Xiaodong Yuan

Jun Li

Cédric Coulouarn

Tao Lin

Laurent Sulpice

Damien Bergeat

Carolina De

La Torre

Roman Liebe

Norbert Gretz

Matthias P A Ebert

Steven Dooley
email: steven.dooley@medma.uni-heidelberg.de

Hong-Lei Weng
email: honglei.weng@medma.uni-heidelberg.de

Theodor-Kutzer Ufer

Theodor -Kutzer

SOX9 expression decreases survival of patients with intrahepatic cholangiocarcinoma by conferring chemoresistance

Keywords: SOX9, cholangiocarcinoma, gemcitabine, checkpoint kinase 1, multidrug resistance CHEK1, Checkpoint kinase 1 CK19, cytokeratin 19 CSC, cancer stem cell DFS, disease free survival DR, ductular reaction EpCAM, epithelial cell adhesion molecule eCCA, extrahepatic cholangiocarcinoma

Introduction

Cholangiocarcinoma (CCA) is the second most common primary liver cancer following hepatocellular carcinoma (HCC), and accounts for approximately 10%-15% of all primary liver malignancies (1). The global incidence and mortality rate for CCA have been increasing over the last decades [START_REF] Bridgewater | Guidelines for the diagnosis and management of intrahepatic cholangiocarcinoma[END_REF][START_REF] Njei | Changing pattern of epidemiology in intrahepatic cholangiocarcinoma[END_REF]. Anatomically, CCA is classified into intrahepatic (iCCA) and extrahepatic cholangiocarcinoma (eCCA) depending on the location of the tumor along the biliary tract (4). To date, curative surgical resection is the most efficient treatment for longterm survival of selected iCCA patients [START_REF] Rizvi | Emerging molecular therapeutic targets for cholangiocarcinoma[END_REF][START_REF] Deoliveira | Cholangiocarcinoma: thirty-one-year experience with 564 patients at a single institution[END_REF]. However, in most cases, the tumors are quite advanced at the time of diagnosis and surgical resection is not possible [START_REF] Rizvi | Emerging molecular therapeutic targets for cholangiocarcinoma[END_REF]. Systemic chemotherapy and radiotherapy regimens remain the only approach to render patient eligible for surgery and palliative treatment [START_REF] Valle | Cisplatin plus gemcitabine versus gemcitabine for biliary tract cancer[END_REF]. However, the response of iCCA to these treatments is very weak [START_REF] Valle | Cisplatin plus gemcitabine versus gemcitabine for biliary tract cancer[END_REF]. Therefore, elucidating the underlying mechanisms of iCCA chemoresistance is one key issue to improve survival of patients.

Sex-determining region Y-box (SRY-box) containing gene 9 (SOX9) belongs to the SOX family of transcription factors [START_REF] Gordon | Long-range regulation at the SOX9 locus in development and disease[END_REF]. It is widely expressed in multiple organs during embryonic development, including the liver [START_REF] Pritchett | Understanding the role of SOX9 in acquired diseases: lessons from development[END_REF][START_REF] Jo | The versatile functions of Sox9 in development, stem cells, and human diseases[END_REF]. In liver embryogenesis, SOX9 expression is the most specific and earliest marker of hepatoblasts and determines the timing of intrahepatic bile duct morphogenesis [START_REF] Antoniou | Intrahepatic bile ducts develop according to a new mode of tubulogenesis regulated by the transcription factor SOX9[END_REF][START_REF] Si-Tayeb | Organogenesis and development of the liver[END_REF]. In normal adult liver, SOX9 is expressed in the periportal small intrahepatic ducts, and peribiliary glands lining the large bile ducts [START_REF] Carpino | Biliary tree stem/progenitor cells in glands of extrahepatic and intraheptic bile ducts: an anatomical in situ study yielding evidence of maturational lineages[END_REF]. SOX9 plays important roles in maintaining liver homeostasis, regulating liver regeneration, and eventually in liver cancer development [START_REF] Kawaguchi | Sox9 and programming of liver and pancreatic progenitors[END_REF]. In acute or chronic liver disease, SOX9 expression robustly manifests in ductular reactions (DRs), which contain putative progenitor cells capable of differentiating into both cholangiocytes and hepatocytes [START_REF] Jors | Lineage fate of ductular reactions in liver injury and carcinogenesis[END_REF]. Moreover, SOX9 positive cells express stem cell markers, such as epithelial cell adhesion molecule (EpCAM), neural cell adhesion molecule, CD133, CXC motif chemokine receptor 4 [START_REF] Carpino | Biliary tree stem/progenitor cells in glands of extrahepatic and intraheptic bile ducts: an anatomical in situ study yielding evidence of maturational lineages[END_REF][START_REF] Cardinale | Multipotent stem/progenitor cells in human biliary tree give rise to hepatocytes, cholangiocytes, and pancreatic islets[END_REF]. In contrast to normal hepatocytes, where SOX9 is not expressed, a subset of HCC cells displayed SOX9 expression. These patients usually demonstrate severe venous cancer invasion, advanced tumor stage and shorter survival [START_REF] Leung | Sox9 confers stemness properties in hepatocellular carcinoma through Frizzled-7 mediated Wnt/beta-catenin signaling[END_REF]. Recent studies reported that SOX9 positive HCC cells exhibit liver cancer stem cell-like features, and that SOX9 in cancer cells confers self-renewal and tumorigenicity by promoting symmetrical cell division [START_REF] Leung | Sox9 confers stemness properties in hepatocellular carcinoma through Frizzled-7 mediated Wnt/beta-catenin signaling[END_REF][START_REF] Liu | Sox9 regulates self-renewal and tumorigenicity by promoting symmetrical cell division of cancer stem cells in hepatocellular carcinoma[END_REF]. To date, only few studies had addressed the role of SOX9 in CCA [START_REF] Matsushima | Sox9 expression in carcinogenesis and its clinical significance in intrahepatic cholangiocarcinoma[END_REF]. Here, we report clinical and functional data supporting an oncogenic role and therapeutic significance of SOX9 expression in iCCA.

Materials and Methods

Patients and liver tissues

This study enrolled 59 iCCA patients from Tübingen, Germany (18 iCCA patients) and Rennes, France (41 iCCA patients) between 2002 and 2010. In addition, 21 liver tissues from patients with chronic hepatitis B infection were enrolled in Mannheim, Germany. Basic characteristics of the enrolled chronic hepatitis B and iCCA patients are shown in Supplementary Table 1 and2. The study protocol fulfilled national laws and regulations and was approved by the local Ethics Committees.

Cell culture and treatment

The following cell lines were investigated in the study: CC-SW-1 and HuCCT-1 (iCCA lines), EGI-1 and TFK-1 (eCCA lines), HCCC-9810 (mix CCA line) and MMNK-1 (normal cholangiocyte line). EGI-1, CC-SW-1 and MMNK-1 were cultured in DMEM (BE12-709F, Lonza) supplemented with 10% FBS (10270-098, Invitrogen), 4mM L-glutamine (17-605C, Lonza) and 100 U/mL penicillin/streptomycin (A2210 Biochrom KG). TFK-1 and HuCCT-1 were cultured in RPMI1640 supplemented with 10% FBS (10270-098, Invitrogen), 4mM Lglutamine (17-605C, Lonza) and 100 U/mL penicillin/streptomycin (A2210 Biochrom KG).

All the cell lines were cultured in a humidified incubator at 37° and with 5% CO 2 atmosphere. Cells underwent starvation without FBS medium for 10 to 16 hours before treatment with gemcitabine and cisplatin (Kindly provided by Prof. Lu LG, Shanghai Jiao Tong University School of Medicine), which was dissolved in phosphate-buffered saline (PBS) to make a 100mM stock solution and diluted with cell culture medium to indicated concentrations during treatment.

Immunohistochemistry and staining evaluation

Tissue microarray assay was performed as previously described [START_REF] Weng | The etiology of liver damage imparts cytokines transforming growth factor beta1 or interleukin-13 as driving forces in fibrogenesis[END_REF]. In brief, formalin-fixed, paraffin-embedded specimens were deparaffinized in serial ethanol dilutions and rehydrated.

After a single PBS wash, heat-induced antigen retrieval was performed with 1mM EDTA solution, pH 8.4 (03677; Sigma-Aldrich, Steinheim, Germany) at 98°C for 10 minutes.

Endogenous peroxidase activity was blocked with Dako dual endogenous enzyme blocking reagent (S2003; Dako,Via Real Carpinteria, California, USA), followed by blocking with 3 % hydrogen peroxidase for 5 minutes at room temperature to prevent unspecific binding of antibodies. The tissue sections were incubated with polyclonal rabbit anti-SOX9 antibody (HPA001758; Sigma-Aldrich) at a dilution of 1:100, or monoclonal mouse anti-CK19 antibody (SC-6287; Santa Cruz) at a dilution of 1:100 overnight at 4°C. The specimens were subsequently washed in PBS for 3 × 5 minutes, and incubated with anti-rabbit or anti-mouse secondary antibody conjugated with horseradish peroxidase for 1 hour at room temperature, and then detected with 3.3'-diaminobenzidine for 7 minutes. The slides were counterstained with hematoxylin. All sections were dehydrated and mounted with malinol mounting medium.

Immunostaining results for SOX9 were scored semi-quantitatively based on the intensity and proportion of positive tumor cell nuclei. In detail, the intensity score of SOX9 nuclear staining was defined as four grades: 0, negative; 1, weak with color yellow; 2, medium with color brown; 3, strong with color black. The number of cells with SOX9 positive nuclei was defined as six grades: 0, no detectable positive cells; 1, positive cells ≤ 1%; 2, positive cells > 1%, and ≤ 10%; 3, positive cells > 10%, and ≤ 33%; 4, positive cells > 33%, and ≤ 66%; 5, positive cells > 66%. The final immune staining scores were calculated as the intensity scores × the proportion scores. The samples with final scores over 10 were defined as "high SOX9 expression", and the remainder as "low SOX9 expression". The representative pictures of SOX9 staining and for semi-quantitative scoring system are presented in Supplementary Figure 1. CK19 expression was categorized into high expression and low expression according to the immunoreactivity in tumor cells. The immunoreactivity of CK19 was defined as four grades: 0, positive cells ≤1%; 1, positive cells > 1% and ≤ 33%; 2, positive cells > 33% and ≤66%; 3, positive cells > 66%. The samples with grade 3 were defined as high CK19 expression, and the others were low CK19 expression.

RNA interference (RNAi)

For transient transfection of short interfering RNA (siRNA), cells were treated with indicated culture medium without penicillin/streptomycin. SiRNA targeting human SOX9 (M-021507-00) and control siRNA (D-001206-14) were purchased from Dharmacon. SOX9 siRNA were transfected with RNAiMAX (13778, Invitrogen). The transfection was performed in six-well cell culture vessels. Tumor cells were plated at a density of 1.5 × 10 5 cells per well with 2ml corresponding growth medium. Briefly, for siRNA transfection, 2μl RNAiMAX was mixed with 20pmol SOX9 siRNA in 200μl Opti-MEM medium. The mixtures were preincubated for 20 min at room temperature before adding to cells. RNA and whole cell protein were extracted 48 and 60 hours after transfection for further examination, respectively.

3-(4, 5-dimethylthiazolyl-2)-2, 5-diphenyltetrazolium bromide (MTT) assay

Cells were incubated with 5mg/ml MTT reagent (M5655, Sigma Aldrich) for 5h. Then, the supernatant was removed carefully and the 100µl solvent solution containing 40µl of 10% SDS, 40µl DMSO and 20µl of 1.2% Acetate acid solution (600µl Acetate acid in 50mL PBS) was added and incubated overnight for measurement. Absorbance was measured at 570 nm with a reference to 630 nm. For cell viability assay and gemcitabine IC50 measurement, cells were incubated in 96-well plate for 48 hours before incubation with MTT.

Cell cycle analysis

Cells were harvested at 48 hours after siRNA treatment and washed with cold PBS, then fixed with 70% cold ethanol. To remove RNA, the cells were re-suspended in solution containing TritonX-100 (0.1%) and 100µg/ml RNase. The samples were stained with propidium iodide (20μg/mL) for 30 minutes in the dark, and then subjected to analysis for DNA content using FACS Calibur (BD Biosciences, Heidelberg, Germany) and data analysis was performed using Flowjo version10 software.

Transwell migration assay

Cell culture inserts with 8μM pore size (Falcon) were used. For tumor cell migration, 2.0 × 10 5 iCCA tumor cells were suspended in RPMI or DMEM medium with 0.5% FBS and plated in the upper chambers. The lower chambers were filled RPMI or DMEM with 10% FBS. After 16 h, the medium in the inserts were removed and washed with PBS. The inserts were filled with 3.7% formaldehyde for 5 minutes. Thereafter, the inserts were incubated in methanol for 30 minutes. The filters were stained with 10% Gimsa (Sigma, St. Louis, MO) for 15 minutes. The inner side was wiped with cotton swabs. Migrated cells were counted under a light microscope.

