Nicolas Queyriaux

Dongyue Sun

Jennifer Fize

Jacques Pécaut

Martin J Field

Murielle Chavarot-Kerlidou

Vincent Artero

Electrocatalytic Hydrogen Evolution with a Cobalt Complex Bearing Pendant Proton

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Introduction

Hydrogen (H2) evolved from water through proton reduction appears as an appealing sustainable energy carrier, potentially capable for substituting current carbon-based fuels. The development of cheap H2-evolving molecular catalysts based on abundant transition metals [START_REF] Du | Catalysts made of earth-abundant elements (Co, Ni, Fe) for water splitting: Recent progress and future challenges[END_REF][START_REF] Artero | Splitting water with cobalt[END_REF][START_REF] Mckone | Earth-abundant hydrogen evolution electrocatalysts[END_REF][START_REF] Zee | Metal-polypyridyl catalysts for electro-and photochemical reduction of water to hydrogen[END_REF][START_REF] Pal | Hydrogen production from water using a bis(imino)pyridine molybdenum electrocatalyst[END_REF][START_REF] Tran | Coordination polymer structure and revisited hydrogen evolution catalytic mechanism for amorphous molybdenum sulfide[END_REF] is thus the subject of intense research activity. Current challenges include the design of new catalytic platforms featuring higher activity together with increased stability. Introduction of proton relays in the second coordination sphere of the metal center [START_REF] Dubois | Development of molecular electrocatalysts for CO2 reduction and H2 production/oxidation[END_REF][START_REF] Dubois | The roles of the first and second coordination spheres in the design of molecular catalysts for H2 production and oxidation[END_REF][START_REF] Chapovetsky | Pendant Hydrogen-Bond Donors in Cobalt Catalysts Independently Enhance CO2 Reduction[END_REF] or redox-active ligands 10- 14 have been proved essential to promote, tune or enhance catalytic activity. Notable examples involve the control of the formation of cobalt-hydride derivatives as the key active species in catalysis. [START_REF] Jurss | Bioinspired design of redox-active ligands for multielectron catalysis: effects of positioning pyrazine reservoirs on cobalt for electro-and photocatalytic generation of hydrogen from water[END_REF][START_REF] Jacobsen | Hydrogen production using cobalt-based molecular catalysts containing a proton relay in the second coordination sphere[END_REF][START_REF] Roubelakis | Proton-coupled electron transfer kinetics for the hydrogen evolution reaction of hangman porphyrins[END_REF][START_REF] Graham | Electrocatalytic H2 Evolution by Proton-Gated Hangman Iron Porphyrins[END_REF][START_REF] Wang | Electronic and Steric Tuning of Catalytic H2 Evolution by Cobalt Complexes with Pentadentate Polypyridyl-Amine Ligands[END_REF][START_REF] Ai | Hydride Transfer Reactions Catalyzed by Cobalt Complexes[END_REF] Recently, cobalt(II) polypyridyl complexes have emerged as promising catalysts for hydrogen evolution. Such ligand platforms offer high robustness thanks to their ability to stabilize low oxidation states of the metal centers involved in the catalytic cycle [START_REF] Queyriaux | Recent developments in hydrogen evolving molecular cobalt(II)-polypyridyl catalysts[END_REF][START_REF] Hickey | Investigating the Role of Ligand Electronics on Stabilizing Electrocatalytically Relevant Low Valent Co(I) Intermediates[END_REF] and their resistance to hydrogenation. Indeed, intramolecular hydride transfers have been proposed as a likely cause for the deactivation of catalysts from the well-known cobaloxime family. [START_REF] Simandi | Activation of Molecular-Hydrogen by Cobaloxime(II) Derivatives[END_REF][START_REF] Simandi | Effect of strong base on the activation of molecular hydrogen by pyridinebis(dimethylglyoximato)cobalt(II)[END_REF] It has recently been stressed out that the optimization of catalytic performances does not only rely on the tuning of the electronic properties of the starting (pre)catalysts but also requires manipulating mechanistic pathways, thus affecting the nature of the active species. [START_REF] Costentin | Homogeneous Molecular Catalysis of Electrochemical Reactions: Manipulating Intrinsic and Operational Factors for Catalyst Improvement[END_REF] By using a complex displaying potential bifurcations in its catalytic pathways due to its proton-and redox-responsive ligand platform, it becomes possible to compare different routes by simply tuning the catalytic conditions (pKa of the acid, applied potential). Herein, we describe how catalytic potential and strength of the proton source affect the reactivity of a cobalt(II)polypyridyl catalyst [START_REF] Queyriaux | A noble metal-free photocatalytic system based on a novel cobalt tetrapyridyl catalyst for hydrogen production in fully aqueous medium[END_REF] combining pendant bases and redox-active ligand. Using analytical electrochemical tools, [START_REF] Lee | Electrochemical and spectroscopic methods for evaluating molecular electrocatalysts[END_REF][START_REF] Sandford | A synthetic chemist's guide to electroanalytical tools for studying reaction mechanisms[END_REF] four distinct catalytic pathways for hydrogen evolution, involving different cobalt-hydride intermediates, have been identified. For the first time, the impact of the nature of these intermediates on both the rate and stability of the catalytic processes has been determined.

Results and discussion

Synthesis, UV-vis properties and structural information.

Metalation of the 6,6'-bis-(2-aminopyridyl)-2,2'-bipyridine (bapbpy) ligand [START_REF] Bonnet | A two-step spin crossover mononuclear iron(II) complex with a [HS-LS-LS] intermediate phase[END_REF][START_REF] Molenbroek | Zinc coordination to the bapbpy ligand in homogeneous solutions and at liposomes: zinc detection via fluorescence enhancement[END_REF] with CoCl2.6H2O in methanol yields the chloride salt of [Co(bapbpy)Cl] + (Figure 1, Co II L) as a yellow powder. This complex is soluble in aprotic polar solvents such as acetone, CH3CN, DMF but only poorly soluble in water. In methanol, the UV-vis absorption spectrum of Co II L is dominated by -* and intra-ligand charge transfer transitions (Figure S1 and Table S1).

Noticeably, it also displays a shoulder at 410 nm, assigned to a d-d transition. [START_REF] Bacchi | Cobaloxime-Based Artificial Hydrogenases[END_REF] The X-ray structure of Co II L shows a strongly distorted trigonal bipyramidal geometry ( = 0.55) [START_REF] Addison | Synthesis, structure, and spectroscopic properties of copper(II) compounds containing nitrogen-sulphur donor ligands; the crystal and molecular structure of aqua[1,7bis(N-methylbenzimidazol-2'-yl)-2,6-dithiaheptane]copper(II) perchlorate[END_REF]33 with one pyridine ring and the bipyridine moiety in the equatorial plane. The coordination sphere is complemented by the second pyridine ring sitting slightly outside the plane and a pseudo-axial chloride ligand (Figure 1 and Table S2 for selected bond lengths and angles). This saddle-shape structure appears similar to the previously reported zinc(II) bapbpy complex. [START_REF] Molenbroek | Zinc coordination to the bapbpy ligand in homogeneous solutions and at liposomes: zinc detection via fluorescence enhancement[END_REF] The amino groups of the bapbpy ligand are not bound to the Co II center and are therefore free to act as proton relays during catalysis.

Electrochemical properties.

The cyclic voltammogram of Co II L recorded at a glassy carbon (GC) electrode in DMF displays quasi-reversible redox processes at -0.11 V (E = 136 mV) and -1.44 V (E = 76 mV) vs Fc + /Fc, respectively assigned to the Co III /Co II and Co II /Co I couples on the basis of previous literature data (Figure 2 and Table S3). [START_REF] Jurss | Bioinspired design of redox-active ligands for multielectron catalysis: effects of positioning pyrazine reservoirs on cobalt for electro-and photocatalytic generation of hydrogen from water[END_REF][START_REF] Lo | Synthesis, Characterization, and Photocatalytic H2-Evolving Activity of a Family of [Co(N4Py)(X)] n+ Complexes in Aqueous Solution[END_REF][START_REF] Tong | Visible Light-Driven Hydrogen Evolution from Water Catalyzed by A Molecular Cobalt Complex[END_REF][START_REF] Deponti | Photoinduced hydrogen evolution by a pentapyridine cobalt complex: elucidating some mechanistic aspects[END_REF] The broadness of the wave associated with the Co III/II couple is an expected feature of this class of compounds [START_REF] Deponti | Photoinduced hydrogen evolution by a pentapyridine cobalt complex: elucidating some mechanistic aspects[END_REF][START_REF] Nippe | Catalytic proton reduction with transition metal complexes of the redox-active ligand bpy2PYMe[END_REF] and is likely due to a slow interfacial electron transfer kinetics related to structural reorganization from high-spin Co II complex to a potentially low-spin Co III compound. [START_REF] Nippe | Catalytic proton reduction with transition metal complexes of the redox-active ligand bpy2PYMe[END_REF] By contrast, the well-defined and reversible Co II /Co I couple supports low structural reorganization: upon reduction, chloride loss is precluded on the timescale of cyclic voltammetry. When swept to more cathodic potentials, a poorly reversible wave is observed at -1.99 V vs Fc + /Fc. We assign this redox event to a ligand-based process from comparison with the CV of the bapbpy complex of the redoxinactive zinc(II) ion that displays a similar wave at -1.96 V vs Fc + /Fc (Figure 2). DFT calculations further support this assignment to a Co I (L •-) species, the spin density of this S= ½ species being mainly centered on the bipyridine moiety of the bapbpy ligand (Tables S4-S6).

