Keywords: Polymeric nanoparticles, MART1, poly (benzyl glutamate), melanoma, biopolymers, liposomes, β-lapachone, wound healing Nanopartículas poliméricas, MART1, poli (benzil glutamato), melanoma, biopolímeros, lipossomas, β-lapachona, cicatrização

Development and characterization of targeted MART-1-nanoparticles for melanoma treatment and β-lapachone-loaded liposomal in hydrogel

Titre : Développement et caractérisation de nanoparticules conjuguées à MART1 pour le traitement du mélanome et de liposomes contenant de la bêta-lapachone et incorporés dans un hydrogel destiné à la cicatrisation des plaies.

Mots clés : Nanoparticules polyméres, MART1, poly(glutamate de benzyle), mélanome, biopolymères, liposomes, β-lapachone, cicatrisation.

Résumé : L'objectif principal de cette travail a été le développement, la caractérisation et l'évaluation in vitro et in vivo de différents nanovecteurs avec, d'une part, des nanoparticules spécifiques pour le traitement de la mélanome et des liposomes de β-lapachoneincorporés dans des hydrogels de biopolymère pour la cicatrisation de blessures topiques. La première partie de cette thèse présente une revue de la littérature et les récentes avancées dans le domaine du ciblage des cellules circulantes mésenchymateuses issues des mélanomes. Les principaux biomarqueurs de ces cellules ont été passes en revue afin de définir les caractéristiques qu'il est souhaitable de conférer aux nanovecteurs. La partie expérimentale du travail a consisté à preparer des nanoparticules (non sphériques) par nanoprécipitation de copolymères dérivés du poly(γ-benzyl-Lglutamate). Ces nanoparticules, taille comprise entre 20 et 100 nm et porteuses d'une charge négative (-3 à -30 mV). Elles ont ensuite été combinées avec l'anticorps MART-1 spécifique de la membrane des cellules de mélanome, en mettant en oeuvre des liaisons de type biotinestreptavidine. La combinaison de l'anticorps en surface des nanoparticules a été évaluée par western blot. L'affinité des immunenanoparticules pour des cellules modèles du mélanome (lignée B16-GFP) et pour des cellules endothéliales de la veine ombilicale humaine (HUVECs) a ensuite été évaluée in vitro par cytométrie de flux. Par ailleurs, étant destinées à une injection intra-veineuse, il était important d'évaluer le degré d'activation du système du complément induit par les nanoparticules. La technique d'immunoélectrophorèse 2D mise en oeuvre a permis de conclure à une activation limitée, favorable à une augmentation du temps de circulation sanguine après injection IV.

Par ailleurs, les nanoparticules présentaient une faible cytotoxicité (test au MTT) vis à vis des cellules de mélanome ou des cellules endothéliales. En terme de capture cellulaire, les immuno-nanoparticules fonctionnalisées par MART1, anticorps spécifique pour la reconnaissance de l'antigène surexprimé dans des cellules de mélanome, a été augmentée de 40 à 50% par rapport au contrôle. La deuxième partie de cette thèse a été consacrée au développement, à la caractérisation et l'évaluation in vivo de l'activité cicatrisante de la β-lapachone encapsulée dans des liposomes multilamellaires, eux-mêmes incorporés dans un hydrogel d'un biopolymère produit par la bactérie Zoogloea sp. Ces formulations originales (β-lap-Lipo/ZBP/HEC) présentaient un pH et un comportement rhéologique approprié pour l'application topique, ainsi que la capacité de ralentir la liberation de la βlapachone à partir de l'hydrogel. Une étude hystopathologique détaillée de l'activité cicatrisante a été conduite dans un modèle in vivo et a permis de montrer que l' hydrogel de biopolymère était capable de stimuler la reparation tissulaire, d'augmenter la cellularité locale, de favoriser les fibroblastes, les cellules inflamatoires, les vaisseaux sanguins et la production de fibrilles de collagène pendant la phase proliférative de la cicatrisation. De plus, la formulation β-lap-Lipo/ZBP/HEC a favorisé l'angiogenèse locale et a permis de diminuer l'inflammation de la blessure, démontrant le potentiel de cette formulation originale de β-lap-Lipo/ZBP/HEC dans la thérapie des lesions cutanées. Pour conclure, les nanovecteurs développés constituent des outils intéressants en vue d'intercepter les cellules circulantes du mélanome, tandis que les formulations liposomales associant des biopolymers originaux présentent un potential intéressant dans la cicatrisation des blessures.

The main aim of this work was the development, characterization and in vitro and in vivo evaluation of different nanocarriers with specific nanoparticles for the treatment of melanoma and β-lapachone liposomes incorporated in biopolymer hydrogels for the healing of topical wounds. The first part of this thesis presents a review of the literature and recent advances in the field of targeting mesenchymal circulating cells derived from melanomas. The main biomarkers of these cells have been reviewed to define the suitable characteristics of this nanocarriers. The experimental part of the work consisted of development of nanoparticles (nonspherical) by nanoprecipitation of copolymers derived from poly (γ-benzyl-Lglutamate). These nanoparticles, size between 20 and 100 nm and carrying a negative charge (-3 to -30 mV) were then combined with the MART-1 antibody, specific for the melanoma cell membrane, by biotin-streptavidin binding. The binding of the antibody on the surface of nanoparticles was evaluated by Western blot. The affinity of immuno-nanoparticles for melanoma cells (B16-GFP line) and for endothelial cells of human umbilical vein (HUVECs) was then evaluated in vitro by flow cytometry and, being intended for intravenous injection, it was important to evaluate the degree of activation of the complement system induced by the nanoparticles.

The 2D immunoelectrophoresis technique used made it possible to conclude that the activation was limited and favourable to increase the blood circulation time of nanoparticles, after intravenous injection. The nanoparticles exhibited low cytotoxicity (MTT assay) against melanoma cells or endothelial cells.

In terms of cellular uptake, the immunonanoparticles functionalized with MART1, a specific antibody for the recognition of the overexpressed antigen in melanoma cells, was increased by 40 to 50% compared to control. The second part of this thesis was dedicated to the development, characterization and in vivo evaluation of the wound healing activity of β-lapachone encapsulated in multilamellar liposomes and incorporated in a hydrogel of a biopolymer produced by the bacterium Zoogloea sp. These original formulations (β-lap-Lipo/ ZBP/HEC) had a pH and rheological behavior suitable for topical application, as well as the ability to slow the release of βlapachone from the hydrogel. A detailed histopathological study of the wound healing activity was conducted in an in vivo model and showed that the biopolymer hydrogel was able to stimulate tissue repair, increasing the local cellularity, fibroblasts, cells inflammatory, blood vessels and the production of collagen fibrils during the proliferative phase of healing. In addition, the β-lap-Lipo/ZBP/HEC formulation promoted local angiogenesis and reduced inflammation of the wound, demonstrating the potential of this original formulation of β-lap-Lipo/ZBP/HEC in cutaneous lesions therapy. To conclude, the developed nanocarriers are interesting approachs for intercepting the circulating melanoma cells, while liposomal formulations combining with original biopolymers have an interesting potential for wound healing applications.

O objetivo principal deste trabalho foi o desenvolvimento, caracterização e avaliação in vitro e in vivo de diferentes nanocarreadores: nanopartículas específicas para o tratamento de melanoma e lipossomas contendo β-lapachona incorporados em hidrogéis de biopolímero para a cicatrização de feridas tópicas. A primeira parte desta tese apresenta uma revisão da literatura e avanços recentes no campo do direcionamento de fármacos através do uso de nanocarreadores para a células circulantes mesenquimatosas derivadas de melanomas. Os principais biomarcadores presentes nestas células foram descritos com o objetivo de definir as características adequadas para o desenvolvimento de nanocarreadores. A parte experimental do trabalho consistiu no desenvolvimento de nanopartículas (não esféricas) por nanoprecipitação de copolímeros derivados de poli (γ-benzil-Lglutamato). Essas nanopartículas, tamanho entre 20 e 100 nm e carga negativa (-3 a -30 mV) foram então combinadas com o anticorpo MART-1, específico para a membrana das células do melanoma, pela ligação biotinaestreptavidina. A ligação do anticorpo na superfície das nanopartículas foi avaliada por Western blot. A afinidade das imunonanopartículas para células de melanoma (linha B16-GFP) e para células endoteliais de veia umbilical humana (HUVECs) foi então avaliada in vitro por citometria de fluxo e, sendo destinada a injeção intravenosa, era importante avaliar o grau de ativação do sistema complemento induzido pelas nanopartículas. A técnica utilizada de imunoeletroforese 2D permitiu concluir que a ativação foi limitada e favorável para aumentar o tempo de circulação sanguínea das nanopartículas, após a injeção intravenosa. As nanopartículas apresentaram baixa citotoxicidade (teste MTT) contra células de melanoma ou células endoteliais. Em termos de absorção celular, as imuno-nanopartículas funcionalizadas com MART1, um anticorpo específico para o reconhecimento do antígeno super-expresso em células de melanoma, foi aumentada em 40 a 50% em comparação com o controle. A segunda parte desta tese foi dedicada ao desenvolvimento, caracterização e avaliação in vivo da atividade de cicatrização da β-lapachona encapsulada em lipossomas multilamelares e incorporada em um hidrogel de um biopolímero produzido pela bactéria Zoogloea sp. Estas formulações originais (βlap-Lipo/ZBP/HEC) apresentaram um pH e comportamento reológico adequados para aplicação tópica, bem como a capacidade de retardar a liberação de β-lapachona do hidrogel. Um estudo histopatológico detalhado da atividade de cicatrização de feridas foi conduzido em um modelo in vivo e mostrou que o hidrogel de biopolímero foi capaz de estimular o reparo tecidual, aumentando a celularidade local, fibroblastos, células inflamatórias, vasos sanguíneos e a produção de fibrilas de colágeno durante a fase proliferativa. Além disso, a formulação βlap-Lipo/ZBP/HEC promoveu angiogênese local e reduziu a inflamação da ferida, demonstrando o potencial desta formulação original de β-lap-Lipo/ZBP/HEC na terapia de lesões cutâneas. Para concluir, os nanocarreadores desenvolvidos demonstraram ser abordagens interessantes para interceptar as células de melanoma circulantes, enquanto as formulações lipossomais combinada com biopolímero têm um potencial interessante para aplicações de cicatrização de feridas.

