The owners and would-be developers of a tract of California real estate appeal from various rulings of the district court in a long-running dispute arising out of a local, government’s rejection of a proposed housing development. For the reasons which follow we affirm each of the district court’s rulings.
FACTS AND PRIOR PROCEEDINGS1
These consolidated appeals represent another round in a protracted legal contest involving John and David Herrington (“Her-ringtons”) and the County of Sonoma, California (“County”). In 1970 the Herringtons inherited a 540-acre farm (“Property”) near Sebastopol, California. Six years later they began work on a proposal to develop the Property for residential use. Over the course of the following three years the Her-ringtons, working with the County’s planning staff, scaled back their original plans for a 103-unit residential development to a final proposal for a 32-unit development.
*903The Herringtons filed their 32-unit application with the County in 1979.- The County rejected the application as being inconsistent with both its General Plan2 and a pending Special Plan for West Sebastopol. In 1980, after exhausting their administrative remedies, the Herringtons filed a civil rights action under 42 U.S.C. § 1983 against the County in federal district court. The district court stayed the proceedings pending resolution of a parallel state court action, and the Herringtons appealed. We reversed the district court, denied the Herringtons’ petition for a writ of mandamus, and imposed costs on the County. Herrington v. County of Sonoma (Herrington I), 706 F.2d 938, 940 (9th Cir.1983).
The Herringtons’ case went to trial in 1985. The jury awarded them $2.5 million in damages and, pursuant to a stipulation between the parties, the court entered an injunction declaring the County’s inconsistency determination to be invalid. Before appealing the judgment against it the County moved to disqualify the trial judge for bias. The district judge denied the motion but recused himself from further participation in the case. The County then appealed, ássert-ing a host of issues. In that appeal, we rejected most of the County’s arguments and affirmed all but the damages award, which we vacated as excessive and remanded for a new trial solely on the issue of damage's. Herrington v. County of Sonoma (Herrington II), 834 F.2d 1488, 1503-1506 (9th Cir.1987), opinion amended & reh’g denied, 857 F.2d 567 (1988), cert. denied, 489 U.S. 1090, 109 S.Ct. 1557, 103 L.Ed.2d 860 (1989).3
Following a lengthy period of time devoted largely to the question of deciding just how much the County had to pay the Herring-tons’ attorneys,4 the district court retried the damages claim. In a careful and searching analysis of the parties’ respective positions, the district court found that the Herringtons were entitled to damages of $52,123.50 and interest in the amount of $69,348.56 for a total judgment of $121,472.06. Herrington v. County of Sonoma, 790 F.Supp. 909, 925 (N.D.Cal.1991) (as amended). In addition, the district court separately ruled that, because the damages awarded were substantially less than the amount offered in settlement by the County before retrial, Fed. R.Civ.P. 68 precluded the Herringtons from receiving the full amount of their requested attorney’s fees under 42 U.S.C. § 1988. Finally, the trial court rejected the Herring-tons’ demand for enforcement of injunctive relief, ie., a request that the County act on the court’s prior declaration that the County’s inconsistency determination was invalid, holding that to do so would require the court to make additional factual determinations and fashion a hew form of relief. The Her-ringtons have timely appealed from each of these three rulings.
• ANALYSIS '
I. Computation of Damages Award [93-15884]
A. Standard of Review
A district court’s computation of damages to be awarded is a factual finding examined for clear error. Stephens v. City of Vista, 994 F.2d 650, 655 (9th Cir.1993).
*904B. Discussion
The question of whether the district court properly calculated the amount of damages caused by the County and suffered by the Herringtons constitutes the very heart of the three consolidated cases now before us. The gist of our prior decision to set aside the damages award of $2.5 million was based on our conclusion that the award was “so grossly excessive that it shocks the conscience.” Herrington II, 834 F.2d at 1503. In reaching this conclusion we noted three things: First, the amount of damages awarded was outrageously speculative and nearly double the estimated sale value of the Property with a 32-unit development potential (estimated at between $1.3 million and $1.5 million), an obviously disproportionate figure since the Herringtons still retained possession of the land and never had a right to construct a 32-unit development thereon, anyway. Id. at 1504-05. Second, and even assuming that the Herringtons’ proposed development had been approved by the County, the harm they suffered involved only a temporary delay in receiving the difference between the developed and undeveloped values of their land. Id. at 1505. Third, the damages were improperly cumulative because they included profits from both the sale and the development of the land; ie., if the Herringtons sold their land they could not also realize profits from its development by the purchaser. Id. at 1506.