The poor reversibility is likely due to chloride elimination upon reduction, a process that was calculated as thermodynamically favorable.

In order to obtain a full picture of the system, we sought to obtain quantitative data best describing its electrochemical behavior. Using the Randles-Sevcik equation, the catalyst diffusion coefficient was determined through a variable scan rate study, affording a value of 3 ×10 -6 cm 2 .s -1 , in line with related molecular cobalt complexes (see the Supporting Information for more details). [START_REF] Roubelakis | Proton-coupled electron transfer kinetics for the hydrogen evolution reaction of hangman porphyrins[END_REF][START_REF] Elgrishi | Reaction Parameters Influencing Cobalt Hydride Formation Kinetics: Implications for Benchmarking H-2-Evolution Catalysts[END_REF][START_REF] Wiedner | Electrochemical Detection of Transient Cobalt Hydride Intermediates of Electrocatalytic Hydrogen Production[END_REF][START_REF] Elgrishi | Terpyridine complexes of first row transition metals and electrochemical reduction of CO2 to CO[END_REF] Interestingly, when plotting the variation of the anodic and cathodic peak potentials (Ep,a and Ep,c) for the two cobalt-centered processes versus the logarithm of the scan rate (ranging from 100 mV.s -1 to 2000 mV.s -1 ), linear trends are observed. Equivalent slopes of opposite signs (+20 and -20 mV.dec -1 ) were determined for the Co II/I couple (Figure S2). By contrast, unbalanced results were obtained in the case of the Co III/II couple (+90 and -35 mV.dec -1 ), highlighting an asymmetry of the energy barrier for electron transfer.

Acid-base properties.

Knowing the pKa values associated with Co II L is important for better understanding the protonation states that can be reached by the catalyst in the absence of any reductive process, i.e. the nature of the entry point in the catalytic cycle. We thus performed UV-Vis titration of Co II L in the presence of increasing amounts of trifluoromethanesulfonic acid (TfOH) in DMF (See Figure S3-S4 and associated paragraph in the Supporting Information for more details).

The total dissociation of TfOH in DMF allowed direct measurements of the two first pKas associated with the cobalt catalyst at 6.3 (Co II L/Co II LH + ) and 0.8 (Co II LH + /Co II LH2 2+ ).

Relevant ligand pKa values to compare with are rare in the literature. pKa of well-studied nickel-phosphine compounds for which the role of pendant bases has been widely investigated by DuBois and Bullock are ranging from 6 to 14 in CH3CN, depending on the protonation site and overall conformation of the complex. [START_REF] Raugei | The Role of Pendant Amines in the Breaking and Forming of Molecular Hydrogen Catalyzed by Nickel Complexes[END_REF] Similarly, Marinescu and Miller recently reported pKa values ranging between 2 and 4 for related cobalt polypyridyl complexes capable to selectively electro-assist CO2-to-CO conversion in CH3CN. [START_REF] Chapovetsky | Pendant Hydrogen-Bond Donors in Cobalt Catalysts Independently Enhance CO2 Reduction[END_REF] It should be noted that the pKa scale is shrunk in DMF compared to CH3CN, with for example the pKa of HNEt3 + being 9 units higher in CH3CN than in DMF. [START_REF] Izutsu | Acid-Base Dissociation Constants in Dipolar Aprotic Solvents[END_REF] Therefore the bapbpy ligand in Co II L can behave as a base of similar strength than trifluoroacetate in DMF. [START_REF] Fourmond | A nickel-manganese catalyst as a biomimic of the active site of NiFe hydrogenases: a combined electrocatalytical and DFT mechanistic study[END_REF] Electrocatalytic properties and mechanistic investigations.

Investigation of the electrocatalytic activity of Co II L toward hydrogen evolution was then performed in DMF. While a similar complex has already been shown active for H2 evolution under electro-assisted conditions in aqueous media, [START_REF] Queyriaux | A noble metal-free photocatalytic system based on a novel cobalt tetrapyridyl catalyst for hydrogen production in fully aqueous medium[END_REF] the use of an organic solvent allows for precise monitoring of the proton delivery to the catalytic system by the control of the pKa and the concentration of the proton sources. This approach enables easier investigations on the mechanism pathways followed by the catalytic system. Three proton sources with distinct pKa values in this solvent were thus selected: [START_REF] Izutsu | Acid-Base Dissociation Constants in Dipolar Aprotic Solvents[END_REF] tetrafluoroboric acid HBF4 (pKa = 3.4), triethylammonium HNEt3 + (BF4 -salt, pKa = 9.3) and acetic acid CH3COOH (pKa = 13.5). CVs of those acids in DMF have been recorded and are displayed in Figure S5-7. Whereas tetrafluoroboric acid and triethylammonium are not known to undergo homoconjugation in DMF, [START_REF] Fourmond | H2 evolution and molecular electrocatalysts: determination of overpotentials and effect of homoconjugation[END_REF] such process could potentially impact acetic acid and slightly reduce the apparent acidity of this proton source. Interestingly, switching from one acid to another allowed very distinct electrocatalytic behaviors to be observed, as detailed in the following.

Electrocatalytic properties and mechanistic investigations: weak acids.

Figure 3 shows the cyclic voltammograms of Co II L in the presence of increasing amounts of HNEt3 + . The addition of acid triggers the development of a "peak-shaped" catalytic wave with a half-wave potential of -1.90 V vs Fc + /Fc, regardless of acid concentration. The catalytic peak current is both proportional to the acid concentration and the square root of the scan rate, indicating that the catalytic process is diffusion-limited (Figure S8). Bulk electrolysis confirmed this wave to correspond to catalytic H2 evolution (Table 1 andS10). In parallel with the appearance of this electrocatalytic wave, it is worth noting that the Co II /Co I couple is progressively shifted towards anodic potential upon acid addition (Figure 3, process 1). This observation clearly evidences a coupled chemical reaction, here a protonation. [START_REF] Savéant | Coupling of Electrode Electron Transfers with Homogeneous Chemical Reactions[END_REF] Although the pKa of triethylammonium does not allow the protonation of Co II L, the increased electronic density of the one-electron reduced species Co I L on N2 and N5 (Tables S4 andS5) is likely to trigger bapbpy ligand protonation. Interestingly, the electrochemical system remains reversible, indicative of a fast and equilibrated protonation reaction. Under these conditions, a linear increase of the reduction potential of 32 ± 2 mV per decade was observed (Figure S9). In the absence of any modifications in the shape and intensity of the wave associated with the Co II /Co I couple, such behaviour cannot be ascribed to a 2e -/1H + process. Although not canonical, such a non-ideal behavior is not unprecedented for a proton-coupled electron transfer (PCET) [START_REF] Efros | Towards a functional model of hydrogenase: electrocatalytic reduction of protons to dihydrogen by a nickel macrocyclic complex[END_REF] and may reflect a low equilibrium constant or complex equilibria that occur in non-aqueous solvent. [START_REF] Fourmond | H2 evolution and molecular electrocatalysts: determination of overpotentials and effect of homoconjugation[END_REF][START_REF] Mccarthy | Decoding Proton-Coupled Electron Transfer with Potential-pKa Diagrams[END_REF] From these results, we propose that catalysis follows an ECEC mechanism (where E corresponds to an electron transfer step and C to a chemical reaction, here protonation): once reduced to Co I L, protonation rapidly occurs at an amino group of the bapbpy ligand to form Co I (LH + ). [START_REF] De ; Savéant | Elements of Molecular and Biomolecular Electrochemistry[END_REF] The latter is easier to reduce than Co I L, accounting for the 100 mV anodic shift of the catalytic onset potential compared to the ligand-centered process measured in the absence of acid (Figure 3, process 2). Further reduction of Co I (LH + ) to Co I (LH • ), followed by protonation to form the bis-protonated Co II H(LH + ) species, the nature of which will be discussed below, finally allows H2 to be evolved. Based on such a mechanism, kinetically controlled by the second protonation step, foot-of-the-wave analysis (FOWA) [START_REF] Costentin | Turnover numbers, turnover frequencies, and overpotential in molecular catalysis of electrochemical reactions. Cyclic voltammetry and preparative-scale electrolysis[END_REF][START_REF] Costentin | Multistep Molecular Catalysis of Electrochemical Reactions: Benchmarking of Homogeneous Catalysts[END_REF] allowed the determination of a second order kinetic rate constant of 1.3 ± 0.1 10 5 M -1 .s -1 for the protonation of Co I (LH • ) (see the Supporting information, Table S9 and Figure S10). Controlled-potential electrolysis (CPE) performed at -2.00 V vs Fc + /Fc (Table 1 andS10) allowed a second-order rate constant value of 2.9 ± 0.1 10 4 M -1 .s -1 , associated with the protonation of Co I (LH • ) to be evaluated. [START_REF] Roy | Molecular Cobalt Complexes with Pendant Amines for Selective Electrocatalytic Reduction of Carbon Dioxide to Formic Acid[END_REF] This value is ~4.5 times lower than that the one calculated from the FOWA. We relate this apparent discrepancy to ohmic drop, which more significantly biases CPE than CV measurements. When the weaker acetic acid is used as the proton source (pKa = 13.5 in DMF), CVs only display a slight increase in current in the vicinity of the ligand-centered process (Figure S11).

