This appeal follows the jury trial of a civil rights action under 42 U.S.C. § 1983. Plaintiffs, prisoners incarcerated at the Missouri Training Center for Men (MTCM) in Moberly, Missouri, sued MTCM and Missouri State Penitentiary (MSP) officials, claiming that beatings by prison guards following a prison riot violated plaintiffs’ constitutional rights. After a five-day trial, the jury returned a verdict for plaintiffs, awarding both compensatory and punitive damages. We affirm the district court1 in all respects.
I.
On July 3, 1983, thirty to thirty-five inmates at the MTCM forced their way into the rotunda of Housing Unit # 2 (HU# 2) where they fought correctional officers who were attempting to remove an intoxicated prisoner from Wing B of HU# 2 to the rotunda. During the fight one officer was fatally stabbed and several other officers were wounded. When the disturbance quieted down, prison officials deadlocked the inmates in their rooms, administered medical treatment to those prisoners requiring it, and searched the rooms for weapons and intoxicating substances. The following day, correctional officers bused prisoners allegedly involved in the incident to the state penitentiary.
*1346The constitutional violations complained of in plaintiffs’ civil rights suit did not occur during the disturbance, but after prison officials had restored order to the facility. According to plaintiffs’ testimony, the assaults occurred after the disturbance had subsided and the inmates had been returned to and locked down in their cells. Plaintiffs testified that they were removed from their cells by two or more officers, thrown to the floor, kicked and hit, handcuffed and taken to the rotunda. In the rotunda, prison officials forced plaintiffs to lie on their stomachs over a laundry table while officers kicked them and beat them with night sticks, fists and slap jacks. Prison officials then removed plaintiffs from the rotunda. Plaintiff McWorthy was taken to another location in HU# 2 and was beaten on the way; plaintiffs Bolin and Wilson were taken to MTCM’s medical facility, where they were subjected to further beatings by officers. The next morning, plaintiffs and other inmates were transported by bus from MTCM to MSP, a distance of sixty miles. During that trip, prison guards inflicted further injury on the inmates.
Plaintiffs filed a complaint against R. Dale Riley, Assistant Director of the Division of Adult Institutions, Bill Armontrout, Associate Warden of MSP, and various supervisory and nonsupervisory personnel at both MTCM and MSP.2 The jury found in favor of plaintiffs, assessed damages against each defendant and set limits on the supervisory defendants’ joint and several liability. The district court affirmed the various damage awards, except for the damages awarded plaintiff Bolin against defendant Halley.3 Judgment was entered accordingly, reflecting the following total damage amounts for each plaintiff: Bolin —$9,500 actual and $56,000 punitive; McWorthy — $14,500 actual and $74,000 punitive; Wilson — $14,000 actual and $73,000 punitive.
The prison officials appeal from the judgment and from the district court’s order denying their motions for judgment n.o.v., or in the alternative, new trial, or remit-titur.
II.
Defendants raise six grounds for error:
(1) the district court erred in submitting claims against three supervisory officials because the evidence did not demonstrate their personal involvement in or tacit authorization of the excessive force;
(2) the district court erred with respect to the jury instruction on supervisor liability because the instruction failed to require a finding that a supervisor either knew of and authorized his subordinates’ acts or failed to take remedial steps to prevent the acts;
(3) the district court’s instructions erroneously placed the burden of proof on the prison officials;
(4) the district court erred in giving an excessive force instruction that did not require the jury to find the intentional infliction of unnecessary and wanton pain and suffering and damage to plaintiffs;
(5) the district court erred when it failed to conduct an evidentiary hearing regarding one juror’s failure to disclose essential information during voir dire; and
*1347(6) the district court erred in not reducing the punitive damage awards.
We address each of appellants’ arguments.
A. Sufficiency of Evidence
Plaintiffs predicate their claims against Riley, Armontrout and Halley on the failure of those defendants to protect plaintiffs from the excessive use of force by other corrections officers. Appellants maintain that the district court allowed submission and recovery of punitive damages against Riley, Armontrout and Halley on evidence supporting nothing more than a respondeat superior theory. Plaintiffs disagree.
It is well settled that the doctrine of respondeat superior is insufficient to allow recovery in a § 1983 action. Monell v. Department of Social Services, 436 U.S. 658, 694, 98 S.Ct. 2018, 2037, 56 L.Ed.2d 611 (1978); Harris v. Pirch, 677 F.2d 681, 685 (8th Cir.1982). This court has held that a cause of action predicated on a supervisor’s failure to supervise or control his subordinates may be maintained “only if [a defendant] demonstrated deliberate indifference or tacit authorization of the offensive acts.” Wilson v. City of North Little Rock, 801 F.2d 316, 322 (8th Cir.1986). We must analyze separately the case against each of the three defendants to determine the sufficiency of the evidence.