Caspase 3 assay

Caspase 3 assay was performed as previously described [START_REF] Meyer | Caveolin-1 abrogates TGF-beta mediated hepatocyte apoptosis[END_REF]. In brief, cells were lysed in 80µl of lysis buffer (50mM HEPES, 100mM NaCl, 0.1% CHAPS, 1mM DTT, 0.1mM EDTA, pH 7.4). Then, 20μl of cell lysate were incubated in 70μl reaction buffer (50mM HEPES, 100mM NaCl, 0.1% CHAPS, 10mM DTT, 0. mM EDTA, 10% (w/v) glycerol, pH 7.4) and 10μl AC-DEVD-AFC caspase 3 fluorimetric substrate (Biomol, Hamburg, Germany) for 90 min at 37 °C. Subsequently, Caspase 3 activity was detected by fluorometric measurement using Tecan infinite M200 (excitation 400 nm; emission 505 nm). The caspase 3 activity was normalized to protein levels and reported as relative fluorescent units per minute per mg protein.

Immunoblotting

Immunoblotting assay was performed as previously described [START_REF] Weng | The etiology of liver damage imparts cytokines transforming growth factor beta1 or interleukin-13 as driving forces in fibrogenesis[END_REF]. Briefly, total cell protein was extracted on ice using radio immunoprecipitation assay (RIPA) buffer with freshly added protease and phosphatase inhibitors. Protein concentrations were assessed with a Bio-Rad protein assay. 20μg of total cell protein extracts were subjected to 10% or 12% sodium dodecyl sulfate polyacrylamide electrophoresis (SDS-PAGE) gel and transferred to nitrocellulose membranes. 5% nonfat milk in Tris-buffered saline with Tween 20 (TBST) was used to block nonspecific binding. Membranes were probed with primary and secondary antibodies in TBST according to manufacturer's instructions. HRP-linked anti-mouse and anti-rabbit Abs were used as secondary antibodies. Alpha-tubulin and GAPDH were used as loading control. Signal was visualized by incubating the blots in Supersignal Ultra (Pierce, Hamburg, Germany).

RNA isolation and quantitative real-time reverse transcription polymerase chain reaction

Total cell RNA was extracted using the InviTrap spin universal RNA mini kit (Stratec, Berlin, Germany), according to the manufacturer's instructions. For first strand cDNA synthesis, reverse transcription of 500ng RNA was performed with random primers (Thermo Scientific)

and RevertAid H Minus M-MuLV reverse transcriptase (Thermo Scientific) according to the manufacturer's instructions and subsequently diluted with nuclease-free water (Invitrogen) to 10ng/µl cDNA. For PCR amplification, 10.4µl mixtures contained 5µl (50ng) template cDNA, 5µl SYBR Green (4367659, Life Technologies), and 4µM forward and reverse primer PCRs were run in triplicate and performed on a StepOnePlus Real-time PCR (Applied Biosystems). PCR amplification cycling conditions comprised 10 min polymerase activation at 95 °C and 40 cycles at 95°C for 15s and 60°C for 1 min. A melting curve analysis was performed for each PCR analysis. Relative quantification of target genes was normalized against the house keeping gene PPIA.

Microarray

Gene expression profiling was performed using arrays of human HuGene-2_0-st-type from Affymetrix. Biotinylated antisense cDNA was prepared according to the Affymetrix standard labelling protocol with the GeneChip® WT Plus Reagent Kit and the GeneChip® Hybridization, Wash and Stain Kit (both from Affymetrix, Santa Clara, USA). Subsequently, the hybridization on the chip was performed on a GeneChip Hybridization oven 640, then dyed in the GeneChip Fluidics Station 450 and thereafter scanned with a GeneChip Scanner 3000. All of the equipment used was from the Affymetrix-Company (Affymetrix, High Wycombe, UK).

Bioinformatic analyses

A Custom CDF Version 21with ENTREZ based gene definitions was used to annotate the arrays [START_REF] Dai | Evolving gene/transcript definitions significantly alter the interpretation of GeneChip data[END_REF]. The Raw fluorescence intensity values were normalized applying quantile normalization and RMA background correction. Differential expressed genes were identified by using a commercial software package SAS JMP10 Genomics, version 6, from SAS (SAS Institute, Cary, NC, USA). A false positive rate of a=0.05 with FDR correction was taken as the level of significance.

Statistical analyses

Variables were summarized as means ± standard deviation (S.D) and depicted graphically as means ± S.D. P values were calculated using the chi-square test or calculated using a twosided (unpaired) Student's t test. Kaplan-Meier survival curve and uni-variate Cox analysis was used to evaluate overall survival rates and disease free survival rate of iCCA patients. P values were calculated using the log-rank test. P< 0.05 was considered significant.

Results

SOX9 has distinct expression patterns in chronic liver disease and CCA

First, we compared expression of SOX9 and CK19, two classic markers of biliary tree, in 80 patients with chronic liver disease or iCCA. Among 21 patients with chronic liver disease, 17 showed SOX9 positive immunoreactivity whereas 4 were negative (Figure 1A shows representative patients.). In contrast to SOX9, CK19 immunostaining was positive in all patients (data not shown). The results suggest that SOX9 expression in cholangiocytes is unstable in chronic liver disease compared to CK19. Distinct from CK19, which localized in cytoplasm of cholangiocytes, SOX9 was expressed in the nuclei of cells in the canals of Hering, reactive ductules and bile ducts (Patient 1 and 2, Figure 1 A). As in chronic liver disease, SOX9 expression was observed in the nuclei of iCCA tumor cells, while CK19 localized in the cytoplasm of cancer cells (Figure 1B). Expressions of SOX9 and CK19 in cancer cells were heterogeneous. Figure 1B display four patterns of SOX9 and CK19 expression in iCCA: SOX9 high CK19 high , SOX9 high CK19 low , SOX9 low CK19 high and SOX9 low CK19 low . There was no significant correlation between expression of SOX9 and CK19 in iCCA tumor cells (P > 0.05). In all examined tissue specimens, neither SOX9 nor CK19 were detected in hepatocytes.

Expression of SOX9 predicts poor clinical outcome of iCCA

Next, we analyzed the correlation of SOX9 and CK19 expression with clinical parameters of the iCCA patients, including age, gender, vascular invasion, existence of cirrhosis and AJCC classification. Among these clinical parameters, CK19 expression was associated with AJCC classification of CCA, while SOX9 did not show any association with these clinical parameters (Supplementary Table 3). However, multivariate analysis showed that among the analyzed variables, only SOX9 expression significantly influenced the overall survival of iCCA patients (Hazard ratio=3.614, 95% Confidence Interval=1.493-9.076 , P=0.006, Supplementary Table 4). Furthermore, Kaplan-Meier analysis and the log-rank test showed that patients with high SOX9 expression had shorter overall survival (OS) and disease free survival (DFS) rates than those with low SOX9 expression (P < 0.01 and P < 0.05, respectively, Figure 2A and2C). The median OS time in patients with SOX9 low expression was 62 months, whereas the value in those patients with high SOX9 expression was only 22 months (Figure 2A). In contrast to SOX9, there was no association between CK19 expression and survival time in these patients (P > 0.05, Figure 2B and2D).

This cohort of iCCA included 9 patients who received chemotherapy (e.g. gemcitabine and Cisplatin) (Supplementary Table 5). Among them, 6 patients had low SOX9 expression and 3 patients showed high levels of SOX9. Survival analyses revealed that patients with high SOX9 expression had shorter OS time (P < 0.05, Figure 3A). In patients who received chemotherapy, the mean survival time in patients with SOX9 low expression was 62 months, whereas the value in those patients with high SOX9 expression was only 22 months (Figure 3A). Except 1 patient who received chemotherapy following surgery, additional 8 patients, 5 with low and 3 with high SOX9 expression, received chemotherapy due to the recurrence of iCCA. The survival times until the end of the follow-up in 5 patients with low SOX9 levels were 16, 19, 20, 29 and 34 months, whereas the values in 3 with high SOX9 expression were 13, 14 and 16 months, respectively (Supplementary Table 5). Survival difference between the two groups was significant (P < 0.01, Figure 3A). Notably, 4 out of 5 patients with low SOX9 were still survival when the follow-up ended (Supplementary Table 5). However, all 3 patients with high SOX9 expression were dead during follow-up (Supplementary Table 5).

CK19 expression did not show any correlation with chemotherapy response (P > 0.05, data not shown).

SOX9 inhibition sensitizes CCA cells to gemcitabine

To investigate how SOX9 expression in CCA cell might modify their response to chemotherapy, we performed microarray analysis in iCCA CC-SW-1 cells after SOX9 silencing. We found that gene expression associated with drug metabolism and ABC transporters such as ABCB1 (MDR1) and ABCC4 (MRP4) was decreased, while genes related with the p53 signaling pathway were increased when SOX9 was knocked down with siRNA (Supplementary Figure 2A). Western blot and qPCR further confirmed that the expression of multidrug resistance genes ABCC4 and ABCB1 was markedly reduced when SOX9 was inhibited in CC-SW-1 cells (Supplementary Figure 2B-C).

Next, we treated different types of CCA cells with gemcitabine, an analog of deoxycytidine, which is widely used in the treatment of CCA. Notably, basal expression of SOX9 in CCA cells was significantly higher than in normal cholangiocytes (Figure 4A). More impressively, expression of SOX9 protein was further increased upon gemcitabine treatment in both iCCA CC-SW-1 and eCCA EGI-1 cells (Figure 4B). To examine the function of SOX9 in gemcitabine-treated CCA cells, we knocked down SOX9 expression using siRNA in CC-SW-1 and EGI-1 cells, followed by treatment with gemcitabine for 24 hours (Figure 4C). MTT assay showed that when SOX9 expression was inhibited, the IC50 of gemcitabine-treated cells significantly decreased from 7.1 ± 0.15nM to 2.0 ± 0.23nM in CC-SW-1 cells and from 380.3 ± 249.1nM to 46.3 ± 21.9nM in EGI-1 cells, respectively (Figure 4D).

Given the key role of phosphorylation of checkpoint kinase 1 (CHEK1) in coordinating the DNA damage response and inhibiting the expression of multidrug resistance genes (23), we examined whether disruption of SOX9 impacted CHEK1 activation. Immunoblot analysis showed that SOX9 siRNA remarkably inhibited gemcitabine-dependent pCHEK1 in both CC-SW-1 and EGI-1 cells and MRP4 expression in CC-SW-1 cells (Figure 4D-F). Consistent with reduced expression of MRP4 and pCHK1, analyses based on immunoblot and cleaved Caspase 3 activity assay revealed marked increases in cleaved Caspase 3 and 8 expression and Caspase activity in CCA cells with SOX9 knockdown, indicating that gemcitabine-induced apoptosis was increased when SOX9 expression was inhibited (Figure 4G-H).

In addition to gemcitabine, we examined the role of SOX9 in cisplatin-treated CC-SW-1 and EGI cells. MTT assay showed that knockdown of SOX9 did not have impact on cisplatin-inhibited cell viability in both cells (Supplementary Figure 3A-B). In contrast to gemcitabine, administration of cisplatin and/or knockdown of SOX9 did not influence the expression of pCHEK1 (Supplementary Figure 3C-D).

SOX9 is essential for CCA cells proliferation, stemness and migration

Next, we examined the role of SOX9 in CCA cell proliferation, stemness and migration. MTT analyses showed that knockdown of SOX9 expression significantly inhibited cell proliferation in 4 types of CCA cells (Figure 5A). In CC-SW-1 and EGI-1 cells, SOX9 inhibition significantly decreased the proportion of cells staying in G1 phase and increased those in G2/M phase (Figure 5B). The results suggest that SOX9 is required for maintaining CCA cell proliferation.

Subsequently, we investigated the effects of SOX9 on stemness of CCA cells. We found that knockdown of SOX9 expression decreased EpCAM expression at both RNA and protein levels in CC-SW-1 cells (Figure 5C). In HCC, EpCAM is considered as a crucial factor in the maintenance of cancer stem cell (CSC)-like features in cancer cells [START_REF] Yamashita | EpCAM-positive hepatocellular carcinoma cells are tumor-initiating cells with stem/progenitor cell features[END_REF]. To investigate whether SOX9 is implicated to the CSC features of CCA cells, we performed tumor sphere formation assay, a widely recognized method to evaluate cancer stem cell self-renewal and differentiation at the single cell level in vitro [START_REF] Pastrana | Eyes wide open: a critical review of sphereformation as an assay for stem cells[END_REF]. SOX9 knockdown significantly inhibited the capacity of tumor sphere formation in CC-SW-1 (Figure 5D).

In addition, we also investigated the role of SOX9 in CCA cell migration. Transwell assay showed that knockdown of SOX9 expression significantly inhibited cell migration in CC-SW-1 cells (Figure 5E).

