record, on which the verdicts were returned and the 16 special interrogatories were answered in the affirmative, was such that the jury could have and did find that the Qity of Joliet maintained a policy of employing a police force with police officers like Berry, and others, who regularly used excessive force and lethal violence in the discharge of their duties; and that Breen, the police chief, executed the policy of the city. Dr. Fyfe, an expert on police practices testified, as he had in the first trial, without being contradicted or refuted, that from records in the possession of Breen and the city, he was able to conclude that of the 54 largest cities in the United States, Joliet had the highest rate of incidents in which police officers fired their guns at citizens in the course of discharging their duties. Breen was present at both trials, and testified in both. Yet he never said a word either questioning or denying the assertions of Dr. Fyfe; and when Dr. Fyfe was cross-examined, Tr. 746-886, defense counsel never asked him about this aspect of his testimony. In sum, what this widely respected expert told the jury was that Joliet, an obscure Illinois municipality, had earned the dubious distinction of being the city, among the 54 largest in the country, where citizens could expect to be fired on by police officers with greater regularity than anywhere else in the country.
*210For these reasons, and without dwelling more on the ample evidence in the record, this court concludes that the jury’s verdicts finding that Breen and the City of Joliet maintained a police force that used excessive and lethal force against citizens are supported by evidence as required by law. The [affirmative] answers to the 16 special interrogatories [reproduced in Defendants’ App. 65 and in Plaintiff’s Supp.App. 7-21] find support in the evidence; the jury found in them that Breen and the City of Joliet maintained Berry and other police officers who used excessive force on citizens; and appropriate ones found that the policy and practice of Breen and the City of Joliet were factors that caused the death of Ronald Sherrod on December 8, 1979.
That is a fair summing up of the evidence which we adopt as our own. Defendants’ sufficiency argument must be rejected.
The judgments are affirmed.5
. The dissent makes much of the fact that Berry believed he was responding to a "stick-up” or armed robbery. It is important to note that none of the radio dispatches ever used the words "stick-up" or "armed robbery." Rather Berry testified that he inferred from the dispatches that there had been a "stick-up.”
. The dissent emphasizes that it took three commands before Ronald and Gary Duckworth raised their hands in the air. It is not clear from the record how much time elapsed between each of these commands. Officer Klepfer testified that Berry was screaming at them to "get their motherfucking hands up" (Tr. 406-407). Moreover, Berry himself testified that Ronald and Gary were sitting "peacefully” in the car and gave no indication that they would offer any resistance at all (Tr. 502, 1203-1204).
. The dissent recounts that Officer Klepfer testified that Sherrod made a "quick” movement with his hand into his coat and that there was no doubt in Klepfer’s mind that when Ronald started to move, “he was going to reach for a weapon of some type.” On cross-examination, however, Klepfer admitted that Ronald’s movement was not “a sudden movement” and that "it wasn’t as if someone were reaching for something of some type” (Tr. 1263-1265). Klepfer also testified that he did not feel any particular danger when he saw Ronald’s hand move (Tr. 1280-1281).
. In its discussion of this issue, the dissent cites two cases, Young v. City of Killeen, 775 F.2d 1349 (5th Cir.1985), and Davis v. Freels, 583 F.2d 337 (7th Cir.1978), neither of which discusses the admissibility of evidence regarding whether the victim of a police shooting was armed or unarmed. Nevertheless, the dissent claims that our treatment of the present case is in conflict with these cases. In Tennessee v. Garner, 471 U.S. 1, 105 S.Ct. 1694, 85 L.Ed.2d 1, the Supreme Court held that a police officer may use deadly force to prevent the escape of an apparently unarmed suspected felon only if the officer has probable cause to believe that the suspect poses a significant threat of death or serious physical injury to the officer or others. Whether the officer has such probable cause is a question of fact, which the jury in this case answered in the negative. By contrast, in Young after a trial before the court, the district judge found that the defendant officer was justified in shooting the plaintiff as a result of the plaintiffs sudden movements but erroneously concluded that the officer had created a dangerous situation by his disregard of prudent police procedure and hence could not escape liability for the plaintiffs death. Similarly, in Davis, the jury found that the defendant possessed a reasonable belief that the plaintiff posed an imminent danger to him of death or bodily harm. The jury here simply did not find that "[Ronald's] movements gave [Berry] cause to believe that there was a threat of serious physical harm." Young, 775 F.2d at 1353.
. The dissent charges that by upholding a jury verdict finding that Berry did not have a reasonable belief that the use of deadly force against Ronald was necessary to prevent death or great bodily harm to himself or others, we are handcuffing law enforcement officers in their efforts to protect the public. We of course agree with the dissent that police officers perform a vital function in our society and often must perform this function in conditions of constant peril. Nevertheless, this fact does not give them license to engage in lawless behavior. As the Fifth Circuit aptly recognized in Grandstaff v. City of Borger, 767 F.2d 161, 166 (5th Cir.1985), certiorari denied, — U.S.-, 107 S.Ct. 1369, 94 L.Ed.2d 686:
Peace officers stand at the front of law and the ordering processes of society. They restrain the violator, protect the compliant, and represent constituted authority in the scenes of both peace and turbulence of community life. We depend heavily upon their skill and disposition. They deserve and require the understanding and support of judges as well as of all citizens. Where any officer fails— whether for lack of courage, judgment, integrity, or humaneness — all the community suffers. We suffer because vested authority has failed to prevent some harm or because authority has been sullied and abused. With any abuse of authority the entire ordering process is weakened. The public trusts the entire process less, and antagonism to all figures of authority rises. No one should be more alert to the cost of failure than responsible law enforcement officers and we who work in the courts. We must do what we can to avoid the failures, to prevent their reoccurrence, and — at all times — stay true to the requisites of honesty and accountability imposed upon all who are at once representative of the law and subject to it.