Sarah Samadi

Nicolas Puillandre

Eric Pante

Marie-Catherine Boisselier

Wei-Jen Chen

Laure Corbari

Philippe Maestrati

Ralph Mana

Justine Thubaut

Dario Zuccon

Stéphane Hourdez

Patchiness of deep-sea communities in Papua New Guinea and potential susceptibility to anthropogenic disturbances illustrated by seep organisms

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Despite some significant exploration efforts notably during the last decade (e.g. [START_REF] Costello | A census of marine biodiversity knowledge, resources, and future challenges[END_REF], the deep-sea realm remains among the least known biodiversity compartments on Earth, even for macro-organisms. Because it is difficult to protect what we do not know exists, this knowledge gap makes the deep sea particularly vulnerable to human disturbances. Pioneering work in the 1960's with the development of appropriate tools revealed that deep-sea biodiversity was greatly underestimated and rivals that of shallow tropical communities (e.g. [START_REF] Hessler | Faunal diversity in the deep-sea[END_REF]. Later studies also revealed that deepsea assemblages are far from homogenous, with strong between-site differences, and some endemism [START_REF] Brandt | Deep-sea isopod biodiversity, abundance, and endemism in the Atlantic sector of the Southern Ocean--Results from the ANDEEP I-III expeditions[END_REF][START_REF] Levin | Environmental influences on regional deep-sea species diversity[END_REF]. Despite ongoing efforts to describe deep-sea biodiversity, working in this environment is challenging and much remains to be learned.

As a result of this knowledge gap, many common assumptions on the structure of deep-sea biodiversity, on which many environmental policies are based, are questionable.

Among them, the seemlingly uniform nature of the deep sea ('everything is everywhere') is potentially one of the most damaging assumptions. This assumption is mainly based on the observation that many deep-sea organisms have a cosmopolitan distribution. However, most of these cosmopolitan species are patchily distributed over their wide geographic range (McClain & Hardy, 2010 and references therein). More generally, recent studies revealed an unexpected heterogeneity of deep-sea habitats linked to geomorphological, geochemical and hydrographic features and stressed the resulting vulnerability of associated communities to human activities (e.g. [START_REF] Levin | Ecological theory and continental margins: where shallow meets deep[END_REF][START_REF] Levin | Understanding continental margin biodiversity: a new imperative[END_REF]. Thus, considering that 'everything is everywhere' minimizes the potential effect of the destruction of discrete habitats. In this communication we illustrate this problem by reporting the discovery of two chemosynthetic sites in Papua New Guinea (PNG), one of them being impacted by the tailings rejected by a nickel mine refinery.

The first chemosynthetic site was discovered during the BIOPAPUA cruise off the Sepik River [START_REF] Pante | Exploration of the deep-sea fauna of Papua New Guinea[END_REF]. The discovery of the impacted site, in the Basamuk canyon near the Astrolabe Bay, occurred during the deep-sea leg of the 'Papua Niugini Biodiversity Expedition,' (hereafter PAPUA NIUGINI) the latest expedition of 'Our planet reviewed' programme (www.laplaneterevisitee.org). This programme is an international scientific project whose goals are to provide better estimates of the magnitude of biodiversity worldwide, to provide baseline data for several sites that are particularly rich or vulnerable on our planet. The most recent expedition of the programme targeted PNG and the surrounding Bismarck and Solomon Seas. PNG is located in the Coral Triangle, a region of exceptional zooxanthellate coral diversity [START_REF] Veron | Delineating the coral triangle. Galaxea[END_REF]. Its forests count among the most important in the world in terms of diversity of plant species [START_REF] Joppa | Biodiversity hotspots house most undiscovered plant species[END_REF]. The marine part of the expedition included a shallow-water sampling workshop and a deep-sea cruise. Over 3 months, the marine expedition involved a total of 88 scientists from 18 countries, in addition to 51 trainees, media, logistics coordinators and visitors. Prior to this expedition, exploration of PNG's marine biodiversity has been minimal, and has mostly focused on shallow-water coral reefs (e.g. [START_REF] Drew | Biodiversity inventories and conservation of the marine fishes of Bootless Bay, Papua New Guinea[END_REF]) and hydrothermal vents in the Manus Basin (e.g. Levin et al. 2009). Other deep-sea habitats of PNG waters were particularly poorly explored, including during great historical expeditions [START_REF] Pante | Exploration of the deep-sea fauna of Papua New Guinea[END_REF] The BIOPAPUA cruise and the deep-sea part of the 2012 marine expedition used the RV Alis for deploying hauls from 100 to 1000 meters depth. The BIOPAPUA cruise covered a wide area surrounding the Bismarck Sea and the Solomon Sea [START_REF] Pante | Exploration of the deep-sea fauna of Papua New Guinea[END_REF]. The 2012 (PAPUA NIUGINI) cruise surveyed the deep-sea environments along the coasts of New Guinea Island in the Bismarck Sea, from the Vitiaz Strait to the border between PNG and Irian Jaya. Both expeditions aimed at covering at best the diversity of deep-sea habitats and had three main targets. First, seamounts were one of the targets because they generally offer hard bottoms with a fauna dominated in terms of biomass by suspension-feeding organisms that shelter an important diversity of associated organisms. The second goal was to sample the organisms associated with soft bottoms that can be found in accumulation basins separating islands, or, when the topography allows the accumulation of sediments, on islands slopes.