Discussion

The standard treatment for advanced-stage iCCA is systemic chemotherapy with gemcitabine and cisplatin [START_REF] Valle | Cisplatin plus gemcitabine versus gemcitabine for biliary tract cancer[END_REF]. However, the median overall survival time is less than 12 months [START_REF] Valle | Cisplatin plus gemcitabine versus gemcitabine for biliary tract cancer[END_REF]. In patients treated with gemcitabine alone, the survival time is less than 8 months [START_REF] Valle | Cisplatin plus gemcitabine versus gemcitabine for biliary tract cancer[END_REF], thus improving the sensitivity of cholangiocarcinoma cells to chemotherapy is a key to prolonging the survival of iCCA patients. In the current study, we found that SOX9, the earliest cholangiocyte marker during embryonic liver development [START_REF] Antoniou | Intrahepatic bile ducts develop according to a new mode of tubulogenesis regulated by the transcription factor SOX9[END_REF][START_REF] Si-Tayeb | Organogenesis and development of the liver[END_REF], plays a crucial role in iCCA cells' resistance to chemotherapy. We examined the expression of SOX9 in 59 iCCA patients who received surgery. High expression of SOX9 in the nuclei of iCCA cancer cells was significantly associated with shorter survival time (P = 0.0039). Of 9 patients treated with chemotherapy following surgery, the median survival time reached to 62 months in 6 patients who had low levels of SOX9 expression, whereas survival time was only 22 months in the 3 patients who had high SOX9 levels. Although the sample size was small in this study, the difference in survival time between both groups was significant (P = 0.017). Among the 9 patients, 8 received chemotherapy because of the recurrence of cancer. The 5 patients with low SOX9 levels survived between 16 and 34 months, however, the longest survival time in 3 patients with high SOX9 expression was only 16 months. These results suggest that SOX9 expression correlates with cholangiocellular cancer cells' response to chemotherapy. Further in vitro studies provided the following mechanistic explanations of the observed differences:

(1) Microarray, qPCR and western blot analyses showed that disruption of SOX9 with siRNA significantly decreased expression of genes/proteins associated with drug metabolism and multidrug resistance and increased the abundance of genes associated with p53 signaling pathway; (2) Knockdown of SOX9 markedly inhibited gemcitabine-induced activation of CHK1, a key cell cycle checkpoint protein that coordinates the DNA damage response, and expression of multiple drug resistance protein MRP4, and thus increased cancer cell apoptosis; (3) Gemcitabine dose-dependently induced expression of SOX9, indicating that CCA cells increase SOX9 as a defensive mechanism against treatment with chemotherapeutics such as gemcitabine. Figure 6 depicts the gemcitabine-induced loop in CCA cells.

The role of SOX9 in the chemo-resistance has been reported in multiple tumors, e.g. chondrosarcoma, breast cancer, glioblastoma, cervical cancer, gastric cancer and lung cancer [START_REF] Chen | MALAT1-miR-101-SOX9 feedback loop modulates the chemo-resistance of lung cancer cell to DDP via Wnt signaling pathway[END_REF][START_REF] Feng | SOX9/miR-130a/CTR1 axis modulates DDP-resistance of cervical cancer cell[END_REF][START_REF] Garros-Regulez | mTOR inhibition decreases SOX2-SOX9 mediated glioma stem cell activity and temozolomide resistance[END_REF][START_REF] Liang | Downregulation of microRNA-206 promotes invasion and angiogenesis of triple negative breast cancer[END_REF][START_REF] Riemenschnitter | Stability and prognostic value of Slug, Sox9 and Sox10 expression in breast cancers treated with neoadjuvant chemotherapy[END_REF][START_REF] Wang | Upregulation of microRNA-524-5p enhances the cisplatin sensitivity of gastric cancer cells by modulating proliferation and metastasis via targeting SOX9[END_REF]. In cervical cancer cells, SOX9 was found to increase cancer cell chemo-resistance through inhibiting miR-130a [START_REF] Feng | SOX9/miR-130a/CTR1 axis modulates DDP-resistance of cervical cancer cell[END_REF]. However, the detailed mechanisms of how SOX9 contributes to chemo-resistance have not been clarified to date. Our observation that SOX9 confers chemo-resistance to cholangiocarcinoma through the activation of CHK1 and the expression of multiple drug resistance proteins might provide a light pointer for further investigation of this aspect.

The current study also investigated the role of SOX9 in CCA cells receiving cisplatin. In contrast to gemcitabine, knockdown of SOX9 did not impact the efficiency of cisplatin. Administration of cisplatin did not alter expression of pChk1 in CCA cells. The discrepancy between the two compounds might be due to their different mechanisms of action.

Gemcitabine results in cell death mainly through the inhibition of DNA synthesis and the inhibition of enzymes relevant to deoxyribonucleotide metabolism [START_REF] De Sousa Cavalcante | Gemcitabine: metabolism and molecular mechanisms of action, sensitivity and chemoresistance in pancreatic cancer[END_REF]. Thus, Gemcitabine exerts these actions through impacting multiple pathways, including regulating checkpoint kinases. Different from gemcitabine, cisplatin causes cell death through the formation of [PtCl(guanine-DNA)(NH3)2]+, which inhibits DNA repair and activates apoptosis [START_REF] Wang | Cellular processing of platinum anticancer drugs[END_REF].

The defensive effects of SOX9 for CCA cells are not limited to cancer cells facing chemotherapy. Knockdown of SOX9 in both iCCA and eCCA cells remarkably inhibited the capacity of cancer cell proliferation and migration, decreased CSC stemness and increased apoptosis. These results provide an explanation why SOX9 expression is associated with survival time of patients receiving chemotherapy, but also in those who are treated by surgery alone. On the other hand, it should be kept in mind that these in vitro findings are not always consistent with the observation obtained in patients. For example, there was no correlation between the expression of SOX9 and cell proliferation markers, e.g. Ki67 and PCNA, in iCCA patients (data not shown). Given that there are multiple growth factors and proliferative signals governing cancer cell expansion in patients with iCCA, it is not surprising that low levels of SOX9 expression alone does not have a significant impact on the proliferation of cancer cells. In addition, although there is a crucial role of SOX9 in maintaining cell identity, we did not observe a significant correlation between SOX9 expression and histological differentiation in this cohort of iCCA patients (data not shown). The result might have two explanations: (1) SOX9 expression does not impact on differentiation of iCCA, and (2) SOX9 might have a subtle influence on cancer cell differentiation, but the association is to small to be detected in the currently small number of specimens.

In diseased liver, high levels of SOX9 occur not only in CCA, but also in HCC. However, in contrast to cholangiocytes, normal hepatocytes do not express SOX9. Expression of SOX9 in HCC reflects a cancer stem cell/progenitor cell [START_REF] Liu | Sox9 regulates self-renewal and tumorigenicity by promoting symmetrical cell division of cancer stem cells in hepatocellular carcinoma[END_REF]. Given that SOX9 is the earliest and dominant phenotype marker of normal cholangiocytes, the induction of a cancer stem cell-like phenotype should not be attributed to the expression of SOX9. However, the current results do suggest that SOX9 plays a role in the maintenance of cancer stem cell phenotypes. Like CCA patients, HCC patients with high levels of SOX9 had poor prognosis. As in CCA, SOX9 in HCC is implicated in maintaining proliferation and self-renewal of cancer cells [START_REF] Liu | Sox9 regulates self-renewal and tumorigenicity by promoting symmetrical cell division of cancer stem cells in hepatocellular carcinoma[END_REF]. In the future, it will be interesting to find out whether SOX9 dependent control of check point protein activation may also play a role in chemo-resistance of HCC.

Besides SOX9, this study analyzed the association between CK19 expression and the clinical outcome of iCCA. CK19 is a classical marker for cholangiocytes. It has been reported that CK19 contributes to the differentiation of iCCA from metastatic adenocarcinoma and is associated with the histological differentiation of iCCA (34). In the current cohort of iCCA patients, CK19 expression was correlated with AJCC classification of iCCA, whereas it did not show any association with the survival of iCCA patients.

Taken together, SOX9 expression is a sensitive marker that predicts the survival time of iCCA patients, particularly in those receiving chemotherapy. Our study demonstrates that SOX9 is a key transcription factor that prevents iCCA cells from apoptosis when the cells are attacked by drugs such as gemcitabine. SOX9 exerts the observed effects on CCA cells, at least in part, through the activation of Chk1 and upregulation of multidrug resistance genes.

Limitations of this study are: 1) the low number of samples precludes more general conclusions; 2) it is not clear how SOX9 exerts the observed effects on the expression of multidrug resistance genes; 3) we cannot conclude whether SOX9 expression has a similar effect in eCCA. Our study also enrolled 5 eCCA patients who demonstrate a similar biological behavior as iCCA (data not shown). Results in eCCA cell line EGI-1 indicate that the SOX9 expression has similar effects as observed in iCCA. Further investigation based on a large size of patient cohorts is required in the future.

34.

Shimonishi T, Miyazaki K, Nakanuma Y. Cytokeratin profile relates to histological subtypes and intrahepatic location of intrahepatic cholangiocarcinoma and primary sites of metastatic adenocarcinoma of liver. Histopathology. 2000;37(1):55-63.

Fig. 1 .

 1 Fig. 1. Expression of SOX9 and CK19 in patients with iCCA. (A) SOX9 express in cells of bile ducts (Patient 1) and reactive ducts (Patient 2) of liver tissues with chronic HBV infection. Patient 3 shows negative SOX9 immune reaction in the liver. (B) Four patterns of SOX9 and CK19 expression in iCCA patients.

Fig. 2 .

 2 Fig. 2. SOX9, but not CK19, is associated with survival time of iCCA patients. Kaplan-Meier survival analysis shows the association between SOX9 (A and C)/CK19 expression (B and D) and survival time in 59 iCCA patients.

Fig. 3 .

 3 Fig. 3. SOX9 is associated with survival time of iCCA patients receiving chemotherapy. (A) Kaplan-Meier plot for OS rate of 9 iCCA patients received chemotherapy before or after cancer recurrence. (B) Kaplan-Meier plot for OS rate of 8 iCCA patients received chemotherapy after cancer recurrence.

Fig. 4 .

 4 Fig. 4. Disruption of SOX9 increases gemcitabine-induced iCCA cell apoptosis. (A) The expression of SOX9 in normal cholangiocytes and different CCA cells. (B) Administration of gemcitabine dose-dependently induced expression of SOX9 in both CC-SW1 and EGI-1 cells. (C) SOX9 was knocked down by siRNA. (D) MTT analyses showed cell viability in gemcitabine-treated CC-SW1 and EGI-1 cells with or without SOX9 knockdown. (E-G) Phosphorylation of CHK1, MRP4, cleaved Caspase 3 and 8 were measured in gemcitabinetreated CC-SW1 and EGI-1 cells with or without SOX9 knockdown. (H) Caspase 3 assay was used to measure Caspase 3 activity in gemcitabine-treated CC-SW1 and EGI-1 cells with or without SOX9 knockdown. All Western blot analyses in A-C and E-G were performed at least for three times. Quantification of western blot is shown by the numbers between bands.

Fig. 5 .

 5 Fig. 5. Disruption of SOX9 impact biological behaviors of CCA cells. (A) MTT analyses were performed to measure cell viability in different CCA cells with or without SOX9 knockdown. (B) FACS analyses were used to examine the impact of SOX9 on cell cycle of CC-SW1 and EGI-1 cells. (C) mRNA and protein expression of EpCAM were measured by qPCR and Western blot in CC-SW1 and EGI-1 cells with or without SOX9 knockdown. (D) 18

Table 3 .

 3 Patient characteristics in iCCA patients with different levels of SOX9/CK19

	Supplementary Clinicopathological		SOX9 expression			CK19 expression
	factors	low expression (n=42)	high expression (n=17)	P value	low expression (n=40)	high expression (n=19)	P value
	Age (year)	64.55 ± 9.60	60.82 ± 10.62	0.20	60.00 ± 10.70	65.13 ± 9.30	0.064
	60>	12	7	0.34	10	9	0.086
	≥60	30	10		30	10
	Gender
	male	30	14	0.58	31	13	0.454
	female	12	3		9	6
	Vascular invasion
	Yes	4	4	0.32	5	3	0.730
	No	38	13		35	16
	Cirrhosis
	Yes	11	7	0.26	11	7	0.466
	No	31	10		29	12
	AJCC classification
	I+II	26	13	0.44	30	9	0.036
	II+III	16	4		10	10

Table 5 .