However, bulk electrolysis experiments demonstrated that H2 evolution can also be observed.

The absence of any significant anodic shift of the Co II /Co I process when the concentration of acid is progressively increased appears as an evidence of the inability of AcOH to protonate the bapbpy ligand in Co I L. On the contrary, a ~10 mV cathodic shift is observed, which may point to some ligand exchange process in Co II L, as similarly identified on related pentapyridine cobalt(II). [START_REF] King | A mechanistic study of proton reduction catalyzed by a pentapyridine cobalt complex: evidence for involvement of an anation-based pathway[END_REF] To further investigate this hypothesis, the CV of Co II L in the presence of sodium acetate (10 mM) have been recorded (Figure S11), displaying a ~150 mV cathodic shift of the potential associated with the Co II /Co I process. The partial dissociation of acetic acid in DMF results in the presence of acetate ions thus capable to efficiently bind Co II L complex. Unlike what is observed in the case of triethylammonium addition, no anodic shift of the catalytic onset potential was identified upon addition of acetic acid. Although acetate binding will likely impact the Co I L/Co I (L •-) redox potential, the shift should remain similar to that of the Co II /Co I process until a significant amount of acetate is produced concomitantly to H2 evolution.

Collectively, these data suggest that the increased electronic density of Co I L is insufficient to allow bapbpy ligand protonation by acetic acid. An EECC mechanism passing through the double protonation of the two-electron reduced Co I (L •-) species can thus be assumed in the presence of such a poor proton source in DMF.

Electrocatalytic properties and mechanistic investigations: strong acid.

A drastic change in the electrocatalytic response is observed when HBF4 is used as the proton source (Figure 4). The Co II /Co I process now becomes irreversible and its cathodic peak potential is gradually shifted towards more positive potentials (Figure 4, process 1). Plotting this variation as a function of the logarithm of the acid concentration provides a linear trace with a slope of 37 ± 1 mV per decade (Figure S12). Associated with the loss of reversibility, this value, in line with the expected value of 29 mV per decade, strongly evidenced an irreversible EC process with fully displaced protonation equilibrium. [START_REF] Costentin | Concertedness in proton-coupled electron transfer cleavages of carbon-metal bonds illustrated by the reduction of an alkyl cobalt porphyrin[END_REF] Considering the relative pKa values of the unreduced complex Co II L and HBF4, this 1e -/1H + sequence occurs on an already protonated Co II (LH + ) complex, thus either leading to the formation of a doubly ligandprotonated Co I complex Co I (LH2 2+ ) or a protonated Co III hydride species Co III H(LH + ). [START_REF] Costentin | Concertedness in proton-coupled electron transfer cleavages of carbon-metal bonds illustrated by the reduction of an alkyl cobalt porphyrin[END_REF] There are two important elements to preferentially support protonation at the metal center and thus the generation of Co III H(LH + ): (i) the appearance of a new catalytic process at slightly more cathodic potential (vide infra) and (ii) calculations performed on cobaloximes that concludes Co III hydrides formation is thermodynamically favored over protonated Co I derivatives (see figure 2 of ref. [START_REF] Solis | Effects of Ligand Modification and Protonation on Metal Oxime Hydrogen Evolution Electrocatalysts[END_REF] ). In addition to the first process described above, the cyclic voltammograms of Co II L in the presence of HBF4 (Figure 4) display two proton-dependent waves with half-wave potentials at -1.65 and -1.90 V vs Fc + /Fc. The latter is reminiscent of the ligand-centered catalytic process observed with HNEt3 + as proton source. The first one develops at a potential intermediate between those of the Co II /Co I couple and the ligand-based process. Such a catalytic wave likely involves the reduction of the ligand-protonated Co III hydride Co III H(LH + ) formed after the first reduction process) into a ligand-protonated Co II hydride species, Co II H(LH + ), capable of H2 evolution. The potential of the Co III H(LH + )/Co II H(LH + ) couple can thus be estimated to be close to -1.65 V vs Fc + /Fc, i.e. more cathodic than the Co II /Co I couple. This behavior contrasts with the situation found for cobaloximes, [START_REF] Muckerman | Theoretical studies of the mechanism of catalytic hydrogen production by a cobaloxime[END_REF][START_REF] Solis | Theoretical analysis of mechanistic pathways for hydrogen evolution catalyzed by cobaloximes[END_REF] but is similar to what was established for cobalt complexes bearing ligands that are able to stabilize low Co oxidation states, [START_REF] Koelle | Electrochemical reduction of protonated cyclopentadienylcobalt phosphine complexes[END_REF][START_REF] Marinescu | Molecular mechanisms of cobalt-catalyzed hydrogen evolution[END_REF][START_REF] Wiedner | Thermochemical and mechanistic studies of electrocatalytic hydrogen production by cobalt complexes containing pendant amines[END_REF] a property shared by polypyridine ligands. [START_REF] Queyriaux | Recent developments in hydrogen evolving molecular cobalt(II)-polypyridyl catalysts[END_REF][START_REF] Hickey | Investigating the Role of Ligand Electronics on Stabilizing Electrocatalytically Relevant Low Valent Co(I) Intermediates[END_REF][START_REF] Zee | Metal-Polypyridyl Catalysts for Electro-and Photochemical Reduction of Water to Hydrogen[END_REF] Due to the close proximity of the irreversible Co II (LH + )/Co III H(LH + ) PCET process at the foot of the catalytic wave, FOWA could not be safely applied. Maximal turnover frequencies of ~760 ± 20 s -1 and a second order rate constant of 7.6 ± 0.2 10 3 M -1 .s -1 were therefore extracted from CPE experiments (See the Supporting Information, Tables 1 andS10). [START_REF] Costentin | Turnover numbers, turnover frequencies, and overpotential in molecular catalysis of electrochemical reactions. Cyclic voltammetry and preparative-scale electrolysis[END_REF][START_REF] Costentin | Multistep Molecular Catalysis of Electrochemical Reactions: Benchmarking of Homogeneous Catalysts[END_REF][START_REF] Roy | Molecular Cobalt Complexes with Pendant Amines for Selective Electrocatalytic Reduction of Carbon Dioxide to Formic Acid[END_REF][START_REF] Costentin | Molecular Catalysis of H2 Evolution: Diagnosing Heterolytic versus Homolytic Pathways[END_REF] The ligand-protonated Co II hydride species Co II H(LH + ) thus revealed as a potentially crucial intermediate, involved in the electro-assisted H2 evolution triggered by the presence of HNEt3 + or HBF4 (either through the protonation of Co I (LH • ) at the metal center or the reduction of Co III H(LH + )). DFT calculations of such a bis-protonated species [START_REF]DFT calculations showed that in cobalt hydride derivatives, the hydride ligand destabilizes the trans Co-Cl bond, resulting in the elimination of the chloride ligand from the cobalt coordination sphere[END_REF] have been undertaken, highlighting the potentially prominent role played by the amino groups to bring together the cobalt hydride and a proton (Figure 5 and Tables S7-S8). Intramolecular protonation of the cobalt hydride by a pendant amino group is indeed favored in Co II H(LH + ) by a short Co-H•••H-N distance (1.57 Å, Figure 5 and Table S11), reminiscent of that found in the active site of hydrogenases [START_REF] Reijerse | Direct Observation of an Iron-Bound Terminal Hydride in [FeFe]-Hydrogenase by Nuclear Resonance Vibrational Spectroscopy[END_REF][START_REF] Katz | Vibrational spectroscopy reveals the initial steps of biological hydrogen evolution[END_REF] and some of their mimics that bear a pendant basic site. [START_REF] Ding | Hemilabile Bridging Thiolates as Proton Shuttles in Bioinspired H2 Production Electrocatalysts[END_REF][START_REF] Tang | Biomimetics of [NiFe]-Hydrogenase: Nickel-or Iron-Centered Proton Reduction Catalysis?[END_REF][START_REF] Carroll | Synthetic Models for the Active Site of the [FeFe]-Hydrogenase: Catalytic Proton Reduction and the Structure of the Doubly Protonated Intermediate[END_REF][START_REF] Huynh | Protonation of Nickel-Iron Hydrogenase Models Proceeds after Isomerization at Nickel[END_REF][START_REF] Liu | Heterolytic cleavage of hydrogen by an iron hydrogenase model: an Fe-HH-N dihydrogen bond characterized by neutron diffraction[END_REF] A quick look at the different DFT-calculated structures seems to indicate a certain flexibility of the bapbpy ligand that can accommodate various coordination geometries. We provide in the Supporting Information an analysis of the various geometries found in the calculated structures. Although all structures are highly distorted, a major change in coordination mode is found in Co II H(LH + ).