Preliminary work during the BIOPAPUA cruise allowed us to focus in 2012 the sampling of the fauna of soft bottoms in the Astrolabe Bay. Additionally, starting refinery activities had already been identified as a potential threat to the marine fauna but little biological data were available (e.g. [START_REF] Reichelt-Brushett | Risk assessment and ecotoxicology: limitations and recommendations for ocean disposal of mine waste in the Coral Triangle[END_REF]. Finally, we targeted chemosynthetic habitats as they are among the most productive environments in the deep sea. In PNG, hydrothermal vents are easily accessible, and have been intensively studied for many years (e.g. [START_REF] Erickson | Evidence for a chemoautotrophically based food web at inactive hydrothermal vents (Manus Basin)[END_REF], Levin at al. 2009). For the same reason of accessibility, the hydrothermal deposits characterized by a high grade of precious ore, in the Bismarck Sea represent a prime target for deep-sea refinery activities ( [START_REF] Dover | Mining seafloor massive sulphides and biodiversity: what is at risk?[END_REF]. Other types of deep-sea chemosynthetic environments (i.e. cold seeps and accumulations of decaying terrestrial plant debris) remain poorly surveyed in PNG, even if geological surveys in this active area provided sporadic records of cold-seep environments (e.g. [START_REF] Tappin | The Sissano, Papua New Guinea tsunami of July 1998-offshore evidence on the source mechanism[END_REF]. The BIOPAPUA cruise allowed us to discover a site off the Sepik River (facing Broken Water Bay), at about 400 meters depth, with fauna typically associated with cold-seep environments. This area, which we revisited during the 2012 cruise, is close to the area detected by the JAMSTEC cruise [START_REF] Tappin | The Sissano, Papua New Guinea tsunami of July 1998-offshore evidence on the source mechanism[END_REF] In these poorly-explored deep-sea regions, chemosynthetic organisms are perhaps the best studied, especially at vents in the Manus Basin. The two seep sites sampled in 2012, in combination with the data available from the Japanese geological surveys and the 2010 data from BIOPAPUA cruise, offered the opportunity of comparing the specific diversity of bathymodioline mussels and siboglinid tubeworms at a relatively small spatial scale (the Astrolabe Bay and Broken Water Bay sites are separated by only about 250 km). Moreover, operations off the Ramu refinery revealed the presence of a brick-red deposit. The activities of the Ramu refinery started by the end of November 2012, and tailings were announced to be released from the refinery at 200 meters depth. The deposits observed off the Ramu refinery have been found nowhere else during either the BIOPAPUA cruise or the PAPUA NIUGINI expedition. Dredging operations brought back a dark gray mud (as found in many other locations) but overlaid by a brick-red deposit, consistent with the release of tailings from the refinery, suggesting that the newly discovered site is potentially threatened by rapidly accumulating tailings from the refinery.

The bathymodiolione mussels and siboglinid worms sampled in the two discovered seep sites in PNG waters offer the opportunity of illustrating the patchiness of distribution of organisms, even at small scales, and to question some common misconceptions about the pattern of diversity in the deep sea that result from the paucity of available knowledge.

MATERIAL AND METHODS

Sampling

During the deep-sea leg of the 2012 expedition, 137 hauls (92 trawling and 45 dredging operations) were conducted. These samples were added to the ones collected the during BIOPAPUA cruise [START_REF] Pante | Exploration of the deep-sea fauna of Papua New Guinea[END_REF]. Out of all 2012 sampling operations (Fig 1), five contained fauna typically found in cold-seep environments, and in particular representatives of the Siboglinidae (Polychaeta) and Bathymodiolinae (Bivalvia). Three were conducted in Broken Water Bay, off the Sepik River (stations CP4040, CP4042, CP4043, depth range 400-800 meters), a site were cold seeps were discovered during a previous expedition in 2010 (BIOPAPUA,stations CP3703 and CP3704,, and two in the Basamuk canyon facing the Ramu refinery, near the Astrolabe Bay (stations CP4020 and CP4081, depth range 400-860 meters). During the 2012 cruise, a total of 16 Lamellibrachia (siboglinid tubeworms), and 52 bathymodioline mussels (Table 1), were preserved in 90% alcohol for molecular analyses. Out of the 11 Siboglinidae in the small aggregation sampled in the Basamuk Bay, however, only 4 provided DNA that could be amplified. These animals are thought to have been dead, burried under an accumulation of tailings. To facilitate preservation of tissues in alcohol, mussels were first microwaved following the procedure described in [START_REF] Galindo | Using microwaves to prepare gastropods for DNA Barcoding[END_REF].

DNA amplification and sequencing

DNA from mussel specimens was extracted using the QIAmp® DNA Micro Kit (Qiagen, Valencia, CA). After removal of excess ethanol, total DNA from tubeworms was isolated following a CTAB + PVPP extraction protocol [START_REF] Doyle | A rapid DNA isolation procedure for small quantities of fresh leaf tissue[END_REF]. For the Siboglinidae, only the barcode fragment of the COI gene was amplified using the Folmer primers LCO1490 and HCO2198 (Folmer et al. 1994). For the Bathymodiolinae both the barcode fragment of the COI gene and a fragment of the rDNA 28S gene were amplified using the Folmer primers and the C1 and D2 primers (Jovelin & Justine, 2001), respectively. All PCR reactions were performed in 20 µl, using 3 ng of DNA, 10X reaction buffer containing 15 mM MgCl2, 0.26 mM dNTP, 0.3 µM of each primer, 5% DMSO, 1 mg/ml BSA, and 1 unit of QBiotaq (MP Biomedicals, Illkirch Graffenstaden, France). Amplification consisted of an initial denaturation step at 95°C for 5 min, followed by 40 cycles of denaturation at 95°C for 1 min, annealing at 48°C for the COI gene and 55°C for the 28S gene for 30 s, followed by extension at 72°C for 30 s. The final extension was at 72°C for 5 min. PCR products were purified and sequenced in both forward and reverse directions by the Genoscope or the Eurofins sequencing facilities.

Phylogenetic analyses

Sequences were aligned automatically using Muscle 3.8.31 (Edgar 2004). Both maximum likelihood (ML) and Bayesian analyses (BA) were performed for the inference of phylogenetic relationships. ML analyses were performed using RAxML 7.0.4 (Stamatakis, 2006), with a GAMMAI model. Three partitions were defined for the COI gene, corresponding to each position of the codon. RaxML analyses were performed on the Cipres Science Gateway (http://www.phylo.org/portal2/) using the RAxML-HPC2 on TG Tool [START_REF] Miller | Creating the CIPRES Science Gateway for inference of large phylogenetic trees[END_REF]. Node support was estimated by bootstrapping (1000 replicates). For the Bathymodiolinae, a concatenated dataset (COI+28S) was also analysed, with four partitions defined (the three codon position of the COI, and one for the 28S). Bayesian Analyses (BA) were performed running two parallel analyses in MrBayes (Huelsenbeck et al. 2001), consisting each of four Markov chains of 30 million generations each with a sampling frequency of one tree each thousand generations. The number of swaps was set to two, and the chain temperature to 0.02. Similarly to the ML approach, unlinked models (each with six substitution categories, a gamma-distributed rate variation across sites approximated in four discrete categories and a proportion of invariable sites estimated during the analysis) were applied for each partition. Convergence of each analysis was evaluated using Tracer 1.4.1 (Rambaut and Drummond, 2007), and analyses were terminated when ESS values were all superior to 200. A consensus tree was then calculated after omitting the first 25% trees as a burn-in.

