SELYA, Circuit Judge.

This appeal follows a jury verdict awarding a grand total of $1,300,000 in compensatory and punitive damages to plaintiffs Pedro Casillas-Diaz (Casillas) and the Estate of E'liomar Lopez-Claudio (Lopez). In reaching this verdict, the jury found that four police officers, defendants Romualdo Palau, Juan Martinez, Liz Diaz, and Gabriel De Leon, had used excessive force in an encounter with Casillas and Lopez. In this venue, the officers, as appellants, maintain that the lower court erred in denying their motion for judgment as a matter of law and their alternative motion for a new trial or a remittitur. Discerning no preserved error, we affirm the judgment below.

I. BACKGROUND

We rehearse the facts as the jury supportably could have found them, guided by the rule that "when the losing party protests the sufficiency of the evidence, the court of appeals must take both the facts and the reasonable inferences therefrom in the light most hospitable to the jury's verdict." Correa v. Hosp. San Francisco, 69 F.3d 1184, 1188 (1st Cir.1995).

On September 18, 2000, at approximately 4:30 a.m., the defendants encountered Lopez, Casillas, and Casillas's bride-to-be, Barbara Lee Camacho, while the three civilians were making the rounds of San Juan's casinos. Officers Palau and Martinez stopped Lopez's vehicle, in which the trio was riding, at the Ambassador Hotel. Without incident, they ticketed the operator, Casillas, for driving against traffic and failing to wear a seat belt.

Sometime later that morning, the plaintiffs and the defendants met again at the Condado Plaza Hotel. The defendants claim that they confronted Casillas and Lopez after receiving information that the pair had been involved in a "hit-and-run" incident in the parking lot of the Ambassador Hotel. Although all parties agree that an altercation ensued, the testimony as to exactly what transpired is widely divergent.

The plaintiffs say that at the time of the confrontation Casillas was speaking to a cabdriver outside the lobby of the Condado Plaza, while Lopez was asleep in the back seat of his parked car. According to their version of the events, the police berated the two men and then manhandled them for no apparent reason. The defendants tell a vastly different tale. They claim that they approached Casillas near the hotel lobby, that he pushed one of the officers, and that Lopez physically interfered with the officers' attempt to subdue Casillas. In the officers' view, Casillas and Lopez were the aggressors in the fracas that ensued; in the plaintiffs' view, the officers were the aggressors.

It is undisputed that, during the melee, Casillas and Lopez sustained injuries, and that the defendants proceeded to take the two men into custody. They charged Casillas with eight misdemeanor traffic infractions. In addition, they swore out felony charges against both Casillas and Lopez for alleged acts of violence aimed at public officials.

The criminal charges came to naught. The case against Lopez was rendered moot when he committed suicide; the charges against Casillas were dropped. In the meantime, the battleground shifted from the criminal courts to the civil courts: on September 18, 2001, Casillas and Lopez's parents, representing his estate, sued the officers under 42 U.S.C. § 1983 and Puerto Kico law.

The plaintiffs' complaint alleged, vark ously, use of excessive force, false arrest, false imprisonment, and malicious prosecution. The defendants denied the material allegations of the complaint, and, in due course, a jury trial commenced. At the close of the plaintiffs' case in chief, the defendants moved for judgment as a matter of law, see Fed.R.Civ.P. 50(a), citing an ostensible lack of evidence. The district court reserved decision. At the close of all the evidence, the court granted judgment as a matter of law in favor of the defendants on the claims of false arrest, false imprisonment, and malicious prosecution but allowed the excessive force claims to go forward.

The jury found that the defendants had used excessive force against both Casillas and Lopez. It awarded Casillas $50,000 in compensatory damages ($25,000 each against Palau and Martinez) and awarded Lopez's estate $250,000 in compensatory damages ($125,000 each against Diaz and De Leon). As a coup de grace, the jury awarded a total of $1,000,000 in punitive damages ($250,000 against each of the four defendants).

The defendants filed a timely post-trial motion for judgment as a matter of law or, in the alternative, for a new trial or a remittitur. See Fed.R.Civ.P. 50(b), 59(a). The district court denied the motion. This appeal ensued.

II. LIABILITY

We divide our discussion of the defendants' challenge to the liability verdict into three segments.