A pyridine ring indeed adopts an axial position, leaving the equatorial position to the hydride ligand (Figure 5) and thus allowing for a significantly reduced distance between the hydride and the proton born by the amine function of the bapbpy ligand. It is worth noting that the peak current related to the Co II (LH + )/Co III H(LH + ) couple measured in the presence of HBF4 is ~1.5 higher than that of the Co II /Co I process measured in the absence of acid. This value is significantly higher than the expected 1.11 increase for a transition from a pure electron transfer E to an irreversible EC process and might conceal some catalytic activity. [START_REF] Savéant | Coupling of Electrode Electron Transfers with Homogeneous Chemical Reactions[END_REF] Bulk electrolysis was thus run at -1.35 V vs Fc + /Fc and demonstrated that H2

evolution is observed at this potential (Table 1), whereas blank experiment deprived from the cobalt catalyst were unable to yield any hydrogen gas. This catalytic process supports the formation of the Co III hydride species Co III H(LH + ). Homolytic H2 evolution was formerly described by Gray and coworkers for a related cobalt catalyst. [START_REF] Marinescu | Molecular mechanisms of cobalt-catalyzed hydrogen evolution[END_REF] However, the current measured at the peak/plateau linearly depends on the catalyst concentration in the 0.3-3.0 mM range (Figure S13) whereas the mid-wave potential remains unaffected by variation of the catalyst concentration, ruling out homolytic H2 evolution with non-steady-state concentration of the catalytic intermediate (HON zone). [START_REF] Costentin | Multistep Molecular Catalysis of Electrochemical Reactions: Benchmarking of Homogeneous Catalysts[END_REF][START_REF] Costentin | Molecular Catalysis of H2 Evolution: Diagnosing Heterolytic versus Homolytic Pathways[END_REF] The catalytic current is also not significantly affected by the acid concentration, ruling out both classical heterolytic mechanisms [START_REF] Costentin | Multistep Molecular Catalysis of Electrochemical Reactions: Benchmarking of Homogeneous Catalysts[END_REF] and homolytic mechanism with steady-state concentration of the catalytic intermediate (HOS zone). [START_REF] Costentin | Molecular Catalysis of H2 Evolution: Diagnosing Heterolytic versus Homolytic Pathways[END_REF] A possible mechanism to explain such a behavior involves intramolecular proton-hydride coupling as the rate-determining step, as already observed for cobaloximes [START_REF] Rountree | Linear Free Energy Relationships in the Hydrogen Evolution Reaction: Kinetic Analysis of a Cobaloxime Catalyst[END_REF] and DuBois' nickel complexes. [START_REF] Dubois | Development of Molecular Electrocatalysts for Energy Storage[END_REF] In good agreement, DFT calculation revealed a long Co-H•••H-N distance of 2.463 Å in Co III H(LH + ) (Figure 5 and Table S11), accounting for the low rate of H2 evolution measured under these conditions (a TOF of 110 ± 20 s -1 , in this case directly equating with the rate-determining first-order constant, was determined from the bulk electrolysis data, Table 1 and Table S10).

Depending on the strength of the acid used and the applied potentials, four distinct catalytic pathways are therefore proposed for Co II L, respectively triggered by the Co II (LH + )/Co III H(LH + ), Co III H(LH + )/Co II H(LH + ), Co I (LH + )/Co I (LH • ) and Co I L/Co I (L •-) processes (Figure 6). They all involve cobalt-hydride intermediates. Such intermediates may exist as tautomeric forms [START_REF] Estes | The Reaction of Cobaloximes with Hydrogen: Products and Thermodynamics[END_REF][START_REF] Bhattacharjee | Combined Experimental-Theoretical Characterization of the Hydrido-Cobaloxime [HCo(dmgH)2(P n Bu3)[END_REF] and are often difficult to isolate [START_REF] Wiedner | Electrochemical Detection of Transient Cobalt Hydride Intermediates of Electrocatalytic Hydrogen Production[END_REF][START_REF] Lacy | The Cobalt Hydride that Never Was: Revisiting Schrauzer's "Hydridocobaloxime[END_REF] because of their intrinsic transience and reactivity. Metal hydrides are nevertheless considered as ubiquitous active species for H2 evolution, not only in synthetic molecular systems, but also in biology [START_REF] Reijerse | Direct Observation of an Iron-Bound Terminal Hydride in [FeFe]-Hydrogenase by Nuclear Resonance Vibrational Spectroscopy[END_REF] and materials science. [START_REF] Copéret | Isolated Surface Hydrides: Formation, Structure, and Reactivity[END_REF] Figure 6. Proposed mechanistic pathways followed by complex Co II L during electro-assisted H2evolution in the presence of acids of various strengths (blue: AcOH at -2.0 V vs Fc + /Fc; green: HNEt3 + and HBF4 at -1.9 V vs Fc + /Fc; orange: HBF4 at -1.65 V vs Fc + /Fc; purple: HBF4 at -1.35 V vs Fc + /Fc) in DMF.

The "catalytic Tafel plot" relating turnover frequency (TOF) and driving force (or overpotential, ) [START_REF] Artero | Toward the Rational Benchmarking of Homogeneous H2-Evolving Catalysts[END_REF] for H2 evolution shown in Figure 7 benchmarks the performance of the Co II L complex for H2 evolution with those of previously reported catalysts. To build this plot, we used catalytic rate constants extracted from FOWA or bulk electrolysis experiments as detailed previously. The overpotential requirement [START_REF] Fourmond | H2 evolution and molecular electrocatalysts: determination of overpotentials and effect of homoconjugation[END_REF] measured at the half-wave potential of the catalytic process is reported in Table 1, with errors estimated to 50 mV because of the various tabulated values for the standard potential for the reduction of protons in the solvent considered. [START_REF] Fourmond | H2 evolution and molecular electrocatalysts: determination of overpotentials and effect of homoconjugation[END_REF][START_REF] Roberts | Direct Determination of Equilibrium Potentials for Hydrogen Oxidation/Production by Open Circuit Potential Measurements in Acetonitrile[END_REF] The overpotential requirement roughly corresponds to that of change in slope in catalytic Tafel plots. Although not competitive with cobaloximes [START_REF] Razavet | Proton Electroreduction Catalyzed by Cobaloximes: Functional Models for Hydrogenases[END_REF] or DuBois' bioinspired Ni complexes, [START_REF] Helm | A Synthetic Nickel Electrocatalyst with a Turnover Frequency Above 100,000 s -1 for H2 Production[END_REF] Co II L compares well with other noble-metal free electrocatalysts, such as Fe(TPP) [START_REF] Bhugun | Homogeneous Catalysis of Electrochemical Hydrogen Evolution by Iron(0) Porphyrins[END_REF] (TPP: tetraphenylporphyrin). Stability is another crucial factor for an electrocatalyst, in addition to kinetic and overpotential requirements. Savéant and co-workers described how to determine stability-related constants of electrocatalytic systems from the fit of the charge passed during bulk electrolysis experiments using equation 1, in which qlim reflects the charge passed when the catalysts has reached its total turnover number. [START_REF] Costentin | Benchmarking of homogeneous electrocatalysts: overpotential, turnover frequency, limiting turnover number[END_REF] The tchem values associated with three of the proposed mechanisms are reported in Table 1. tchem can be construed as the lifetime of the catalyst in the diffusion layer, under the considered operating conditions.

𝑞(𝑡) = 𝑞 𝑙𝑖𝑚 [1 -𝑒 -𝑡 𝑡 𝑐ℎ𝑒𝑚 ⁄ ] equation 1
In the presence of strong acid, tchem is more than three times lower at -1.65 V vs Fc + /Fc (tchem = 230 ± 15 min) than at -1.35 V vs Fc + /Fc (tchem = 830 ± 25 min). When the weaker HNEt3 + is used as the proton source, the time constant proves to be even lower (tchem = 90 ± 10 min).

Interestingly, the catalyst stability appears as directly related to the potential required to form the active species driving catalysis. The generation of highly-reduced reactive intermediatessuch as Co II hydrides or ligand-reduced speciessignificantly alters the overall stability of the catalytic system. On the other hand, the process requiring the lowest driving force confers a higher stability to the molecular systems, in addition to high-energy conversion yields. This is however at the expense of the catalytic rate. In thermal catalysis, the most (re)active catalysts are more likely to deactivate. [START_REF] Crabtree | Deactivation in homogeneous transition metal catalysis: causes, avoidance, and cure[END_REF] To the best of our knowledge the present study provides for the first time a quantitative demonstration that this trend also occurs for electro-assisted processes (low tchem values correlate with high rate constants in Table 1). It is also generally observed in thermally-activated processes that, for a given catalyst, deactivation becomes more severe as substrates more difficult to activate are targeted. [START_REF] Crabtree | Iridium compounds in catalysis[END_REF] The catalytic rate indeed drops while the deactivation process remains unchanged. This trend is also observed here, but for the different reasons, specific to electro-assisted catalysis: we indeed observe a lower stability of the catalyst when it catalyzes the reduction of the weakest HNEt3 + acid compared to HBF4, but catalytic intermediates with distinct oxidation state are involved. In particular switching to catalytic intermediates with lower (more negative) redox potential allows activating weaker acids, but this translate in higher rate for deactivation and therefore lower stability. ), overpotential requirements were determined at the mid-wave potential of the catalytic process with 10 mM proton source [START_REF] Fourmond | H2 evolution and molecular electrocatalysts: determination of overpotentials and effect of homoconjugation[END_REF] and rate constants were extracted from initial current densities (see the supporting information and Table S10). a the constant systematic error in the determination of H2 amount due to the online detection method used here translates into lower faradic efficiencies (%) for lower catalytic current.

b additional simulations were performed to estimate the uncertainty on the fit performed to get the tchem values.