Before retrial the district court held that certain matters had been disposed of on appeal and were not subject to redetermination. Citing “the principles of the law of the case, the mandate rule, or estoppel!,]” the trial court declared that, because the appellate court had found the Herringtons’ claimed lost value figure of $810,000.00 to be too high, the district court was barred from awarding them an amount equal to or greater than that sum.5 Herrington, 790 F.Supp. at 912-13.
The Herringtons contend that the district court erred by (1) concluding that the law of the case doctrine, the rule of mandate, and/or the principle of estoppel applied to limit their recovery; (2) refusing to consider their new appraisal evidence; and (3) erroneously calculating their damages by (a) deducting the same figure twice, (b) using amounts from the wrong appraisal, and (c) incorrectly determining the delay period to be l1/ years instead of 9)4 years. Each of these issues will be discussed in its turn.
1. Law of the Case/Mandate Rule
“The law of the case doctrine states that . the decision of an appellate court on a legal issue must be followed in all subsequent proceedings in the same case.” Maag v. Wessler, 993 F.2d 718, 720 n. 2 (9th Cir.1993) (citations and internal quotation omitted). The doctrine
is a judicial invention designed to aid in the efficient operation of court affairs. Under the doctrine, a court is generally precluded from reconsidering an issue previously decided by the same court, or a higher court in the identical case. For the doctrine to apply, the issue in question must have been decided explicitly or by necessary implication in [the] previous disposition.
Milgard Tempering, Inc. v. Selas Corp. of America, 902 F.2d 703, 715 (9th Cir.1990). Accord Thomas v. Bible, 983 F.2d 152, 154 (9th Cir.), cert. denied, — U.S. -, 113 S.Ct. 2443, 124 L.Ed.2d 661 (1993).
Although broader than the law of the case doctrine, the so-called rule of mandate allows a lower court to decide anything not foreclosed by the mandate. See Lindy Pen Co. v. Bic Pen Corp., 982 F.2d 1400, 1404 (9th Cir.) (“An order issued after remand may deviate from the mandate ... if it is not counter to the spirit of the circuit court’s decision!, but t]he district court must be given a meaningful opportunity to follow the directive of the circuit in resolving the issues.”) (internal citation omitted), cert, de*905nied, — U.S. -, 114 S.Ct. 64, 126 L.Ed.2d 34 (1993).
Because the mandate in the instant ease provided for a new trial on the issue of damages, the rule of mandate did not in and of itself serve to preclude the district court from considering the possibility that the extent of those damages might have exceeded $810,000.00 in lost valué damages. However, we emphasized in Herrington II that all three components6 of the $2.5 million damages award were grossly excessive, and that the $810,000.00 figure in particular was “an excessive award' for lost value because the $1.3 million dollar [sic] figure is too high and the $490,000 figure is too low.” Herrington II, 834 F.2d at 1504. We hold that the law of the case doctrine applies here, and that the district court did not err in concluding that our rejection of the $810,000.00 lost value damages figure set an upper limit on the amount of the Herringtons’ recoverable damages.7
2. New Appraisal Evidence
In light of our holding that the district court did not err by applying the law of the case doctrine to preclude consideration of evidence supporting a lost value damages claim in excess of $810,000.00, it follows that the Herringtons’ contention that the district court erred by failing to consider their subsequent appraisal, which set the Property’s value at $2.3 million, $1 million higher than the figure we had previously found to be “grossly excessive”, must also.fail.
3. Formula for Damages Calculation
In an extremely careful and detailed series of steps, the district court set out its formula for calculating the actual lost value suffered by the Herringtons as the result of the delay caused by the County’s inconsistency determination with respect to their 32-unit application., Boiled to its essence, the formula may be summarized as follows: The court first multiplied the maximum value of the Property (i.e., assuming approval of the 32-unit application, $1.3 million) by one-third (i.e., the district court’s calculation of the probability of the application’s approval). The court then subtracted from the product the purported undeveloped value of the Property ($490,000.00). Next, the court multiplied the difference by the product of the interest rate (i.e., based on the amount the money could have earned) and the duration of the delay. Finally, the court added to the product whatever increased development costs were likely to have resulted from the delay.8 The Herringtons challenge the correctness of this formulation by arguing, inter alia, that the maximum value is wrong, the minimum value is deducted twice, and the delay period is too short.