 5 SOX9 and CK19 expression and the clinical outcome of iCCA patients that received chemotherapy

	SOX9 IHC intensity	CK19 IHC intensity	(1=survival, 0=death) Death or survival	(months) Survival time	(months) Survival time after CTx	Treatment
	low	low	0	23	19	Gemcitabine and XELOX
	low	strong	0	62	62*	Gemcitabine plus Cisplatin and
						XELOX
	low	strong	1	33	29	Gemcitabine
	low	low	1	36	34	Palliative chemotherapy
	low	strong	1	57	16	Cisplatin and gemcitabine
	low	low	1	51	20	Palliative chemotherapy
	strong	strong	0	14	13	Erlotinib
	strong	strong	0	22	16	Gemcitabine and cisplatin
	strong	strong	0	41	14	Palliative chemotherapy
	CTx: Chemotherapy; XELOC: Capecitabine plus Oxaliplatin.

* The patient received chemotherapy following surgery.

Acknowledgements:

We are grateful to Drs. Kobayashi N, Rückert F and Lu LG for providing normal cholangiocyte and CCA cell lines and gemcitabine. We thank the core facilities of Biosit and Biogenouest (CRB Santé BB-0033-00056 and H2P2) for TMA production.

Conflict of interest:

The authors disclose no conflicts.

Financial support:

The study was supported by Chinese-German Cooperation Group project, grant number: GZ

Supplementary tables

Supplementary Table 1. Clinicopathological features of patients with chronic liver disease

Supplementary Table 2. Clinicopathological features of validating set iCCA

Supplementary Table 3


Han Bleichrodt

Christophe Courbage

Béatrice Rey

The Value of a Statistical Life Under Changes in Ambiguity

Keywords: Value of a statistical life, ambiguity, prudence, smooth ambiguity model, neoadditive preferences JEL codes: D81, I18, Q51 Value of a statistical life, ambiguity, prudence, smooth ambiguity model, neoadditive preferences. JEL classification: D81, I18, Q51

The value of a statistical life (VSL) is a key parameter in the analysis of government policy. Most policy decisions are made under ambiguity. This paper studies the effect of changes in ambiguity perception on the value of a statistical life (VSL). We propose a definition of increases in ambiguity perception based on Ekern's (1980) definition of increases in risk. Ambiguity aversion alone is not sufficient to lead to an increase in the VSL when the decision maker perceives more ambiguity. Our results highlight the importance of higher order ambiguity attitudes, particularly ambiguity prudence.

Introduction

Reductions in mortality risk are the main benefit of government regulation in the US and other developed countries [START_REF] Viscusi | Pricing Lives: Guideposts for a Safer Society[END_REF]. This makes the value of a statistical life (VSL), the most common metric to assess the effectiveness of such regulation, a key parameter in the analysis of public policy. 1 A large literature has explored how the VSL changes with, amongst others, wealth [START_REF] Hammitt | Valuing Mortality Risk: theory and practice[END_REF], the baseline mortality risk [START_REF] Pratt | Willingness to pay and the distribution of risk and wealth[END_REF], background financial risk [START_REF] Eeckhoudt | Background risks and the value of a statistical life[END_REF], age [START_REF] Viscusi | The value of a statistical life: A critical review of market estimates throughout the world[END_REF], health status [START_REF] Hammitt | QALYs versus WTP[END_REF], and altruism [START_REF] Andersson | Benevolence and the value of road safety[END_REF]. All these studies take the mortality risk (and changes therein) as known. However, in most real-life decisions mortality risks are not precisely known and subject to ambiguity. For example, experts have conflicting opinions about the exact risks to human life posed by climate change. Most experts agree that these risks exist, but they disagree about their exact size leading to ambiguous estimates.

Ellsberg's famous paradox (1961) showed that ambiguity affects people's behavior.

Ambiguity aversion can account for several empirical puzzles that traditional economics has difficulty explaining. Examples are the stock market participation puzzle (the finding that many people do not buy stocks even though they should according to standard portfolio theory 2 ), home bias (the finding that people invest too much in stocks from their own country 3 ), and the low take-up of freely available genetic tests [START_REF] Hoy | Take-up for genetic tests and ambiguity[END_REF]).

1 [START_REF] Viscusi | The value of individual and societal risks to life and health[END_REF] provides an overview of government practices and lists VSL amounts used in different regulatory analyses.

2 See e.g. [START_REF] Mankiw | The consumption of stockholders and nonstockholders[END_REF] and [START_REF] Haliassos | Why do so few hold stocks?[END_REF] 3 [START_REF] French | Investor diversification and international equity markets[END_REF] Ellsberg's paradox has spurred the development of a variety of new ambiguity theories [START_REF] Ryan | Generalizations of SEU: a geometric tour of some non-standard models[END_REF][START_REF] Gilboa | Ambiguity and the Bayesian Paradigm[END_REF][START_REF] Machina | Ambiguity and ambiguity aversion[END_REF]. Of these new theories, the smooth model of Klibanoff, Marinachi, and Mukerji (KMM) (2005) is most widely used in economic applications.

In this paper we explore the impact of ambiguity on the VSL under KMM's smooth model. [START_REF] Treich | The value of a statistical life under ambiguity aversion[END_REF] showed that when one decision situation is ambiguous but the other is not ambiguity leads to an increase in the VSL of an ambiguity averse (smooth) decision maker compared with the situation in which there is no ambiguity. However, in real-world decisions both alternatives are usually ambiguous, although they may differ in their degree of ambiguity. For example, more research into climate change may lead to more precise estimates of the risks to human life even when the exact risks remain unknown. Our main results (Results 2 and 3) show that in this more general case it is no longer true that an ambiguity averse decision maker will always have a higher VSL when he perceives more ambiguity. The effect of general changes in perceived ambiguity is more complex than the comparison between ambiguity and no ambiguity may suggest and requires information about higher order ambiguity preferences. In particular, our results show the importance of ambiguity prudence.

The paper is structured as follows. In Section 2 we give background and derive the VSL under risk. Section 3 then introduces ambiguity and derives the VSL under KMM's smooth model. Section 4 considers mean-preserving spreads in ambiguity perception. Result 1 shows that a mean-preserving spread in ambiguity perception can actually lead to a lower VSL of an ambiguity averse decision maker. Result 2 shows that a general mean-preserving spread in ambiguity perception always leads to a higher VSL if ambiguity prudence is not too high. It highlights the importance of ambiguity prudence in the evaluation of mortality risk benefits.

Section 5 extends the analysis to higher order changes in ambiguity. We define the concept of 𝑛 𝑡ℎ order increases in ambiguity and show that the effect of these increases on the VSL depends on higher order ambiguity preferences under the smooth model.

The remainder of the paper explores the robustness of our findings. In Section 6, we consider the case of ambiguity seeking. While most of the theoretical literature has focused on ambiguity aversion, [START_REF] Ellsberg | Risk, ambiguity, and the Savage axioms[END_REF] already pointed out people may sometimes be ambiguity seeking, particularly for unlikely events and when losses are involved (for an overview of empirical evidence supporting Ellsberg's conjecture see Trautmann and van de Kuilen 2016; Wakker 2010). As the risk of death is typically small, the case of ambiguity seeking cannot be ignored. Result 5 shows that the combination of ambiguity seeing and ambiguity prudence leads makes it impossible to sign the effect of changes in ambiguity perception on the VSL. This is noteworthy as most ambiguity models imply ambiguity prudence [START_REF] Baillon | Prudence with respect to ambiguity[END_REF].

Most of our analysis uses KMM's smooth model. The smooth model incorporates ambiguity aversion by using a different utility function for uncertainty than for risk. A different strand of the ambiguity literature models ambiguity aversion through a difference in the weighting of events between risk and uncertainty. The main models in this class are Choquet expected utility [START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF]) and the multiple priors models (Gilboa and Schmeidler 1989). [START_REF] Baillon | Testing Ambiguity Models Through the Measurement of Probabilities for Gains and Losses[END_REF] found that Choquet expected utility better described ambiguity preferences than the smooth model. In Section 7, we extend our analysis to [START_REF] Chateauneuf | Choice under uncertainty with the best and worst in mind: Neo-additive capacities[END_REF] neo-additive preferences, which have intuitive interpretations in terms of Choquet expected utility and in terms of multiple priors. We show that our conclusions remain valid under neo-additive preferences: ambiguity aversion alone does not necessarily lead to an increase in the VSL when perceived ambiguity increases and ambiguity prudence plays a key role.

Background

The standard VSL model [START_REF] Drèze | L'utilité sociale d'une vie humaine[END_REF]) assumes that a decision maker (DM) evaluates decisions involving a fatality risk by (state-dependent) expected utility:

𝑉 0 = (1 -𝑝)𝑈 𝑙 (𝑤) + 𝑝𝑈 𝑑 (𝑤).
(1)

In Eq. ( 1), 𝑝 is the probability that the DM dies during the current period, 𝑈 𝑙 (𝑤) is the DM's utility of wealth if he survives the period and 𝑈 𝑑 (𝑤) is his utility of wealth if he does not survive (i.e. his utility of a bequest). If the DM has no bequest motive then 𝑈 𝑑 (𝑤) is zero for all wealth levels.

It is common to assume that the DM prefers more wealth to less (𝑈 𝑙 ′ (𝑤) > 0 and 𝑈 𝑑 ′ (𝑤) > 0 for all 𝑤), that he is risk averse (𝑈 𝑙 ′′ (𝑤) < 0 and 𝑈 𝑑 ′′ (𝑤) < 0 for all 𝑤), and that both the utility of wealth and the marginal utility of wealth are always higher when alive than when dead (𝑈 𝑙 (𝑤) > 𝑈 𝑑 (𝑤), 𝑈 𝑙 ′ (𝑤) > 𝑈 𝑑 ′ (𝑤) for all 𝑤). The VSL is the marginal rate of substitution between wealth and mortality risk. It is obtained by totally differentiating Eq.

(1) with respect to 𝑝 and 𝑤 holding expected utility constant:

𝑉𝑆𝐿 𝑟 = 𝑑𝑤 𝑑𝑝 = 𝑈 𝑙 (𝑤) -𝑈 𝑑 (𝑤) (1 -𝑝)𝑈 𝑙 ′ (𝑤) + 𝑝𝑈 𝑑 ′ (𝑤) (2)
The subscript 𝑟 serves as a reminder that Eq. (2) concerns the case of risk where the mortality risk is objectively known. Under the assumptions made, 𝑉𝑆𝐿 𝑟 is strictly positive and increases with wealth and with the mortality risk 𝑝. The positive relation between the VSL and the mortality risk has been coined the "dead anyway" effect by [START_REF] Pratt | Willingness to pay and the distribution of risk and wealth[END_REF]. It expresses that, abstracting from bequest considerations, a DM who faces a high probability of death will be inclined to spend as much as he can on mortality risk reduction as he is unlikely to survive anyway.

The smooth ambiguity model

We will now explore the impact of ambiguity. Suppose the baseline mortality risk 𝑝 is no longer objectively known, but is ambiguous. We express this by adding a random variable 𝜀, which reflects the DM's perceived ambiguity, to 𝑝 so that the new mortality risk becomes: 𝑝 ̃= 𝑝 + 𝜀. To avoid negative probabilities or probabilities larger than 1, the support of 𝜀̃ is restricted to [-𝑝, 1 -𝑝]. We assume that the DM evaluates the mortality risk according to the smooth model of KMM (2005):

𝑊 𝜀 ̃= 𝜑 -1 (𝐸(𝜑((1 -𝑝 ̃)𝑈 𝑙 (𝑤) + 𝑝 ̃𝑈𝑑 (𝑤)))).

(3)

In the smooth model the DM's ambiguity perception is modeled by a second order distribution 𝑝 ̃, which reflects his beliefs about the mortality risk. The (increasing) function 𝜑 reflects the DM's ambiguity attitudes. If 𝜑 is everywhere concave (𝜑 ′′ < 0) then the DM is (uniformly) ambiguity averse. If 𝜑 is everywhere convex (𝜑 ′′ > 0) then the DM is (uniformly) ambiguity seeking, If 𝜑 is linear (𝜑 ′′ = 0), the DM is ambiguity neutral and the smooth model is equivalent to subjective expected utility. Then the DM behaves according to Eq.( 2) with 𝑝 = 𝐸(𝑝 ̃). In Eq.(3) we use the subscript 𝜀̃ to emphasize that utility depends on the DM's ambiguity perception.

We assume that 𝐸[𝜀] = 0 and, thus, 𝐸[𝑝 ̃] = 𝑝. In other words, ambiguity leads to an increase in the spread of the distribution of mortality risks that the DM perceives as possible, but it does not lead to a systematic bias in the perceived risks. We discuss the effects of a systematic bias in the perceived risks in Appendix A.