Conclusion

Polypyridyl ligands have proved instrumental in the design of stable and water-soluble Cobased H2-evolving catalysts. [START_REF] Mckone | Earth-abundant hydrogen evolution electrocatalysts[END_REF][START_REF] Queyriaux | Recent developments in hydrogen evolving molecular cobalt(II)-polypyridyl catalysts[END_REF] The cobalt(II) polypyridyl complex [Co(bapbpy)Cl] + , Co II L, is one of the rare catalysts for H2 evolution that features both a redox-active moiety and proton relays, an original combination of features in the Co polypyridyl family of catalysts. Depending on the acidity of the medium and the operating potential, one or both these features can be exploited to promote catalytic activity, resulting in four identified pathways for H2 evolution involving either metal-centered and/or ligand-assisted processes. Importantly, we also established a direct relationship between the stability of the catalytic system and the operating potential required to generate the active species, showing that stable intermediates not only confer low overpotential requirement but also higher stability. As Co II L is active for H2 evolution in aqueous media, [START_REF] Queyriaux | A noble metal-free photocatalytic system based on a novel cobalt tetrapyridyl catalyst for hydrogen production in fully aqueous medium[END_REF] future work in our group will target the implementation of such a catalytic platform into dye-sensitized photoelectrochemical cells for overall water splitting. 84- 85

ASSOCIATED CONTENT

The following files are available free of charge.

Electronic supporting information (PDF): experimental and computational section, additional spectroscopic, electrochemical and structural data, methodology used to extract kinetic and thermodynamic information from electrochemical and spectroscopic data.

Figure 1 .

 1 Figure 1. Structure of [Co(bapbpy)Cl] + (Co II L) in [Co(bapbpy)Cl]Cl . MeOH with thermal ellipsoids at the 70% probability level. Chloride counterion and non-coordinating solvent molecules were omitted for sake of clarity.

Figure 2 .

 2 Figure 2. Cyclic voltammograms of Co II L, (1 mM, red trace) and [Zn(bapbpy)Cl] + (1 mM, grey trace) recorded in DMF (0.1 M nBu4NBF4) at a glassy carbon electrode and 100 mV.s -1 .

Figure 3

 3 Figure 3 Cyclic voltammograms of Co II L (1 mM) recorded in the absence (blue trace) or in the presence of HNEt3BF4: 1 equivalent (green trace), 3 equivalents (yellow trace), 5 equivalents (orange trace) and 10 equivalents (red trace) in DMF (0.1 M nBu4NBF4) at a glassy carbon electrode and 100 mV.s -1 .

Figure 4 .

 4 Figure 4. Cyclic voltammograms of Co II L (1 mM) recorded in the absence (blue trace) or in the presence of HBF4: 2 equivalent (green trace), 6 equivalents (yellow trace) and 10 equivalents (red trace) in DMF (0.1 M nBu4NBF4) at a glassy carbon electrode and 100 mV.s -1 .

Figure 5 .

 5 Figure 5. DFT-calculated structures of the relevant bis-protonated Co III H(LH + ) and Co II H(LH + ) intermediates. For sake of clarity, aromatic hydrogens have been omitted.

Figure 7 .

 7 Figure 7. Catalytic Tafel plots relating turnover frequency (s -1 ) and driving force of H2 evolution of Co II L in the presence of 1M HNEt3 + (bold green trace) or 1 M HBF4 (bold red and purple traces for the processes observed at -1.65 V and -1.35 V vs Fc + /Fc, respectively) with previously reported H2evolving electrocatalysts: [Co(dmgH)2(py)] (grey trace), [Ni(P2 Ph N2 C6H4CH2P(O)(OEt)2 )2] 2+ (blue trace) and [Fe(TPP)Cl] (orange trace). The colored stars indicate TOF values calculated from CPE experiments.

Table 1 .

 1 Bulk electrolysis experiments for H2 evolution catalyzed by Co II L (1 mM) in DMF (0.1 M n-

	Bu4NBF4) in the presence of 100 mM acid, coupled to gas chromatography analysis (determination of
	total TON). Working electrode: Hg pool (1.23 cm 2 ). Counter-electrode: Pt wire. Reference electrode:
	Ag/AgCl. tchem values were determined using equation 1 to fit the charge passed over the course of bulk
	electrolysis experiments (Figures S14-16

ACKNOWLEDGEMENTS. This work was supported by the European Research Council and European Commission's Seventh Framework Program (FP7/2007-2013) under grant agreement n° 306398 (project PhotocatH2ode) and the French National Research Agency (Labex Program, ArCANE, ANR-11-LABX-0003-01 and CBH-EUR-GS, ANR-17-EURE-0003). We also thank Dr. Nicolas Kaeffer for fruitful discussion regarding this study.

AUTHOR INFORMATION

Corresponding Authors

nicolas.queyriaux@lcc-toulouse.fr vincent.artero@cea.fr

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.


R A Dana
email: dana@ceremade.dauphine.fr

C Le Van
email: levan@univ-paris1.fr

Overlapping sets of priors and the existence of efficient allocations and equilibria for risk measures

Keywords: Overlapping sets of priors, collective absence of arbitrage, equilibria with short-selling, measures of risk. JEL Classification: C62, D50

The overlapping expectations and the collective absence of arbitrage conditions introduced in the economic literature to insure existence of Pareto optima and equilibria with short-selling when investors have a single belief about future returns, is reconsidered. Investors use measures of risk. The overlapping sets of priors and the Pareto equilibrium conditions introduced by Heath and Ku for coherent risk measures are respectively reinterpreted as a weak no-arbitrage and a weak collective absence of arbitrage conditions and shown to imply existence of Pareto optima and Arrow Debreu equilibria.

Introduction

The problem of the existence and characterization of Pareto optima and equilibria in markets with short-selling, has recently been addressed by Barrieu and El Karoui [START_REF] Barrieu | Inf-convolution of risk measures and optimal risk transfer[END_REF], Jouini et al [START_REF] Jouini | Optimal risk sharing for law invariant monetary utility functions[END_REF], Filipovic and Kupper [START_REF] Filipovic | Equilibrium Prices for monetary utility functions[END_REF] and Burgert and Rüschendorf [START_REF] Burgert | On the optimal risk allocation problem[END_REF] for convex measures of risk in infinite markets. Existence of an equilibrium for finite markets with short-selling is an old problem in the economic literature. It has first been considered in the early seventies by Grandmont [START_REF] Grandmont | Temporary General Equilibrium Theory[END_REF], Hart [START_REF] Hart | On the Existence of an Equilibrium in a Securities Model[END_REF] and Green [START_REF] Green | Temporary General Equilibrium in a Sequential Trading Model with Spot and Future Transaction[END_REF] since Debreu's standard theorems on existence of equilibrium could not be applied, investors' sets of portfolios being unbounded below. In these early papers, investors were assumed to hold a single homogeneous or heterogeneous probabilistic belief and be von Neumann-Morgenstern risk averse utility maximizers. Two sufficient conditions for existence of an equilibrium were given: -the overlapping expectations condition which expresses that investors are sufficiently similar in their beliefs and risk tolerances so that there exists a non empty set of prices (the no-arbitrage prices) for which no agent can make costless unbounded utility nondecreasing purchases -the no unbounded utility arbitrage condition, a collective absence of arbitrage condition, which requires that investors do not engage in mutually compatible, utility nondecreasing trades. These conditions have later been weakened and shown to be equivalent under adequate conditions and under further assumptions, necessary for existence of equilibrium (see e.g. Page [START_REF] Page | Arbitrage and Asset Prices[END_REF], Page and Wooders [START_REF] Page | A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium[END_REF]). They have been generalized to abstract economies (see Werner [START_REF] Werner | Arbitrage and the existence of competitive equilibrium[END_REF] and Nielsen [START_REF] Nielsen | Asset market equilibrium with short-selling[END_REF]). Other sufficient conditions were given. For a review of the subject in finite dimension, see Allouch et al [START_REF] Allouch | The geometry of arbitrage and the existence of competitive equilibrium[END_REF], Dana et al [START_REF] Dana | On the different notions of arbitrage and existence of equilibrium[END_REF], Page [START_REF] Page | On equilibrium in Hart's securities exchange model[END_REF], [START_REF] Page | Inconsequential arbitrage[END_REF]. The theory has also been developed for infinite markets but the conditions given above do not generalize to the infinite dimension (see for example Brown and Werner [START_REF] Brown | Arbitrage and Existence of Equilibrium in Infinite Asset Markets[END_REF]) and it is in general difficult to provide sufficient conditions on the primitives of an economy to have an equilibrium.