There are many things to be grateful at this moment and the conclusion of my PhD certainly will be a big landmark in my life. When I was a child I had the willness to be a scientist, discovery new things and travel the world changing experiences and knownlegde. Nevertheless, the reality was not that simple as seemed to be. During my graduation, master's degree and PhD, many difficult tasks have come into my way, impelling me to be strong, resilient and hardworking.

On the other hand, great moments of joy, reciprocity and passion also accompanied me on this journey. Naturally, there is still a long path to go towards the construction of scientific knowledge.

I would like to thank the wonderful people who crossed my path and who have helped me to achieve my goals and always encouraging me to be the best version of me. First, I will be eternally grateful to my family, my parents and brothers, for supporting me in all moments. We are a very close-knit family, full of love and compasion. An specially thank for my father who is my mentor and main supporter.

What would life be without friends as well? So, my special thank will be to my childhood friends that knows me more than myself: Elisa, Duda, Tatiane, Rivanda, Pinelli, Renata Lima, Camila Farias and Marcela. Thank you also to my college friends: Amanda, Larissa Morgana, Giovanna, Dea, Rebecca, Vinicius e Fernanda. Still talking about great friends, but now the new ones, I really must thanks to my colleagues from Maison du Brésil for all the friendship and support during my stay in Paris: Taty, Bárbara, Priscila, Renata, Nayara, Rodrigo, Igor, Júlia, Gustavo, Fabiano etc. Thank you very much for being part of one of the great moments of my life.

I would like to greatly thanks to my supervisors and co-superviors: Nereide Stela Santos Magalhães, Gilles Ponchel and Christine Vauthier for provide amazing opportunities of knowledge and for supporting me in all my needs and requirements. You are for me the biggest examples of competence, intelligence and resilience. I would like to express my huge gratitude for my teammates from Brasil and France, that make this journey less labored and more joyful.

Any Taylor (my mexican sister) and Mariane Lira (my roomate), I have no words to thank you girls for all your support and friendship, you were a key part of my development as a person and professional. Any, thank you for teaching me and helping me to construct this thesis. Isabella Macário, Rafaela Ferraz, Marília Evelynn, Daniel Charles, Jean Baptist Coty, Francisco Junior Xavier, thank you for all the priceless contribuitions in this work and for trusted and believed in me. Finally, but not least, thanks for all my SLC's (Milena, Marcela, Dani, Vanderval, Clara, Victor etc) and Institut Galien/Paris-sud teammates (André, Henrique, Cassiana, Gabriela, Andreza etc) and professors (Juliette Verganaud, Kawtar Bauchemal, Valerie Nicolas etc) that helped me enormously with all knowledge and friendship.

"If you are distressed by anything external, the pain is not due to the thing itself but to your own estimate of it; and this you have the power to revoke at any moment."

Marcus Aurelius Antoninus

GENERAL INTRODUCTION

The nanotechnology has been widely applied in diagnosis and treatment of several diseases [START_REF] Angeli | Nanotechnology applications in medicine[END_REF]. Nanotechnology applications in biomedical field, including vaccination, diagnostics and drug-delivery [START_REF] Boulaiz | Nanomedicine: Application areas and development prospects[END_REF][START_REF] Chowdhury | Cancer nanotheranostics: Strategies, promises and impediments[END_REF][START_REF] Reed | Vaccination with melanoma helper peptides induces antibody responses associated with improved overall survival[END_REF]. In pharmaceutical field, the nanocarriers has been used to improve the efficacy of therapeutic and/or diagnosis agents and to In this context, one of the cancer types with greatest metastatic potential and chemotherapeutic resistance is the melanoma. This tumor is characterized as the malignant transformation of melanocytes of neural crest origin and it is considered the deadliest skin cancer, with a 5-year-survival rate for distant melanoma metastasis of less than 20% (ALBINO et al., 1992;CHEN, et al., 2013). Owing to the clinical relevance of this skin cancer, the melanoma treatment has been widely explored for nanotechnology applications, including the use of polymeric nanoparticles (ANTÔNIO et al., 2017;JAIN;THANKI;JAIN, 2013;[START_REF] Li | Nanobiotechnology for the Therapeutic Targeting of Cancer Cells in Blood[END_REF].

The first section of this thesis was dedicated to the nanotechnological approaches for metastatic melanoma treatment. The first chapter consisted in a literature review about the role of CSCs and CTCs in the development of metastatic melanoma; the current biomarkers of these distinct population of melanoma cells and the advances in polymer based nanoparticles for the metastatic melanoma treatment. In this perspective, the influence of architectural properties of polymeric nanoparticles in the effectiveness of passive and active-melanoma targeting were widely discussed. Thus, the goal of this review was to present and discuss the up-to-date status of melanoma biomarkers and to evaluate the advances in polymeric nanoparticles strategies in order to develop effective drug delivery systems for the treatment of metastatic melanoma.

Thereafter, the second chapter comprised of the development and in vitro evaluation of polymeric nanoparticles for targeting melanoma cells. In this study, the melanoma antigen recognized by T-cells (MART-1) was selected as a target, due to this overexpression in melanoma cell lines (TAZZARI et al., 2015). Immunonanoparticles, based on poly (γ-benzyl-L-glutamate) (PBLG) and coupled with antibody against MART-1, were developed, characterized and tested in vitro on melanoma and endothelial lineages.

The use of nanotechnology for wound healing treatment also offers a number of advantages when compared to the conventional cutaneous therapies, such as occlusive dressings (JACKSON; KOPECKI; COWIN, 2013). These advantages comprise on the protection of active principles, enhanced drug penetration, promotion of localized drug effects and reduced unwanted systemic absorption (CARNEIRO et al., 2010). In this context, nanoemulsions and liposomes are the most utilized nanocarriers for cutaneous applications (WU; GUY, 2009).

Liposomes are spherical vesicles formed by one or more concentric phospholipid bilayers with an aqueous core. The main advantage of this nanocarrier is the ability to encapsulate lipophilic, amphiphilic and hydrophilic substances, due to their biphasic character (TORCHILIN, 2005). Lipophilic substances, such as β-lapachone, a naphthoquinone that presents important biological properties including wound healing, can be encapsulated into liposomes with high drug loads (CAVALCANTI et al., 2015;[START_REF] Fu | ß-Lapachone Accelerates the Recovery of Burn-Wound Skin[END_REF]KUNG et al., 2008). Nevertheless, the application of liposomal dispersion directly to the skin is limited specially due to their low viscosity characteristic (MOURTAS et al., 2008). In this way, the use of thickening agents, including hydrogels, can improve the liposomes rheological properties for topical drug-delivery and also exhibit biological properties (CIOBANU et al., 2014;MOURTAS et al., 2007).

Thus, the second section of this thesis aimed to develop a liposomal-hydrogel containing β-lapachone for wound healing applications. The liposomal-hydrogel formulation consisted of β-lapachone-loaded multilamellar liposomes incorporated in a polymeric blend, containing a bacterial cellulose hydrogel produced by Zoogloea sp. Both bacterial cellulose and β-lapachone are expected to have wound healing properties. This study evaluated the in vitro kinetics and rheological properties of the liposomal-hydrogels containing β-lapachone, as well as their in vivo wound healing activity.

In general, this thesis aimed to contribute for the development of polymeric and lipidic nanocarriers with different biological applications and administration routes, for instance systemic treatment of melanoma and topical action in wound healing.

The first section of this thesis was dedicated to site-specific polymeric nanoparticles for melanoma treatment. In the first chapter, it was conducted an overview of recent literature about polymeric nanoparticles for the therapeutic targeting of melanoma cells (m-CSCs and CMCs) with high metastatic potential. Based on literature review, it is clear the key role of these cells at the physiopathology of metastasis. The reported biomarkers expressed in m-CSCs and CMCs can be used as potential targets of site-specific theranostic nanoparticles for melanoma. In addition, the optimal design of polymeric nanoparticles for passive and active tumor targeting were also discussed take into account the size, shape and surface properties of the nanoparticles.

In the second chapter, surface modified polymeric nanoparticles (stealth, fluorescent label and site-specific) based on PBLG-derivatives were prepared and characterized. The nanoparticles showed small sizes, homogenous population, slightly elongated shapes and negative surface potential. Besides, the PBLG-based nanoparticles and immunonanoparticles were not recognized by the complement system and were not cytotoxic for normal endothelial or melanoma cells. In addition, the developed immunonanoparticles containing MART-1 antibody (PBLG-PEG-Bt-MART-1) showed a specific cellular uptake by B16-GFP cells, that overexpress the MART-1 receptor.

However, further studies should be performed in order to optimize these immunonanoparticles and to enhance their specific recognition by melanoma cells, for example, increasing the antibody ratio conjugated at the surface of nanoparticles, as well as the incorporation of the drugs into these particles to increase the cytotoxicity against melanoma cells. In summary, these studies highlighted the potential of PBLG-based nanoparticles to be used as systemic drug-delivery systems for melanoma targeting approach. As discussed in the literature review, other promising antibodies against CMC and/or m-CSCs biomarkers can be essayed for melanoma therapeutic and diagnosis purposes and the PBLG-based nanoparticles can be used as versatile and modifiable nanoplatforms for this aim.