With respect to the first point, the Her-ringtons insist that the $1.3 million maximum value figure is based on an incorrect reading of the record. The Herringtons argue that the former appraiser actually set a maximum value of the property at $2.7 million, and only mentioned the $1.3 million figure as a maximum value after factoring in the probability of the County’s acceptance of the application. We implicitly rejected this argument when it was previously raised in the Herringtons’ petition for rehearing from the decision in Herrington II, the district court rejected it at retrial, and it is inconsistent with other estimates. See e.g., Herrington II, 834 F.2d at 1504 & n. 16. Accordingly, we find no *906clear error in the district court’s ruling on this point.
The Herringtons’ argument that the minimum value is improperly deducted twice must also be rejected. The district court’s formula does not actually deduct the so-called “after” value twice; rather, it adds two-thirds of the $490,000.00 “after” value to one-third of the $1.3 million “before” value in order to arrive at an average figure, then subtracts the “after” value to account for the fact that the Herringtons retained possession of their land, developed or not. Under these circumstances, we hold that the district-court did not err in its treatment of the minimum value formulation.
Finally, the Herringtons insist that the V/¿ year delay used in th,e formula misrepresents the actual delay of 9¡4 years. The gist of their argument is that the time that should be used in the formula is not how long the delay should have taken, but the amount of time that actually elapsed between the County’s initial rejection of their application on December 11, 1979 and the 1988 remand following the amendment to our opinion in Herrington II.
The task for the district court on remand was to assess the damages for the delay in processing . the Herringtons’ application. There was no way for the court to determine whether, absent the erroneous ground for denial, the proposed subdivision would have been denied on other grounds. Accordingly, the district court utilized a probability factor and took testimony to assess what that probability was. In doing so, the court determined the amount of damages that then existed as the result of the' County’s failure to process the Herririgtons’ application. Having been awarded full damages for the failure to process their application, the Herringtons were not entitled to have that application processed. The probabilities for success were assessed-and the full measure of damages was awarded on that basis.
In light of these facts, and because of the unresolved dispute over the question of whether the Herringtons do, in fact, even have a completed application on file, we find that the district court did not err by setting the period of delay at Vk years.
II. Rule 68 and Attorney’s Fees [92-16915]
A. Standard of Review
“[Ijssues involving, construction of Rule 68 offers are reviewed de novo, [while] disputed factual findings concerning the circumstances under which the offer was made are usually reviewed for clear error.” Erdman v. Cochise County, 926 F.2d 877, 879 (9th Cir.1991) (footnote omitted). Accord Simon v. Intercontinental Transp. (ICT) B.V., 882 F.2d 1435, 1439 (9th Cir.1989) (“[W]e review de novo whether [defendant’s] offer of judgment, [plaintiffs] rejection of the offer, and the judgment following the trial satisfied the requirements of Rule 68.”).
B. Discussion
Prior to retrial on the damages claim the County offered to settle the case by promising to pay the Herringtons $501,000.00 and' all costs of the action, including the Herringtons’ attorney’s fees.9 The Herring-tons ignored the offer and proceeded to trial, where their judgment, including interest, amounted to less than one-fourth of the offered settlement figure. Citing Fed.R.Civ.P. 68, the district court declined to award the Herringtons that portion of their costs and fees related to work performed after the date of the County’s settlement offer. The Herringtons challenge the district court’s ruling on the ground that the County’s offer did not meet the requirements of Rule 68.
Rule 68 says, in relevant part:
At any time more than 10 days before the trial begins, a party defending against a claim may serve upon the adverse party an offer to allow judgment to be taken against the defending party for the money or property or to the effect specified in the offer, with costs then accrued.'... If the judgment finally obtained by the offeree is not *907more favorable than the offer, the offeree must pay the costs incurred after the making of the offer.... When the liability of one party to another has been determined by verdict or order or judgment, but the amount or extent of the liability remains to be determined by further proceedings, the party adjudged liable may make an offer of judgment, which shall have the same effect as an offer made before trial if it is served within a reasonable time not less than 10 days prior to the commencement of hearings to determine the amount or extent of liability.
Fed.R.Civ.P. 68.
Rule 68 is a cost-shifting provision designed to encourage settlement of legal disputes by forcing a plaintiff to weigh the risk of incurring post-offer costs and fees he may not be able to recover even if successful on his claim. Because attorney’s fees are included as costs of a federal civil rights action under 42 U.S.C. § 1988, Corder v. Gates, 947 F.2d 374, 381 n. 9 (9th Cir.1991), the cost-shifting provision of Rule 68 applies to limit a prevailing plaintiffs recovery of fees in a section 1983 action if the plaintiff rejects an offer that exceeds his damages award. Marek v. Chesny, 473 U.S. 1, 9, 105 S.Ct. 3012, 3016-17, 87 L.Ed.2d 1 (1985).