A. Standard of Review.

The district court's denial of a motion for judgment as a matter of law poses a question of law, engendering de novo review. See Colasanto v. Life Ins. Co. of N. Am., 100 F.3d 203, 208 (1st Cir.1996). Where, as here, such a motion contests the sufficiency of the proof, "the court of appeals must examine the evidence and the inferences reasonably to be extracted therefrom in the light most hospitable to the nonmovant." Sanchez v. P.R. Oil Co., 37 F.3d 712, 716 (1st Cir.1994). In performing this tamisage, "we may not consider the credibility of witnesses, resolve conflicts in testimony, or evaluate the weight of the evidence." Wagenrnann v. Adams, 829 F.2d 196, 200 (1st Cir.1987).

Judgment as a matter of law should be approved, or the denial of such a judgment reversed, "only when the evidence, viewed from this perspective, is such that reasonable persons could reach but one conclusion." Id. It follows that we "may reverse the denial of such a motion only if reasonable persons could not have reached the conclusion that the jury embraced." Correa, 69 F.3d at 1191 (quoting Sanchez, 37 F.3d at 716).

Our review of the district court's disposition of a Rule 59(e) motion is also narrowly circumscribed. A district court may set aside the jury's verdict and order a new trial only if the verdict is against the law, against the weight of the credible evidence, or tantamount to a miscarriage of justice. See Sanchez, 37 F.3d at 717. A district court's disinclination to disturb a jury's verdict "can be reversed solely for abuse of discretion." Id.

B. Waiver.

As a threshold issue, the plaintiffs contend that the defendants have waived their sufficiency of the evidence challenge by not raising it until their post-trial motion for judgment as a matter of law. The premise on which this contention rests is impeccable: a renewed motion for judgment as a matter of law under Rule 50(b) cannot assert new grounds but, rather, is "bounded by the movant's earlier Rule 50(a) motion." Correa, 69 F.3d at 1196. In other words, a party cannot use a Rule 50(b) motion as an instrument for introducing a neoteric legal theory — one not distinctly articulated in his end-of-the-evidence motion for judgment as a matter of law — into the case. See id.; see also James W. Moore, 5A Moore's Federal Practice ¶ 50.08 (2d ed.1994) (explaining that "any argument omitted from the motion made at the close of the evidence is waived as a ground for judgment under Rule 50(b)").

Despite the correctness of the plaintiffs' premise, their conclusion is awry. The record here reflects that the defendants began the oral presentation of their Rule 50(a) motion at the close of the plaintiffs' case with the assertion that the plaintiffs had failed "to put forth sufficient admissible evidence" to allow a reasonable jury to "find in their favor on a variety of crucial factual issues." When the defendants renewed the Rule 50(a) motion at the close of all the evidence, they made transparently clear their belief "that plaintiffs have failed to provide any evidence of a use of excessive force." Based on these and other pertinent references contained in the record, we are satisfied that the defendants raised their sufficiency of the evidence challenge in a timely manner. Consequently, the issue has been preserved for appeal.

C. Use of Excessive Force.

Turning to the merits, we inquire first whether a reasonable juror could have reached a conclusion favorable to the plaintiffs concerning the use of excessive force. See Wagenrnann, 829 F.2d at 200. The record here demands an affirmative answer to that inquiry.

At trial, Casillas and Camacho testified about the events of September 18, 2000. Their testimony, taken at face value, indicated that Casillas and Lopez were attacked without any provocation and savagely beaten. The defendants presented a markedly different version of the events surrounding the arrests. They depicted Casillas and Lopez as the aggressors, and characterized their own use of force as reasonably necessary under the circumstances.

In the last analysis, the issue boiled down to an assessment of the comparative credibility of the witnesses. When, as now, the credibility of witnesses comprises the crux of the matter, a reviewing court must take special care not to intrude upon the jury's domain. See Tennant v. Peoria Pekin Union Ry. Co., 321 U.S. 29, 35, 64 S.Ct. 409, 88 L.Ed. 520 (1944); Wagenmann, 829 F.2d at 200. It is the proper province of the jury, not the court of appeals, to separate wheat from chaff, resolve inconsistencies in the witnesses's accounts, and determine what testimony is or is not worthy of credence. See Correa, 69 F.3d at 1194.

Here, the plaintiffs' evidence showed that both Casillas and Lopez were brutally assaulted and beaten into unconsciousness, without legitimate reason or provocation. Viewing the record, as we must, in the light most favorable to the plaintiffs and making credibility calls to their behoof, we cannot say that the liability verdict was irrational or unresponsive to the proof. The short of it is that the plaintiffs' evidence, which included photographs depicting the physical injuries inflicted by the defendants and medical records describing those injuries, was enough to ground the liability verdict. From that evidence, a reasonable jury could well have concluded — as this jury did — that the defendants used extravagant force in arresting Casillas and Lopez. Accordingly, the defendants were not entitled to judgment as a matter of law.