Additional sequences from GenBank and outgroups

Within the Siboglinidae, vestimentiferans constitute a monophyletic group (e.g. Eichinger at al. 2013). We thus downloaded all the COI sequences of vestimentiferans from GenBank.

Four sequences (GU059239.1, GU059172.1, GU059250.1 and GU059230.1), obviously contaminated or misidentified (they clustered within groups of specimens with different species and genus names) were removed from the dataset. To reduce computational times, 215 Ridgeia piscesae sequences (EU190494.1-EU190709.1, all highly similar) were removed from the dataset (one sequence from the popset was kept). In total 283 COI sequences from GenBank were included in the alignment (Table 1). COI sequences of Sclerolinum were used as closely related outgroups, and a sequence of Osedax roseus was used to artificially root the tree, following the results of Eichinger at al. (2013). Also, two sequences obtained from two escarpiid collected in 2010 during the BIOPAPUA cruise, collected in the Broken Water Bay, off the Sepik River (stations CP3703 and CP3704) were added to the dataset (Table 1).

For the Bathymodiolinae, we used the dataset from [START_REF] Thubaut | The contrasted evolutionary fates of deep-sea chemosynthetic mussels (Bivalvia, Bathymodiolinae)[END_REF], representative of the known diversity the sub-familly (Table 1). Modiolus modiolus was used as an outgroup.

RESULTS AND DISCUSSION

New data and collections of deep-sea habitats in PNG

Due to a very active tectonic regime, the PNG bottom topography is very uneven. Bathymetric datasets are also scarce. The 2012 deep-sea expedition largely took advantage of the deep-sea results of the BIOPAPUA cruise that both allowed us to identify the most interesting spots and provided preliminary bathymetric maps that were added to during the 2012 expedition. During each cruise, when the topology of the bottom allowed, each selected spot was explored over a depth range of 100 to 1000 meters. Depending of the topology and the kind of substrates (i.e. hard or soft bottom), either a trawl or a dredge was used. Each collection was sieved on board, using seawater to separate animals from the substrate in different size classes. Specimens were then quickly sorted and preserved. All invertebrates were then integrated into the collections of Muséum national d'histoire naturelle (MNHN) in Paris and then distributed over an international network of taxonomists for study [START_REF] Bouchet | A quater-century of deep-sea malacological exploration in the South and West Pacific: Where do we stand? How far to go?[END_REF][START_REF] Richer De Forges | The MUSORSTOM-TDSB deep-sea benthos exploration programme (1976-2012): An overview of crustacean discoveries and new perspectives on deep-sea zoology and biogeography[END_REF].

Seamounts were particularly difficult to localize and to sample, but a diversified fauna was obtained when operations were successful. On island slope, from the Sepik River to Vanimo, collections were characterized by soft bottom associated with low diversity vastly dominated in biomass by irregular sea-urchins (morphologically resembling Sarsiaster greigii, SH pers. obs.), most of which contained an Amphinomidae (Annelida) in the terminal portion of their digestive tube. Sunken plant remains associated with their typical fauna were present in most of the operations. The typical fauna associated with cold-seep environments was only found in a very restricted area off Broken Water Bay. From Manam Island to the Vitiaz Strait, i.e. the limit between the Bismark and Solomon Seas, island slopes are very steep. Consequently hauls mainly contained round stones suggesting that strong currents do not allow settlement of sessile organisms or sediment accumulation. When terrestrial plant remains were found, these were large pieces of wood, in early stages of decomposition but greatly abraded. Accumulations of sediments and/or of terrestrial plant remains were mostly found in small canyons and in large bays, notably in the Astrolabe Bay.

Fauna sampled off Broken Water Bay (Sepik River)

In a very restricted area at the Sepik site, off Broken Water Bay, at stations dominated by irregular sea urchins (cf. Sarsiaster greigii), a fauna typical of cold seeps was collected (figure 2), including Bathymodiolinae mussels and siboglinid tubeworms (sampled during both the BIOPAPUA cruise (2010) and Niugini (2012) Expedition). Irregular sea urchins are often found at seeps, although rarely properly identified (e.g. [START_REF] Cordes | Macro-ecology of Gulf of Mexico cold seeps[END_REF][START_REF] Roy | Cold seep communities in the deep eastern Mediterranean Sea: composition, symbiosis and spatial distribution on mud volcanoes[END_REF]). The other bivalves sampled at this location were typical of seep environments. Among these, Rudo Von Cosel (pers. com.) identified two species of Vesicomyidae, one species of Thyasiridae attributed to the genus Conchocele, and one species of Nuculidae attributed to the genus Acila. Among gastropods, predatory snails attributed to the genera Phymorhynchus (Raphitomidae) and Manaria sp. (Buccinidae closely related to Eosipho) are also typical of chemosynthetic environments [START_REF] Warén | Molluscs on Biogenic Substrates[END_REF], at al. 2009[START_REF] Kantor | Deep-water Buccinidae (Gastropoda: Neogastropoda) from sunken wood, vents and seeps: molecular phylogeny and taxonomy[END_REF]). Among the Nassariidae, this location was the only one where a specimen attributed to the genus Profundinassa was sampled. For the crustacean fauna, a hippolytid shrimp typical of deep-sea reducing environments, attributed to the genus Lebbeus, was identified [START_REF] Komai | Records of species of the hippolytid genus Lebbeus White, 1847 (Crustacea: Decapoda: Caridea) from hydrothermal vents in the Pacific Ocean, with descriptions of three new species[END_REF][START_REF] Nye | A new species of Lebbeus (Crustacea: Decapoda: Caridea: Hippolytidae) from the Von Damm Vent Field, Caribbean Sea[END_REF], while other specimens were not typical of chemosynthetic environments.

The ethusid crab Ethusa curvipes, a species typical of muddy deep-sea bottoms (P. Castro, pers. com.), was also found among the animals collected.