Defendant Riley was the Assistant Corrections Director at MTCM at the time of the riot. He came to MTCM at about 10:30 p.m. on July 3, after he learned of the disturbance. According to his testimony, he went to HU# 2 to check out the situation and stayed there observing for twenty to thirty minutes, during which time he “saw no reprisals taking place.” Riley admitted, however, that he saw plaintiff Roberts placed on the laundry table. Plaintiff McWorthy testified that he was placed on the laundry table next to Roberts and that both he and Roberts were severely beaten by defendant Williams. Furthermore, each inmate who testified described being taken to the laundry table in handcuffs, laid across the table and then beaten by the corrections officers. Alan Personette, a maintenance engineer at MTCM, was in the rotunda with Riley. Personette testified that he did not see the guards beating the inmates because he “didn’t want to see it.” While Riley was in the rotunda, Personette asked him, “Got a pair of blinders?” implying that they should ignore the beatings that were taking place.
The above testimony places Riley in clear view of the beatings taking place in the rotunda. Riley took no action; he did not give orders to stop the beatings or attempt to calm the situation. Riley’s conduct is an example of deliberate indifference to plaintiffs’ constitutional rights. The beatings at least of Roberts and McWorthy took place within Riley’s view, but he did nothing to stop the beatings. His action reflects tacit approval of the beatings as well as authorization to continue physical abuse of the plaintiffs.
Defendant Armontrout was the Associate Warden of MSP and commanded the emergency squad summoned to Moberly. The evidence establishes that Armontr-out knew or should have known that flaring tempers among the prison guards would lead them to retaliate against the inmates. During an interview with plaintiff Bolin, Armontrout made a reference to the possibility of retaliatory punishment by a reference to defendant Borghardt “getting out his black gloves,” alluding to Bolin getting a whipping from Armontrout. Ar-montrout took no steps to avoid the possibility of retaliatory punishment. On direct examination by plaintiffs’ counsel, Ar-montrout stated that he chose not to ride on the inmate bus and supervise it despite his knowledge that a guard had been killed and that the entire staff — including the officers riding with the inmates — was upset. This evidence supports an inference that Armontrout knew there could be trouble on the inmate bus and did nothing to prevent it. Armontrout also testified that he and others followed the inmate bus to assist the guards on that bus in the event of a disturbance. These facts give rise to an inference that Armontrout was available to exercise direct control over the guards on the inmate bus. As in the case of Riley, the *1348evidence adduced at trial supports the conclusion that Armontrout acted with, at minimum, deliberate indifference to the plaintiffs’ constitutional rights.
At the time of the laundry table beatings, Captain Halley was either the ranking officer at the institution or in control of HU# 2. Plaintiff Bolin testified that Halley was present when guards removed Bolin from his cell, pushed him to the ground and beat him up. Plaintiff McWorthy testified that either Halley or Dennis was present when he was removed from his cell. Halley himself testified that he saw five or six inmates brought to the rotunda and laid on the laundry table, and that he “gave the order that I didn’t want anybody killed.” His later statement to Highway Patrol investigators that “I’ve seen a lot more ass kicking than was done in there” implies that he knew of the beatings. In addition, Personette testified that he observed Halley strike an inmate with a baseball bat as the inmate (not one of the plaintiffs) was brought into the rotunda.
The above testimony is sufficient to support the jury’s finding that Halley actively participated in violating Bolin’s and McWorthy’s constitutional rights by inflicting physical injury. His direct abuse of some inmates demonstrated his deliberate disregard for plaintiffs’ constitutional rights and, by virtue of his supervisory authority, invited other corrections officers to follow his example. His conduct rose at least to the level of “tacit authorization.”
Based on our review of the record, we find the evidence sufficient to allow submission to the jury of plaintiffs’ claims against Riley, Armontrout and Halley, and to support the jury’s verdict against each of those defendants.
B. Jury Instructions
Í. Supervisory Liability
Appellants argue that the jury instructions were deficient because they did not require the jury to find that a supervisor knew of and authorized a subordinate's inappropriate acts or that the supervisor failed to take steps to prevent such conduct. The district court gave the following verdict directing instruction against all the supervisors:
Your verdict must be for Plaintiff _ and against Defendant _, on Plaintiff’s claim regarding the failure of Defendant_to protect Plaintiff_and to supervise his subordinates, if you believe:
First, that Plaintiff_was intentionally kicked, beaten or hit by an employee of the Missouri Department of Corrections, and
Second, the use of such force was excessive under the circumstances as that phrase is explained in Instruction No. 11, and
Third, that Defendant _ had a duty to protect Plaintiff_ and to properly supervise the individual who hit, kicked, or beat Plaintiff_, and
Fourth, that Defendant _ knew or was deliberately indifferent to Plaintiff _ need to be protected from such injuries, and
Fifth, that Defendant_with deliberate indifference to or conscious disregard for the rights of Plaintiff _failed or refused to properly supervise his subordinates to prevent such injuries. (Appellants’ Appendix p. 33).
