The plaintiffs, Barbara Gavoni, Angela Ro-sendale and Lela Jordan, worked together as manicurists and hair dressers at a salon in La Crosse, Wisconsin and attended a cosmetology convention at a Chicago-area hotel owned and operated by the defendant, Dobbs House. On the morning of March 29, 1993, the plaintiffs and another convention attendee boarded an elevator on the eighth floor hoping and expecting to go down to the lobby. Their hopes were dashed — literally and figuratively. The elevator went up to the eleventh floor, stopped briefly, descended to the lower lobby at an uneven rate and abruptly stopped. When the doors did not open, Jordan pressed the alarm button and tried calling for help on the emergency telephone. The riders eventually made enough noise to alert the hotel staff. A maintenance employee, Edward Johnstone, arrived within minutes and rescued the riders. Johnstone testified that when he opened the elevator doors he found “three young ladies standing there laughing and joking around.” He also claimed that each held a drinking glass with liquid in it. The plaintiffs admitted that Jordan made a sarcastic comment when asked if they wanted to be rescued but denied both that they were jocular and that they had drinks in hand. The plaintiffs told their rescuers that they were not injured, sat down for several minutes and then proceeded to the convention. That afternoon, Jordan and Rosendale competed in a fingernail decoration competition while Gavoni returned to their room. The plaintiffs later dined together and attended a cocktail party and ham show. Each had several alcoholic drinks; Jordan and Rosendale were dancing.
*1074The next day, on the bus back to La Crosse the plaintiffs discussed the elevator incident and decided to go to the emergency room together that same night. Each complained of neck, shoulder and upper back pain. Over the next several years, each of the plaintiffs continued to experience pain and discomfort.
The plaintiffs brought this diversity suit against Dobbs House and the elevator manufacturer, Westinghouse. Pursuant to Fed. R.Civ.P. 68, Dobbs House made an offer of $10,000 “to be divided among all three plaintiffs, with costs then accrued.” The plaintiffs rejected the offer. Prior to trial, the plaintiffs settled their claims against Westinghouse for a total of $105,000: Gavoni received $17,850; Rosendale got $44,100; and Jordan received $43,050.
The case against Dobbs House proceeded to trial. Dobbs House never denied that the elevator had malfunctioned; experts from both sides testified that the incident was likely caused by a faulty electrical connection. Instead, Dobbs House argued that the plaintiffs had inflated their injuries and consequent damages, making a federal case out of a minor accident. The plaintiffs presented testimony from a single doctor that the elevator incident caused each plaintiff a variety of ailments, from sore knees to cracked teeth to chronic back pain. Dobbs House presented expert testimony contradicting this conclusion. At closing, the plaintiffs sought $825,000 in damages: $230,000 for Gavoni, $320,000 for Rosendale and $275,000 for Jordan. The jury found against Dobbs House on liability but awarded the plaintiffs a relatively paltry $6500 — $2000 for Gavoni, $2000 for Rosendale and $2500 for Jordan.
Following the verdict, the plaintiffs moved for costs as the prevailing party under Fed. R.Civ.P. 54(d). They also moved under Fed. R.Civ.P. 59 for a new trial on various grounds. Dobbs House, for its part, moved for costs pursuant to Fed.R.Civ.P. 68. The court denied all three motions.
The plaintiffs raise five separate issues on appeal. Their first four complaints were originally included in their Rule 59 motion, and the plaintiffs do little, if anything, to develop these arguments on appeal. Although we might be free to ignore these undeveloped arguments, cf. Indurante v. Local 705, Int’l Bhd. of Teamsters, AFL-CIO, 160 F.3d 364, 366-67 (7th Cir.1998) (citing cases), we will nonetheless briefly consider them. The plaintiffs also appeal the denial of their Rule 54(d) motion for costs and the defendant cross-appeals from the denial of its Rule 68 motion for costs.