This paper provides sufficient conditions for existence of Pareto optima and equilibria when agents use convex measures of risk in finite markets with shortselling. In contrast, with the papers of Barrieu and El Karoui [START_REF] Barrieu | Inf-convolution of risk measures and optimal risk transfer[END_REF] who deal with families of ρ-dilated risk measures and Jouini et al [START_REF] Jouini | Optimal risk sharing for law invariant monetary utility functions[END_REF] who consider law invariant convex monetary utilities, it makes no specific assumptions on the risk measures. However it assumes that there is a finite number of states of the world and uses finite dimensional convex analysis techniques. It builds on one hand, on the economic literature on equilibrium with short-selling and on the other hand on a paper by Heath and Ku [START_REF] Heath | Pareto Equilibrium with coherent measures of risk[END_REF]. Heath and Ku [START_REF] Heath | Pareto Equilibrium with coherent measures of risk[END_REF]introduced a condition now on denoted HKPE that they called the Pareto equilibrium con-dition which requires that if investors do engage in mutually compatible, utility nondecreasing trades, then those trades do not increase their utilities. They showed, for a subclass of measures of risk, the equivalence between HKPE and an overlapping sets of priors condition (see their proposition 4.2). They however did not address the question of existence of Pareto optima and equilibria. This paper makes two main contributions. The first is to relate Heath and Ku's HKPE and overlapping sets of priors condition to a weakening of the no unbounded utility arbitrage condition and to a weakening of the no-arbitrage price condition mentionned above. The second is to show that these conditions are sufficient conditions for existence of Pareto optima and Arrow Debreu equilibria. Two types of proof are provided, one uses the sup-convolution, the other applies standard results in the theory of equilibrium with short-selling. Following Heath and Ku [START_REF] Heath | Pareto Equilibrium with coherent measures of risk[END_REF], the case of constraints is also considered.

The paper is organized as follows. Section 2 presents the model and recall concepts in equilibrium theory. Two concepts of Pareto optima are introduced, one for complete preferences represented by measures of risk, the other for incomplete preferences associated to agents' priors. Section 3 contains the main results of the paper, the equivalence between HKPE and an overlapping sets of priors condition. These conditions are then shown to be sufficient for existence of an efficient allocation. A first proof of existence of efficient allocations is given by using the sup-convolution. Finally necessary conditions for existence of an efficient allocation are given. Section 4 relates the overlapping sets of priors condition and HKPE to the theory of arbitrage and equilibrium. Another proof of existence of efficient allocations and equilibria based on general equilibrium techniques is provided. Section 5 deals with the case of constraints on trades.

The model

We consider a standard Arrow-Debreu model of complete contingent security markets. There are two dates, 0 and 1. At date 0, there is uncertainty about which state s from a state space Ω = {1, ..., k} will occur at date 1. At date 0, agents trade contingent claims for date 1. The space of contingent claims is the set of random variables from Ω → R. The random variable X which equals x 1 in state 1, x 2 in state 2 and x k in state k, is identified with the vector X = (x 1 , . . . , x k ). Let = {π ∈ R k + :

k s=1 π s = 1} be the prob- ability simplex in R k and π ∈ . We note E π (X) := k l=1 π l x l and for p ∈ R k , p • X := k l=1 p l x l .
There are m agents indexed by i = 1, . . . , m. Agent i has an endowment

E i ∈ R k of contingent claims. Let E = m i=1 E i denote aggregate endowment.
We assume that each agent has a preference order i over R k represented by a monetary utility function V i where we recall that Definition 1 A function V : R k → R is a monetary utility function if it is concave monotone and has the cash invariance property

V (X + C) = V (X) + C, for any X ∈ R k , C constant
A positively homogeneous monetary utility function is a monetary utility function that is positively homogeneous.

Monetary utility functions can be identified with convex measures of risk (see Föllmer and Schied [START_REF] Föllmer | Stochastic finance. An introduction in discrete time[END_REF]) and positively homogeneous monetary utility functions with coherent risk measures(see Artzner et al [START_REF] Artzner | Coherent measures of risk[END_REF]) by defining ρ = -V . We recall that monetary utility functions have the following representation

V (X) = min π∈ E π (X) + c(π) (1)
where

c(π) = sup X∈R k V (X) -E π (X) ∈ R ∪ {+∞} (2)
which is convex, lower semi-continuous, is the conjuguate function of V . Let

P = dom c = {π ∈ | c(π) < ∞} (3)
be the set of effective priors associated with V . Clearly, we also have:

V (X) = min π∈P E π (X) + c(π) (4)
Positively homogeneous monetary utility functions are obtained when c is an indicator function δ P (in other words, c(π) = 0 if π ∈ P and c(π) = ∞ otherwise). In that case, P = {π ∈ : c(π) = 0} is a convex compact subset of and we have V (X) = min π∈P E π (X).

We next recall standard concepts in equilibrium theory. An allocation

(X i ) m i=1 ∈ (R k ) m is attainable if m i=1 X i = E. A trade (W i ) m i=1 ∈ (R k ) m is feasible if m i=1 W i = 0.

The set of individually rational attainable allocations A is defined by

A = (X i ) m i=1 ∈ (R k ) m | m i=1 X i = E and V i (X i ) ≥ V i (E i ) for all i .
Definition 2 An attainable allocation (X i ) m i=1 is Pareto optimal if there exists no feasible trade

(W i ) m i=1 such that V i (X i + W i ) ≥ V i (X i
) for all i with a strict inequality for some i. It is individually rational Pareto optimal if it is Pareto optimal and

V i (X i ) ≥ V i (E i ) for all i. Definition 3 A pair (X * , p * ) ∈ A × R k \{0} is a contingent Arrow-Debreu equilibrium if 1. for each agent i and X i ∈ R k , V i (X i ) > V (X i * ) implies p * • X i > p * • X i * , 2. for each agent i, p * • X i * = p * • E i .
Assertions 1 and 2 express that X i * solves investor's i maximization problem at price p * . Markets clear since X * is attainable.

We also define a weaker concept of Pareto optimality under incomplete preferences. Let P be a set of priors. Consider the following incomplete preferences on pairs (X, Y ) ∈ R k × R k defined by

X P Y iff E π (X) ≥ E π (Y ) for all π ∈ P (5)
Given P = (P i ) m i=1 a family of set of priors, an attainable allocation (X i ) m i=1 is P-Pareto optimal if there exists no feasible trade (W i ) m i=1 such that E π (X i + W i ) ≥ E π (X i ) for all i and all π ∈ P i with a strict inequality for some i and some π ∈ P i . Equivalently: HKPE: there exists no feasible trade (W i ) m i=1 such that E π (W i ) ≥ 0 for all i and all π ∈ P i with a strict inequality for some i and some π ∈ P i . Hence, for m incomplete preferences defined by ( 5), either HKPE is fulfilled and any attainable allocation is P-Pareto optimal or HKPE is not fulfilled and there exists no P-Pareto optimal allocation.

Existence of efficient allocations and equilibria

Given a convex subset A ⊆ R p , the relative interior of A, ri A, is the interior which results when A is regarded as a subset of its affine hull aff A.

Heath and Ku's Pareto equilibrium condition

Heath and Ku [START_REF] Heath | Pareto Equilibrium with coherent measures of risk[END_REF] introduced the HKPE condition for a subclass of risk measures and called it the Pareto equilibrium condition. They showed the equivalence between HKPE and the non-emptyness of the intersection of the relative interiors of agents' sets of priors (see their proposition 4.2). They however did not address the question of existence of Pareto optima and equilibria in the sense of definitions 2 and 3. The next theorem which contains the main result of the paper, may be viewed as an elaboration of Heath and Ku's [START_REF] Heath | Pareto Equilibrium with coherent measures of risk[END_REF] proposition 4.2. It establishes that HKPE is a sufficient condition for existence of a Pareto allocation or equivalently of an equilibrium for monetary utilities.

Theorem 1 Let V i fulfill (4) for each i. Then the following are equivalent:

1. ∩ i ri P i = ∅,
2. there exists no feasible trade W 1 , . . . , W n , with E π (W i ) ≥ 0 for all π ∈ P i and all i with a strict inequality for some i and π ∈ P i , 3. Any attainable allocation is P-Pareto optimal. Any of the previous assertions imply any of the following assertions:

4. there exists an individually rational Pareto optimal allocation, 5. there exists an equilibrium.

The equivalence between 2 and 3 follows from the definition of P-Pareto optima, that between 1 and 2 is proven in lemma 1 below. 5 implies 4 follows from the first welfare theorem. In the remainder of the paper, we shall provide two types of proofs for 1 implies 4 or 5. The first uses the sup-convolution, the second builds on general equilibrium techniques.

Lemma 1 Let (P i ) m
i=1 be a family of convex sets of probabilities. Then the following are equivalent:

1. ∩ i ri P i = ∅,
2. there exists no feasible trade W 1 , . . . , W n , with E π (W i ) ≥ 0 for all π ∈ P i and all i with a strict inequality for some i and π ∈ P i .