The requirements for a valid Rule 68 offer are simple: “An offer of judgment should ... [be] served on the adverse party in the manner provided by [Fed.R.Civ.P.] 5(b)_ The offer must specify a definite sum for which judgment may be entered, which plaintiff can either accept or reject. It must be unconditional and must include ‘costs then accrued.’ ” 12 Charles A. Wright & Arthur R. Miller, Federal Practice & Procedure: Civil § 3002 (1973). The consequences of rejecting such an offer are clear: “If the judgment finally obtained by the offeree is not more favorable than the offer, the offeree must pay the costs incurred after the making of the offer.” ■ Id. at § 3005.
In order to determine whether an offer of settlement comports with the requirements of Rule 68, a court will ordinarily apply the usual rules of contract construction. Erdman, 926 F.2d at 880. Thus, ambiguities will be construed against the of-.feror as the drafting party and, where such ambiguities are found to exist, extrinsic evidence of the parties’ actual intentions will be examined to clarify those ambiguities and arrive at the meaning of the offer’s material terms. Id.
The offer submitted by the County to the Herringtons reads as follows:
Pursuant to Federal Rules of Civil Procedure, Rule 68, defendant COUNTY OF SONOMA, hereby offers to allow judgment to be taken against it and in favor of plaintiffs as follows:
(1) In the amount of $501,000; plus,
(2) Costs incurred by plaintiffs which shall include attorneys’ fees in an amount to be determined by the Court.
This offer is subject to the terms and conditions of Federal Rules of Civil Procedure, Rule 68.
This offer is clear on its face, and we reject the Herringtons’ efforts to inject ambiguity into the settlement offer by listing those things it does not promise, e.g., that the County will not proceed with any other appeals and that the Herringtons’ application will be (re)processed. The County’s offer was properly and timely served on the Her-ringtons, and it comports with the strict requirements of Rule 68. Accordingly, the district court did not err in concluding that Rule 68 precluded the Herringtons from recovering that portion of their costs and fees related to work performed after thé date of the County’s offer.
III. Denial of Enforcement of Injunction [92-17012]
A. Standard of Review
Because we view .the lower court’s refusal to grant the Herringtons’ motion to enforce the injunction as being tantamount to a denial of injunctive relief, we examine that decision de novo. See Inland Empire Pub. Lands Council v. Schultz, 992 F.2d 977, 980 (9th Cir.1993). Our review of the decision is limited, however, and we will reverse it only if we determine “that the district court abused its discretion or based its decision on an erroneous legal standard or on clearly *908erroneous findings of fact.” Metro Publishing, Ltd. v. San Jose Mercury News, 987 F.2d 637, 639-40 (9th Cir.1993). We examine de novo the lower court’s underlying legal determinations. See id.
B. Discussion
The parties’ final bone of contention revolves around the meaning of the injunction entered' by the district court in 1986 and upheld on a prior appeal. Pursuant to a stipulation proposed'by the district court and agreed to by both sides, the trial court issued the following injunction: “The Sonoma County Board of Supervisors’ December 11, 1979, denial of plaintiffs’ subdivision, as described in Resolution Number 65069, is invalid.” We upheld that order on appeal when we declared, “[W]e AFFIRM the district court’s injunctive order invalidating the County’s inconsistency determination.” Herrington II, 834 F.2d at 1507.
The Herringtons contend that the stipulation upon which the injunction was based provided that the County could not claim that the Herringtons’ 32-unit application was inconsistent with the General Plan. They concede, however, that the injunction did not prohibit the County from reviewing — and perhaps rejecting — the application on grounds other than that of inconsistency. The County insists that the injunction only prevented the County from reviving and relying on its December 11,1979 inconsistency determination as a basis for rejecting the Herringtons’ application. The County further argues that there is nothing to review with respect to this issue, anyway, because the Herringtons’ application was never properly prepared or filed and was returned to them long ago.
Although the Herringtons’ contentions have much superficial appeal, they fly in the face of the district court’s explicit statement that “[Tlhe ,judgment did not enjoin the County from finding the •proposed subdivision inconsistent with the General Plan; it [merely] invalidated the earlier inconsistency finding by the Board of Supervisors.” Herrington, 790 F.Supp. at 922 (emphasis added).