Fauna sampled below the Ramu refinery

Off the Ramu refinery, specimens were sampled from 200 to 1000 meters depth. The samples contained fauna associated with both soft bottom and plant remains. Sampling off the Ramu refinery showed a heavy release of tailings (figure 3) although refining activities had only started 10 days prior to our sampling operations. These tailings formed a layer of color and texture consistent with the known nickel refining protocols used at the Ramu refinery. The tailings overlay the otherwise grey sediment, very similar to sediment sampled elsewhere during the expedition, including the Broken Water Bay cold-seep site. Shallower operations (ca. 250-400 m) brought back sediments only slightly stained by the tailings (figure 3A), while deeper collections (400-700 m), sediments were covered by a thick layer of red sediment (figure 3B). In the impacted sediments, we notably sampled polychaete tubes that had incorporated red particles over a few centimeters. The topography of the Basamuk canyon, off the Ramu refinery rendered the operations difficult, with many instances of damage to the trawls for the deeper sites (hard bottoms). Although the catch was not very abundant in this canyon we unexpectedly captured organisms typical of cold-seep environments (Bathymodiolinae mussels and Siboglinidae tubeworms; Figure 4). As for the Broken Water Bay, other fauna typical of seep environments was identified, notably bivalves of the families Thyasiridae and Nuculidae (genus Acila). The crustacean fauna was mostly composed of species that were not specific to reducing environments (e.g. Ethusa curvipes as in the Broken Water Bay), with the exception of the hippolytid shrimp Lebbeus sp. (also found in the Broken Water Bay). The squat-lobster Munidopsis andamanica, largely distributed in the Indo-Pacific and frequently found associated with sunken wood [START_REF] Hoyoux | Wood-based diet and gut microflora of a galatheid crab associated with Pacific deep-sea wood falls[END_REF]), was also found in these samples.

Bathymodioline mussel diversity

All phylogenetic trees (COI, 28S and COI+28S) are very congruent. Specimens collected in the Broken Water Bay (Sepik) and in the Basamuk canyon (Astrolabe Bay) cluster into four distinct groups (Figure 5), some of which also include sequences from GenBank. The specimens from Broken Water Bay (Sepik station CP4042), are divided into three groups.

One well-supported clade (three specimens) is sister to Gigantidas taiwanensis. This new clade is closely related but distinct from G. taiwanensis which is only known from a shallow vent off Taiwan. The second group clusters 39 newly-collected specimens, including one collected during the BIOPAPUA cruise in the Broken Water Bay (labeled in [START_REF] Thubaut | The contrasted evolutionary fates of deep-sea chemosynthetic mussels (Bivalvia, Bathymodiolinae)[END_REF] as "Gigantidas sp 1 Broken Bay"). Finally, one specimen clustered with one specimen collected in 2010 (labeled "Gigantidas sp. 2 Broken Bay" in [START_REF] Thubaut | The contrasted evolutionary fates of deep-sea chemosynthetic mussels (Bivalvia, Bathymodiolinae)[END_REF]. Interstingly this cluster also included one sequence from a specimen collected in the Basamuk canyon (CP4042). Based on COI data, this species was also collected earlier in a site after the nearby Aitape village, at 420 m depth [START_REF] Kyuno | Dispersal and differentiation of deep-sea mussels of the genus Bathymodiolus (Mytilidae, Bathymodiolinae)[END_REF]). The 2012 expedition also explored the Aitape area but did not bring back any catch of fauna typical of seep environments. In this area, two additional undescribed Gigantidas species (assignation based on COI data) were collected deeper during the Japanese cruise (1600-1900 m, [START_REF] Kyuno | Dispersal and differentiation of deep-sea mussels of the genus Bathymodiolus (Mytilidae, Bathymodiolinae)[END_REF]). These species were not collected again during our expeditions, probably because they live deeper than the explored bathymetric range. The eight other specimens from Basamuk canyon (CP4020) cluster into a distinct group and form a well-supported group, sister to Nypamodiolus longissimus. These specimens are thus attributed to a potential new species of the genus

Nypamodiolus.

If we consider the high genetic distances found within and among these clusters, and the fact that the phylogenetic information from mitochondrial and nuclear genes is congruent, we can conclude that four putative species were found at the Broken Water Bay site and in the Basamuk canyon, and two of these were previously collected during the BIOPAPUA cruise.

Overall, the three explorations in this area (JAMSTEC cruise, BIOPAPUA cruise and PAPUA NIUGINI expedition) identified five different Gigantidas species and one Nypamodiolus species. Interestingly, only one of these species was found at all locations (including at Aitape), the five others being each restricted to a single location. Similarly, Nipponiomodiolus manusensis (new genus under description, Cosel & Thubaut) has only been found at hydrothermal vent sites in the Manus Basin and was not found in our cold-seep collections [START_REF] Thubaut | The contrasted evolutionary fates of deep-sea chemosynthetic mussels (Bivalvia, Bathymodiolinae)[END_REF].

The genus Nypamodiolus contains species that are very ecologically diverse (i.e. associated to plant remains, whale bones or seep environements over a wide bathymetric range). For example the geographic range N. longissimus coincides with that of the Nypa palm, even if it is sometimes associated with other types of wood; N. simpsoni has been collected in both the Mediterranean and in the Atlantic, over a wide range of depths, and associated with whale carcasses, sunken wood, and cold seeps [START_REF] Thubaut | The contrasted evolutionary fates of deep-sea chemosynthetic mussels (Bivalvia, Bathymodiolinae)[END_REF]. The specimens attributed to Nypamodiolus collected below the Ramu refinery, belong to an unknown species: they have never been found elsewhere in our collections (neither in the Astrolabe Bay in general nor on the abundant plant debris often collected at other sites during our expeditions).

Siboglinidae diversity

Siboglinid tubeworms were only found at two locations (Broken Water Bay and Astrolabe Bay sites). In the Astrolabe Bay, tubeworms were only found in the Basamuk canyon off the Ramu refinery. Part of COI was sequenced for a total of 11 siboglinid specimens collected from these locations. Specimens fall into three distinct lineages (Figure 6) that are not shared between locations: two of these clades are only found at the Sepik River site and the third has only been collected at the Astrolabe site during our expeditions.

Two specimens from the BrokenWater Bay collections (BIOPAPUA cruise) cluster among escarpiids, but clearly form a separate clade. Although clearly distinct, the sequences of the Broken Water Bay escarpiid species most closely resembles that of a distinct isolate of an animal identified as Paraescarpia cf. echinospica (isolate E1) collected near Japan [START_REF] Andersen | Escarpia southwardae sp. nov., a new species of vestimentiferan tubeworm (Annelida, Siboglinidae) from West African cold seeps[END_REF]). Sequences from two other specimens of Paraescarpia cf. echinospica, also collected near Japan, however form a different clade. Closer examination of the morphology of all these worms (if vouchers are available) will be necessary to determine species identities.