Proof : We first have that ∩

i ri P i = ∅ iff cone ∩ i ri P i = ∅, equivalently iff cone ∩ i ri P i = ∩ i cone ri(P i ) = ∩ i ri cone(P i ) = ∅,
the last equality following from Rockafellar's [20] corollary 6.6.1. Let (f i ) m i=1 be a family of convex functions with domain ri coneP i for i = 1, . . . , m. From Rockafellar's [START_REF] Rockafellar | Convex Analysis[END_REF] corollary 16.2.2, the condition

∩ i ri cone P i = ∅ is equivalent to the inexistence of a feasible trade (Z i ) m i=1 such that i sup π∈P i ,λ i ≥0 λ i E π i (Z i ) ≤ 0 (6) i sup π∈P i ,λ i ≥0 λ i E π i (-Z i ) > 0 (7)
Since ( 6) is equivalent to E π i (Z i ) ≤ 0, for all i and π ∈ P i , ∩ i ri P i = ∅ is thus equivalent to the inexistence of a feasible trade (Z i ) m i=1 such that E π i (Z i ) ≤ 0, for all i and π ∈ P i with a strict inequality for some i and π ∈ P i which is assertion 2.

Corollary 1 Let V i fulfill (4 ) for all i and P i be independent of i. Then there exists an individually rational Pareto optimal allocation and an equilibrium.

Proof : Let P denote the common set of priors. Since P is convex, ri P = ∅.

Sup-convolution

We now provide a proof based on the sup-convolution. This approach to efficient sharing has been used by Barrieu and El Karoui [START_REF] Barrieu | Inf-convolution of risk measures and optimal risk transfer[END_REF], Filipovic and Kupper [START_REF] Filipovic | Equilibrium Prices for monetary utility functions[END_REF], Jouini et al [START_REF] Jouini | Optimal risk sharing for law invariant monetary utility functions[END_REF], Burgert and Rüschendorf [START_REF] Burgert | On the optimal risk allocation problem[END_REF] in an infinite dimensional framework. As is well known, from the monetary invariance, an attainable allocation is Pareto optimal for aggregate endowment E if and only if it solves the following problem:

sup m i=1 V i (X i ) subject to m i=1 X i = E. For X ∈ R k , let i V i (X) = sup m i=1 V i (X i ), m i=1 X i = X be the sup-convolution of the V i . Since V i is finite for every i, i V i (X) > -∞ and dom i V i = R k if and only if ∩ i dom c i = ∩ i P i = ∅.
In that case, i V i is a monetary utility (the representative agent's utility when aggregate endowment is X) and i V i and m i=1 c i are conjuguate. Furthermore, from Rockafellar's theorem 16.4 [START_REF] Rockafellar | Convex Analysis[END_REF], a sufficient condition for existence of a Pareto optimum (X 1 , . . . , X m ) is that

∩ i ri dom c i = ∩ i ri P i = ∅. (8)
We have thus proven that assertion 1 in theorem 1 implies assertion 4. Let us now show, that assertion 4 implies existence of an equilibrium. The proof provided does not use a fixed point theorem, contrary to the standard proofs of existence.

Let us first remark that π

∈ ∂ i V i (X) iff π ∈ ∩ i ∂V i (X i ) for any Pareto optimum (X 1 , . . . , X m ) associated with X. Indeed, π ∈ ∂ i V i (X) iff i V i (X) = m i=1 c i (π) + E π (X). Since i V i (X) = m i=1 V i (X i ) for any Pareto optimum (X 1 , . . . , X m ) associ- ated with X and c i (π) + E π (X i ) -V i (X i ) ≥ 0, for all π ∈ , we obtain that V i (X i ) = c i (π) + E π (X i ) for all i, equivalently, π ∈ ∩ i ∂V i (X i ).
Therefore, a pair ((

X * i ) m i=1 , p * ) ∈ A × R k \{0} is a contingent Arrow-Debreu equilibrium, when aggregate endowment is E iff 1. (X * i ) m i=1 is Pareto optimal, 2. p * ∈ λ∂ i V i (E) for some λ > 0, 3. p * • X * i = p * • E i for all i.
As remarked by Filipovic and Kupper [START_REF] Filipovic | Equilibrium Prices for monetary utility functions[END_REF], given a Pareto optimum (X 1 , . . . , X m ) and any p ∈ λ∂ i V i (E), we have that (

X 1 + p • (E 1 -X 1 ), . . . , X m + p • (E m - X m ), p) is an equilibrium. Indeed, (X 1 + p • (E 1 -X 1 ), . . . , X m + p • (E m -X m )) is Pareto optimal since m i=1 V i (X i + p • (E i -X i )) = m i=1 V i (X i ) + m i=1 p • (E i -X i ) = m i=1 V i (X i )
since (X 1 , . . . , X m ) is attainable. By construction p fulfills assertion 2 and

p • (X i + p • (E i -X i )) = p • E i for all i.

Necessary conditions for existence

Theorem 1 provides sufficient conditions for existence of efficient allocations (or of an equilibrium). We next give necessary conditions for existence of an efficient allocation .

Proposition 1 Let V i fulfill (4 ) for each i. If there exists an efficient allocation, then 1.

∩ i P i = ∅,
2. there exists no feasible trade W 1 , . . . , W n fulfilling E π (W i ) > 0 for all π ∈ P i and for all i.

Proof : To prove assertion one, if X * is efficient, then for every i, there exists λ i > 0, such that ∩ i λ i δV i (X i * ) = ∅. As δV i (X i * ) ⊆ P i , for each i, there exists π i ∈ P i such that λ i π i is independent of i. Hence λ i and π i are independent of i and π ∈ ∩ i P i = ∅ as was to be proven. To show the second, if there exists a feasible trade W 1 , . . . , W n fulfilling E π (W i ) > 0 for all π ∈ P i and for all i,

then V i (X i + W i ) > V i (X i
) for all i contradicting the existence of an efficient allocation Remark 1 1. Assertions 1 and 2 of proposition 1 are weaker than assertions 1 and 2 of theorem 1.

2. If V i is coherent for any i, then P i is convex compact for any i. From Samet [START_REF] Samet | Common priors and separation of convex sets[END_REF], assertions 1 and 2 of proposition 1 are then equivalent.

Let us consider the expected utility case with a common prior.

Corollary 2 Let V i fulfill (4 ) for each i. Let P i = {π i } for all i. Then there exists an equilibrium if and only if π i is independent of i.

Proof : The sufficient condition follows from corollary 1 while the necessary condition follows from proposition 1.

Relation with equilibrium with short-selling

In this section, we provide an alternate proof of existence of an equilibrium based on equilibrium with short-selling techniques. We thus recall a number of standard concepts. We first define and characterize for monetaries utilities the useful and useless trading directions. We next define the concept of weak no-arbitrage price as a price giving strictly positive value to any useful and not useless vector. We finally show that HKPE is the concept of collective absence of arbitrage introduced by Hart [START_REF] Hart | On the Existence of an Equilibrium in a Securities Model[END_REF].

Useful vectors

Let C ⊆ R k be a non-empty convex set. The asymptotic cone of C is the set

{W ∈ R k | X + λW ∈ C, for all X ∈ C and λ ≥ 0}.
Let V be a monetary utility and X ∈ R k . Let

Q(X) = {Y ∈ R k | V (Y ) ≥ V (X)}
be the preferred set at X and let R(X) be its asymptotic cone. Since V is concave, by Rockafellar's theorem 8.7 in [START_REF] Rockafellar | Convex Analysis[END_REF], R(X) is independent of X and will simply be denoted by R. It is called the set of useful vectors for V in the economic literature. We thus have

R = W ∈ R k | V (λW ) ≥ V (0), for all λ ≥ 0 .
The lineality space of V or set of useless vectors is defined by

L = {W ∈ R k |V (λW ) ≥ V (0), for all λ ∈ R} = R ∩ (-R).
We first characterize R and L.

Proposition 2 We have

R = {W ∈ R k | E π (W ) ≥ 0, for all π ∈ P } L = {W ∈ R k | E π (W ) = 0, for all π ∈ P } Proof : Let W fulfill E π (W ) ≥ 0 for all π ∈ P . Then V (λW ) = min π∈P E π (λW ) + c(π) ≥ min π∈P c(π) = V (0) for all λ ≥ 0 which implies that W ∈ R. Conversely, let W ∈ R. Then V (λW ) ≥ V (0), for all λ ≥ 0, hence E π (λW ) + c(π) ≥ V (0)
, for all λ ≥ 0, π ∈ P . For a fixed π ∈ P , the map from

R + into R, λ → λE π (W )) is bounded below, hence E π (W ) ≥ 0.
The other assertion is straightforward.

In the following subsections, R i and L i will denote respectively the set of useful and useless vectors for agent i.

Concepts of absence of arbitrage

A no-arbitrage price for agent i is a price giving strictly positive value to any useful vector for i. As the existence of a no-arbitrage price for i is incompatible with the existence of a useless vector for i, we use a weaker no-arbitrage concept due to Werner [START_REF] Werner | Arbitrage and the existence of competitive equilibrium[END_REF]. Definition 4 A price vector p ∈ R k is a " weak no-arbitrage price" for agent i if p • W > 0 for all W ∈ R i \ L i . A price vector p ∈ R k is a "weak no-arbitrage price" for the economy if it is a weak no-arbitrage price for each agent.

The polar of A is defined by A 0 = {p ∈ R p | p • X ≤ 0, for all X ∈ A}.