The plaintiffs first allege that the district court erred by allowing the jury to view one side (the left side) of an expert witness’s chart. Dobbs House’s elevator expert, John Donnelly, referred to a chart which compared the G-forces of a normally functioning elevator (the right side of the chart) with the G-forces of everyday human activity (the left side of the chart). Donnelly had prepared only the right side, but was familiar with the left. After a prolonged argument at side bar and a foundation voir dire by the judge and defense counsel, the district court ruled that the chart was “expert data that is utilized in the field and, therefore, for the limited purpose of his testimony I will permit the inquiry.” We will not disturb a district court ruling on expert testimony unless it is manifestly erroneous, see Deimer v. Cincinnati Sub-Zero Prod., Inc., 58 F.3d 341, 344 (7th Cir.1995), and there is no such error here. The district court made an extensive inquiry into the foundation for the expert’s testimony and admitted it subject to being later stricken. Further, and directly relevant to their present argument, the plaintiffs did not ask that the left side of the chart (not prepared by the witness) be covered or redacted. Defense counsel later displayed the entire chart during closing argument and the plaintiffs now argue that this was error. Failure to demand redaction or that the chart be covered waives these arguments on appeal. See Miksis v. Howard, 106 F.3d 754, 761 (7th Cir.1997); Holmes v. Elgin, Joliet & Eastern Ry. Co., 18 F.3d 1393, 1398 (7th Cir.1994).
The plaintiffs next claim that the district court erred by allowing the defense to argue that the plaintiffs’ presentation of only one doctor as a witness created an inference of collusion and faked injuries. The plaintiffs claim this argument was preju*1075dicial and violated a motion granted in li-mine. The record on appeal, however, contains no evidence of such a motion. And in any event, trial courts have broad discretion to allow or prohibit argument on close, see, e.g., Miksis, 106 F.3d at 764, and we find no abuse here.
The plaintiffs’ third complaint is convoluted but also concerns the defendant’s closing argument. During the plaintiffs’ case in chief, plaintiffs’ counsel attempted to have Gavoni testify about an alleged conversation with a hotel employee, John Cusimano. Dobbs House objected on hearsay grounds. At side bar, the plaintiffs argued that Cusi-mano’s statements were party opponent admissions. The court heard more argument, referred to depositions and sustained Dobbs House’s objection. During his summation, defense counsel, over the plaintiffs’ objection, which was denied, argued that the plaintiffs’ failure to produce the fourth elevator rider supported the defense theory that the plaintiffs’ injuries were illusory. On appeal, the plaintiffs attempt to link these rulings. We do not see the connection — the two rulings were distinct in both time and content. More, the district court was well within its broad discretion both in allowing the argument on close, see, e.g., id., and in refusing to admit the hearsay testimony. See, e.g., Cook v. Navistar Int’l Trans. Corp., 940 F.2d 207, 212-13 (7th Cir.1991); id. at 215.
The plaintiffs’ fourth complaint is that the jury’s award was so minuscule, so unsupported by the evidence that it should have offended the conscience of the court. The district court was not persuaded, and neither are we. We review a district court’s decision whether to set aside an award for an abuse of discretion, and the plaintiffs must show that “there is no rational connection between [the award] and the evidence.” Raybestos Prod. Co. v. Younger, 54 F.3d 1234, 1244 (7th Cir.1995) (internal quotations and citations omitted). This is a heavy burden. Here, because the record on appeal can support the jury’s award, the district court did not abuse its discretion. See Blumenfeld v. Stuppi, 921 F.2d 116, 118 (7th Cir.1990).
The plaintiffs’ fifth complaint— that the district court erred in failing to award their costs pursuant to Fed.R.Civ.P. 54(d) — is their most substantial challenge, but it too fails. Fed.R.Civ.P. 54(d) provides in pertinent part that “costs other than attorneys’ fees shall be allowed as of course to the prevailing party unless the court otherwise directs.” The plaintiffs argue that because they prevailed, the district court was obliged to award them costs: This argument ignores both the language of the Rule and well-settled law in this Circuit. Rule 54(d) expressly grants the trial court discretion in awarding costs — the prevailing party wins “unless the court otherwise directs.” Further, courts have especially broad discretion to award or deny costs in mixed result cases, see, e.g., Testa v. Village of Mundelein, 89 F.3d 443, 447 (7th Cir.1996), including cases in which liability was established but recovery was nominal relative to what was sought. See Northbrook Excess & Surplus Ins. Co. v. Proctor & Gamble Co., 924 F.2d 633, 641-42 (7th Cir.1991). The jury here awarded each plaintiff less than one percent of what she requested. The district court, in denying the plaintiffs’ Rule 54(d) motion for costs, therefore did not abuse its discretion. Thus, on each of the plaintiffs’ appeals, we affirm the district court.