The sequences for five specimens of Lamellibrachia from the Broken Water Bay site most closely cluster with the sequence of a Lamellibrachia specimen collected about 325 km west of the Broken Water Bay collection site (Aitape site Lamellibrachia L6 in [START_REF] Kojima | Molecular taxonomy of vestimentiferans of the western Pacific, and their phylogenetic relationship to species of the eastern Pacific III. Alaysia-like vestimentiferans and relationships among families[END_REF]. Although distinct in sequence, two other species of Lamellibrachia are closely related to this group: L. sp. L4, from the PACMANUS vent site in the Manus Basin (about 650 km east of the Broken Water Bay site), and L. sp. L5, from the cold-seep sites of Kuroshima Knoll, about 3 000 km north-northwest of the Broken Water Bay site [START_REF] Kojima | Molecular taxonomy of vestimentiferans of the western Pacific, and their phylogenetic relationship to species of the eastern Pacific I. Family Lamellibrachiidae[END_REF].

Additional samples from the different locations will be necessary to futher study this lineage and determine whether there is genetic continuity between the groups identified.

Sequences from Basamuk canyon specimens all cluster with different haplotypes of Lamellibrachia juni, including four specimens collected from the DESMOS hydrothermal vent site in the Manus Basin, about 770 km to the East. However, some genetic structure exists in this clade, and the haplotype network produced for this lineage (Figure 7) revealed some geographic patterns that could correspond to different species occupying different locations. Basamuk canyon specimens (n=4) represent 3 distinct haplotypes differing by 2 mutations at the most. These differ from the DESMOS specimens (n=4) by at least 10 fixed mutational steps. Sequences from specimens collected 1600 km to the North (TOTO Caldera, South Mariana) differ from Basamuk canyon sequences by 6 fixed mutational steps. At the Brothers Caldera (4600 km Southeast of the TOTO Caldera, and 3500 km Southeast of the Astrolabe Bay), there are two haplotype groups, one of which only differs from the TOTO haplotype by a single mutational step, possibly corresponding to a single species. The other haplotypic group from Brothers Caldera differs from all the other haplotypes by at least 4 fixed mutational steps.

For at least two clades, even though sequence polymorphism does exist, interspecific variation is no greater than intraspecific variation [START_REF] Miglietta | Species boundaries of Gulf of Mexico vestimentiferans (Polychaeta, Siboglinidae) inferred from mitochondrial genes[END_REF]). This observation does not reflect within-species morphological variation but a reduced divergence of the mitochondrial genome [START_REF] Miglietta | Species boundaries of Gulf of Mexico vestimentiferans (Polychaeta, Siboglinidae) inferred from mitochondrial genes[END_REF][START_REF] Cowart | Restriction to large-scale gene flow vs. regional panmixia among cold seep Escarpia spp. (Polychaeta, Siboglinidae)[END_REF]. The COI marker thus offers a very conservative estimation of specific diversity in Siboglinidae. Based solely on COI sequence data, there is therefore little doubt that the species studied here represent new species that have only been collected in the sampled area so far. The Lamellibrachia specimens collected at Broken Water Bay probably correspond to the same species as the specimen collected at the Aitape site, further west on the PNG coast [START_REF] Kojima | Molecular taxonomy of vestimentiferans of the western Pacific, and their phylogenetic relationship to species of the eastern Pacific I. Family Lamellibrachiidae[END_REF][START_REF] Kojima | Molecular taxonomy of vestimentiferans of the western Pacific, and their phylogenetic relationship to species of the eastern Pacific III. Alaysia-like vestimentiferans and relationships among families[END_REF]. A population genetics study including more variable markers (e.g. microsatellites) could reveal whether these populations separated by 350 km are genetically connected. The number of specimens collected in Basamuk canyon (Lamellibrachia juni clade), and other specimens of the same clade from different locations was sufficient to initiate a study of connectivity. The specimens from the Astolabe Bay were only found at the site below the Ramu refinery. Although close to L. juni, there are some fixed differences in the COI sequences that strongly suggest that this lineage should be separated into at least four species (non-interbreeding populations), two of which are found in Papua-New Guinea. Except for the Brothers Caldera site where two species co-occur, only a single haplotypic group (=species) occurs at each site. Even within the Manus Basin, the sequences of the specimens from the hydrothermal vent site DESMOS are 10 mutational steps distant from the sequences of the specimens collected at the cold-seep site in the Astrolabe Bay, indicating that they belong to reproductively isolated evolutionary units.

CONCLUDING REMARKS

Deep-sea chemosynthetic environments are rare and very patchy [START_REF] Levin | Understanding continental margin biodiversity: a new imperative[END_REF], Van Dover at al. 2012). Some organisms can live in several types of chemosynthetic environments (e.g. Idas washingtonius is present at vents and on sunken wood or bones in the Eastern Pacific; Gigantidas platifrons is present both at vent and seep environments in the western Pacific, [START_REF] Thubaut | The contrasted evolutionary fates of deep-sea chemosynthetic mussels (Bivalvia, Bathymodiolinae)[END_REF]; Pyropelta musaica is found at cold seeps, hot vents and on sunken whale carcasses, [START_REF] Smith | Ecology of whale falls at the deep-sea floor[END_REF], and some have a very wide distribution, sometimes spanning thousands of kilometers (e.g. Rimicaris exoculata on the Mid-Atlantic Ridge, [START_REF] Teixeira | High connectivity across the fragmented chemosynthetic ecosystems of the deep Atlantic Equatorial Belt: efficient dispersal mechanisms or questionable endemism?[END_REF]). In the Atlantic Ocean, the among-site faunal similarity is better explained by depth than by geographic distances [START_REF] Olu | Biogeography and potential exchanges among the Atlantic equatorial belt cold-seep faunas[END_REF]. The potential threat to the fauna associated with deep-sea chemosynthetic habitats should therefore take into account a complex pattern of patchiness and connectivity at small and large geographic scales.

Although intensive, our sampling in the area has probably missed some chemosynthetic sites.

For instance, we re-visited the Aitape site sampled earlier with a remotely operated vehicle [START_REF] Tappin | The Sissano, Papua New Guinea tsunami of July 1998-offshore evidence on the source mechanism[END_REF], and did not find any fauna typical of cold seeps in our collections. The area is characterized by very complex bottom structures that are difficult to sample using dredges and trawls. In addition, the sampled animals formed very small clumps [START_REF] Tappin | The Sissano, Papua New Guinea tsunami of July 1998-offshore evidence on the source mechanism[END_REF]) that are easily missed. It is therefore very likely that there are sites with chemosynthetic fauna elsewhere in the sampled area.