We obtain the VSL under the smooth ambiguity model by totally differentiating Eq. ( 3)

with respect to 𝑝 and 𝑤. This gives:

𝑉𝑆𝐿 𝜀 ̃= 𝑑𝑤 𝑑𝑝 = (𝑈 𝑙 (𝑤) -𝑈 𝑑 (𝑤))𝐸[𝜑 ′ ((1 -𝑝 ̃)𝑈 𝑙 (𝑤) + 𝑝 ̃𝑈𝑑 (𝑤))] 𝐸 [((1 -𝑝 ̃)𝑈 𝑙 ′ (𝑤) + 𝑝 ̃𝑈𝑑 ′ (𝑤))𝜑 ′ ((1 -𝑝 ̃)𝑈 𝑙 (𝑤) + 𝑝 ̃𝑈𝑑 (𝑤))] (4)
Treich (2010) showed that an ambiguity-averse DM will have a higher VSL under ambiguity than under risk: 𝑉𝑆𝐿 𝜀 ̃> 𝑉𝑆𝐿 𝑟 . The intuition is that an ambiguity averse DM concentrates on the higher mortality risks and we noticed before that higher mortality risks increase the VSL as a result of the dead anyway effect. This also implies that an increase in ambiguity aversion (reflected by a more concave 𝜑) increases 𝑉𝑆𝐿 𝜀 ̃.

Changes in perceived ambiguity

We now turn to the comparison between two ambiguous situations. Consider two situations for which the DM's levels of perceived ambiguity are described by the random variables 𝜀1 and 𝜀2. If we write 𝑝 𝑖 ̃= 𝑝 + 𝜀ĩ, 𝑖 = 1,2, Eq. ( 4) becomes:

𝑉𝑆𝐿 𝜀 𝑖 ̃= 𝑑𝑤 𝑑𝑝 = (𝑈 𝑙 (𝑤) -𝑈 𝑑 (𝑤))𝐸[𝜑 ′ ((1 -𝑝 𝑖 ̃)𝑈 𝑙 (𝑤) + 𝑝 𝑖 ̃ 𝑈 𝑑 (𝑤))] 𝐸 [((1 -𝑝 𝑖 ̃)𝑈 𝑙 ′ (𝑤) + 𝑝 𝑖 ̃𝑈𝑑 ′ (𝑤))𝜑 ′ ((1 -𝑝 𝑖 ̃)𝑈 𝑙 (𝑤) + 𝑝 𝑖 ̃𝑈𝑑 (𝑤))] (5)
In this Section we consider the case where 𝜀2 is a mean-preserving spread of 𝜀1. Higher order changes in ambiguity are studied in the next Section. Because 𝜀2 is a mean-preserving [START_REF] Treich | The value of a statistical life under ambiguity aversion[END_REF] showed that if 𝜀1 = 0 𝑉𝑆𝐿 𝜀 ̃2 will exceed 𝑉𝑆𝐿 𝜀 ̃1 . The next example shows that this is no longer true when 𝜀̃1 and 𝜀̃2 are both nondegenerate.

spread of 𝜀1, 𝐸[𝜀1] = 𝐸[𝜀2], but 𝑉𝑎𝑟[𝜀1] < 𝑉𝑎𝑟[𝜀2].

EXAMPLE:

Let 𝜑(𝑥) = - (1 -𝑝 ̃𝑖)𝑈 𝑙 ′ (𝑤) + 𝑝 𝑖 ̃𝑈𝑑 ′ (𝑤), 𝑖 = 1,2. In the special case where one situation is unambiguous (𝜀1 = 0), 𝑍 ̃1 ′ is constant. Then the denominator of Eq. ( 5) becomes 5) gives after some rearranging that

𝐸[𝑍 ̃1 ′ ]𝐸[𝜑 ′ (𝑍 ̃1)]. Because 𝜀 2 ̃ is a mean-preserving spread of 𝜀1, 𝐸[𝑍 ̃1 ′ ] = 𝐸[𝑍 ̃2 ′ ]. Substituting 𝐸[𝑍 ̃1 ′ 𝜑 ′ (𝑍 ̃1)] = 𝐸[𝑍 ̃1 ′ ]𝐸[𝜑 ′ (𝑍 ̃1)] and 𝐸[𝑍 ̃1 ′ ] = 𝐸[𝑍 ̃2 ′ ] into Eq. (
𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 iff 𝐸[𝑍 ̃2 ′ ]𝐸[𝜑 ′ (𝑍 ̃2)] > 𝐸[𝑍 ̃2 ′ 𝜑 ′ (𝑍 ̃2)]
. This last inequality holds if the covariance between 𝑍 ̃2 ′ and 𝜑 ′ (𝑍 ̃2) is negative, which is true for an ambiguity averse DM. However, if

𝜀̃1 is also ambiguous, 𝑍 ̃1 ′ is no longer constant and the above argument can no longer be applied.

To ensure that a mean-preserving spread in ambiguity always leads to an increase in the VSL, we must impose additional conditions on the DM's ambiguity attitudes, in particular ambiguity prudence. Result 2 gives sufficient conditions. [START_REF] Baillon | Prudence with respect to ambiguity[END_REF] gives a model-free definition of ambiguity prudence. Ambiguity prudence is implied by most ambiguity models. It reflects the intuition that a decision maker prefers to spread harms across events rather than to concentrate them in one or a few events. In our decision context, it means that the DM prefers to have ambiguity in the mortality risk in states of the world in which the mortality risk is low rather than in states of the world in which the mortality risk is high. The importance of prudence in explaining economic behavior is widely-documented for decisions under risk where it corresponds to a preference for precautionary saving [START_REF] Kimball | Precautionary saving in the small and in the large[END_REF]). For ambiguity, [START_REF] Guerdjikova | Survival with ambiguity[END_REF] show that ambiguity prudence plays a crucial role in the survival of ambiguity averse agents in the market. [START_REF] Baillon | Prudence with respect to ambiguity[END_REF] illustrates its importance for prevention behavior. Other recent illustrations of the role of ambiguity prudence are [START_REF] Berger | Precautionary saving and the notion of ambiguity prudence[END_REF][START_REF] Berger | The impact of ambiguity and prudence on prevention decisions[END_REF] ii. If 𝜑 ′′′ = 0 then also 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 .

iii. If the DM is ambiguity imprudent then the sign of 𝑉𝑆𝐿 𝜀 ̃2 -𝑉𝑆𝐿 𝜀 ̃1 is indeterminate.

Proofs of all results presented throughout the paper are in the Appendix. There we also explain why in case (iii) the sign of 𝑉𝑆𝐿 𝜀 ̃2 -𝑉𝑆𝐿 𝜀 ̃1 is indeterminate.

In Result 2 part (i), 𝑆 * denotes the minimum value of 𝑆 = 𝑈 𝑙 ′ -𝑈 𝑑 ′ (𝑈 𝑙 -𝑈 𝑑 )(𝑍 i ′ -𝜀 𝑖 (𝑈 𝑙 ′ -𝑈 𝑑 ′ ))
, 𝑖 = 1,2. It is obtained for the minimum value that the random variable 𝜀̃2 takes as we show in the Appendix. Result 2 part (ii) says that a mean-preserving spread in the DM's perceived ambiguity always leads to an increase in the VSL if the DM does not care about the skewness of the distribution of his perceived mortality risks. This is the only case where we can straightforwardly extend Treich's result. If the DM is ambiguity prudent then a mean-preserving spread in ambiguity perception also leads to an increase in the VSL if his ambiguity prudence is not too extreme (it should not exceed 𝑆 * ). The reason why ambiguity prudence cannot be too extreme is to exclude cases as the one in our example.

For 𝜑(𝑥) = -𝑒 -𝛼𝑥 𝛼 the index of ambiguity prudence is equal to 𝛼 and the example showed that if ambiguity prudence becomes too large then Result 2 part (i) no longer holds.

Intuitively, 𝜀2 can be more negatively-skewed than 𝜀1. An ambiguity prudent DM likes this negative skewness. On the other hand, the DM is also ambiguity averse and dislikes the greater ambiguity involved in 𝜀2. If his ambiguity prudence is not too strong the negative effect of ambiguity aversion will dominate and the increase in ambiguity leads to an increase in the VSL. However, if ambiguity prudence is strong then the positive effect of the negative skewness of 𝜀2 can prevail and the VSL will decrease.

Generalization to higher order changes in ambiguity

We will now extend our analysis to more general changes in ambiguity perception. Part (𝑖) of Definition 1 implies that the (𝑛 -1) first moments of 𝐹 1 and 𝐹 2 are equal.

Part (𝑖𝑖) implies that the 𝑛-th moment of 𝐹 2 exceeds the 𝑛-th moment of 𝐹 1 . 4 So if 𝜀2 has more first order ambiguity than 𝜀1 then the mean of 𝜀2 exceeds the mean of 𝜀1. In other words, 𝜀2 has a higher expected mortality risk than 𝜀1. If 𝜀2 has more second order ambiguity than 𝜀1 then 𝜀2 is a mean-preserving spread of 𝜀1, the case we considered in Section 4. If 𝜀̃2 has more third order ambiguity than 𝜀̃1 then 𝜀̃2 can be obtained from 𝜀̃1 by a series of mean-variance-preserving-transformations (Menezes at el., 1980). These transformations do not affect the mean and the variance but they transfer ambiguity from lower to higher values of the mortality risk with the result that the distribution becomes more positively skewed.

Lemma A3 in the Appendix shows that a mixed ambiguity averse DM will indeed always dislike increases in 𝑛 th order ambiguity. The intuition is that a mixed ambiguity averse DM prefers to distribute harms across states of nature rather than to concentrate them in one state. In other words, he prefers to combine good with bad over combining good with good (and bad with bad). An example may clarify. Suppose a DM faces a certain increase 𝑘 in his perceived mortality risk and a zero mean random change in ambiguity perception 𝜀. All ambiguity averse DMs will dislike these two changes and they are both perceived as harmful. A mixed ambiguity averse DM will then prefer a situation in which his ambiguity perception changes to [0.5: 𝑘; 0.5: 𝜀̃ ] to the situation in which it changes to [0.5: 𝑘 + 𝜀̃ ; 0.5: 0]. In the latter situation, the two harms are concentrated in one state, whereas in the former situation they are divided over the two states. The two changes have the same mean (0.5𝑘) and the same variance (0.5𝑘 2 ), but [0.5: 𝑘 + 𝜀̃ ; 0.5: 0] is more positively skewed. Definition 1 reflects that a mixed ambiguity averse DM dislikes such positive skewness.

The assumption of mixed ambiguity aversion is plausible and common in the literature.

Brockett and Golden (1987) have pointed out that all commonly used functions in economic theory with a positive first derivative and a negative second derivative, have successive derivatives that changes sign. Consequently, all functions 𝜑 that reflect ambiguity aversion must be mixed ambiguity averse. Examples of functions that are mixed ambiguity averse are the constant ambiguity aversion function 𝜑(𝑥) = -𝑒 -𝛼𝑥 /𝛼, for 𝛼 > 0, which we used in our example and which was also used by KMM (2005), and the functions used in [START_REF] Ju | Ambiguity, learning and asset returns[END_REF] and in [START_REF] Gollier | Does ambiguity aversion reinforce risk aversion? Applications to Portfolio Choices and Asset Pricing[END_REF].

We can now state the generalization of Result 2 to 𝑛 𝑡ℎ order changes in ambiguity.

RESULT 3. Let the DM be mixed ambiguity averse. For all 𝜀1 and 𝜀2 such that 𝜀2 ≽ 𝑛 𝜀1, if

- 𝜑 (𝑛+1) 𝜑 (𝑛) < 𝑛𝑆* then 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 .
𝑆 * is defined as the minimum of 𝑆 =

𝑈 𝑙 ′ -𝑈 𝑑 ′ (𝑈 𝑙 -𝑈 𝑑 )(𝑍 𝑖 ′ -𝜀 𝑖 (𝑈 𝑙 ′ -𝑈 𝑑 ′ ))
. In words, Result 3 says that if the DM's 𝑛 th order ambiguity aversion is not too extreme, 𝑛 th order increases in ambiguity will lead to an increase in his VSL.

Ambiguity seeking

Until now, we have only analyzed the preferences of an ambiguity averse DM. However, empirical evidence suggests that uniform ambiguity aversion is rare and that there are decision contexts in which ambiguity seeking prevails [START_REF] Kocher | Ambiguity aversion is not universal[END_REF][START_REF] Trautmann | Blackwell Handbook of Judgment and Decision Making[END_REF][START_REF] Wakker | Prospect Theory for Risk and Ambiguity[END_REF]. This holds in particular for unlikely events and losses.

In the smooth model an ambiguity-seeking DM is characterized by 𝜑 ′ > 0 and 𝜑 ′′ > 0.

The introduction of ambiguity increases the utility of an ambiguity seeker and Result 4

shows that it lowers the VSL of an ambiguity seeking DM compared with the situation in which there is no ambiguity:

RESULT 4: Let the DM be ambiguity seeking. Let 𝜀1 and 𝜀2 be two random variables with 𝜀̃1 = 0 and 𝜀̃2 a mean-preserving spread of 𝜀̃1. Then 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 .

Result ii. If 𝜑 ′′′ = 0 then also 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 .

iii. If the DM is ambiguity prudent then the sign of 𝑉𝑆𝐿 𝜀 ̃2 -𝑉𝑆𝐿 𝜀 ̃1 is indeterminate.