We also affirm the denial of Dobbs House’s motion for Fed.R.Civ.P. 68 costs. Under Fed.R.Civ.P. 68, if a plaintiff rejects a defendant’s settlement offer and “the judgment finally obtained by the offeree is not more favorable than the offer,” then the plaintiff “must pay the costs incurred [by the defendant] after the making of the offer.” Dobbs House claimed that its unapportioned $10,000 offer triggered this cost-shifting provision. The district court disagreed, holding that it was impossible to determine whether the offer was more favorable than the individual jury awards. We review the district court’s underlying factual findings for clear error. See Arkla Energy Resources v. Roye Realty & Dev., 9 F.3d 855, 866-67 (10th Cir.1993). To the extent the entitlement to costs rests on an interpretation of the Rule, we review the district court’s legal conclusions de novo. See Herrington v. County of Sonoma, 12 F.3d 901, 906 (9th Cir.1993).
The defendant, must show that the offer was more favorable than the judg*1076ment and that the mandatory cost-shifting provision was therefore triggered. See generally 12 Charles Alan Wright, Arthur R. Miller, & Richard L. Marcus, Federal Practice and Procedure § 3006.1 (2d ed.1997). Defendants should bear this burden for two reasons. First, because Rule 68’s cost-shifting provision is mandatory and applies to individual parties — the “offeree must pay the costs incurred” by the offeror, Fed.R.Civ.P. 68 (emphasis added); see also Webb v. James, 147 F.3d 617, 621 (7th Cir.1998); Mallory v. Eyrich, 922 F.2d 1273, 1279 (6th Cir.1991) — plaintiffs face serious consequences in either accepting or rejecting a Rule 68 offer. See Webb, 147 F.3d at 621. A judgment less favorable than the offer requires that a plaintiff pay the defendant’s usually substantial post-offer costs. There must therefore be a clear baseline from which plaintiffs may evaluate the merits of their case relative to the value of the offer. See id.; Arkla, 9 F.3d at 866 (“the offeree must know what is being offered in order to be responsible for refusing the offer”); Radecki v. Amoco Oil Co., 858 F.2d 397, 402-03 (8th Cir.1988). Cf. Gay v. Waiters’ & Dairy Lunchmen’s Union, Local 30, 86 F.R.D. 500, 502 (N.D.Cal.1980) (Rule 68 offers of judgment function by forcing “an individual of-feree to weigh his own exposure to liability for the offeror’s subsequent costs against his own expected recovery, thereby encouraging a close evaluation of the merits of his claim”) (emphasis added). Further, plaintiffs should not have to speculate how courts will interpret an offer; “a defendant should state his intentions clearly, and any failure to do so will be at his peril.” Chambers v. Manning, 169 F.R.D. 5, 8 (D.Conn.1996) (citation omitted).
Second, courts also need easily comparable sums. In applying Rule 68, courts have “no discretion to alter or modify the parties’ agreement.” Webb, 147 F.3d at 621. Courts have therefore consistently resisted efforts by either party to qualify or explicate the terms of an accepted offer, see, e.g., Herrington, 12 F.3d at 907 (rejecting the plaintiffs’ efforts to “inject ambiguity into the settlement offer”); Shorter v. Valley Bank & Trust Co., 678 F.Supp. 714, 720 (N.D.Ill.1988) (refusing to consider the history of the settlement negotiations to support the defendant’s interpretation of the offer because “[t]he offer was what the written offer said”); Sas v. Trintex, 709 F.Supp. 455, 458 (S.D.N.Y.1989) (rejecting the defendant’s argument that the court should determine the parties’ intentions), or the terms of a rejected offer. See Johnston v. Penrod Drilling Co., 803 F.2d 867, 870 (5th Cir.1986) (rejecting multiple defendants’ arguments that a court should itself divide an unapportioned rejected joint offer because Rule 68 demands that courts compare only “two clearly defined figures”).1 And there is good reason for this narrow approach: after-the-fact efforts to clarify offers “undermine the Rule’s purpose of encouraging settlement and avoiding protracted litigation.” Webb, 147 F.3d at 621 (citing Sas, 709 F.Supp. at 458). On the other hand, clarifying the terms of an offer before it has been accepted or rejected furthers the purpose of the Rule. See Radecki, 858 F.2d at 403.