However, the collections so far clearly reveal not only the great patchiness in the distribution of different environments (cold-seeps, hydrothermal vents, sedimentary areas, rubble, plant debris) that has a direct effect on the distribution of species, but also differences in the species that occupy different sites of the same habitat type, in particular for the cold-seep sites studied so far in PNG.

In other words, even if some species are shared between the two locations sampled, each location is also characterized by unique assemblages. The Broken Water Bay site does share some species with the Aitape site (Lamellibrachia L6 and Gigantidas sp. 2, this latter also found in the Astrolabe Bay) but the Astrolabe Bay site is characterized by some species that have so far only been found there. Interestingly, the Northern sites (Aitape and Broken Water Bay) receive currents from the Northwest (with occasional upwellings) while the Astrolabe Bay receives currents coming from the Southeast through the Vitiaz Strait [START_REF] Hasegawa | Cold water flow and upper-ocean currents in the Bismarck Sea from December 2011 to January 2002[END_REF], suggesting each location could receive larvae from different sources.

Our results however provide clear evidence of the patchiness of the cold-seep habitats in PNG and of the limited connectivity between the two studied sites. Although more exploration remains to be performed, it is clear that the assertions that impact of human activities on deepsea species is limited because the deep sea is uniform and that all species are present everywhere, are inherently wrong. In case the of the Ramu refinery in the Astrolabe Bay, we could not exclude that the major impact it has on the local deep-sea habitat in the Basamuk Bay would actually correspond to local species extinction. These concerns call for major exploration efforts and studies of connectivity in the hotspot of biodiversity that PNG represents. Posterior Probabilities (> 0.85) and Bootstraps (> 75%) are shown for each node. *: Posterior Probabilities > 0.98 and Bootstraps > 95%. For clarity purpose, supports for within species relationships are not shown and clades that included sequences from the same species or the same species complex were collapsed in triangles (as long as they did not include Papua Niugini sequences). Specimens from the PAPUA NIUGINI are in BOLD. Specimens of the species Gigantidas sp. 1 Broken Water Bay were collapsed into a triangle (details in table 1).

FIGURE CAPTION

Figure 7: Sampling locations (left) and TCS haplotype network [START_REF] Clement | TCS: a computer program to estimate gene genealogies[END_REF] of the Lamellibrachia juni clade (right). Circles on the network represent unique mtCOI haplotypes, circle diameter proportional to number of individuals (key: bottom right). Intermediate mutations (i.e. unsampled haplotypes) are represented by small white circles. Site colors on the map correspond to circle colors on the network. The map was prepared based on NOAA's ETOPO1 data using the marmap package in R [START_REF] Pante | marmap: A Package for Importing, Plotting and Analyzing Bathymetric and Topographic Data in R[END_REF].

  , providing the only report of chemosynthetic organisms from cold-seep environments in PNG waters (e.g. Kuyno at al. 2009 for bathymodiolinae, Kojima at al. 2003 for siboglinidae). Additionaly, the deeper operations off the Ramu refinery unexpectedly brought back bathymodioline mussels and siboglinid tubeworms, revealing a new location of chemosynthetic habitat.

Figure 1 :

 1 Figure 1: Map of Papua New Guinea with locations of stations sampled during the

  Other stations are represented by grey circles. The map was prepared based on NOAA's ETOPO1 data using the marmap package in R[START_REF] Pante | marmap: A Package for Importing, Plotting and Analyzing Bathymetric and Topographic Data in R[END_REF]. Stations CP4020 and 4081: Basamuk canyon, canyon below the Ramu refinery. StationsCP3703, 3704, 4040, 4042, and 4043: Broken Water Bay, Sepik site.

Figure 2 :

 2 Figure 2: Broken Water Bay fauna. Photos taken on board. A-Bathymodiolinae mussels,

Figure 4 :

 4 Figure 4: Basamuk Canyon fauna (PAPUA NIUGINI, CP4081). A-B Siboglinidae

Figure 5 :

 5 Figure 5: COI (A), 28S (B) and COI+28S (C) Bayesian phylogenetic trees obtained from 105

Figure 6 :

 6 Figure 6: Bayesian phylogenetic tree obtained from 298 Vestimentifera COI sequences.

  Fig 1

Table 1 :

 1 List of analyzed specimens.

	MNHN #	Group	ID

ACKNOWLEDGMENTS

This project was supported by the network "Bibliothèque du Vivant" funded by the CNRS, the Muséum National d'Histoire Naturelle, the INRA and the CEA (Genoscope). The PAPUA NIUGINI specimens were obtained during the "Our Planet Reviewed" Papua Niugini Biodiversity Expedition organized by Muséum National d'Histoire Naturelle (MNHN), Pro Natura International (PNI), Institut de Recherche pour le Développement (IRD) and University of Papua New Guinea (UPNG), Principal Investigators Philippe Bouchet, Claude Payri and Sarah Samadi. The organizers acknowledge funding from the Total Foundation, Prince Albert II of Monaco Foundation, Fondation EDF, Stavros Niarchos Foundation and Entrepose Contracting, and in-kind support from the Divine Word University (DWU). The expedition operated under a permit delivered by the Papua New Guinea Department of Environment and Conservation. We would like to thank Philippe Bouchet and Rudo Von Cosel for their taxonomic expertise, and the Service de Systématique Moléculaire and the Direction des Collections (especially to Stéphanie Le Prieur, Julien Brisset, Barbara Buge) at MNHN for processing the samples for molecular analyses and the voucher specimens. The 2012 deep-sea cruise was part of the French-Taiwanese project TF-DeepEvo funded by ANR and NSC (ANR 12-ISV7-0005-01).


Aymeric Lardon

A partial characterization of the core in Bertrand oligopoly TU-games with transferable technologies

Keywords: Bertrand oligopoly TU-games, Transferable technologies, Core, Convexity property, JEL Classifications: C71, D43

In this article we study Bertrand oligopoly TU-games with transferable technologies under the α and β-approaches (Aumann 1959). Although the convexity property does not always hold, we show that it is satisfied when firms' marginal costs are not too heterogeneous. Furthermore, we prove that the core of any game can be partially characterized by associating a Bertrand oligopoly TU-game derived from the most efficient technology. Such a game turns to be an efficient convex cover (Rulnick and Shapley 1997) of the original one. This result implies that the core is non-empty and contains a subset of payoff vectors with a symmetric geometric structure easy to compute.