In part (i), 𝑆 * is defined as in Result 2. RESULT 6. Let the DM be ambiguity seeking. For all 𝜀1 and 𝜀2 such that 𝜀2 ≽ 𝑛 𝜀1 with 𝑛 ≥ 3:

i. If 𝜑 (𝑛) > 0 and 𝜑 (𝑛+1) < 0 for 𝑛 odd and if -𝜑 (𝑛+1) 𝜑 (𝑛) < 𝑛𝑆* then 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1.

ii. If 𝜑 (𝑛) > 0 and 𝜑 (𝑛+1) < 0 for 𝑛 even and if -𝜑 (𝑛+1) 𝜑 (𝑛) < 𝑛𝑆* then 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 .

with 𝑆 * as in Result 3.

In words, we get conclusive results for an ambiguity seeker if the higher order derivatives change sign and if the ratio of the derivatives is not too extreme. In all other cases no conclusive results can be derived. Analogously to the case of mixed risk seeking (Crainich et al. 2013), we say that a DM is mixed ambiguity seeking if the derivatives of his ambiguity function 𝜑 are always positive: 𝜑 (𝑛) > 0 for all 𝑛. Mixed ambiguity seeking can be explained by a preference for combining good with good. Result 6 shows that we can make no clear predictions about the effect of 𝑛-th order increases in ambiguity for mixed ambiguity seekers.

Neo-additive preferences

The empirical literature gives no clear answer which ambiguity model best describes people's preferences. While the results in Cubitt et al. ( 2016) are consistent with the smooth model, [START_REF] Baillon | Testing Ambiguity Models Through the Measurement of Probabilities for Gains and Losses[END_REF] and [START_REF] Chew | Partial Ambiguity[END_REF] observed that models like Choquet expected utility [START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF]) and 𝛼-maxmin [START_REF] Ghirardato | Differentiating ambiguity and ambiguity attitude[END_REF] could better explain their data. In this Section we explore the robustness of our results

under the neo-additive model of [START_REF] Chateauneuf | Choice under uncertainty with the best and worst in mind: Neo-additive capacities[END_REF]. Neo-additive preferences are a special case of Choquet expected utility and they also have an interpretation in terms of 𝛼-maxmin (see for instance [START_REF] Baillon | Learning under Ambiguity: An Experiment Using Initial Public Offerings on a Stock Market[END_REF] for details). Under neo-additive preferences, the DM's evaluation of the mortality risk is equal to:5

𝑊 𝜀 ̃= (1 -𝑎)𝐸((1 -𝑝 ̃)𝑈 𝑙 (𝑤) + 𝑝 ̃𝑈𝑑 (𝑤)) + 𝑎 -𝑏 2 𝑚𝑎𝑥[(1 -𝑝 ̃)𝑈 𝑙 (𝑤) + 𝑝 ̃𝑈𝑑 (𝑤)] + 𝑎 + 𝑏 2 𝑚𝑖𝑛[(1 -𝑝 ̃)𝑈 𝑙 (𝑤) + 𝑝 ̃𝑈𝑑 (𝑤)]. ( 7
)
In Eq. ( 7 to the minimum (expected) utility that he can obtain.

Expected utility is the special case of Eq. ( 7) with 𝑎 = 𝑏 = 0. Baillon (2017) shows that under neo-additive preferences his model-free definition of ambiguity aversion is equivalent to 𝑏 > 0 and his definition of ambiguity prudence to 𝑎 > 0. Hence, neo-additive preferences provide an easy way to capture ambiguity aversion and ambiguity prudence.

For a random variable 𝜀̃ the maximum expected utility is obtained for the lowest mortality risk, i.e. for the lowest value of 𝜀. Denote the absolute value of this by 𝜀 𝑚𝑖𝑛 .

Similarly, the minimum expected utility is obtained for the highest mortality risk, i.e. for the highest value of 𝜀. Denote the absolute value of this by 𝜀 𝑚𝑎𝑥 . As we consider changes in ambiguity perception with mean zero, Eq. ( 7) can then be written as:

(1 -𝑎)

RESULT 7:

Suppose that the DM has neo-additive preferences and is ambiguity averse (𝑏 > 0). Let 𝜀1

and 𝜀2 be two random variables with 𝜀2 a mean-preserving spread of 𝜀1.

i. if 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 ≤ 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 then 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 . ii. if 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝑒 + 𝑘 > 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒 then 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 if 𝑎 < 2𝑒+𝑘 𝑘 𝑏.
Result 7 shows that the effect of an increase in ambiguity perception depends on the skewness of the change in ambiguity perception and on the DM's ambiguity prudence. If 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 ≤ 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 then VSL will always increase after an increase in ambiguity perception. If 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 exceeds 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 , but the DM's is not too ambiguity prudent (𝑎 is not too high), then the VSL will increase after an increase in ambiguity perception. If ambiguity prudence is strong compared to ambiguity aversion and 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 exceeds 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 then it is possible that an increase in ambiguity perception actually leads to a decrease in the VSL. Result 7 also shows that the probability that 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 increases if the DM becomes more ambiguity averse (𝑏 increases ceteris paribus).

Treich (2010) studied the special case of Result 7 where 𝜀1 = 0. By setting 𝜀 2,𝑚𝑎𝑥 = 𝜀 𝑚𝑎𝑥 , 𝜀 2,𝑚𝑖𝑛 = 𝜀 𝑚𝑖𝑛 , 𝜀 1,𝑚𝑎𝑥 = 𝜀 1,𝑚𝑖𝑛 = 0, it follows that an ambiguity averse DM will always have a higher VSL under ambiguity than under no ambiguity if 𝑎 < 𝜀 𝑚𝑎𝑥 +𝜀 𝑚𝑖𝑛 𝜀 𝑚𝑖𝑛 -𝜀 𝑚𝑎𝑥 𝑏. 7

Result 8 states the effects of changes in ambiguity perception for an ambiguity seeking DM.

RESULT 8:

Suppose that the DM has neo-additive preferences and is ambiguity seeking (𝑏 < 0). Let 𝜀1

and 𝜀2 be two random variables with 𝜀2 a mean-preserving spread of 𝜀1.

i. if 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 ≥ 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 then 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 . ii. if 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝑒 < 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒 + 𝑘 then 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 if 𝑎 < - 2𝑒+𝑘 𝑘 𝑏.
Hence, we observe, as in Result 5, that an ambiguity seeking DM's VSL will decrease when his ambiguity perception increases if he is not too ambiguity prudent.

Conclusion.

The VSL is a key parameter in the evaluation of government regulation. The properties of the VSL have mainly been studied under risk where probabilities are objectively known. However, in most real-world decisions probabilities are at best vaguely known. The empirical literature shows that people are not neutral towards such ambiguity.

7 If 𝜀̃ is symmetric (𝜀 𝑚𝑖𝑛 = 𝜀 𝑚𝑎𝑥 ) or 𝜀 𝑚𝑖𝑛 < 𝜀 𝑚𝑎𝑥 then it is of course always true that 𝑉𝑆𝐿 𝜀 ̃> 𝑉𝑆𝐿 𝑟 .

Treich (2010) derived that an ambiguity averse DM who behaves according to the smooth ambiguity model of KMM (2005) will have a higher VSL under ambiguity than under no ambiguity. We have shown that this cannot be generalized to general increases in ambiguity. An ambiguity averse DM's VSL may actually be lower in more ambiguous decision situations. To sign the effect of increases in ambiguity, information on higher order ambiguity attitudes is required. We have particularly highlighted the important role of ambiguity prudence. Our results confirm Baillon's (2017) conclusion that ambiguity prudence plays a key role in explaining economic behavior.

The implications of our results for cost-benefit analysis depend on whether deviations from ambiguity neutrality are considered normative or not. Cost-benefit analysis is a prescriptive exercise but the VSL is estimated by eliciting people's preferences for mortality risk reductions, which is a descriptive task. If ambiguity aversion is viewed as irrational then our results indicate what the bias in the estimated VSL will be. If an increase in perceived ambiguity is positively skewed, i.e. people perceive more situations with a high risk to human life, then they will react too strongly to the ambiguity leading to estimates of the VSL that are too high. An example is the approval of new pharmaceuticals [START_REF] Viscusi | Regulating ambiguous risks: the less than rational regulation of pharmaceuticals[END_REF]. If the distribution is negatively skewed, i.e. people perceive more situations in which the risk to human life is lower, and they are not too ambiguity prudent, their estimated VSL will be too low. On the other hand, if ambiguity aversion is considered rational (see for example Gilboa and Marinacchi 2013) our results guide policy as to how the VSL used in policy evaluations should be adjusted to changes in ambiguity.

Empirical research on ambiguity prudence is still thin on the ground. [START_REF] Baillon | Measuring Higher Order Ambiguity Preferences[END_REF] tested ambiguity prudence and found support for it. However, they only obtained qualitative support for ambiguity prudence and did not quantify its intensity. Our results highlight that such quantification is required to understand the effects of (increases in)

ambiguity on the VSL. [START_REF] Ebert | Joint measurement of risk aversion, prudence, and temperance[END_REF] showed how the intensity of risk prudence can be measured. Extending their research to ambiguity is a worthwhile topic for future research.

than 𝜀1, i.e., 𝑔 2 > 𝑔 1 then the conditions for the VSL to increase with an increase in ambiguity become less stringent. The intuition is that if 𝑔 2 > 𝑔 1 , the DM expects a higher mortality risk under 𝜀2 and we know from the dead anyway effect that a higher (expected) mortality risk increases the VSL. However, if 𝜀̃2 is less upward biased than 𝜀̃1, i.e., 𝑔 2 < 𝑔 1 then the conditions for the VSL to increase with an increase in ambiguity become more stringent because of the dead-anyway effect and even in the symmetric case (i) ambiguity aversion alone is not enough to ensure that the VSL will increase.

Result A2 states the results for an ambiguity seeking DM. It is the immediate counterpart of Result A1.

RESULT A2:

Suppose that the DM has neo-additive preferences and is ambiguity seeking (𝑏 < 0). Let 𝜀̃1

and 𝜀̃2 be two random variables with the support of 𝜀̃1 contained in the support of 𝜀̃2. (5),

𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 if 𝐸[𝜑 ′ (𝑍 ̃2)] 𝐸[𝜑 ′ (𝑍 ̃1)] > 𝐸[𝑍 ̃2 ′ 𝜑 ′ (𝑍 ̃2)] 𝐸[𝑍 ̃1 ′ 𝜑 ′ (𝑍 ̃1)] . (𝐴1)
A sufficient condition to obtain Eq. ( A1) is

𝐸[𝜑 ′ (𝑍 ̃2)] 𝐸[𝜑 ′ (𝑍 ̃1)] > 1 > 𝐸[𝑍 ̃2 ′ 𝜑 ′ (𝑍 ̃2)] 𝐸[𝑍 ̃1 ′ 𝜑 ′ (𝑍 ̃1)] . ( 𝐴2
)
Let 𝑍 0 = (1 -𝑝)𝑈 𝑙 (𝑤) + 𝑝𝑈 𝑑 (𝑤), 𝑍 0 ′ = (1 -𝑝)𝑈 𝑙 ′ (𝑤) + 𝑝𝑈 𝑑 ′ (𝑤), ∆𝑣 = 𝑈 𝑙 -𝑈 𝑑 , and

∆𝑣 ′ = 𝑈 𝑙 ′ -𝑈 𝑑 ′ .
Define the following functions: 𝑔(𝜀) = 𝜑 ′ (𝑍 0 -𝜀∆𝑣), 𝑧(𝜀) = 𝑍 0 ′ -𝜀∆𝑣 ′ , and 𝐻(𝜀) = 𝑔(𝜀)𝑧(𝜀). Eq. (A2) can then be rewritten as:

𝐸[𝑔(𝜀̃2)] 𝐸[𝑔(𝜀1)] > 1 > 𝐸[𝐻(𝜀̃2)] 𝐸[𝐻(𝜀1)] . (𝐴3) 𝐸[𝑔(𝜀̃2)] 𝐸[𝑔(𝜀1)] = 1 > 𝐸[𝐻(𝜀̃2)] 𝐸[𝐻(𝜀1)] ,
which also implies that 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 .

If the DM is imprudent (𝜑 ′′′ < 0) then the second derivative of 𝑔 is negative and We first show that mixed ambiguity averse DMs will dislike 𝑛 th order increases in ambiguity.

LEMMA A3: For all changes in ambiguity perception 𝜀1 and 𝜀2 and for all 𝑛: 𝜀2 ≽ 𝑛 𝜀1 iff every mixed ambiguity averse DM prefers 𝜀1 to 𝜀2.