Dobbs House failed to carry its burden. Dobbs House urges us either to compare its $10,000 offer with the total value of the jury’s verdict, $6500, or to use $3333, one-third of its offer, as a point of comparison with the individual verdicts. Alternatively, Dobbs House correctly points out that, mathematically, its offer would have had to be more favorable than the individual judgment of at least one of the plaintiffs. These varied constructions of the single offer only underscore its fatal problem: imprecision. The plaintiffs simply could not have evaluated the individualized values of the offer. Similarly, without two precise figures to compare, the district court was in no position to resolve the lack of precision. And neither are we. Cf. Webb, 147 F.3d at 621; Johnston, 803 F.2d at 870. The “lump sum” case which Dobbs House cites, Blumel v. Mylander, 165 F.R.D. 113 (M.D.Fla.1996), and others like it, are not to the contrary. Those cases descend from Marek v. Chesny, 473 *1077U.S. 1, 105 S.Ct. 3012, 87 L.Ed.2d 1 (1985), in which the Supreme Court held that Rule 68 did not require a defendant to “itemize the respective amounts being tendered for settlement” between damages for the substantive claim and costs. Id. at 6, 105 S.Ct. 3012. The Court reasoned that such a requirement would “not in any way help plaintiffs know in advance whether the judgment at trial will exceed a defendant’s offer.” Id. at 7, 105 S.Ct. 3012. In the instant situation, these predictive considerations' — whether precision in the offer aids a plaintiff’s evaluation of the offer — cut the other way; an offer itemized by individual plaintiff would have helped Ga-voni, Rosendale and Jordan independently evaluate the offer.
Dobbs House also pins its hopes on ■ a policy argument. It argues that Rule 68 should be interpreted to further the Rule’s underlying purpose of promoting settlement. Rule 68 is designed to encourage parties to evaluate objectively the' strength of then-cases; financial and judicial economy are at its core. See Marek, 473 U.S. at 5, 105 S.Ct. 3012. These important policies do frequently animate Rule 68 decisions, see, e.g., Delta Air Lines, Inc. v. August, 450 U.S. 346, 352-56, 101 S.Ct. 1146, 67 L.Ed.2d 287 (1981), but they are not subverted here.
Requiring a defendant to apportion a settlement offer made to multiple plaintiffs would not undermine Rule 68’s policy toward settlement. A requirement of plaintiff-specific offers would not discourage a defendant from tendering a Rule 68 offer. A defendant with a total figure in mind could simply apportion the figure to reflect the relative values of the plaintiffs’ alleged injuries. For example, Dobbs House eould have gauged the plaintiffs’ individualized offers using either the Westinghouse settlement (“because you, Gavoni, got 17% of the Westinghouse settlement, we offer you $1700”) or the plaintiffs’ requested damages (“because you, Ga-voni, are going to ask for 28% of the total requested damages, we offer you $2800”). Alternatively, if a defendant did not have such a readily accessible benchmark or was reluctant to specify the relative values of the individual claims, it could offer each plaintiff an equal share of the fixed figure. In the instant case, Dobbs House could have offered Gavoni, Rosendale and Jordan $3333 each. At the very least, the precisely apportioned offers might precipitate further negotiations between the defendant and all or some of the plaintiffs.
Similarly, a precision requirement would not deter plaintiffs from accepting individualized offers. One or more of a group of plaintiffs might accept an individualized offer. For every settling plaintiff, the pressure on the remaining plaintiffs .to settle increases, since they become potentially responsible for a greater share of the post-offer costs should their individual judgment be less favorable than the precise offer. For example, if Dobbs House had offered $3333 , to. each plaintiff and Gavoni had accepted, then Ro-sendale and Jordan would have been responsible for one-half, rather than one-third, of Dobbs House’s post-offer costs. They would have risked more by going to trial. Had Gavoni and Rosendale both accepted such an individualized offer, Jordan would have faced an even bigger gamble. In any event, agreement of some plaintiffs to settle would probably at least stimulate further negotiations by the nonsettling plaintiffs. Individualized offers, then, would probably increase the incentives for both sides to settle. In short, we can see no reason why divided offers to multiple plaintiffs are more likely to frustrate settlement than Rule 68 offers to a single plaintiff.