No useful purpose would be served by a separate discussion of the new trial motion insofar as it pertains to liability. After all, the district judge, who saw and heard the witnesses, declined to meddle with the jury's factfinding. Given the sharply divergent accounts presented by the witnesses, we are bound to defer to the trial judge's first-hand appraisal of which side had the better of its argument. See, e.g., id.

That ends this aspect of the matter. On this variegated record, a finding that the officers resorted to excessive force was contrary to neither the law nor the weight of the evidence. By the same token, that finding did not in any way herald a miscarriage of justice. Accordingly, we hold that the district court's ratification of the jury's liability verdict did not constitute an abuse of discretion.

III. COMPENSATORY DAMAGES

The Civil Rules permit a party aggrieved by a jury verdict to move within a stipulated time frame for a new trial. See Fed.R.Civ.P. 59(e). In their timely post-trial motion, the defendants challenged, inter alia, the size of the compensatory damage awards, shaping this aspect of their motion as a request for a remittitur. The district court rejected the argument that the awards were overly munificent, declaring that neither award was "so grossly excessive or inordinate to warrant setting it aside." The defendants now renew this argument on appeal.

It is trite, but true, that "[translating legal damage into money damages — especially in cases which involve few significant items of measurable economic loss — is a matter peculiarly within a jury's ken." Wagenmann, 829 F.2d at 215. Accordingly, only infrequently — and then, for compelling reasons — will we, from the vantage point of an algid appellate record, override the jury's judgment as to the appropriate amount of non-economic damages to which a plaintiff is entitled. See id.; Brown v. Freedman Baking Co., 810 F.2d 6, 11 (1st Cir.1987). That is especially so where, as here, the trial judge, who saw and heard the fray fought out in real time, has endorsed the award. See Ruiz v. Caraballo, 929 F.2d 31, 34 (1st Cir.1991). It follows logically that a verdict approved by both the jurors and the trial judge will be pared "only if it is shown to exceed any rational appraisal or estimate of the damages that could be based upon the evidence before the jury." Dopp v. Pritzker, 38 F.3d 1239, 1249 (1st Cir.1994) (quoting Segal v. Gilbert Color Sys., Inc., 746 F.2d 78, 81 (1st Cir.1984)).

This standard of review is extremely deferential. That deference ordinarily is expressed in terms of abuse of discretion. See Borges Colon v. Roman-Abreu, 438 F.3d 1, 20 (1st Cir.2006); RiveraTorres v. Velez, 341 F.3d 86, 102 (1st Cir. 2003), cert, denied, 541 U.S. 972, 124 S.Ct. 1875, 158 L.Ed.2d 466 (2004). The defendant bears the heavy burden of demonstrating that a challenged award is "grossly excessive, inordinate, shocking to the conscience of the court, or so high that it would be a denial of justice to permit it to stand." Correa, 69 F.3d at 1197 (quoting Segal, 746 F.2d at 81).

The district court instructed the jury, without objection, that in assessing damages it could consider any physical, mental, or emotional harm sustained by the injured parties, their reasonable medical expenses, and the extent and duration of their injuries. The court cautioned that the damages must be "reasonable" and not "based on speculation or guesswork." The defendants take no exception to the court's instruction. The question, then, is fact-bound — but the defendants do not deal with the facts in any meaningful way.

Few principles are more a part of the warp and woof of appellate practice than the principle that "issues adverted to in a perfunctory manner, unaccompanied by some effort at developed argumentation, are deemed waived." United States v. Zannino, 895 F.2d 1, 17 (1st Cir.1990). We have parroted this principle with a regularity bordering on the monotonous. See, e.g., Fradera v. Mun'y of Mayaguez, 440 F.3d 17, 21 (1st Cir.2006); Cytyc Corp. v. DEKA Prods., Ltd. P'ship, 439 F.3d 27, 32 (1st Cir.2006); Goldman, Antonetti, Ferraiuoli, Axtmayer Hertell v. Medfit Int'L, Inc., 982 F.2d 686, 687 (1st Cir.1993). These reiterations are not meant to be regarded as empty words: our adherence to this principle imposes on litigants an unflagging obligation to spell out their contentions "squarely and distinctly, or else forever hold [their] peace." Zannino, 895 F.2d at 17. "[I]t is not enough merely to mention a possible argument in the most skeletal way, leaving the court to do counsel's work." Id.