Proof. The proof is very similar to [START_REF] Ekern | Increasing Nth degree risk[END_REF] . Repeated application of integration by parts gives:

∫ 𝜑(𝑝)𝑑𝐹 2 (𝑝) -∫ 𝜑(𝑝)𝑑𝐹 1 (𝑝) = 𝑑 𝑐 𝑑 𝑐 𝜑(𝑑)(𝐹 2 (𝑑) -𝐹 1 (𝑑)) + ∑ (-1) 𝑘-1 𝜑 (𝑘-1) (𝐹 2 𝑘 (𝑑) -𝐹 1 𝑘 (𝑑)) 𝑛 𝑘=2 + ∫ (-1) 𝑛 𝜑 (𝑛) (𝐹 2 𝑛 (𝑝) -𝐹 1 𝑛 (𝑝)) 𝑑 𝑐 𝑑𝑝 (𝐴5).
Suppose the DM is mixed ambiguity averse. Then it follows immediately from Definition

1 that ∫ 𝜑(𝑝)𝑑𝐹 2 (𝑝) -∫ 𝜑(𝑝)𝑑𝐹 1 (𝑝) > 0 𝑑 𝑐 𝑑 𝑐
and thus that 𝜀1 ≽ 𝜀2 where ≽ denotes the DM's preference relation.

For the reverse implication, suppose that every mixed ambiguity averter prefers 𝜀1to 𝜀2, but that not 𝜀̃2 ≽ 𝑛 𝜀̃1. Then there must be some 𝑚 < 𝑛 such that 𝐹 2 𝑚 (𝑑) -𝐹 1 𝑚 (𝑑) < 0. Then mixed ambiguity averse DMs for whom -𝜑 𝑛 𝜑 (𝑚-1) converges to zero will prefer 𝜀2 to 𝜀1, a contradiction. Q.E.D.

Define the functions 𝑔, 𝑧, and 𝐻 as in the proof of Result 2. It follows that for every 𝑛 ≥ 2 𝑔 (𝑛) = (∆𝑣) 𝑛 𝜑 (𝑛+1) (𝑍 0 -𝜀∆𝑣), 𝑧 (𝑛) = 0, and 𝐻 (𝑛) = 𝑛𝑧 ′ (𝜀)𝑔 (𝑛-1) (𝜀) + 𝑧(𝜀)𝑔 (𝑛) (𝜀) = 𝑛(∆𝑣 ′ )𝑔 (𝑛-1) (𝜀) + 𝑧(𝜀)𝑔 (𝑛) Now, 𝐻 (𝑛) = 𝑛(∆𝑣 ′ )𝑔 (𝑛-1) (𝜀) + 𝑧(𝜀)𝑔 (𝑛) (𝜀) = 𝑛(∆𝑣 ′ )(∆𝑣) 𝑛-1 𝜑 (𝑛) (𝑍 0 -𝜀∆𝑣) + (𝑍 0 ′ -𝜀∆𝑣 ′ )(∆𝑣) 𝑛 𝜑 (𝑛+1) (𝑍 0 -𝜀∆𝑣). If 𝑛 is odd this expression should be positive. That is, (𝑍 0 ′ -𝜀∆𝑣 ′ )(∆𝑣) 𝑛 𝜑 (𝑛+1) (𝑍 0 -𝜀∆𝑣) > -𝑛(∆𝑣 ′ )(∆𝑣) 𝑛-1 𝜑 (𝑛) (𝑍 0 -𝜀∆𝑣). As 𝜑 (𝑛) and ∆𝑣 are both positive, it follows that -𝜑 (𝑛+1) 𝜑 (𝑛) < 𝑛(∆𝑣 ′ ) (∆𝑣)(𝑍 0 ′ -𝜀∆𝑣 ′ ) = 𝑛𝑆 * .

If 𝑛 is even 𝐻 (𝑛) should be negative. That is, (𝑍 0 ′ -𝜀∆𝑣 ′ )(∆𝑣) 𝑛 𝜑 (𝑛+1) (𝑍 0 -𝜀∆𝑣) < -𝑛(∆𝑣 ′ )(∆𝑣) 𝑛-1 𝜑 (𝑛) (𝑍 0 -𝜀∆𝑣). As 𝜑 (𝑛) is negative and ∆𝑣 is positive, it follows that This follows straightforwardly from the proof of Result 9 by setting 𝑏 < 0.

  0 . We can measure the intensity of the DM's ambiguity aversion as -𝜑 ′′ 𝜑 ′ , much like the well-known Arrow-Pratt index measures the DM's risk aversion. The function 𝜑(𝑥) = -𝑒 -𝛼𝑥 𝛼 has the attractive property that ambiguity aversion is constant and equal to 𝛼. Normalize wealth to 1 and let 𝑈 𝑙 = (1 -𝑒 -.1𝑥 ) and 𝑈 𝑑 = 0.5 * (1 -𝑒 -.05𝑥 ). Then 𝑈 𝑙 (𝑥) > 𝑈 𝑑 (𝑥) and 𝑈 𝑙 ′ (𝑥) > 𝑈 𝑑 ′ (𝑥) for all positive wealth levels 𝑥. We write (𝑝 1 : 𝑥 1 , … , 𝑝 𝑛 : 𝑥 𝑛 ) for the random variable that gives 𝑥 𝑗 with probability 𝑝 𝑗 , 𝑗 = 1, … , 𝑛. Consider two random variables 𝜀1 = (0.5: -0.2, 0.5: 0.2) and 𝜀2 = (0.125: -0.5, 0.375: -0.1, 0.5: 0.2). Then 𝐸[𝜀̃1] = 𝐸[𝜀̃2] = 0 and 𝜀̃2 is a mean-preserving spread of 𝜀1. Figure 1 displays the relation between 𝑉𝑆𝐿 𝜀 ̃1 -𝑉𝑆𝐿 𝜀 ̃2 and 𝛼. The figure shows that initially 𝑉𝑆𝐿 𝜀 ̃2 exceeds 𝑉𝑆𝐿 𝜀 ̃1 , but as 𝛼 increases the difference becomes smaller and for 𝛼 sufficiently large the relation is reversed and 𝑉𝑆𝐿 𝜀 ̃2 is less than 𝑉𝑆𝐿 𝜀 ̃1 . Because the intensity of the DM's ambiguity aversion is equal to 𝛼, the figure shows that more ambiguity aversion does not necessarily lead to a higher VSL when the DM's ambiguity perception increases.RESULT 1: Let 𝜀1 and 𝜀2 be two nondegenerate random variables with 𝜀2 a meanpreserving spread of 𝜀̃1. Then the sign of 𝑉𝑆𝐿 𝜀 ̃2 -𝑉𝑆𝐿 𝜀 ̃1 is indeterminate.

Figure 1 :

 1 Figure 1: The relationship between 𝑽𝑺𝑳 𝜺 ̃𝟏 -𝑽𝑺𝑳 𝜺 ̃𝟐 and ambiguity aversion (measured by the parameter 𝜶) in the example.

Finally,

  Result 6 summarizes what happens in the case of higher order increases in ambiguity. While Results 4 and 5 are rather straightforward counterparts of Results 1 and 2, Result 6 is a bit different. The intuition underlying this difference is that, unlike for an ambiguity averse DM, for an ambiguity seeking DM the first and second derivative of 𝜑 do not change sign.

  ) 𝑎 ∈ [0,1] and 𝑏 ∈ [-𝑎, 𝑎]. A neo-additive DM gives weight (1 -𝑎) to the expected utility of a random variable 𝑝 ̃, weight 𝑎-𝑏 2 to the maximum (expected) utility that he can obtain and weight 𝑎+𝑏 2

Because

  𝜀̃1 is degenerate (𝜀̃1 = 0), 𝑍 ̃1 ′ is constant. Then the denominator of Eq. (A6)becomes𝐸[𝑍 ̃1 ′ ]𝐸[𝜑 ′ (𝑍 ̃1)]. Because 𝜀2 is a mean-preserving spread of 𝜀1, 𝐸[𝑍 ̃1 ′ ] = 𝐸[𝑍 ̃2 ′ ]. Substituting 𝐸[𝑍 ̃1 ′ 𝜑 ′ (𝑍 ̃1)] = 𝐸[𝑍 ̃1 ′ ]𝐸[𝜑 ′ (𝑍 ̃1)] and 𝐸[𝑍 ̃1 ′ ] = 𝐸[𝑍 ̃2 ′ ] into Eq. (A6) gives 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 iff 𝐸[𝑍 ̃2 ′ ]𝐸[𝜑 ′ (𝑍 ̃2)] < 𝐸[𝑍 ̃2 ′ 𝜑 ′ (𝑍 ̃2)].The last inequality is holds if the covariance between 𝑍 ̃2 ′ and 𝜑 ′ (𝑍 ̃2) is positive, which holds because the DM is ambiguity seeking. 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 ))] ⋚ 0.Part (i). Suppose 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒. Then [ 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 ))] = -𝑏𝑒 < 0. Hence, 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 .Suppose 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝑒, 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒 + 𝑘, 𝑘 > 0. Then [ Hence, 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 .Part (ii). Denote 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝑒 + 𝑘, 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒, 𝑘 > 0. Then [ This follows straightforwardly from the proof of Result 7 by setting 𝑏 < 0. ∎Proof of Result A1.Eq. (9) shows that we have 𝑉𝑆𝐿 𝜀 ̃2 ⋛ 𝑉𝑆𝐿 𝜀 ̃1 iff (𝑔 1 -𝑔 2 . Suppose 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒. Then (𝑔 1 -𝑔 2 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 ))] = (𝑔 1 -𝑔 2 ) -𝑏𝑒. Hence, 𝑉𝑆𝐿 𝜀 ̃2 ⋛ 𝑉𝑆𝐿 𝜀 ̃1 iff 𝑏 ). Suppose 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝑒 < 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒 + 𝑘. 𝑘 > 0. Then (𝑔 1 -𝑔 2 ). Suppose 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝑒 + 𝑘 > 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒. 𝑘 > 0. Then (𝑔 1 -𝑔 2

  on saving and prevention,[START_REF] Gierlinger | Saving for an ambiguous future[END_REF] on the socially efficient discount rate, and Peter and Ying (2017) on insurance. The index reflects the extent to which the DM cares about the skewness of his ambiguity perceptions. If 𝜑 ′′′ = 0, the DM does not care about skewness. If 𝜑 ′′′ < 0, the DM is ambiguity imprudent.

	Under the smooth model, ambiguity prudence is equivalent to 𝜑 ′′′ > 0. We can define
	an index of ambiguity-prudence as -	𝜑 ′′′ 𝜑 ′′ with higher values of the index corresponding to
	more ambiguity prudence.

RESULT 2: For all 𝜀1 and 𝜀2 with 𝜀2 a mean-preserving spread of 𝜀1, i. If the DM is ambiguity prudent and -𝜑 ′′′ 𝜑 ′′ < 2𝑆* then 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1.

  For decision under risk,[START_REF] Ekern | Increasing Nth degree risk[END_REF] gave a definition of more 𝑛 th order risk aversion when risks about wealth are introduced.Caballé and Pomanski (1996) defined a DM as mixed risk averse if his von Neumann Morgenstern utility function 𝑢 has positive odd and negative even derivatives: for all 𝑘 = 1, . . , 𝑛, (-1) (𝑘+1) 𝑢 (𝑘) > 0.[START_REF] Ekern | Increasing Nth degree risk[END_REF] definition implies that a mixed risk averse DM will find any 𝑛 𝑡ℎ order increase in risk undesirable.Under the smooth model, we define a DM as mixed ambiguity averse if his ambiguity function 𝜑 has positive odd and negative even derivatives: for all 𝑘 = 1, . . , 𝑛, (-1) (𝑘+1) 𝜑 (𝑘) > 0. Courbage and Rey (2016) defined a change in ambiguity 𝜀̃2 as more ambiguous than another change 𝜀1 if every mixed ambiguity averse DM prefers 𝑝 ̃1 = 𝑝 + 𝜀1 to 𝑝 ̃2 = 𝑝 + 𝜀2. Using this definition, we can now formally define what it means to have more 𝑛 th order ambiguity in the context of our decision problem.