Introduction

The literature on cooperative games uses three main typologies in order to classify oligopoly TU (Transferable Utility)-games. The first typology concerns competition type namely Cournot (quantity) and Bertrand (price) competition in which firms propose homogeneous or differentiated products. The second typology distinguishes two types of oligopolies according to the possibility or not for cooperating firms to transfer their technologies. 1 When technologies are transferable, cooperating firms are allowed to produce according to the most efficient technology available in the cartel. When technologies are not transferable such a transfer is not possible. The third typology concerns blocking rules used to convert a normal form oligopoly game into an oligopoly TU-game. Among many blocking rules we can cite three main approaches. The first two called the α and β-approaches are suggested by [START_REF] Aumann | Acceptable points in general cooperative n-person games[END_REF].

According to the first, every cartel computes its max-min profit, i.e. the profit it can guarantee regardless of what outsiders do. The second approach consists in computing the min-max cartel profit, i.e. the minimal profit for which external firms can prevent the cartel from getting more. The third approach called the γ-approach is proposed by [START_REF] Chander | A core of an economy with multilateral environmental externalities[END_REF]. The cartel profit is then computed by considering a competition setting in which the cartel faces external firms acting individually.

When technologies are not transferable, Zhao (1999b) proves that the core of Cournot oligopoly TU-games under the α and β-approaches is non-empty if every individual profit function is continuous and concave.2 Furthermore, [START_REF] Norde | Oligopoly games with and without transferable technologies[END_REF] show that such games are convex when the inverse demand function and cost functions are linear, and [START_REF] Driessen | Convexity of oligopoly games without transferable technologies[END_REF] provide sufficient conditions to guarantee the convexity property in a more general setting. Under the γ-approach, [START_REF] Lardon | The γ-core of cournot oligopoly games with capacity constraints[END_REF] shows that Cournot oligopoly TU-games are balanced if every individual profit function is concave and provides a single-valued allocation rule in the core, called NP(Nash Pro rata)-value, when cost functions are linear. Concerning price competition where firms operate at a constant and identical marginal cost, [START_REF] Lardon | Convexity of Bertrand oligopoly TU-games with differentiated products[END_REF] proves that the convexity property holds for Bertrand oligopoly TU-games under the α and β-approaches and shows that the core is non-empty under the γ-approach. When technologies are transferable, Zhao (1999a) provides a necessary and sufficient condition which guarantees the convexity property in Cournot oligopoly TU-games under the α and β-approaches where the inverse demand function and cost functions are linear. Although the convexity property does not hold in general, [START_REF] Norde | Oligopoly games with and without transferable technologies[END_REF] show that these games are totally balanced.

Until now, no work has dealt with Bertrand oligopoly TU-games with transferable technologies. In this article, as a counterpart to this lack of interest, we study such games by following the α and β-approaches. First, we show that these games are well-defined and argue that the α and β-approaches lead to the same class of games by using Zhao's result (1999b). Although the convexity property does not always hold, we prove that it is satisfied when firms' marginal costs are not too heterogeneous. Then we show that the core of any game can be partially characterized by associating a Bertrand oligopoly TU-game derived from the most efficient technology. Such a game turns to be an efficient convex cover [START_REF] Rulnick | Convex Covers of Symmetric Games[END_REF] relative to the original one. This result implies that the core is non-empty and contains a subset of payoff vectors with a symmetric geometric structure, and so easy to compute. This article is organized as follows. Section 2 gives some basic concepts on TU-games. In Section 3 we introduce the model and argue that the α and β-approaches are equivalent. In Section 4 we provide a sufficient condition under which the convexity property holds and propose a partial characterization of the core. Section 6 gives some concluding remarks.

Preliminaries

We denote by |X| the cardinality of a finite set X. Given x ∈ R, ⌈x⌉ is the smallest integer greater than or equal to x. A point x ∈ R n is a convex combination of points x 1 , . . . , x k ∈ R n , k ∈ N, if there exist non-negative real numbers α 1 , . . . , α k with k i=1 α i = 1 such that x = k i=1 α i x i . Given a finite set X, conv(X) denotes the convex hull of X, i.e. the set of points x ∈ R n which are a convex combination of some points in X.

Let N = {1, . . . , n} be a fixed and finite set of players. We denote by 2 N the power set of N and call a subset S ∈ 2 N , S = ∅, a coalition. The size s = |S| of coalition S is the number of players in S. A TU-game on N is a set function v : 2 N -→ R with the convention that v(∅) = 0, which assigns a real number v(S) ∈ R to every coalition S ∈ 2 N . The number v(S) is the worth of coalition S. We denote by G N the set of TU-games on N where v is a representative element of G N .

In a TU-game v ∈ G N , every player i ∈ N may receive a payoff x i ∈ R. A vector x = (x 1 , . . . , x n ) is a payoff vector. A payoff vector x ∈ R n is acceptable if for every coalition S ∈ 2 N , i∈S x i ≥ v(S), i.e. the payoff vector provides a total payoff to the members of coalition S that is at least as great as its worth. A payoff vector x ∈ R n is efficient if i∈N x i = v(N ), i.e. the payoff vector provides a total payoff to all the players that is equal to the worth of the grand coalition N . The core C(v) ⊆ R n of a TU-game v ∈ G N is the set of payoff vectors that are both acceptable and efficient, i.e.

C(v) = x ∈ R n : ∀S ∈ 2 N , i∈S x i ≥ v(S) and i∈N x i = v(N ) .
Given a payoff vector in the core, the grand coalition N forms and distributes its worth as payoffs to its members in such a way that any coalition cannot contest this sharing by breaking off from the grand coalition.

A TU-game v ∈ G N has a non-empty core if and only if it is balanced [START_REF] Bondareva | Some applications of linear programming methods to the theory of cooperative games[END_REF][START_REF] Shapley | On balanced sets and cores[END_REF]. A permutation on the set of players N is a bijection σ : N -→ N which assigns rank number σ(i) ∈ N to any player i ∈ N . We denote by Π N the set of all n! permutations. For every permutation σ ∈ Π N , we denote by

S σ,i = {j ∈ N : σ(j) ≤ σ(i)} the set of predecessors of i with respect to σ, including i himself. Given a TU-game v ∈ G N and a permutation σ ∈ Π N , the marginal vector m σ (v) ∈ R n is defined as ∀i ∈ N, m σ i (v) = v(S σ,i
)v(S σ,i \{i}) and assigns to player i his marginal contribution to the worth of the coalition consisting of all his predecessors with respect to σ. [START_REF] Shapley | Cores of convex games[END_REF][START_REF] Ichiishi | Super-modularity : Applications to convex games and to the greedy algorithm for lp[END_REF]. The core C(v) is then equal to the convex hull of the set of all the marginal vectors, i.e.