Appellants attack this instruction on four grounds: (1) it did not require plaintiffs to prove their case by a preponderance of the evidence; (2) it did not require a finding that the kicking, hitting and beating caused “unnecessary and wanton pain and suffering;” (3) it did not require a finding that the prison officials knew of the acts and, knowing of the acts, tacitly authorized them or failed to take remedial steps to prevent them; and (4) it did not require a finding that the supervisor’s conduct actually injured plaintiffs.
The rule of this circuit is clear: jury instructions are sufficient if they state the governing law fairly when read as a whole, and considering each instruction in light of the whole. After examining the *1349jury instructions in their entirety, we find appellants’ arguments to be without merit.
With respect to the first allegation, Instruction No. 7 instructed the jury as to which party had the burden of proof and how that burden could be met. The burden of proof instruction need not be repeated in each verdict director. Similarly, Instruction No. 11 defines the excessive force required by the verdict director in terms of “unnecessary and wanton infliction of pain.” In effect, Instruction No. 11 did require a finding that the officers’ use of force caused unnecessary and wanton pain and suffering. Appellants’ third criticism ignores the plain language of the verdict director requiring the jury to find that the defendant “knew or was deliberately indifferent to [plaintiffs’] need to be protected,” and thereafter “with deliberate indifference or conscious disregard for [plaintiffs’ rights] failed or refused to properly supervise his subordinates to prevent such injuries.” Appellants’ reading of the instruction is clearly incorrect. Finally, we find no merit in appellants’ fourth argument. Instruction No. 44 limited plaintiffs to compensation for injury sustained “as a direct result of the conduct of that defendant against whom you find * * This forbade the jury from awarding compensatory damages unless they believed a plaintiff suffered damages as the direct result of a given defendant’s conduct.
We also note that appellants failed to object to the verdict directing instructions regarding supervisory liability. This court has steadfastly held that errors in instructions are waived if not properly objected to at trial, unless the instructions constituted plain error. See Beckman v. Mayo Foundation, 804 F.2d 435, 438 (8th Cir.1986). We find no error here.
2. Burden of Proof
Appellants argue that one of the general instructions given by the trial judge improperly placed a burden of proof on the corrections officials. The challenged instruction reads, in part:
Your verdict depends upon whether or not you believe a certain set of facts submitted to you. The burden of causing you to believe certain facts is upon the party whose claim or defense depends upon those facts.
The party who has the burden of proving certain facts must prove they are more likely true than not true. If the evidence does not persuade you that a fact is more likely true than not true, then you must decide the issues dependent on that fact against the party who has the burden of proof on that fact.
Instruction No. 7 (emphasis added). Appellants maintain that the district court improperly cast a burden of proof on them by refusing to either modify the instruction, delete the reference to defense, or give an alternative instruction proffered by appellants. After reviewing the instructions in their entirety, we disagree.
Instruction No. 7’s mention of “defense” is the only such reference in the entire set of jury instructions. The instructions make no other reference to defense or otherwise indicate that defendants had any burden whatsoever. Rather, the instructions specifically require that the jury find that plaintiffs had proved the truth of facts set out in the verdict directors before returning a verdict in favor of plaintiffs. We find that the instructions, taken as a whole, do not tend to impose any burden of proof on defendants.
3. Verdict Directing Instructions
Defendants challenge the instructions given against those defendants accused of using excessive force, claiming that the instructions are in error because they did not expressly require the jury to find (1) that the force applied inflicted unnecessary and wanton pain and suffering and (2) that the plaintiff was damaged.
It is true that the verdict directing instructions did not expressly require a finding that the force used by the corrections officials inflicted unnecessary and wanton pain and suffering. Each verdict director, however, referred the jury to Instruction No. ll’s definition of cruel and unusual punishment, which stated that the force used must involve the unnecessary and *1350wanton infliction of pain. The two instructions taken together correctly advised the jury of the standard to apply with respect to plaintiffs’ claims.
At argument, defendants relied on a recent decision by this court for their argument that the instructions should have required specific findings that the force used was either malicious for the purpose of harm or unnecessary and wanton, and that the force actually damaged plaintiffs. See Cowans v. Wyrick, 862 F.2d 697, 699-700 (8th Cir.1989). Cowans was injured when a prison guard slammed a food slot door shut on his hand. A jury found that the guard had inflicted cruel and unusual punishment on Cowans, but did not assess any damages against the guard. The district court awarded Cowans nominal damages based on the jury verdict. This court reversed, holding, under Whitley v. Albers, 475 U.S. 312, 320-21, 106 S.Ct. 1078, 1084-85, 89 L.Ed.2d 251 (1986), that the guard’s action was a reasonable measure undertaken to resolve a disturbance which posed a significant security threat. Cowans differs from the current case, however, in that Cowans involved a threat to institutional security. In the instant case, the beatings complained of by the prisoners occurred after any threat to institutional security had been quelled. Therefore Cowans does not control.