In the case at hand, the defendants' challenge to the compensatory damage awards flagrantly violates this bedrock principle. In the "Argument" section of their brief, the defendants posit that "even if the Court does not grant a new trial, the Court should at least reduce the amount of the damages awarded by the jury." Appellants' Br. at 14. They then cite general language from a few cases addressing the court's authority to order a new trial if a verdict is unreasonably high. See id. After alluding to this case law, the defendants baldly state, in a completely conclusory fashion, that "after reviewing the evidence and testimony presented during the trial [it] is fundamental that the Court review the amount of damages awarded by the jury." Id. at 16.

That is their entire argument on this issue. They at no point attempt to apply the case law to the facts adduced at trial; they at no point marshal the evidence as to the injuries and damages sustained by either Casillas or Lopez; and they at no point discuss, let alone analyze, the pain, suffering, and emotional distress engendered by those injuries. To cinch matters, their brief is devoid of any citations to the record; they attempt neither to identify the relevant evidence nor to show how that evidence relates to — let alone undermines — the compensatory damage awards.

This slapdash approach is wholly inadequate to preserve the issue of excessiveness for appellate review. A case closely on point is Seahorse Marine Supplies, Inc. v. Puerto Rico Sun Oil Co., 295 F.3d 68 (1st Cir.2002). There, the defendant asserted that damages had been calculated improperly but failed to provide any meaningful record references or otherwise to analyze the pertinent evidence. See id. at 82. In the absence of any particularized showing as to why the awarded damages were incorrect, we deemed the argument waived. See id.

The same holds true in this case. The defendants grouse about the size of the compensatory damage awards but then, in effect, invite us to rummage through the record, unassisted by any semblance of developed argumentation, in order to evaluate their claim of excessiveness. As we have explained above, sound practice demands that we decline this unattractive invitation: if a party does not put enough stock in an argument to flesh it out, a reviewing court should normally dismiss the argument out of hand. See King v. Town of Hanover, 116 F.3d 965, 970 (1st Cir.1997). To do otherwise would be un fair both to the adverse party and to the court itself. We therefore treat the claim as waived.

IV. PUNITIVE DAMAGES

The defendants also challenge the amount of punitive damages. Here, too, their argument is insufficiently developed and subject to rejection on that ground alone. See Zannino, 895 F.2d at 17. We need not rest on that ground, however, because the argument is also patently meritless.

A jury may levy punitive damages in a section 1983 action when a defendant's conduct is "shown to be motivated by evil motive or intent, or when it involves reckless or callous indifference to the federally protected rights of others." Smith v. Wade, 461 U.S. 30, 56, 103 S.Ct. 1625, 75 L.Ed.2d 632 (1983); see also Iacobucci v. Boulter, 193 F.3d 14, 25-26 (1st Cir.1999). In order for punitive damages to be appropriate, there must be "proof that the defendant acted 'in the face of a perceived risk that [his] actions [would] violate federal law.'" Iacobucci, 193 F.3d at 26 (quoting Kolstad v. Am. Dental Ass'n, 527 U.S. 526, 536, 119 S.Ct. 2118, 144 L.Ed.2d 494 (1999) (alterations in original)).

We need not wax long eloquent. The district court instructed the jury, without objection, that a determination to award punitive damages was to be based on "whether you find that the defendants acted willfully, deliberately, maliciously, or with reckless disregard of the plaintiffs' constitutional rights." On the basis of the evidence recounted above, the jury easily could have found that the defendants acted in the face of a perceived (and flatly unacceptable) risk that their actions would compromise Casillas's and Lopez's Fourth Amendment rights. No more was exigible. See Borges Colon, 438 F.3d at 22.

The conclusion that an award of punitive damages was permissible does not end our journey. The defendants' challenge to the amount of punitive damages presents an independent legal issue — one that engenders de novo review. See id. at 21. This review is "informed by principles of fundamental fairness," which "dictate that a person receive fair notice not only of the conduct that will subject him to punishment, but also of the severity of the penalty" that may be imposed. Zimmerman v. Direct Fed. Credit Union, 262 F.3d 70, 81 (1st Cir.2001) (quoting BMW of N. Am., Inc. v. Gore, 517 U.S. 559, 574, 116 S.Ct. 1589, 134 L.Ed.2d 809 (1996)).