Finally, explicitly apportioned offers might avoid potential derivative litigation.' In this case, had the district court granted the defendant’s motion for costs under Rule 68 based on Dobbs House’s unapportioned $10,-000 offer, the plaintiffs (or perhaps only one or two of them) would likely have appealed that order. For example, if the district court had held each plaintiff responsible for one-third of Dobbs House’s post-offer costs, Ga-voni, who received only 17% of the Westinghouse settlement and asked for only 28% of the total requested damages, might have appealed the district court’s apportionment. (And, again, we would have been in no better position than the district court to apportion after the plaintiffs’ rejection of the offer.) Gavoni might have sued her fellow plaintiffs to recover the difference between the one-third portion that, the district judge assigned *1078her, in this hypothetical, and the 17% from the Westinghouse settlement or the 28% from the requested damages. She might even have accused her co-plaintiffs of scuttling a deal she was ready to accept and ask for full reimbursement for costs plus her portion (whatever it might have been) of Dobbs House’s rejected unapportioned $10,-000 offer. Individualized offers might limit the chances of such derivative collateral litigation. Thus, precision is efficient and promotes finality. Cf. Webb, 147 F.3d at 621; Arkla, 9 F.3d at 867; Radecki, 858 F.2d at 402-03; Johnston, 803 F.2d at 870; Chambers, 169 F.R.D. at 8.2
The district court did not err in any of its evidentiary decisions or its post-trial rulings on the parties’ motions for costs. The case is therefore Affirmed.
. Courts generally rely on contract principles to interpret Rule 68 offers. See, e.g., Webb, 147 F.3d at 620. The dissent appears to construe the rejected imprecise offer against the offerees. We believe that the offeror should bear the burden of precision; that is, we believe that the contract must be construed against the drafter.
. Here, there is a major difference between $10,-000 and $6500, making it easy to argue, as does the dissent, that under a broad range of arguable apportionments, all the plaintiffs would have done better with the offer than with the judgment. In a slightly different case, however, where the judgment might more closely approximate the offer, the mode of apportionment would be quite critical in determining which plaintiffs could be penalized under Rule 68. For example, had the jury awarded $3000 to Gavoni (instead of $2000), the sum of the judgments would still be less favorable than Dobbs House’s unappor-tioned offer ($7500 is less than $10,000). In this hypothetical, Gavoni gets: (1) a greater percentage of the total award than she requested (40% is greater than 28%); (2) more than her requested percentage (28%) as applied to the offer would have yielded ($3000 is greater than $2800); (3) a greater percentage than she received from the Westinghouse settlement (40% is greater than 17%) and; (4) a greater percentage of the award than an equal split (40% is greater than 33%); but still (5) less than what an equal split of the offer would have yielded ($3000 is less than $3333). In this scenario, was Gavoni’s judgment more favorable than the offer? Gavoni, angling to get off the hook for costs, would answer in the affirmative, citing calculations (1) through (4); Rosendale and Jordan, potentially on the line for half rather than a third of the costs, would almost certainly disagree, relying on calculation (5). So, a minor deviation in the present numbers would complicate things greatly. Indeed, even as it now stands, an equitable division is not patently obvious; Gavoni's award of $2000 is about 31% of the total award, which is greater than the 17% of the Westinghouse settlement she received and, as we have indicated, only one of many possibilities a court assessing costs might choose. The dissent's assumption that the requested percentages best represent the relative merits of the plaintiffs’ claims conflicts with the plaintiffs' different division of the Westinghouse settlement (as well as with the jury’s award).
An appropriate rule here should not rest solely on the special facts of this case, but should be suitable for broader application. Requiring the offeror, always a defendant, to make an offer precise enough to enable each offeree, always a plaintiff, to assess the merits of her claim relative to the value of the offer is the better rule for this case and others.