Let S i w denote the set of weak no arbitrage prices for i and ∩ i S i w the set of weak no arbitrage prices for the economy. We have:

Proposition 3 Let V i fulfill (4) for each i. Then 1. S i w = ri -(R i ) 0 = cone ri P i .
2. The set of weak no arbitrage prices for the economy is

∩ i S i w = cone ∩ i ri P i . Proof : From proposition 2, R i = {W ∈ R k | E π (W ) ≥ 0, for all π ∈ P i }. From Allouch et al, lemma 2, S i w = ri -(R i ) 0 . Therefore S i w = ri -(R i ) 0 = ri cl cone P i = ri cone P i = cone riP i , (9)
the third and fourth equalities following from Rockafellar's [START_REF] Rockafellar | Convex Analysis[END_REF] theorem 6.3 and corollary 6.6.1. Hence the set of weak no arbitrage prices for the economy is

∩ i S i w = ∩ i cone (riP i ) = cone ∩ i riP i .
We now turn to a concept of collective absence of arbitrage introduced by Hart [START_REF] Hart | On the Existence of an Equilibrium in a Securities Model[END_REF] which requires that any utility nondecreasing feasible trade be useless. Using Rockafellar's corollary 6.6.2., we obtain that

∩ i S i w = ∩ i ri (R M i ) 0 = ∩ i (riconeP i + (M i ) ⊥ ) = ∩ i cone (riP i + (M i ) ⊥ ) hence, ∩ i S i w = ∅ iff ∩ i cone (riP i + (M i ) ⊥ ) = ∅ (10)
Since ( 10) is positively homogeneous, let H = {m ∈ R k | j m j = 1}. The set of weak no-arbitrage price is non empty if and on if there exists µ ∈ H such that µ = λ i π i + m i ⊥ with π i ∈riP i and λ i > 0 and m i ⊥ ∈ (M i ) ⊥ . The vector µ may be interpreted as a signed measure and we have

E µ (X i ) = λ i E π i (X i ), for all X i ∈ i and i (11)
with π i ∈riP i , λ i > 0. Hence the restriction of µ to M i is a non-negative measure proportionnal to a prior in the relative interior of P i . Furthermore,

• if agent i can trade the riskless asset or equivalently if constants belong to M i , then λ i =< µ, 1 >= 1.

• If all agents can trade the riskless asset, then λ i is independent of i. (11) may be rewritten as: there exists a signed measure µ and probabilities π i in the relative interior of P i for each agent such that

E µ (X i ) = E π i (X i ), for all X i ∈ M i and i (12)
• If all agents can trade the riskless asset and if M i = R k for some i, then µ is a probability measure and (11) holds true.

Remark 2 1. Condition ( 10) is equivalent to the WNMA condition: there exists no feasible trade W 1 , . . . , W n , with W i ∈ M i for all i and E π (W i ) ≥ 0 for all π ∈ P i and all i with a strict inequality for some i and π ∈ P i .

The condition

µ = λ i π i + m i
⊥ for all i is very similar to the condition one obtains when writing the no-arbitrage condition for finite financial markets with constraints on portfolios.

Let us summarize the results obtained in a proposition: Proposition 5 Let V i fulfill (4) and agent's i trading set be the subspace M i for each i. Then the following are equivalent:

1. there exists a signed measure µ and positive constants λ i and probabilities π i ∈riP i such that (11) holds true, 2. there exists no feasible trade W 1 , . . . , W n , with W i ∈ M i for all i and E π (W i ) ≥ 0 for all π ∈ P i and all i with a strict inequality for some i and π ∈ P i . Any of the previous assertions implies the existence of efficient allocations or of an equilibrium.

The next two examples show that we cannot dispense with the λ i , i = 1, . . . , m in [START_REF] Green | Temporary General Equilibrium in a Sequential Trading Model with Spot and Future Transaction[END_REF] even if some agent is unconstrained contradicting Heath and Ku's proposition 5.1 and corollary 5.1.

Example 1

There are two states and three agents. Each agent has a unique probability over states: agent 1 has probability π 1 = ( 1 4 , 3 4 ), agent 2 probability π 2 = ( 3 4 , 1 4 ) and agent 3, probability π 3 = (1, 0). Assume that the trading sets are M 1 = {X 1 = (x 1 , x 1 ) | x 1 ∈ R}, M 2 = {X 2 = (x 2 , -x 2 ) | x 2 ∈ R}, M 3 = {X 3 = (x 3 , 0) | x 3 ∈ R}. HKPE is fulfilled since R M 1 = {W 1 = (w, w) | w ≥ 0}, R M 2 = {W 2 = (w, -w) | w ≥ 0} and R M 3 = {W 3 = (w, 0) | w ≥ 0} and i W i = 0 implies W i = 0 for all i. But there exists no solution µ = (µ 1 , µ 2 ) to the following system:

E µ (X 1 ) = (µ 1 + µ 2 )x 1 = E π 1 (X 1 ) = x 1 , for all x 1 ∈ R, E µ (X 2 ) = (µ 1 -µ 2 )x 2 = E π 2 (X 2 ) = 1/2x 2 ,
for all x 2 ∈ R, E µ (X 3 ) = µ 1 x 3 = E π 3 (X 3 ) = x 3 , for all x 3 ∈ R.

since the first and the third equations imply µ 1 = 1, µ 2 = 0 which is incompatible with the second equation. Example 2. There are three agents. The state space and the probabilities are as in Example 1 as well as the trading sets of agents 1 and 2. The trading set of agent 3 is M 3 = R 2 . Hence, R M 3 = {W 3 = (w 3 1 , w 3 2 ) | w 3 1 ≥ 0}. As in the previous example, HKPE is fulfilled. However, there exists no solution µ = (µ 1 , µ 2 ) to [START_REF] Green | Temporary General Equilibrium in a Sequential Trading Model with Spot and Future Transaction[END_REF] with λ i = 1 for all i. This last example provides a counter-example to Heath and Ku's corollary 5.2. When there are constraints, we loose the equivalence between assertions 1 and 2 of theorem 1. Example 3. The state space and the probabilities are as in Example 1, the trading sets are

M 1 = M 2 = M 3 = {(x, x) | x ∈ R}. Since R M 1 = R M 2 = R M 3 = {(w, w) | w ≥ 0}
, HKPE is fulfilled but the intersection of the sets of priors is empty.

support of the ANR project "Croyances" and of the Fondation du Risque, Chaire Groupama, "Les particuliers face au risque".

 Definition 5The economy satisfies the Weak No-Market-Arbitrage condition (WNMA) if i W i = 0 and W i ∈ R i for all i implies W i ∈ L i , for all i.

The following proposition follows directly from proposition 2.

Proposition 4 Let V i fulfill (4) for each i. Then the economy satisfies WNMA if there exists no feasible trades W 1 , . . . , W n with E π (W i ) ≥ 0 for all π ∈ P i and all i with a strict inequality for some i and π ∈ P i .

HKPE is therefore the same concept as WNMA.

Let us now prove theorem 1. Assertion 1 implies 5 follows from Allouch et al [START_REF] Allouch | The geometry of arbitrage and the existence of competitive equilibrium[END_REF] theorems 4 and 5, assertion 5 implies 4 from the first welfare theorem. Assertion 4 implies 5 was proven in section 3.3.

Constraints on exchanges 5.1 The model

Heath and Ku [START_REF] Heath | Pareto Equilibrium with coherent measures of risk[END_REF], Filipovic and Kupper [START_REF] Filipovic | Equilibrium Prices for monetary utility functions[END_REF] and Burgert and Rüschendorf [START_REF] Burgert | On the optimal risk allocation problem[END_REF] have considered constraints on exchanges when agents use measures of risk.

We now assume that trades are only possible in linear subspaces

The definitions of attainable, individual rational and Pareto optimal allocations and equilibria are extended by imposing the constraint that X i ∈ M i for all i. In particular, the set of constrained useful vectors for i is defined as

is the unconstrained set of useful vectors for i.

Weak no-arbitrage prices under constraints

For a subset M ⊆ R k , let M ⊥ be its orthogonal. In order to characterize weak no-arbitrage prices for this new economy, let us first characterize R 0 M i the polar of the set of constrained useful vectors for i. From Rockafellar's corollary 16.4.2., (R

and from Rockafellar's theorem 6.3 and corollary 6.6.2.

ri (R M

Sup-convolution and constraints

Assuming to simplify that agents can all trade the riskless asset, let us return to the inf-convolution's approach. Define for each i

The function V M i : R k → R ∪ {-∞} is concave, upper semi-continuous, cash invariant but fails to be monotone. We may still use duality methods but the domain of the conjuguate function is larger than the probability simplex. Let m ∈ R k and

be the conjuguate of V M i . Clearly we have

From the cash invariance of V M i , we also have

and m i=1 c M i are conjuguate. From Rockafellar's theorem 16.4 [START_REF] Rockafellar | Convex Analysis[END_REF], a sufficient condition for existence of a Pareto optimum (X 1 , . . . , X m ) is that

We are thus back to the weak no-arbitrage condition [START_REF] Grandmont | Temporary General Equilibrium Theory[END_REF].