In evaluating the reasonableness of a punitive damage award ancillary to a violation of section 1983, we consider "(1) the degree of reprehensibility of a defendant's conduct; (2) the ratio between punitive and actual and potential damages; and (3) a comparison of the punitive damages figure and other civil and criminal penalties imposed for comparable conduct." Davis v. Rennie, 264 F.3d 86, 116 (1st Cir.2001) (quoting Romano v. U-Haul Int'l, 233 F.3d 655, 672-73 (1st Cir.2000)). We will not disturb such an award "unless we find it certain that the amount in question exceeds that necessary to punish and deter the alleged misconduct." Borges CoIon, 438 F.3d at 21 (citation and internal quotation marks omitted).

The record here satisfies the ancillary criteria that ordinarily are thought to be relevant to the sustainability of punitive damage awards. The reprehensibility of the defendants' misconduct is the most salient of these criteria. See BMW, 517 U.S. at 575, 116 S.Ct. 1589; Romano, 233 F.3d at 673. On the plaintiffs' version of events — a version that the jury evidently believed and that, therefore, we must credit — the defendants' behavior reflects a high degree of culpability. That misconduct was well outside the acceptable norms of police work. See Nydam v. Lennerton, 948 F.2d 808, 811 (1st Cir.1991) (discussing the conduct of police in assaulting a citizen who supposedly was resisting arrest).

To add fuel to the fire, a punitive damage award may be "justified not only by defendants' actions on [the date in question] but also by their subsequent behavior." Davis, 264 F.3d at 115 (quoting Hall v. Ochs, 817 F.2d 920, 927 (1st Cir.1987)). Here, the jury reasonably could have concluded that the subsequent filing of felony charges against Casillas and Lopez demonstrated the officers' improper motives and callous indifference to the plaintiffs' rights. Comparing the testimony of the defendants with that of the plaintiffs' witnesses, "a factfinder might infer that the stark clash could not have resulted from innocent misrecollection and that its intentional quality intensified any need the jury may have found for punishment and deterrence." Id. at 116 (quoting Hall, 817 F.2d at 928).

The punitive damage award also fits comfortably with the remaining BMW factors. For one thing, the award was reasonable in proportion to the compensatory damage awards. Assuming that the punitive damage award is to be divided evenly in two, see supra note 2, the ratio between punitive and compensatory damages is 10:1 for Casillas and 2:1 for Lopez's estate. In Romano, we upheld a 19:1 ratio between punitive and compensatory damages, noting that the Supreme Court has "dismissed any simple, mathematical formula in favor of general inquiry into reasonableness." 233 F.3d at 673. Given that the record supports a finding of substantial physical and emotional harm, the ratio between punitive and compensatory damages presents no cause for concern in this case.

The final BMW factor requires us to view the punitive damage award "in light of the complex of statutory schemes developed to respond to the same sort of underlying conduct." Zimmerman, 262 F.3d at 82. "[A] reviewing court engaged in determining whether an award of punitive damages is excessive should accord substantial deference to legislative judgments concerning appropriate sanctions for the conduct at issue." Id. at 82-83. In drafting section 1983, Congress did not make any reference to the quantum of damages. We may, therefore, consider awards in similar cases to help determine if particular punitive damages in a given case appear excessive. See Davis, 264 F.3d at 117.

A canvass of punitive damage awards upheld in comparable section 1983 cases makes it nose-on-the-face plain that the punitive damages granted in this case are not out of line. See, e.g., Estate of Moreland v. Dieter, 395 F.3d 747, 751 (7th Cir.2005) (approving award of $27,500,000 in punitive damages in an excessive force case); Davis, 264 F.3d at 116-17 (affirming award of $1,025,000 in punitive damages in excessive force case); Gutierrez-Rodriguez v. Cartagena, 882 F.2d 553, 581 (1st Cir.1989) (upholding a punitive damage award of $600,000); see also Nydam, 948 F.2d at 811 (characterizing as "restrainfed]" an award of $65,000 in compensatory damages and $200,000 in punitive damages for excessive force). In fine, there is no indication that the last BMW factor undermines the integrity of the present punitive damage award.

To sum up, the jury supportably found that the defendants brutally beat both Casillas and Lopez, and thereby violated their civil rights. It awarded substantial punitive damages — but the amount, though generous, was not beyond the outer limit of what may have been reasonably necessary to punish the offenders and deter others from similar misconduct. Consequently, we descry no principled basis on which we might either jettison or trim the award.

V. CONCLUSION

We need go no further. For the reasons elucidated above, we reject the appeal and uphold the judgment in all respects.

Affirmed.