The district court’s reading of its own in-junctive order makes sense. To hold otherwise would require us to order the County to process, and effectively approve, an application that may not be compléte or even on file.10 Under these circumstances, and in light of the above, we find no clear error in the district court’s decision denying the Her-ringtons’ motion to enforce the injunction.
CONCLUSION
The district court’s calculation of damages in our No. 92-15834 is AFFIRMED. The district court’s denial of attorney’s fees in our No., 92-16915 is AFFIRMED. The district court’s rejection of the Herringtons’ demand to enforce the injunction in our No. 92-17012 is AFFIRMED. In light of these rulings, and because we find no merit to any of the Herringtons’ remaining contentions, their request for attorney’s fees on appeal is DENIED.
. Most of the underlying facts in these consolidated appeals are not in serious dispute and have been set out in our prior opinions in Herrington v. County of Sonoma, 706 F.2d 938 (9th Cir.1983); Herrington v. County of Sonoma, 834 F.2d 1488 (9th Cir.1987), opinion amended & reh'g denied, 857 F.2d 567 (1988), cert. denied, 489 U.S. 1090, 109 S.Ct. 1557, 103 L.Ed.2d 860 (1989); and Herrington v. County of Sonoma, 883 F.2d 739 (9th Cir.1989), as well as in those of the district court in Herrington v. County of Sonoma, 655 F.Supp. 1111 (N.D.Cal.1987), aff'd in part, modified in part, 883 F.2d 1024 (9th Cir.1989) (TABLE), and Herrington v. County of Sonoma, 790 F.Supp. 909 (N.D.Cal.1991) (as amended).
. The County’s General Plan is a statement of development policies required by California law that would govern all future development in the County. The General Plan, which set a maximum density of 35 residential units for the Property, was subject to downward revision if deemed necessary to preserve the County's forest and agricultural resources.
. Our subsequent amendment of the opinion in Herrington II was limited to a clarification of our holding on the issue of ripeness. Herrington, 857 F.2d at 568-71 (order amending opinion, denying petition for rehearing, and rejecting suggestion for en banc review). The amendment did not affect the merits of the original opinion and is not an issue on,this appeal.
. In Herrington v. County of Sonoma, 655 F.Supp. 1111 (N.D.Cal.1987), the district court awarded the Herringtons $363,832.70 in attorney's fees and an additional $42,629.12 in costs. By unpublished memorandum we reduced the fee award but left the costs award intact. Herrington v. County of Sonoma, 883 F.2d 1024 (9th Cir.1989) (TABLE). In a separately filed but published order we also found the County liable for an additional $41,854.00 in fees and $655.00 in costs. Herrington, 883 F.2d at 747.
. During the course of the retrial on the Herring-tons’ damages claim the district court allowed the Herringtons to place into evidence documentation supporting their contention that each of the three components of the jury’s $2.5 million damages award was (in 1991) greater than the figure we found (in 1987) to be grossly excessive. The district court noted, however, that it was doing so only in the hope of avoiding yet another evidentiary proceeding on this issue. Herrington, 790 F.Supp. at 913, 916.
.The Herringtons argued that the $2.5 million damages award was based on three components: (1) $810,000,00 in lost value (i.e., the difference 'between $1.3 million, representing the value of the Property if the 32-unit application had been approved, and $490,000.00, representing the value of the Property without such approval); (2) $910,000.00 in lost potential profit (i.e., profits derived from the sale and development of the Property itself); and (3) $800,000.00 in loss of return (i.e., interest on (1) and perhaps (2), supra). See Herrington II, 834 F.2d at 1504 & n. 18.
. Because of our conclusion that the district court correctly applied the law of the case doctrine to this issue, we need not reach the question of the applicability vel non of collateral estoppel to these facts.
. Viewed in algebraic form, with variables substituted for the above values, the formula looks like this: [ (aX + bY) — Y]Rt + ac. See Herrington, 790 F.Supp. at 916.
. The Herringtons insist, and the County concedes, that the County’s subsequent settlement offer of $751,000.00 is not at issue here.
. The district court did not reach the question of whether the Herringtons’ application was complete and on file. Herrington, 790 F.Supp. at 921. Although we declared that, for the purpose of appellate jurisdiction, the Herringtons had an application on file with the County as-of 1979, that application may not have been complete. See Herrington, 857 F.2d at 569 ("[W]e do not believe that the Herringtons’ development application was formally complete”). For the purposes of this appeal, however, we need not resolve this question.