On the second section of this thesis, a topical polymeric hydrogel containing βlap-loaded multilamellar liposomes (β-lap-Lipo/ZBP-HEC) was developed for in vivo wound healing application. The liposomal-hydrogels showed suitable sizes, pH and rheological characteristics for topical applications. In addition, the release kinetic profile of β-lap from liposomal-hydrogels at the wound site followed the Higuchi model and was 1.9 folds slower than β-lap released from hydrogel. ZBP/HEC hydrogel enhanced the in vivo wound healing activity, increasing the density of specific cells involved in the wound repair. In addition, β-lap-Lipo/ZBP-HEC increased around 2-folds the formation of new vessels at wound site and decreased the inflammatory process during the proliferative phase of skin repair. These results suggest that β-lap-Lipo and ZBP/HEC hydrogel had a synergic effect. These promising findings also shed light to future researches using the ZBP/HEC hydrogel as a bioactive vehicle for incorporation of other drugs with topical actions.

Taking into account the overall results, this present work contributed for the development of promising polymeric and lipidic nanocarriers with different biological applications and administration routes: immunonanoparticles containing antibody targeted for MART-1 receptor for systemic treatment of melanoma and β-lapachone encapsulated in multilamellar liposomes and incorporated in a biopolymer hydrogel for topical application in wound healing.

Les nanotechnologies permettent de concevoir les outils thérapeutiques nouveaux, conçus pour résoudre les différents problèmes de nature physico-chimique ou biologique qui surviennent lors de l'administration des molécules actives. Les pathologies concernées sont très variées. Toutefois, au cours de la dernière décennie, c'est certainement en cancérologie que les efforts ont été les plus importants (BRYS et al., 2016 ;[START_REF] Sutradhar | Nanotechnology in Cancer Drug Delivery and Selective Targeting[END_REF] et al., 1992 ;CHEN, et al., 2013). En raison de la pertinence clinique de ce cancer de la peau, le traitement du mélanome a été largement étudié pour des applications de la nanotechnologie, y compris l'utilisation de nanoparticules polymériques (ANTÔNIO et al., 2017 ;JAIN ;THANKI;JAIN, 2013;[START_REF] Li | Nanobiotechnology for the Therapeutic Targeting of Cancer Cells in Blood[END_REF].

La première partie de cette thèse a été consacrée aux approches Cette naphtoquinone possède un caractère lipophile favorisant son encapsulation dans des liposomes avec des charges élevées (CAVALCANTI et al., 2015 ;[START_REF] Fu | ß-Lapachone Accelerates the Recovery of Burn-Wound Skin[END_REF]KUNG et al., 2008). Cependant, l'application des dispersions liposomales directement sur la peau n'est pas optimale en raison de leur faible viscosité (MOURTAS et al., 2008).

Pour cette raison, la dispersion des liposomes au sein d' hydrogels permet d'atteindre des caractéristiques rhéologiques favorables à leur administration topique. Simultanément, le choix judicieux des polysaccharides utilisés en tant qu'agents épaississants permet d'associer d'autres propriétés biologiques à la préparation (CIOBANU et al., 2014 ;MOURTAS et al., 2007).

Figure 1 .

 1 Figure 1. Schematic illustration of tumor site derived circulating melanoma cells (CMC)

Figure 2 .Figure 1 .

 21 Figure 2. Schematic illustration of passive targeting (a) and active targeting (b) for

Figure 2 .

 2 Figure 2. FT-IR spectra of PBLG-Bz polymerization film recorded at 0 min (a), 58 min

Figure 3 .

 3 Figure 3. FT-IR spectra of PBLG derivatives: PBLG-Bz (I), PBLG-PEG (II), PBLG-

  viii

Figure 4 .

 4 Figure 4. 1 H RMN spectra of PBLG derivates: PBLG-Bz (I), PBLG-Rhod (II), PBLG-

Figure 5 .

 5 Figure 5. MALDI-TOF MS spectra of PBLG-Rhod (a) and PBLG-Bz (b).

Figure 6 .

 6 Figure 6. TEM photographs of nanoparticles obtained from the following PBLG

Figure 7 .

 7 Figure 7. Cellular uptake of PBLG-Rhod (100%) and PBLG/PBLG-Rhod (10%)

Figure 8 .

 8 Figure 8. Western blot analysis of the remaining free antibodies found in supernatant of

Figure 9 .

 9 Figure 9. Cellular uptake of PBLG-PEG/PBLG-Rhod, PBLG-PEG-Bt/PBLG-Rhod and

Figure 10 .

 10 Figure 10. Electrophoregram peaks of complement activation for different PBLG

Figure 11 .

 11 Figure 11. Evaluation of the MART-1 expression on B16-GFP cells and HUVECs cells

Figure 12 .

 12 Figure 12. B16-GFP cells viability percentage after treatment with PBLG-derived

Figure 13 .

 13 Figure 13. HUVECs cells viability percentage after treatment with PBLG-derived

Figure 14 .Figure 1 .Figure 2 .

 1412 Figure 14. Cellular uptake of PBLG-PEG/PBLG-Rhod, PBLG-PEG-Bt/PBLG-Rhod and

Figure 3 .

 3 Figure 3. The histograms of cellular densities and collagen fibers on day 3, 7 and 14 post-

Figure 4 .

 4 Figure 4. Representative histopathological images of skin in the dermis layer:

  nanotechnologiques pour le traitement du mélanome métastatique. Le premier chapitre consiste en une revue bibliographique. Pour des raisons de simplicité, elles a été divisée en deux parties. Ainsi, dans un premier temps, le rôle et l'importance des cellules souches cancéreuses du mélanome (m-CSC) et des cellules de mélanome circulantes (CMC) dans le développement de métastases ont été décrits et précisés. En particulier, les principaux biomarqueurs connus des m-CSCs et des CMC ont été recensés et présentés sous l'angle de l'importance clinique de ces cellules, de leur utilisation en tant que biomarqueurs pour le diagnostic, le pronostic et le traitement du mélanome. Cette analyse a également examiné le modèle d'expression des m-CSC et des biomarqueurs CMC, leurs fonctions biologiques et les mécanismes de résistance de ces cellules. La deuxième partie de cette analyse a été consacrée à décrire et discuter les approches nanotechnologique actuelles ayant pour objectif de cibler les cellules de mélanome, en mettant l'accent sur les études portant sur les nanoparticules polymériques. Les stratégies de ciblage passives et actives développées ont été présentées et les caractéristiques des nanoparticules qui influencent l'efficacité de ciblage de m-CSCs et CMC a été discutée. Ainsi, l'objectif de cette analyse était de présenter et de discuter l'état actuel des connaissances des biomarqueurs du mélanome et d'évaluer les avantages et les inconvénients des nanoparticules polymériques proposés à ce jour. Elle a permis d'identifier quelles devraient être les principales caractéristiques désirables de nanomédecines efficaces en vue du traitement du mélanome métastatique. Le deuxième chapitre a été consacré au développement rationnel et à la caractérisation des nanoparticules polymères constituées par l'autoassemblage de poly(gamma-L-benzyle glutamate) (PBLG) fonctionnalisées pour le ciblage des cellules du mélanome et spécialement les cellules tumorales circulantes. Ce chapitre expérimental présente le développement et la caractérisation d'immuno-nanoparticules de PBLG obtenues par le greffage en surface d'un anticorps dirigés contre l'antigène MART-1 du mélanome, reconnu par les anticorps de lymphocytes T (MART-1). L'antigène MART-1 a été sélectionné comme cible en raison de sa surexpression dans les cellules du mélanome (TAZZARI et al., 2015) et de l'existence de lignées cellulaires permeettant l'établissement de modèles in vitro et in vivo chez l'animal.

  Ainsi, le troisième chapitre décrit l'élaborationet la caractérisation d'un système liposomes-hydrogel contenant une molécule active, la β-lapachone (β-lap-Lipo/ZBP-HEC). Son activité cicatrisante a été établie in vivo sur un modèle animal. L'hydrogel utilisé consiste en un mélange de biopolymères, comprenant notamment une cellulose d'origine bactérienne obtenue à partir de la mélasse de la canne à sucre. Ces biopolymères ont déjà démontré leur intérêt dans plusieurs applications biomédicales, déjà décrites dans la littérature. Toutefois, ce travail apporte une nouvelle approche pour l'utilisation biomédicale de ce biopolymère produit par Zoogloea sp. puisqu'il a été utilisé ici non seulement comme hydrogel capable de moduler la libération de composés bioactifs, mais aussi pour son action cicatrisante intrinsèque. Cette étude a évalué la cinétique in vitro et les propriétés rhéologiques du liposomale-hydrogels contenant la β-lapachone, ainsi que leur cicatrisation in vivo , et le deux, la cellulose bactérienne et la β-lapachone, étaient censées avoir des propriétés de cicatrisation. Des hydrogels contenant des liposomes chargés en β-lapachone (β-lap-Lipo/ZBP-HEC) ont été préparés à partir de liposome de taille appropriée. Après dispersion, le pH et les caractéristiques rhéologiques des préparations ont été ajustées pour qu'elles soient adaptées à une application topique. L'incorporation de liposomes dans un système d'hydrogel permet de contrôler la libération du principe actif, en fonctionnant comme un réservoir de médicaments. Ainsi, les cinétiques de libération de la β-lapachone à partir des hydrogels a été établie in vivo . L'analyse des cinétiques a montré que la libération obéissait au modèle d'Higuchi et que dans le cas des systèmes β-lap-Lipo/ZBP-HEC la libération était ralentie d'un facteur 2 environ par rapport à l'hydrogel simple contentant de la β-lapachone. Des études de cicatrisation ont été conduites in vivo sur des rats Wistar. L'analyse histologique des tissus a permis de montrer que les animaux ayant reçu seulement l'hydrogel ZBP/HEC présentaient une augmentation de la densité des cellules spécifiques impliqués dans la cicatrisation de la plaie. En revanche, les animaux appartenant aux groupes traités avec le système β-lap-Lipo/ZBP-HEC (hydrogels contenant les liposomes) ont présenté une augmentation d'un facteur 2 de la formation de nouveaux vaisseaux dans la zone de cicatrisation et ont simultanément diminué le processus inflammatoire au cours de la phase proliférative de réparation de la peau. Pris simultanément, ces résultats suggèrent que les liposomes de β-lapachone (βlap-Lipo) d'une part, et l'hydrogel ZBP/HEC, d'autre part ont eu un effet synergique. Ces résultats prometteurs ouvrent aussi la voie à de futures recherches utilisant l'hydrogel ZBP/HEC comme un transporteur bioactif auquel d'autres molécules à action topique pourraient être associées, de façon à obtenir des effets synergiques intéressants. En conclusion, ce travail doctoral a permis de montrer, dans deux pathologies différentes, l'intérêt de concevoir les formulations à l'échelle nanométrique, notamment des immuno-nanoparticules de PBLG fonctionnalisées par des anticorps dirigés contre MART-1 en vue de l'interception des cellules tumorales circulantes (CTCs) du mélanome dans le sang et/ou le traitement systémique du mélanome et des liposomes multilamellaires encapsulant de la β-lapachone et dispersés dans un hydrogel de biopolymères destinés à favoriser la cicatrisation des plaies cutanées.