We also find without merit the objection based on a lack of finding with respect to damages. Although the verdict director did not require an express finding that plaintiffs were damaged, Instruction No. 44 instructed the jury on how to determine the amount of compensatory, or actual, damages sustained by plaintiffs. The jury awarded each plaintiff compensatory damages pursuant to Instruction No. 44. Defendants never objected to this instruction, nor do they contend it was erroneous. Furthermore, the constitutional violation may be vindicated by the award of nominal damages even when no actual damages are proven. Tatum v. Houser, 642 F.2d 253, 254-55 (8th Cir.1981). Because proof of actual damages is unnecessary to establish a constitutional violation, it follows that damages need not be an element to be proved in the verdict director.
C. Juror Misconduct
During voir dire in the instant case, the presiding magistrate asked the prospective jurors:
Do any of you have family members or close friends or yourselves who have been or are now, and I know none of you are now, incarcerated, any of you have got family members or friends who are now incarcerated?
In response, juror Barber stated that her brother was “serving time in Moberly” for violating parole. None of the other jurors responded to the question. Juror Barber was removed for cause due to her testimony that information she received from her brother would lead her to believe one side was more entitled to relief than the other. After the jury returned its verdict, appellants discovered that juror Kramer had not disclosed during voir dire that her thirty-five-year-old son was on probation at the time. Appellants argued juror misconduct in their motion for judgment n.o.v. or, alternatively, for new trial or remittitur. The district court found that the verdict was in no way prejudiced by juror misconduct. Appellants now maintain that the court erred in making such a determination without benefit of an evidentiary hearing.
After reviewing the record, we find no error. Juror Kramer responded to the question asked in a responsive and complete manner by remaining silent. The question asked whether she or any of her family had been incarcerated. “Incarcerated” is commonly understood to mean “in prison,” and juror Kramer’s son was never incarcerated. Because juror Kramer responded truthfully to the question asked, we find no deliberate concealment on her part. See Robinson v. Monsanto Company, 758 F.2d 331, 334-35 (8th Cir.1985). Therefore, defendants failed to make a pri-ma facie case of misconduct and the district court was not obligated to hold an eviden-tiary hearing sua sponte.
*1351D. Reduction of Damages
The district court affirmed all compensatory and punitive damage awards except the award against defendant Halley for his failure to protect plaintiff Bolin.4 The jury found Halley directly liable to plaintiff Bo-lin for $2,500 based on Halley’s actual participation in the beatings. The jury set the limit of Halley’s joint and several liability for failure to protect at $5,000 and set punitive damages at $15,000, twice the amount of Halley’s potential total liability. The magistrate reduced the compensatory award to $1,000 because Halley is only potentially liable for injuries inflicted by Williams, whom Halley supervised, and the jury found only $1,000 actual damages against Williams. The judge also reduced punitive damages against Halley to $7,000, twice Halley’s revised potential liability.
We find no basis for reduction of the amount of damages awarded to plaintiffs. The punitive damages awarded in this case range from $2,000 to $10,000. Such amounts are not excessive, given the constitutional violations found by the jury.
Accordingly, we affirm the district court.
. The Honorable William A. Knox, United States Magistrate for the Western District of Missouri, presided, pursuant to consent of the parties. See 28 U.S.C. § 636(c).
. The district court found the following chain of command at MTCM on the night in question: defendant R. Dale Riley had supervisory authority over all employees and activities when he was present; defendant Willie J. Dennis, Chief Custody Officer, who was at MTCM the night in question although he was not scheduled for duty, had supervisory authority when he was present over defendant Denver Halley, the Senior Officer in Charge; and the other defendant officers at MTCM — William Arney, Clarence Williams and Robert Wilson — were essentially of equal rank. With respect to the bus trip from Moberly to Jefferson City, Missouri, the district court found the following chain of command: defendant Bill Armontrout was overall supervisor; defendant Robert Borghardt was second in command; and all other defendant officers on the bus — James Eberly, Thomas Davis, Robert Malone and Michael Plemmons — were of equal rank.
. The court reduced the compensatory damage award for failure to protect from $5,000 to $1,000 and likewise reduced the punitive damage award from $15,000 to $7,000.
. Appellants allege that the district court reduced actual damages by removing actual damage awards against the supervisors as duplica-tive. We agree with plaintiffs, however, that what appellants now claim to be a reduction in compensatory damages was no more than a mathematical computation of the damages due under the instructions and verdict forms given to the jury.