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Abstract This study investigates the design of the royalty rate in a first-price auction across three types of investments: incremental and lumpy with or without an exogenously given intensity. A bidder’s investment cost comprises private information. This, together with the stochastic evolution of the price of the output generated from the auctioned project, precludes the seller from setting the exact dates of investment with the winner. However, the seller can set the royalty rate to equate the winner’s royalty payment with the winner’s information rent so that the winner acts as if to maximize the seller’s revenue. We derive two main conclusions. First, compared with the case in which investment is lumpy with an exogenously given intensity, the seller can set a lower royalty rate on incremental investment because she can collect additional royalty payments from the winner, who has the option to later expand capacity. Second, the impact of output price uncertainty on the optimal royalty rate for the three types of investments exhibits two different patterns. When investment is either incremental or lumpy with an exogenously given intensity, greater output price uncertainty reduces the royalty rate. When investment is lumpy with variable intensity, greater output uncertainty raises the royalty rate. Our results imply that auctioneers may charge differential royalty rates for different types of investments.

BackgroundMinimally invasive transforaminal lumbar interbody fusion (MIS TLIF) is a surgical technique frequently used to treat symptomatic lumbar spondylolisthesis. We aim to investigate the safety and efficacy of using a biplanar expandable cage in the treatment of symptomatic lumbar spondylolisthesis using a MIS TLIF approach.MethodsA retrospective review of patient records was performed on patients who underwent MIS TLIF for symptomatic lumbar spondylolisthesis using the FlareHawk cage over a 12-month period. Patient demographics, as well as preoperative and postoperative clinical and radiographic outcome measures were recorded and analyzed.ResultsA total of 13 consecutive patients underwent MIS TLIF for symptomatic spondylolisthesis during the study period. The mean age was 60.2 ± 13.9 years, and 61.5% were female. The mean preoperative and postoperative slippage was 7.0 ± 3.0 mm and 1.0 ± 1.9 mm, respectively. The preoperative mean segmental lordosis was 5.1° ± 6.0°, mean anterior, posterior disc, and foraminal height were 9.1 ± 3.9 mm, 5.7 ± 1.5 mm, and 11.0 ± 2.0 mm, respectively. The postoperative mean segmental lordosis was 6.8° ± 4.7°, and mean anterior, posterior disc, and foraminal height were 11.4 ± 2.2 mm, 7.8 ± 1.0 mm, and 12.3 ± 1.3 mm. There was improvement in all radiographic parameters postoperatively. The mean Visual Analog Scale (VAS) back pain, VAS leg pain improved from 7.0 ± 2.9 and 5.1 ± 3.0 preoperatively to 3.1 ± 2.9 and 1.1 ± 1.7 at the latest clinic follow-up visit, respectively (P = .0081). The mean EuroQol-Five Dimensions (EQ5D) score improved from 0.37 ± 1.7 to 0.66 ± 0.23 after surgery. There was no subsidence, endplate violation, cage migration, or other implant-related complications. No patient required reoperation.ConclusionsThe biplanar expandable cage is both safe and efficacious in treating symptomatic lumbar spondylolisthesis using the MIS TLIF approach. Spine surgeons should be familiar with the biplanar expandable cage technology and keep it in their armamentarium in surgical treatment of lumbar spondylolisthesis.Level of evidence4.

Posterior cervical osteotomies are powerful techniques for the correction of rigid cervical deformity. These include a variety of subtypes including partial facet joint resection, complete facet joint resection, opening wedge osteotomy (OWO), and closing wedge osteotomy (CWO). The partial facet joint resection provides limited lordosis but can be applied across multiple levels and provides bony surface for fusion. Complete facet joint resection can also be performed across multiple segments for a cumulative effect and like the partial facet joint resection requires mobility of the anterior column. The OWO is traditionally performed at C7 and involves complete a complete laminectomy, facetectomy, and pediculectomy with special care to fully decompress the C8 nerve roots prior to osteotomy closure. The osteotomy utilizes a fulcrum of rotation in the middle column with shortening of the posterior column and lengthening of the anterior column with an osteoclastic fracture that must be performed with significant care. The CWO is similar to an OWO with the addition of an osteotomy into the vertebral body that is closed like a pedicle subtraction osteotomy (PSO). The goal of this review article is to summarize posterior osteotomy techniques for cervical deformity correction.