Table 1 .

 1 Current biomarkers detected in m-CSC and CMC

Table 2 .

 2 In vivo and in vitro studies with drugs-loaded polymeric nanoparticles for passive and active tumor targeting in advanced melanoma treatment.

	CHAPTER II

Table 1 .

 1 Characteristics of PBLG derivatives.

Table 2 .

 2 Morphological analysis and zeta potential of nanoparticles.

	xii

  . Dans ce domaine, la principale stratégie développée a consisté à utiliser une très grande variété de systèmes de taille nanométrique conçu pour véhiculer plus les molécules antitumorales vers les cellules cancéreuses, plus spécifiquement que ne peuvent le faire les molécules actives seules, afin d'augmenter l'index thérapeutique de ces molécules. Toutefois, cette stratégie ne permet pas de déjouer tous les obstacles. Ainsi, l'hétérogénéité et la plasticité cellulaire au sein des tumeurs aboutissent à la dissémination des métastases dans des organes distincts que les nanomédecines, conçues pour atteindre les tumeurs primaires, ne peuvent pas rejoindre efficacement. les CTCs comme étant des biomarqueurs intéressant. Bien que leur détection représente un véritable défi, en raison de leur rareté, un nombre croissant de techniques est actuellement développé.Dans ce contexte, nous nous sommes focalisés sur le mélanome qui présente un fort potentiel métastatique et fait souvent l'objet de résistance aux traitements chimiothérapeutiques. Le mélanome provient de la transformation maligne de mélanocytes cutanés. Il s'agit d'un type de cancer particulièrement meurtrier, avec un taux de survie de 5 ans pour des métastases de mélanome inférieures à 20 % (ALBINO

	De manière générale, le traitement des métastases est considéré comme étant un
	défi majeur pour l'amélioration des traitements du cancer (BROOKS et al., 2015). Plus
	précisément, deux types de cellules cancéreuses sont directement impliqués dans
	l'hétérogénéité tumorale et le développement des métastases : les cellules souches
	cancereuses (CSCs) et les cellules tumorales circulantes (CTCs). Les CSCs représentent
	une quantité minoritaire de cellules dans la tumeur mais qui ont la particularité d'être
	pluripotentes et de posséder une forte capacité métastatique (BROOKS ; BURNESS ;
	WICHA, 2015 ; CSERMELY et al., 2014). A côté de ces cellules, les CTCs sont des
	cellules tumorales propagées par le sang et/ou la circulation lymphatiques (XU ; ZHONG,
	2010), capables de s'implanter dans un organe distant de la tumeur primaire. Des niveaux

élevés de CSCs et CTCs ont été associés à la progression tumorale, à la chimiorésistance et aux métastases

(LA PORTA ; ZAPPERI, 2013; PORES et al., 2016)

, ce qui a conduit récemment à considérer

ACKNOWLEDGMENTS


Leonardo Pejsachowicz
email: leonardo.pejsachowicz@polytechnique.edu.

Stochastic Independence under Knightian Uncertainty

Keywords: Bewley preferences, stochastic independence, product equivalents. JEL classification: D81. Titre en Francais: Independence Stochastique et incertitude à la préférences à la Bewley, indépendance stochastique, équivalent produit. Classification JEL: D81

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

The distinction introduced by Knight (1921) between risky events, to which a probability can be assigned, and uncertain ones, whose likelihoods are not precisely determined, cannot be captured by the standard subjective expected utility (SEU) model. This paradigm in fact posits a unique probability distribution over the states of the world, the agent's prior, that is used to assign weights to each contingency when evaluating a given course of action.

Of the many models that have been proposed to accommodate Knightian uncertainty, the one pioneered in [START_REF] Bewley | Knightian Uncertainty Theory: Part I[END_REF] has recently been proved very useful, both in economic applications of uncertainty1 and as a foundational tool for the systematic analysis of non-SEU preferences. 2 [START_REF] Bewley | Knightian Uncertainty Theory: Part I[END_REF] allows the agent to hold a set of priors. Choices are then performed using the unanimity rule: an action will be preferred to an alternative one only if its expected utility under each prior of the agent is higher. Since the rank of two options might be reversed when considering different priors, under this paradigm the agent's preferences will typically be incomplete.

The introduction of this type of incompleteness raises a slew of interesting modeling questions, but the one we will be concentrating on in this paper regards the characterization of stochastic independence (from now on s-independence). Specifically, suppose a Bewley decision maker must choose between bets that depend on two different experiments, for example the tosses of two coins. When can we say, based on the observation of his choices, that he considers the two tosses independent? An answer to this question is clearly of great interest, both for applications of the Bewley model to game theory, since independence of beliefs is a central tenet of Nash equilibrium, and as a benchmark for the development of a theory of updating, which is essential in applications to dynamic environments.

In the SEU model, s-independence is captured by the intuitive idea that the preferences of an agent over bets that depend only on one of the tosses should not change when he receives information about the other (see [START_REF] Blume | Lexicographic Probabilities and Choice under Uncertainty[END_REF]), a property we dub conditional invariance. As we show in an example in Section 2.2, such requirement is unable to eliminate all of the forms of correlation between experiments that the multiplicity of priors in the Bewley model introduces. As a consequence under conditional invariance an agent's preferences are no longer uniquely determined by his marginal beliefs, which we argue is essential for a useful definition of s-independence. To overcome this problem we introduce the idea of product equivalent of an act, which is close in spirit to that of certainty equivalent of a risky lottery, although adapted to the product structure of the state space. We show that if a Bewley decision maker treats product equivalents as if they where certainty equivalents, his set of beliefs must coincide with the closed convex hull of the pairwise product of its marginals over each experiment.

An important consequence of our result is that it provides a characterization of sindipendence for MAxMin preferences that coincides with the definition proposed in [START_REF] Gilboa | Objective and Subjective Rationality in a Multi-Prior Model[END_REF]. Such definition has been applied in the characterization of independent beliefs for Nash equilibria under ambiguity, for example in [START_REF] Lo | Equilibrium in Beliefs under Uncertainty[END_REF] and more recently in [START_REF] Riedel | Ellsberg Games[END_REF]. Neverthelss, as [START_REF] Lo | Correlated Nash equilibrium[END_REF] complains, the behavioral implications of Gilboa and Schmeidler's definition are still poorly understood. It is our hope that the present work will provide a first step towards a clarification of these implications. [START_REF] Blume | Lexicographic Probabilities and Choice under Uncertainty[END_REF] are the first to provide a decision theoretic axiom for s-independence in the SEU model, based on the insight of conditional invariance, of which we use a stronger version in Section 2.3. Bewley provides an early definition of s-independence for his model in the original 1986 paper, though he gives no behavioral characterization. His definition is weaker than the model we obtain. The remaining related literature is mostly concerned with other non-SEU models. [START_REF] Gilboa | Objective and Subjective Rationality in a Multi-Prior Model[END_REF], as we already said, define a concept of independent product of relations for MaxMin preferences and characterize it, though their characterization uses directly the representation instead of the primitive preference. [START_REF] Klibanoff | Stochastically Independent Randomization and Uncertainty Aversion[END_REF] gives a definition of an independent randomization device, which he uses to evaluate different types of uncertainty averse preferences in the Savage setting. [START_REF] Bade | Stochastic Independence with Maxmin Expected Utilities[END_REF] explores various possible forms of s-independence for events under the MaxMin model of Gilboa and Schmeidler, providing successively stronger definitions. Bade (2011) contains a characterization of s-independence for general uncertainty averse preferences that is particularly useful for the way in which the paper introduces uncertainty in games. [START_REF] Ghirardato | On Independence for Non-Additive Measures with a Fubini Theorem[END_REF] studies products of capacities, and proposes a restriction on admissible products based on the Fubini theorem. We share with this paper the intuition of using the iterated integral property to characterize product structures outside of the standard model. Finally, [START_REF] Epstein | Symmetry of Evidence without Evidence of Symmetry[END_REF], who study alternative versions of the De Finetti theorem for MaxMin preferences, provide an axiom, dubbed orthogonal independence, which achieves a weaker form of separation in beliefs.

The rest of the paper is organized as follows: Section 2 introduces the model, provides a motivating example and discusses the limits of conditional invariance as a characterization of s-independence. In Section 3 we define product equivalents and give our main characterization result, and some further corollaries, one of which is the afore mentioned characterization of MaxMin s-independence. Section 4 concludes with a brief discussion on the quality of our main assumption.