	Definition 1: 𝜀2 has more 𝑛 th order ambiguity than 𝜀1, written 𝜀2 ≽ 𝑛 𝜀1 if
	i. 𝐹 2 𝑘 (𝑑) = 𝐹 1 𝑘 (𝑑) for 𝑘 = 1, … , 𝑛
	ii. If 𝑛 is odd, 𝐹 2 𝑛 (𝑝) ≤ 𝐹 1
	which 𝐹 2 𝑛 (𝑝) < 𝐹 1 𝑛 (𝑝). If 𝑛 is even, 𝐹 2 𝑛 (𝑝) ≥ 𝐹 1 𝑛 (𝑝) for all 𝑝 ∈ [𝑐, 𝑑] and there
	exists a 𝑝 ∈ [𝑐, 𝑑] for which 𝐹 2 𝑛 (𝑝) > 𝐹 1 𝑛 (𝑝).
	Rewrite 𝐹 𝑖 0 = 𝑝 ̃𝑖, 𝐹 𝑖 1 = 𝐹 𝑖 , 𝑖 = 1,2 and define repeated integrals 𝐹 𝑖 𝑘 for 𝑘 ≥ 1 by:
	𝑝
	𝐹 𝑖 𝑘 (𝑝) = ∫ 𝐹 𝑖 𝑘-1 (𝑞)𝑑𝑞.	(6)
	𝑐

Consider, as before, two random variables 𝜀̃1 and 𝜀̃2 with 𝑝 ̃𝑖 = 𝑝 + 𝜀̃𝑖, 𝑖 = 1,2. Let 𝐹 𝑖 be the cumulative probability distribution of 𝑝 ̃𝑖. [𝑐, 𝑑] denotes the support of 𝑝 ̃1 ∪ 𝑝 ̃2, i.e. the DM believes that mortality risks outside the interval [𝑐, 𝑑] are impossible and, thus, 𝐹 𝑖 (𝑐) = 0 and 𝐹 𝑖 (𝑑) = 1, 𝑖 = 1,2. Obviously, [𝑐, 𝑑] is a subset of [0,1], as probabilities cannot be negative or exceed 1. 𝑛 (𝑝) for all 𝑝 ∈ [𝑐, 𝑑] and there exists a 𝑝 ∈ [𝑐, 𝑑] for

  Totally differentiating Eq. (8) with respect to 𝑝 and 𝑤 gives the VSL under neo-𝑎 = 𝑏 and an ambiguity averse DM will always have a higher VSL when his perception of ambiguity increases.Let us now consider what happens if the DM's ambiguity perception increases.Consider two random variables 𝜀1 and 𝜀2 with 𝜀2 a mean-preserving spread of 𝜀1. We denote the absolute values of the minima of 𝜀1 and 𝜀2 by 𝜀 1,𝑚𝑖𝑛 and 𝜀 2,𝑚𝑖𝑛 and their maximum values by 𝜀 1,𝑚𝑎𝑥 and 𝜀 2,𝑚𝑎𝑥 .

	additive preferences:
	𝑉𝑆𝐿 𝜀 ̃𝑖 =	𝑑𝑤 𝑑𝑝
	=	𝑈 𝑙 (𝑤) -𝑈 𝑑 (𝑤) ′ (𝑤) + (𝑈 𝑙 ′ (𝑤) + 𝑝𝑈 𝑑 (1 -𝑝)𝑈 𝑙 ′ (𝑤) + 𝑈 𝑑 ′ (𝑤)) [	𝑎 -𝑏 2	𝜀 𝑚𝑖𝑛 -	2 𝑎 + 𝑏	𝜀 𝑚𝑎𝑥 ]	(9)
			𝑎-𝑏 2	𝜀 𝑚𝑖𝑛 -
						𝑎-𝑏 2	𝜀 𝑚𝑖𝑛 -	𝑎+𝑏 2	𝜀 𝑚𝑎𝑥 ] is negative and that ambiguity leads
	to an increase in the VSL. Second, under Schmeidler's (1989) definition of ambiguity
	aversion, 6
		((1 -𝑝)𝑈 𝑙 (𝑤) + 𝑝𝑈 𝑑 (𝑤)) +	𝑎 -𝑏 2	[(1 -(𝑝 -𝜀 𝑚𝑖𝑛 ))𝑈 𝑙 (𝑤) + (𝑝 -𝜀 𝑚𝑖𝑛 )𝑈 𝑑 (𝑤)]
			+	𝑎 + 𝑏 2	[(1 -(𝑝 + 𝜀 𝑚𝑎𝑥 ))𝑈 𝑙 (𝑤) + (𝑝 + 𝜀 𝑚𝑎𝑥 )𝑈 𝑑 (𝑤)].	(8)

As 𝑎 ∈ [0,1] and 𝑏 ∈ [-𝑎, 𝑎], it follows that if the DM is ambiguity averse (𝑏 > 0) the sign of the term [ 𝑎+𝑏 2 𝜀 𝑚𝑎𝑥 ] is indeterminate and depends on the relative magnitudes of 𝜀 𝑚𝑖𝑛 and 𝜀 𝑚𝑎𝑥 . Hence, under neo-additive preferences we cannot replicate Treich's result that an ambiguity averse DM will always have a higher VSL under ambiguity than under no ambiguity. Two points are worth making. First, if 𝜀 𝑚𝑖𝑛 = 𝜀 𝑚𝑎𝑥 then ambiguity aversion implies that [

  𝜀1 and 𝜀2 are equally biased then Result 8 still holds. If 𝜀2 is more upward biased than 𝜀1 (𝑔 2 > 𝑔 1 ) then the conditions for the VSL to decrease with an increase in ambiguity become more stringent. The intuition, again, is that the dead anyway effect goes in the direction of a larger value of 𝑉𝑆𝐿 𝜀 ̃2 compared to 𝑉𝑆𝐿 𝜀 ̃1. If 𝜀2 is less upward biased than 𝜀̃1 (𝑔 2 < 𝑔 1 ) then the conditions for the VSL to increase with an increase in ambiguity become less stringent. Parts (i) and (iii) then always hold and the restriction on prudence in Part (ii) becomes weaker.

	i. if 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒 then 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 if 𝑏 < ii. if 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝑒 < 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒 + 𝑘 then 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 if 𝑎 < 𝑔 1 -𝑔 2 𝑒 . -(2𝑒+𝑘)𝑏-2(𝑔 1 -𝑔 2 ) 𝑘 . iii. if 𝜀 2,𝑚𝑖𝑛 -𝜀 1,𝑚𝑖𝑛 = 𝑒 + 𝑘 > 𝜀 2,𝑚𝑎𝑥 -𝜀 1,𝑚𝑎𝑥 = 𝑒 then 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 if 𝑎 > (2𝑒+𝑘)𝑏-2(𝑔 1 -𝑔 2 ) 𝑘 . Proof of Result 2. For an ambiguity seeker, if B: Proofs Define 𝑍 𝑖 ̃= (1 -𝑝 ̃𝑖)𝑈 𝑙 (𝑤) + 𝑝 𝑖 ̃𝑈𝑑 (𝑤) and 𝑍 𝑖 ′ ̃= (1 -𝑝 ̃𝑖)𝑈 𝑙 ′ (𝑤) + 𝑝 𝑖 ̃𝑈𝑑 ′ (𝑤), 𝑖 = 1,2. By Eq.

  (in which case 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 ) are possible and the sign of 𝑉𝑆𝐿 𝜀 ̃2 -𝑉𝑆𝐿 𝜀 ̃1 is indeterminate.

	𝐸[𝑔(𝜀 ̃2)] 𝐸[𝑔(𝜀 ̃1)] < 1. Imprudence implies that -	𝜑 ′′′ 𝜑 ′′ < 0 and, consequently, that -	𝜑 ′′′ 𝜑 ′′ < 2𝑆 so also
	𝐸[𝐻(𝜀 ̃2)] 𝐸[𝐻(𝜀 ̃1)] < 1. Hence, both	𝐸[𝐻(𝜀 ̃2)] 𝐸[𝐻(𝜀 ̃1)] <	𝐸[𝑔(𝜀 ̃2)] 𝐸[𝑔(𝜀 ̃1)] < 1 (in which case 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃1 ) and	𝐸[𝑔(𝜀 ̃2)] 𝐸[𝑔(𝜀 ̃1)] <
	𝐸[𝐻(𝜀 ̃2)] 𝐸[𝐻(𝜀 ̃1)] < 1

∎

Proof of Result 3.

  derivation on p. 331. First note that even though 𝜑 is defined over expected utility in the smooth model, in our setup with only two utilities 𝑈 𝑙 and 𝑈 𝑑 there is a one-to-one negative relationship between 𝑝 and expected utility. Suppose first that 𝜀2 ≽ 𝑛 𝜀1. The DM will prefer 𝜀1 to 𝜀2 if ∫ 𝜑(𝑝)𝑑𝐹 2 (𝑝) -

	𝑑
	𝑐
	𝑑 ∫ 𝜑(𝑝)𝑑𝐹 1 (𝑝) > 0 𝑐

  (𝜀). From Lemma A3, 𝐸[𝐻(𝜀2)] < 𝐸[𝐻(𝜀1)] will hold if 𝐻 is a function that has 𝐻 (𝑛) > 0 if 𝑛 is odd and 𝐻 (𝑛) < 0 if 𝑛 is even.

	It is still true that Eq. (A3) is a sufficient condition to get 𝑉𝑆𝐿 𝜀 ̃2 > 𝑉𝑆𝐿 𝜀 ̃2 . Equation

(A3) holds if 𝐸[𝑔(𝜀̃2)] > 𝐸[𝑔(𝜀̃1)] and 𝐸[𝐻(𝜀̃2)] < 𝐸[𝐻(𝜀̃1)]. Because 𝑔 is a function for which 𝑔 (𝑛) > 0 if 𝑛 is even and 𝑔 (𝑛) < 0 if 𝑛 is odd, it follows from Eq. (A5) that 𝐸[𝑔(𝜀2)] > 𝐸[𝑔(𝜀1)].

We introduce the distinction between 𝑛 odd and 𝑛 even to make sure that a decision maker who is mixed ambiguity averse always prefers 𝜀̃1 to 𝜀̃2. Our definition is slightly different from that of[START_REF] Ekern | Increasing Nth degree risk[END_REF] who considers random changes in wealth with more wealth always desirable while we consider random changes in the mortality risk, increases of which are undesirable.

We interpret neo-additive preferences in a setting where the possible values of 𝜀̃ are the states of the world.

[START_REF] Schmeidler | Subjective probability and expected utility without additivity[END_REF] defines ambiguity aversion as a preference for hedging: if the DM is indifferent between two random variables 𝑝 ̃ and 𝑞 ̃ then he prefers their mixture 𝜆𝑝 ̃+ (1 -𝜆𝑞 ̃), 0 < 𝜆 < 1, to each of these variables.

Appendix:

A. Biased beliefs under neo-additive preferences

Throughout the paper we have assumed that ambiguity aversion does not lead to a bias in the DM's beliefs. That is the random variables 𝜀1 and 𝜀2 have expectation zero. In this subsection we will briefly consider the case where ambiguity leads to a bias in the DM's beliefs. Let 𝐸[𝜀̃1] = 𝑔 1 and 𝐸[𝜀̃2] = 𝑔 2 . If 𝑔 1 ≠ 𝑔 2 , 𝜀̃2 cannot be a mean-preserving spread of 𝜀̃1. We define 𝜀̃2 as more ambiguous than 𝜀̃1 if the interval of possible values that the mortality risk takes under 𝜀1 is a subset of the interval of possible values under 𝜀2. 8

RESULT A1:

Suppose that the DM has neo-additive preferences and is ambiguity averse (𝑏 > 0). Let 𝜀̃1 and 𝜀̃2 be two random variables with 𝜀̃2 more ambiguous than 𝜀̃1.

Result A1 shows that if 𝜀̃1 and 𝜀̃2 are equally biased then Result 7 still holds. Part (ii) holds because - According to Theorem 2 in Rothschild and Stiglitz (1970, p. 237),

if the second derivative of 𝑔 is positive. Hence, 𝑔 ′′ = (-∆𝑣) 2 𝜑 ′′′ (𝑍 0 -𝜀∆𝑣) > 0, which holds if 𝜑 ′′′ (𝑥) > 0 for all 𝑥, i.e. if the DM is ambiguity prudent. Theorem 2 in Rothschild and Stiglitz's also implies that

if the second derivative of 𝐻 is negative. Differentiating 𝐻 twice and rearranging terms gives:

Proof of Result 5.

The proof is very similar to that of Result 2. Define the functions, 𝑔(𝜀), 𝑧(𝜀), and 𝐻(𝜀) as in the proof of Result 2. By Eq. ( 5), a sufficient condition to have, 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 is

Equation (A7) holds when 𝑔(𝜀) has a negative second derivative and 𝐻(𝜀) has a positive second derivative. 𝑔 ′′ (𝜀) = (∆𝑣) 2 𝜑 ′′′ (𝑍 0 -𝜀∆𝑣) < 0, which holds if 𝜑 ′′′ (𝑥) < 0 for all 𝑥, i.e.

if the DM is ambiguity imprudent. 𝐻 ′′ (𝜀) = (∆𝑣)𝜑 ′′′ (𝑍 0 -𝜀∆𝑣)(𝑍 0 ′ -𝜀∆𝑣 ′ ) + 2(∆𝑣 ′ )𝜑 ′′′ (𝑍 0 -𝜀∆𝑣), which is positive if Eq. (A4) holds.

If 𝜑 ′′′ = 0, then 𝐻 ′′ (𝜀) > 0 and 𝑔 ′′ (𝜀) = 0 and, thus,

which also implies that 𝑉𝑆𝐿 𝜀 ̃2 < 𝑉𝑆𝐿 𝜀 ̃1 . ∎

Proof of Result 6.

The proof is identical to the proof of Result 3. ∎

Proof of Result 7.

It is immediate from Eq. ( 9) that the comparison between 𝜀1 and 𝜀2 depends on the relative