Preliminaries

We consider a finite state space Ω with a product structure Ω = X × Y endowed with an algebra of events Σ which, for simplicity of exposition, we assume through the paper to coincide with 2 Ω . Notice that the collections Σ

X = {A × Y | A ⊆ X} and Σ Y = {X × B | B ⊆ Y } are proper sub-algebras of Σ under the convention ∅ × Y = X × ∅ = ∅.
States, elements of Ω, are denoted ω or alternatively through their components (x, y). Elements of X and Y represent the outcomes of two separate experiments. Sets of the form {x} × Y , which we call X-states, will be indicated, with abuse of notation, x, and a similar convention applies to Y-states.

A prior p is an element of (Ω), the unit simplex in R Ω . For a subset S of Ω we let p(S) = ω∈S p(ω). Let p X ∈ (X) and p Y ∈ (Y ) be the marginals of p over X and Y respectively. The notation p X ×p Y indicates the product prior in (Ω) uniquely identified by p X × p Y (x, y) = p X (x)p Y (y). Prior p has a product structure if p = p X × p Y . For a set P ⊆ (Ω), we let P X and P Y stand for the sets of marginals of elements of P over each experiment. The set of pairwise products of P X and P

Y , namely {p X × p Y | (p X , p Y ) ∈ P X × P Y } is denoted P X ⊗ P Y .
We work with a streamlined version of the Anscombe-Aumann model. An act is a map from Ω to K ⊆ R, a non-trivial interval of the real line. The set of all acts is F = K Ω with generic elements f and g. The interpretation of f is that it represents an action that delivers (utility) value f (ω) if state ω is realized. For every λ ∈ (0, 1) and f, g ∈ F, the mixture λf + (1 -λ)g is the state-wise convex combination of the acts. 3 For any finite set S, let 1 S be the indicator function of S. The constant act k1 Ω that delivers k ∈ K in every state of the world is denoted, with abuse of notation, k. We say that f is an X-act if f (x, y) = f (x, y ) for all y, y ∈ Y , namely if f is constant across the realizations of Y -states. The set of X-acts is F X with generic elements f X , g X . The unique value that act f X ∈ F X assumes over {x} × Y is indicated f X (x). We can then see f X as the sum x∈X f X (x)1 x . Similar considerations and notation apply to Y -acts.

Bewley preferences

Our primitive is a reflexive and transitive binary relation -a preference -on F. Through most of the paper we assume that there exist a non-empty convex and closed set P ⊆ (Ω) such that f g if and only if ω∈Ω f (ω)p(ω) ≥ ω∈Ω g(ω)p(ω) for all p ∈ P.

(

We say then that has a (non-trivial) Bewley representation or that it is a Bewley preference. 4 An axiomatic characterization of (2.1) can be found in [START_REF] Gilboa | Objective and Subjective Rationality in a Multi-Prior Model[END_REF]. 5The set P above is uniquely determined by . Because P is the only free parameter in the model, we say that P represents when it satisfies (2.1). We note that any two subsets of (Ω) induce the same Bewley preference via (2.1) as long as their closed convex hulls, which we denote co(P ) for a generic P , coincide. The SEU model corresponds to P = {p}, in which case is complete. Hence Bewley preferences are a generalization of SEU in which completeness is relaxed. At the same time, any extension of a Bewley preference to a complete relation over acts that is SEU corresponds to some prior p in 3 The classical Anscombe-Aumann environment posits an abstract consequence space C and defines acts as maps from Ω to the set s (C) of simple lotteries over C. One then goes on to show that, under standard assumptions (in particular Risk Independence, Monotonicity and Archimedean Continuity), there exists a Von Neumann-Morgenstern utility U : s (C) → R such that two acts f and g are indifferent whenever U (f (ω)) = U (g(ω)) for all ω. Thus one can think of our approach as one that considers acts already in their "utility space" representation. 4 In the model of [START_REF] Bewley | Knightian Uncertainty Theory: Part I[END_REF], preferences satisfy a version of equation (2.1) in which and ≥ are replaced by their strict counterparts and >. The two models are close but distinct, and correspond to the weak and strong versions of the Pareto ranking in which different priors take the role of different agents.

its representing set of priors P . We will say that is a SEU preference whenever it has a Bewley representation with a singleton set {p} of priors.

A motivating example

In this section we illustrate the need for a novel characterization of s-independence with a simple example. Consider an agent who is betting on the results of the tosses of two different coins. All he knows about these is that they have been coined by two separate machines, each of which produces either a coin that comes up heads α% of the times, or one that comes up heads β% of the times. The two machines have no connection to each other, and no information on the mechanism that sets the probability of heads in either machine is given, so that no unique probabilistic prior can be formed. Given this description, it seems agreeable that a Bewley decision maker, facing acts on the state space {H 1 , T 1 }× {H 2 , T 2 } (where H i corresponds to the i-th coin coming up heads), would consider the set of priors

P 1 = {p α × p α , p β × p α , p α × p β , p β × p β } where p α = (α, 1 -α) ∈ ({H 1 , T 1 }) and p β = (β, 1 -β) ∈ ({H 2 , T 2 }).
Does P 1 reflect an intuitve notion of independence between the tosses? One way to answer this question is to ask our agent to compare a particular type of acts which we will call, for lack of a better term, conditonal bets. Namely, assume we ask the agent to decide between bets f 1 and g 1 , where

f 1 H 2 T 2 H 1 1 0 T 1 k k g 1 H 2 T 2 H 1 0 1 T 1 k k
Both f 1 and g 1 pay the same amount k if the first coin turns up tails, and provide opposing bets on the second toss if the first turns up heads. Whichever ranking the agent provides, it stands to reason that, if he treats the tosses as independent, he should rank in the same way the acts f 2 and g 2 in which the opposing bets on the second coin are provided conditional on the first coming up tails, namely:

f 2 H 2 T 2 H 1 k k T 1 1 0 g 2 H 2 T 2 H 1 k k T 1 0 1
This form of invariance of the preferences over one experiment to information on the result of the other can be shown to characterize, for the SEU model, an agent whose unique prior p has a product structure. We notice that P 1 satisfies this invariance with regards to the pairs f 1 , g 1 and f 2 , g 2 . In fact if f 1 g 1 then we must have

α 2 ≥ α(1 -α) and βα ≥ β(1 -α) ⇔ α ≥ (1 -α)
for p α × p α and p β × p α , and also

αβ ≥ α(1 -β) and β 2 ≥ β(1 -β) ⇔ β ≥ (1 -β)
for p α × p β and p β × p β . It is immediate to check that the same two conditions ensure f 2 g 2 . This is in line with our intuition that the situation described above, and its related set of priors P 1 , reflect a natural notion of independence between tosses.

But now imagine the agent comes to learn that the two machines have a common switch. This switch is the one that decides whether the coins produced will be of the α or β variety, thus whenever the first machine produces a coin of a certain kind so does the other. Here the natural set of priors is

P 2 = {p α × p α , p β × p β }.
The preferences induced by this set also satisfy the invariance we discussed between pairs f 1 , g 1 and f 2 , g 2 . In fact f 1 g 1 if and only if α ≥ (1 -α) and β ≥ (1 -β), which also implies f 2 g 2 . Nevertheless we would be hard pressed to argue that this situation reflects the same degree of independence of the first. The priors in P 2 in fact contain information about a certain kind of correlation between the tosses. This correlation, which is novel, regards the mechanism that determines the probabilistic model assigned to each coin.

As we will see in the next section, the conditional invariance requirement we loosely described is unable, even in its strongest form, to eliminate this sort of correlation. One can understand this failure as stemming from the lack of aggregation that is characteristic of the Bewley model. Correlations in the mechanism that selects models for each toss are reflected in the shape of the whole set of priors. On the other side, when a Bewley decision maker compares two acts, he uses priors one by one, hence only the structure of each single distribution comes into play.

Conditional Invariance

Here we formalize and extend the discussion of the previous section. Before we do so, we will need an additional definition:

Definition 1. An event S ⊆ Ω is -non-null if k > l implies k1 S + l1 Ω\S l for any k, l ∈ K.
The definition corresponds to that of a Savage non-null set. For a Bewley preference represented by P , a set S is -non-null if and only if p(S) > 0 for some p ∈ P . Now consider the following axiom:

Conditional Invariance For all acts h ∈ F, f X , g X ∈ F X and any pair of -non-null events R, S ∈ Σ Y ,

f X 1 R + h1 Ω\R g X 1 R + h1 Ω\R =⇒ f X 1 S + h1 Ω\S g X 1 S + h1 Ω\S (2.2)
and the same holds when we switch the roles of X and Y in the above statement.

This is a stronger version of the Stochastic Independence Axiom (Axiom 6) of [START_REF] Blume | Lexicographic Probabilities and Choice under Uncertainty[END_REF]. 6 If we let X = Y = {H, T } with Y representing the first coin toss and X the second, setting R = {H, T } × {H}, S = {H, T } × {T } and choosing f X , g X and h equal to

f X H T H 1 0 T 1 0 g X H T H 0 1 T 0 1 h H T H k k T k k we obtain from (2.2) the implication f 1 g 1 =⇒ f 2 g 2 .
Hence acts of the form f X 1 R + h1 Ω\R are the general version of the conditional bets we discussed in Section 2.2 and carry the same interpretation.

The next proposition highlights the limits of Conditional Invariance as a behavioral characterization of s-independence: Proposition 1. For any P ⊆ (X) ⊗ (Y ), the Bewley preference induced by co(P ) satisfies Conditional Invariance.

Proof: See the Appendix.

Thus the type of correlations embodied by a set such as P 2 from the previous section are in general not excluded by Conditional Invariance. More than that, the degree of non-uniqueness that the assumption allows for in the formation of priors is problematic for any definition of s-independence.

To see this, consider for a moment an agent with SEU preferences . Suppose we elicit his information on each separate experiment by "asking him questions", i.e. proposing him comparisons of acts, that depend either only on X or only on Y . His answers correspond to the restrictions X and Y of to F X and F Y respectively. By the SEU representation theorem, we know that these are uniquely determined by the marginals p X and p Y of his prior. Now if his prior p has a product structure (which in this case, as we show below, is true if and only if satisfies Conditional Invariance), the inverse is also true. Namely, because in this case p = p X ×p Y , we can uniquely determine his preferences over F, and hence his information about the whole set of possible results in X × Y , using X and Y . Thus once we learned about each experiment in isolation we know all that can be known about the pair, a key aspect of the description of s-independence in a single prior environment.

Turning our attention to the Bewley case, we find that the information on each experiment is now subsumed into the sets P X and P Y of marginals, which uniquely determine X an Y . But now suppose we take two sets of priors P and Q inside (X) × (Y ) whose marginals coincide with P X and P Y and such that co(P ) = co(Q) (P 1 and P 2 from Section 2.2 are one such pair, for P X = P Y = {p α , p β } ). Proposition 1 ensures that the preferences induced by P and Q satisfy Conditional Invariance, and by our assumption their restrictions to F X and F Y coincide. But the two preferences will differ, by the uniqueness part of the Bewley representation theorem, hence the condition that characterizes s-independence under SEU does not allow us to uniquely determine the agent's global preferences from X and Y . The information we are lacking is precisely the one on correlations in the mechanism that matches priors on one experiment to priors on the other. To recover the desired degree of uniqueness, we propose in the next section a stronger requirement on preferences.

Stochastic Independence via product equivalents

In this section we propose a new concept, that of the product equivalent of an act, and use it to give a characterization of s-independence for Bewley preferences. In order to illustrate the logic behind product equivalents we first introduce them in the simpler SEU environment.

Product equivalents under SEU

Throughout this section we will consider a Bewley decision maker whose representing set of priors P is a singleton. Thus his preferences are complete and he is a subjective expected utility maximizer. In this case it can be shown7 that every act f ∈ F has a certainty equivalent, namely that there exists some constant act k such that f ∼ k. We denote such act ce(f ). Now notice that any act f ∈ F can be seen as a collection of bets on Y delivered conditional on the outcomes in X. Namely, we can find a collection {f x Y } x∈X of Y -acts, uniquely identified by

f x Y (y) = f (x, y), such that f = x∈X f x Y 1
x . This particular way of seeing an act suggest the following definition, which is partly inspired by that of certainty equivalent:

Definition 2. An act f X ∈ F X is the X-product equivalent of f ∈ F, denoted pe X (f ), if for all x ∈ X f X (x) = ce(f x Y ).
Notice that pe X (f ) need not be indifferent to f . 8 This is intuitive, since evaluating pe X (f ) requires a different thought process than the one used for f , in which first the value of each conditional bet the act induces on Y is determined in isolation, and then these are aggregated using the information the agent has over X. Nevertheless we would think it is precisely when X and Y are independent, and hence information about the aggregate value of f is completely embedded in the agents preferences over each individual experiment, that the two approaches will lead to the same result, and consequently f ∼ pe X (f ). The next theorem vindicates such view: Theorem 1. Let be a SEU preference over F represented by {p}. Then the following are equivalent 1) There are distributions p X ∈ (X) and p Y ∈ (Y ) such that p = p X × p Y .

2)

satisfies Conditional Invariance.

3) f ∼ pe X (f ) for all f ∈ F .

Proof: See the Appendix.

1) ⇔ 2) is well known and easily proved using the uniqueness properties of the SEU representation. Since the definition of product equivalent is novel, the equivalence of 1) and 3) is a new result, although it is an elementary application of separation arguments. We can better understand this part of the result in light of Fubini's celebrated theorem. The latter gives conditions under which the integral of a function through a product measure can be obtained as an iterated integral. Now notice that when is SEU it must be that pe

X (f )(x) = y∈Y f (x, y)p Y (y) (3.1)
and hence the value of pe X (f ) is nothing but x∈X y∈Y f (x, y)p Y (y) p X (x). Thus 1) ⇒ 3) is equivalent to (a very simple version of) the Fubini theorem, while 3) ⇒ 1) provides an inverse of that result.

Uniform product equivalents

Theorem 1 suggest an alternative route for the characterization of s-independence under Bewley preferences, one that goes through the extension of the definition of a product equivalent to the multiple-priors case. The first obstacle we find along this way is that in general even certainty equivalents of acts need not exist for a Bewley decision maker. 9 In order to sidestep this issue, Ghirardato et al. ( 2004) consider a set of constant acts, which for each f ∈ F we will denote Ce(f ), that behave as certainty equivalents do for complete preferences, namely:

Ce(f ) = {k ∈ K | c f implies c k and f d implies k d for all c, d ∈ K}. (3.2)
9 An easy geometric intuition of this fact is the following. If we look at acts in F as elements of R Ω , we can see that the indifference curves induced by an SEU preference with prior p correspond to the restriction to K Ω of the hyperplanes that are perpendicular to p. On the other side, the indifference curve through f of a Bewley decision maker with a set of priors P is given by the intersection of a set of hyperplanes, one for each p ∈ P . Obvious dimensionality considerations suggest then that in general the only act indifferent to f is f itself. For example if |Ω| = 2 and is incomplete, there are at least two non collinear priors in P , hence indifference curves are points and no two acts f = g are indifferent.

They also provide the following characterization result, which illustrates the parallel between Ce(f ) and ce(f ) under Bewley and SEU preferences: We would then hope that substituting Ce(f x Y ) for ce(f x Y ) in the definition of product equivalent would lead to a generalization that retains the intuition behind pe X (f ). Nevertheless here we stumble on a second issue. Both the parallel with Fubini's theorem and equation (3.1) suggest that for a given f , the relevant collection of X-acts in this case should be of the form

{f X ∈ F X | ∃ p Y ∈ P Y such that f X (x) = y∈Y f (x, y)p Y (y) for all x ∈ X}, (3.3)
which is the set of X-acts obtained by evaluating, for each prior model p ∈ P , the conditional bets on Y induced by f via its marginal p Y , reflecting the information in prior p about the outcomes of Y in isolation. But since Ce(f x Y ) is in general an interval, an X-act f X such that f X (x) ∈ Ce(f x Y ) for all x ∈ X need not be of the form (3.3). In fact for each x ∈ X, f X (x) might correspond to an evaluation of f x Y performed using a different p Y ∈ P Y .

Thus we turn to a more indirect approach. First, notice that from any act f X , and α ∈ (X), we can obtain the "reduction"11 of f X via α by taking the mixture x∈X α x f X (x). Looking back, once again, at the SEU setting, we can see that pe X (f ) can be alternatively identified as the unique X-act f X such that, for all elements α of (X), we have

x∈X α x f X (x) = ce( x∈X α x f x Y ).
In fact if f X = pe X (f ) the equality is always true, since

x∈X α x pe X (f )(x) = x∈X α x y∈Y f (x, y)p Y (y) = x∈X y∈Y f (x, y)α x p Y (y) = y∈Y p Y (y) x∈X α x f (x, y) = ce( x∈X α x f x Y ).
The inverse is immediately obtained taking the α's that correspond to degenerate distributions over X. Notice that the equalities above hold exactly because each pe X (f )(x) is found using the same marginal p Y , which also coincides with the marginal used to evaluate every ce(f x Y ). This motivates the following definition:

Definition 3. An act f X ∈ F X is a X-uniform product equivalent of f ∈ F if x∈X α x f X (x) ∈ Ce( x∈X α x f x Y ) (3.4)
for all α ∈ (X). The set of all such acts for given f is denoted U pe X (f ).

Armed with this, we are ready to give the main result of the paper:

Theorem 2. Let be a Bewley preference over F represented by P . Then the following are equivalent

1) P = co(P X ⊗ P Y ) 2) For all f ∈ F and f X ∈ U pe X (f ), c f ⇒ c f X and f d ⇒ f X d for all c, d ∈ K
while at the same time, for all c, d ∈ K,

c f X for all f X ∈ U pe X (f ) ⇒ c f and f X d for all f X ∈ U pe X (f ) ⇒ f d.
Proof: See the Appendix.

The property in item 2) is the requirement that X-uniform product equivalents behave as elements of Ce(f ), the generalized version of certainty equivalents of Ghirardato et al. (2004). As can be seen in item 1), it provides a characterization of s-independence that recovers the desired uniqueness, in the sense that is uniquely determined by X and Y . This is done by building the largest set of product priors consistent with P X and P Y , the set P X ⊗ P Y in which all possible matches of models consistent with X and Y are considered. Obviously, by Proposition 1, when either of the conditions hold, satisfies Conditional Invariance.

Remark: Given the theorem, we would expect that the set U pe X (f ) could be shown to coincide with (3.3). In fact this is not true, and U pe X (f ) is in general larger. The intuition is the following. The sets {f X ∈ F X | ∃p Y ∈ P Y such that f X (x) = y∈Y f (x, y)p Y (y) for all x ∈ X} are clearly convex, and hence as is well known they can be identified by the intersection of the half-spaces that contain them. This is what we are de facto doing when we define uniform products using condition (3.4), with the α's playing the role of normals to the hyperplanes defining such half-spaces. Nevertheless we can do this only up to a point, because the α's have to be positive and normalized, a restriction that binds when trying to separate sets of acts. Hence we are left with less hyperplanes than those needed to "cut out" the right set, and with a larger U pe X (f ). This does not affect the result though because the additional acts cannot be distinguished from those in {f X ∈ F X | ∃p Y ∈ P Y such that f X (x) = y∈Y f (x, y)p Y (y) for all x ∈ X} as long as we evaluate f X acts using a distribution in (X).

Independent acts and independent events

A series of modeling questions concerning s-independence do not lend themselves immediately to representation through a state space with a product structure. Here we propose an approach that leverages Theorem 2 to answer two such questions: when are two acts f, g ∈ F independent according to a Bewley preference? When does such preference consider two events A, B ⊆ Ω independent? We will start as before from a finite state space Ω, though we note that as long as we restrict our attention to simple acts (which assume a finite number of values) the whole discussion can be extended to the case |Ω| = ∞. Now assume we are given a Bewley preference over K Ω . An act f induces a finite partition Π f over Ω given by Π

= {f -1 (k) | k ∈ f [Ω]}. For any two acts f, g ∈ F, let Π f ⊗ Π g = {A ∩ B | A ∈ Π f and B ∈ Π g },
which is once again a partition of Ω. Let Σ f be the algebra generated by Π f and Σ f ×g the one generated by Π f ⊗ Π g . Finally, for a set P of priors over (Ω), let P f be the set of restrictions of elements of P to Σ f , namely:

P f = {p : Σ f → [0, 1] | ∃ p ∈ P such that p (A) = p(A) for all A ∈ Σ f }
and define similarly P f ×g for the restrictions of P to Σ f ×g . We are now ready to give the first definition of this section Definition 4. Say that acts f and g are independent according to if the set P representing satisfies co(P f ×g ) = co(P f ⊗ P g ).

Notice that when P is a singleton {p} this reduces to the usual condition that p(A ∩ B) = p(A) × p(B) for all sets A and B in the algebras generated by f and g respectively. A characterization of independent acts is immediately deduced from Theorem 2. Let f ×g be the restriction of to Σ f ×g measurable acts, let F f be the set of acts that are Σ f measurable, and say that h f ∈ F f is the f -Uniform product equivalent, U pe f (h), of the Σ f ×g measurable act h if:

A∈Π f α A h f (A) ∈ Ce   A∈Π f α A h A Πg   for all α ∈ (Π f ), where h A Πg is the Σ g measurable act identified by h A Πg (B) = h(A ∩ B) for each B ∈ Π g . An
f ∈ U pe f (h), c f ×g h ⇒ c f ×g h f and h f ×g d ⇒ h f f ×g d for all c, d ∈ K
while at the same time, for all c, d ∈ K,

c f ×g h f for all h f ∈ U pe f (h) ⇒ c f ×g h and h f f ×g d for all h f ∈ U pe f (h) ⇒ h f ×g d.
One can now also easily derive a definition of independent events. Fix two elements {k 1 , k 2 } of K such that k 1 > k 2 ,and let f A = k 1 1 A + k 2 1 Ω\A . We will then say that A and B are independent events according to if f A and f B are independent acts according to . It follows that two events are independent under this definition if and only if p(A ∩ B) = p(A)p(B) for all p in the set P representing .

A characterization of s-independence for MaxMin preferences

In their pioneering work, Gilboa and Schmeidler (1989) provide a characterization of independent product of relations for MaxMin preferences that is strictly connected to the representation in Theorem 2. Recall that a preference over F is MaxMin if there is a closed convex set of priors P ⊆ (Ω) such that is represented by the concave functional V (f ) = min p∈P ω∈Ω f (ω)p(ω). Gilboa and Schmeidler (1989) define a notion of independent product of preferences which is equivalent to a MaxMin relation X×Y on F represented by

V (f ) = min p∈co(P X ×P Y ) ω∈Ω f (ω)p(ω)
where P X and P Y are the priors representing two original MaxMin preferences X and Y over K X and K Y respectively. The link with our representation is clear, as is the fact that the definition of Gilboa and Schmeidler also satisfies the requirement of being completely identified by it's marginal preferences.

In fact more can be said on the relation between the two models. Ghirardato et al. ( 2004) introduce the concept of unambiguous preference. This is a sub-relation * of a complete preference over acts that is identified as follows:

f * g ⇔ λf + (1 -λ)h λg + (1 -λ)h for all λ ∈ (0, 1) h ∈ F.
For a large class of preferences, which includes MaxMin, * can be shown to be a Bewley relation. 12 Moreover, the sets of priors representing a MaxMin preference and it's unambiguous sub-relation * coincide. Thus we can state the following corollary of Theorem 2: Corollary 4. For any MaxMin preference over F and its unambiguous sub-relation * , letting U pe * X (f ) stand for the X-uniform product equivalents of f under * , the following are equivalent: 1) There are nonempty, closed and convex sets P X ⊆ (X) and P Y ⊆ (Y ) such that is represented by the functional V : F → R given by

V (f ) = min p∈co(P X ×P Y ) ω∈Ω f (ω)p(ω)
2) For all f ∈ F and f X ∈ U pe X (f ),

c * f ⇒ c * f X and f * d ⇒ f X * d for all c, d ∈ K
while at the same time, for all c, d ∈ K,

c * f X for all f X ∈ U pe X (f ) ⇒ c * f and f X * d for all f X ∈ U pe X (f ) ⇒ f * d.
This provides a characterization of Gilboa Schmeidler independence based on the model primitives ( and the derived relation * ) instead of on elements of the representation, as the one given in the 1989 paper (although see on this our comments in the next section).

Remark: One might be tempted to try to extend this result to other classes of preferences under ambiguity,given that Cerreia-Vioglio et. al. ensures that all MBA preferences have an unambiguous sub-relation with a Bewley representation. Nevertheless, as the following two examples illustrate, this might lead to issues with existence and uniqueness. It is our opinion that Corollary 3 is a natural extension of Theorem 2 precisely because there is a deep link, at a mathematical and interpretational level, between the Bewley and MaxMin models, as illustrated for example in [START_REF] Gilboa | Objective and Subjective Rationality in a Multi-Prior Model[END_REF]. Definitions and characterizations of s-independence for alternative models should be built based on their individual structure and interpretation.

Example 1. Existence: Consider Multiplier preferences, which are represented by the functional U (f ) = min p∈ (Ω) ω∈Ω f (ω)p(ω) + θr(p||q) where q is a reference distribution, r(p||q) the relative entropy of p w.r.t. q and θ a non-negative real number. Ghirardato and Siniscalchi (2010) show that the set of priors representing the unambiguous part * of a multiplier preference must coincide, when Ω is finite, with (Ω). Hence if Ω = X × Y and both X and Y contain at least two elements, no multiplier preference can satisfy condition 2) of Corollary 4, since in this case (X) × (Y ) is strictly included in (X × Y ). Thus no multiplier preference over K Ω can reflect s-independence according to our definition.

Example 2. Uniqueness: A Choquet Expected Utility preference can be represented, assuming for simplicity K ∈ R + , by the functional V (f ) = v({ω | f (ω) ≥ t})dt, where v : 2 Ω → [0, 1] is a capacity, i.e. a normalized, monotone function over sets. As is well known, the Choquet integral can be alternatively expressed as V (f ) = ω∈Ω f (ω)p f (ω), where p f is an additive probability distribution derived from the capacity by assigning to ω the probability

p f (ω) = v({ω | f (ω ) ≥ f (ω)}) -v({ω | f (ω ) > f (ω)}).
There are as many such probabilities as there are orders ≥ f over the state space induced by the rule ω ≥ f ω if and only if f (ω) ≥ f (ω ). [START_REF] Ghirardato | Differentiating Ambiguity and Ambiguity Attitude[END_REF] show that the priors P representing the unambiguous part of a CEU preference coincide with co{p f | f ∈ F}. Hence if we let X = {x 1 , x 2 } and Y = {y 1 , y 2 }, and consider two CEU preferences X and Y on K X and K Y respectively, we can conclude that the number of extreme points of P X and P Y is of at most two each, as there are only two orders over a set of two elements. Thus P X ⊗P Y has at most 2×2 = 4 extreme points. That means that if we wish to define the independent product of X and Y as the CEU preference over K X×Y whose unambiguous preference is represented by P X ⊗ P Y , we will be unable to do so uniquely as we need to assign 4 distributions to 4! = 24 different orders over X × Y . Dropping the requirement that the product be CEU would only worsen the issue, as the number of MBA preferences consistent with priors P X ⊗ P Y is extremely large. Hence our definition of s-independence is unable to uniquely identify the independent product of two CEU preferences.

Some considerations on our results

We conclude with a brief comment concerning falsifiability. Decision theorists in general like, with good reason, to keep what we will call "continuity" and "behavioral" axioms separated. The distinction between the two is sometimes vague, but it can be made precise using finite falsifiability as a litmus test. With this we mean that, starting from the primitive of our model, we should always be able to obtain a violation of a behavioral assumption in a finite number of steps. In this sense the classic Independence axiom is behavioral, since it is negated in two steps, by finding three acts f, g, h and a weight λ ∈ (0, 1) such that f g but λg + (1 -λ)h λf + (1 -λ)h. On the other hand the Archimedean axiom, which asks that for any three acts f, g, h the sets {λ ∈

[0, 1] | λf + (1 -λ)h
g} and {λ ∈ [0, 1] | g λf + (1 -λ)h} be closed, is typically not. In fact to violate it we must be able to check that, for example, λ * f + (1 -λ * )h g while λ n f +(1-λ n )h g for all {λ n } n∈N of a sequence converging to λ * , which involves verifying an infinite number of positive statements.

When considering a novel representation result, we usually prefer new assumptions to be of the first kind rather than the second, since this allows for a direct test of the validity of the model. At the same time, characterizations based on continuity type assumptions do bring a contribution, as they still allow us to identify the position of a model in the space of possible representations. The Conditional Invariance assumption falls in the behavioral side, as can be easily checked. The requirement we proposed as a characterization of s-independence for Bewley preferences, unfortunately, does not. To see this, notice that, for example, a possible violation of the axiom takes place if we can find a c ∈ K such that f c but c f X for some f X ∈ U pe(f ). But showing this requires us to make sure that f X is an X-uniform product equivalent of f , a process which implies checking that for all α ∈ (X), an infinite set, equation (3.4) is satisfied. For this reason we stop short of stating that we provide a full behavioral characterization of the model P = co(P X ⊗ P Y ), and we believe that additional work is still needed to obtain it. This is the focus of ongoing research, which we hope to report in future work.

A Proofs

Proof of Proposition 1: We prove the proposition only for X-acts conditioned on Y -events, since the argument for the inverse situation is symmetric. Assume f

X 1 R + h1 Ω\R g X 1 R + h1 Ω\R for some h ∈ F, f X , g X ∈ F X and R ∈ Σ Y . This means that for all p ∈ P ω∈R f X (ω)p(ω) + ω∈Ω\R h(ω)p(ω) ≥ ω∈R g X (ω)p(ω) + ω∈Ω\R h(ω)p(ω). (A.1)
Subtract the common term on each side, for each prior, to get ω∈R f X (ω)p(ω) ≥ ω∈R g X (ω)p(ω). Because g X and f X are constant over X and there is some

B ⊆ Y such that R = X × B, we have ω∈R f X (ω)p(ω) = x∈X f X (x) y∈B p(x, y) (A.2)
for all p ∈ P , and similarly for g X . Since for each p ∈ P there are p X ∈ (X) and p Y ∈ (Y ) such that p = p X × p Y , we can rewrite the r.h.s. of (A.

f X (x)p X (x) ≥ x∈X g X (x)p X (x)
for all p X ∈ P X , where the common factor p Y (B) can be canceled on both sides (for those priors in P for which p Y (B) = 0 the inequalities are trivially true). Now, since S = X × B for some B ⊆ Y , we can multiply for each p X ∈ P X the inequality above by p Y (B ), for all p Y ∈ (Y ) such that p X × p Y ∈ P , to obtain

ω∈S f X (ω)p(ω) = x∈X f X (x)p X (x)p Y (B ) ≥ x∈X g X (x)p X (x)p Y (B ) = ω∈S g X (ω)p(ω)
for all p ∈ P . Adding ω∈Ω\S h(ω)p(ω) to both sides of the inequality delivers the desired implication.

Proof of Theorem 1: 1) ⇒ 2) is a direct consequence of Proposition 1. The argument for 2) ⇒ 1) is well known, but we provide it here for completeness. To see that 2) ⇒ 1), consider first the case in which X × {y} is -non-null for only one y ∈ Y (at least one such y must exist since otherwise p(Ω) = 0). Then it is clear that p Y = δ y and p = p X × δ y , where δ y is the degenerate distribution at y inside (Y ), namely the element of (Y ) that is 1 at y and zero everywhere else. Now assume that at least two Y -states y, y ∈ Y are -non-null. Denote also by F X the set of maps from X to K, with generic elements f X and g X . Each X-act f X in F X has a corresponding projection f X in this set, identified by f X (x) = f X (x).

For every -non-null Y -state let y be the preference over F X identified by

f X y g X ⇐⇒ f X 1 y + h1 Ω\y g X 1 y + h1 Ω\y . (A.3)
By the usual arguments this preference is independent from h. Moreover, Conditional Invariance requires it to be also independent from y. Because it is still going to be SEU over F X , there is a unique distribution p X ∈ (X) representing each y . On the other side, we can see that for each -non-null y, the second comparison in (A.3) will be satisfied if and only if x∈X p(x,y) = p(x,y) p Y (y) . The latter inequality is clearly an alternative SEU representation of y , hence by the uniqueness of the SEU representation p(.|y) = p X for all -non-null y ∈ Y . But then p = p X × p Y = p X × p Y . 1) ⇒ 3) is an immediate consequence of the characterization of pe X (f ) in (3.1), and elementary distributive properties of the sum of real numbers. To show that 3) ⇒ 1), assume by way of contradiction that 3) holds but p = p X × p Y . Then by an elementary application of the hyperplane separation theorem there is, w.l.o.g., a vector r in R Ω such that ω∈Ω r ω p(ω) > ω∈Ω r ω p X × p Y (ω). Because distributions are positive and normalized to 1, we can multiply both sides of the inequality by a constant and add to both a constant vector without affecting the inequality. Hence we can assume that r corresponds to some f r in F. But clearly (x,y)∈X×Y f r (x, y)p X (x)p Y (y) = x∈X y∈Y f r (x, y)p Y (y) p X (x) is the value of pe X (f r ) under the prior p, hence this implies that f r pe x (f r ), contradicting 3). by the product structure of P . Let p * X and p * Y be the distributions that achieve the above minimum. For any f x ∈ U pe X (f ) we must have, by taking the p X reduction of f X for any p X ∈ P X , 2) ⇒ 1). Suppose first that there is a p ∈ P such that p / ∈ co(P X ⊗ P Y ). Then there must be, by the usual hyperplane separating argument, an f * ∈ F and a c ∈ K such that ω∈Ω f * (ω)p(ω) > c ≥ ω∈Ω f * (ω)p X × p Y (ω) for all p X × p Y ∈ P X ⊗ P Y . Now the last inequality implies that c f X for all f X ∈ U pe X (f ), since otherwise we would have one member fX of such set for which max p X ∈P X fX (x)p X (x) > max p Y ∈P Y ,p X ∈P X y∈Y x∈X f * (x, y)p X (x)p Y (y) in direct contradiction to x∈X fX p X (x) ∈ Ce( x∈X f * x Y p X (x)) for all p X ∈ P X ⊆ (X). But then by assumption c f which implies c ≥ ω∈Ω f * (ω)p(ω) for all p ∈ P . This proves that P ⊆ co(P X ⊗ P Y ).

For the remaining inclusion, assume there are p X ∈ P X and p Y ∈ P Y such that p X × p Y / ∈ P . Then we can find an f * and a c such that

ω∈Ω f * (ω)p X × p Y (ω) > c ≥ ω∈Ω f * (ω)p(ω) (A.4)
for all p ∈ P . Thus c f * . On the other side, we can find an f X ∈ U pe X (f ) such that f X (x) = y∈Y f * (x, y)p Y (y). By assumption c f X , hence c ≥ x∈X y∈Y f * (x, y)p Y (y)p X (x) for all p X ∈ P X , so that the strict inequality in (A.4) cannot hold. This shows that P X ⊗ P Y ⊆ P and hence, since P is closed and convex, co(P X ⊗ P Y ) ⊆ P .

10

 10

Proposition 2 .

 2 (From Proposition 18 in Ghirardato et al. (2004)). For every f ∈ F k ∈ Ce(f ) ⇔ min p∈P ω∈Ω f (ω)p(ω) ≤ k ≤ max p∈P ω∈Ω f (ω)p(ω).

  x∈X f X (x)p(x, y) ≥ x∈X g X (x)p(x, y) ⇐⇒ x∈X f X (x)p(x|y) ≥ x∈X g X (x)p(x|y) where p(x|y) = p(x,y)

Proof of Theorem 2 : 1 )

 21 ⇒ 2). For any c ∈ K, f c if and only if minp∈P ω∈Ω f * (ω)p(ω) ≥ c ⇔ min p Y ∈P Y ,p X ∈P X y∈Y x∈X f * (x, y)p X (x)p Y (y) ≥ c

  x∈X f X (x)p X (x) ≥ min p Y ∈P Y y∈Y x∈X f * (x, y)p X (x) p Y (y) ≥ y∈Y x∈X f * (x, y)p * X (x)p * Y (y)Hence f x c. On the other side, U pe X (f ) contains the set{f X ∈ F X | ∃ p Y ∈ P Y such that f X (x) = y∈Y f (x, y)p Y (y) for all x ∈ X}since for any such f X and any α ∈ (X) we have maxp Y ∈P Y y∈Y x∈X f (x, y)α x p Y (y) p Y ∈P Y y∈Y x∈X f (x, y)α x p Y (y).Hencef x c for all f X ∈ U pe X (f ) implies that min p X ∈P X x∈X   y∈Y f (x, y)p Y (y)   p X (x) ≥ cfor all p Y ∈ P Y , and thus f c.

  immediate corollary of Theorem 2 is then : Corollary 3. Let be a Bewley preference over F. Then f and g are independent if and only if for all h ∈ F f ×g and h

See for example Rigotti and Shannon (2005), Ghirardato and Katz (2006) and Lopomo et al. (2011) for applications in finance, voting, and principal-agent models respectively.

In this regard see Ghirardato et al.(2004), Gilboa et al. (2010) and Cerreia-Vioglio et al.(2011).

While Gilboa et al. (2010) work in the classical Anscombe-Aumann setting, the correspondence to our environment is immediate and can be found in their Appendix B. Ok Ortoleva Riella (2012) give a different axiomatization for a model that corresponds to (2.1) when K is compact.

The difference lies in the fact that in[START_REF] Blume | Lexicographic Probabilities and Choice under Uncertainty[END_REF] the conditioning events are only of the form X × {y}. While the two formulations are equivalent for SEU preferences, it can be shown that for a Bewley decision maker our version is strictly stronger.

Using the Archimedean Continuity, Monotonicity and Completeness properties of the SEU model.

Thus our definition is different from the most intuitive generalization of certainty equivalent that asks for any X-act that is indifferent to f , which one might dub the X-equivalent of the act.

We have adapted Prop. 18 in Ghirardato et al. (2004) to our environment and notation.

[START_REF] Ok | Incomplete Preferences Under Uncertainty: Indecisiveness in Beliefs versus Tastes[END_REF] use this type of reduction in the formulation of an axiom that characterizes two dual representations: Bewley's and the alternative single prior multi-expected utility model.

[START_REF] Cerreia-Vioglio | Rational Preferences under Ambiguity[END_REF] show that this is true for all Monotone Bernoullian Archimedean preferences, a class that includes Variational Preferences, Smooth Ambiguity preferences and many others.

I would like to thank Eric Danan and participants at the THEMA seminar at U.Cergy and Ecole Polytechnique internal seminar for useful comments and discussions. I acknowledge support by a public grant overseen by the French National Research Agency (ANR) as part of the Investissements d'Avenir program (Idex Grant Agreement No. ANR -11-IDEX-0003-02 / Labex ECODEC No. ANR -11-LABEX-0047). Needless to say, all mistakes are my own.