A TU-game v ∈ G N is convex if ∀i, j ∈ N, ∀S ∈ 2 N \{i,j} , v(S ∪ {i}) -v(S) ≤ v(S ∪ {i, j}) -v(S ∪ {j}) (1) A TU-game v ∈ G N is convex if and only if m σ (v) ∈ C(v) for all σ ∈ Π N
C(v) = conv({m σ (v) : σ ∈ Π N }).
3 Bertrand oligopoly TU-games with transferable technologies

In this section, we first define a Bertrand oligopoly situation from which we associate a normal form Bertrand oligopoly game. After formalizing the possibility for cooperating firms to transfer their technologies, we then convert the normal form Bertrand oligopoly game into a Bertrand oligopoly TU-game by using the α and β-approaches [START_REF] Aumann | Acceptable points in general cooperative n-person games[END_REF].

First, a Bertrand oligopoly situation is described by the tuple (N, (D i , C i ) i∈N ) where N = {1, 2, . . . , n} is the finite set of firms,

D i : R n + -→ R, i ∈ N , is firm i's demand function and C i : R + -→ R + , i ∈ N , is firm i's cost function.
Throughout this article, we assume that (a) the demand system is [START_REF] Shubik | Market structure and behavior[END_REF]

, i.e. ∀i ∈ N, D i (p 1 , . . . , p n ) = V -p i -r p i - 1 n j∈N p j ,
where p j is the price charged by firm j, V ∈ R + is the intercept of demand and r ∈ R ++ is the substitutability parameter. 3 The quantity demanded of firm i's brand depends on its price and on the difference between its price and the average price in the industry;

(b) every firm operates at a linear marginal cost

c i ∈ R + , i.e. ∀i ∈ N, C i (q i ) = c i q i ,
where q i ∈ R + is the quantity demanded of firm i's brand.

Without loss of generality we assume that the firms are ranked according to their marginal costs, i.e. c 1 ≤ c 2 ≤ . . . ≤ c n . Given assumptions (a) and (b), a Bertrand oligopoly situation is summarized by the 4-tuple (N, V, r, (c i ) i∈N ).

Then, corresponding to the Bertrand oligopoly situation (N, V, r, (c i ) i∈N ), the associated normal form Bertrand oligopoly game (N, (X i , π i ) i∈N ) is defined as 1. the set of firms is N = {1, . . . , n};

2. for every i ∈ N , the individual strategy set is X i = R + where p i ∈ X i represents the price charged by firm i;

3. the set of strategy profiles is X N = i∈N X i where p = (p i ) i∈N is a representative element of X N ; for every i ∈ N , the individual profit function π i : X N -→ R is defined as

π i (p) = D i (p)(p i -c i ).
Since technologies are transferable, cooperating firms are allowed to produce according to the cheapest technology available in the coalition. As a consequence, for every coalition S ∈ 2 N it holds that (c) every cooperating firm operates at the linear marginal cost of the most efficient firm in S denoted by c S ∈ R + where c S = min{c i : i ∈ S}.

We denote by X S = i∈S X i the coalition strategy set of coalition S ∈ 2 N and X N \S = i∈N \S X i the set of outsiders' strategy profiles for which p S = (p i ) i∈S and p N \S = (p i ) i∈N \S are the representative elements respectively. For every coalition S ∈ 2 N , the coalition profit function π S :

X S × X N \S -→ R is defined as π S (p S , p N \S ) = i∈S D i (p)(p i -c S ).
Finally, we convert the normal form Bertrand oligopoly game (N, (X i , π i ) i∈N ) into a Bertrand oligopoly TU-game. As discussed in the introduction, [START_REF] Aumann | Acceptable points in general cooperative n-person games[END_REF] suggested two approaches of converting a normal form game into a cooperative game called game in α and β-characteristic function form.

Given the normal form Bertrand oligopoly game (N, (X i , π i ) i∈N ), the α and β-characteristic functions are defined for every coalition S ∈ 2 N as

v α (S) = max p S ∈X S min p N \S ∈X N \S π S (p S , p N \S ) (2)
and

v β (S) = min p N \S ∈X N \S max p S ∈X S π S (p S , p N \S ) (3)
respectively. We denote by G N o ⊆ G N the set of Bertrand oligopoly TU-games with transferable technologies.

The following proposition states that the β-characteristic function is well-defined in the framework of Bertrand oligopoly TU-games with transferable technologies. For notational convenience, for any s ∈ {0, . . . , n} we let δ s = 1 + r(ns)/n. Proposition 3.1 Let (N, (X i , π i ) i∈N ) be a normal form Bertrand oligopoly game. Then, for every coalition S ∈ 2 N it holds that

v β (S) = π S (p S , p N \S ),
where (p S , p N \S ) ∈ X S × X N \S is given by ∀i ∈ S, p i = max{c S , (V + δ s c S )/(2δ s )} (4)

and j∈N \S p j = max{0, (n/r)(δ s c S -V )} (5)
Proof: When all the firms operate at a constant and identical marginal cost c ∈ R + , Lardon (2014) solves the optimization problems defined in (3). In our model, although firms operate at possibly distinct marginal costs, we argue that for every coalition S ∈ 2 N , the worth v β (S) can be determined in a similar way. Given any coalition S ∈ 2 N , note that the optimization problems defined in (3) only concern the coalition profit function π S . This implies that outsiders' marginal costs don't count in the computation of the worth v β (S). Moreover, since technologies are transferable cooperating firms in S operate at the same constant and identical marginal cost c S ∈ R + . Thus, we refer to [START_REF] Lardon | Convexity of Bertrand oligopoly TU-games with differentiated products[END_REF] for the technical arguments proving (4) and ( 5).

For every coalition S ∈ 2 N , the computation of the worth v β (S) is consistent with the fact that the quantity demanded of cooperating firm i's brand is positive since for every firm i ∈ S, D i (p) ≥ 0. Moreover, Proposition 3.1 calls for comments which will be useful for the sequel.

Remark 3.2 For every coalition S ∈ 2 N , the computation of the worth v β (S) results from two complementary cases:

1. if V ≤ δ s c S , then by (4) each member i ∈ S charges prices equal to their marginal cost, i.e. p i = c S , and by ( 5) the outsiders charge a non-negative average price, i.e.

j∈N \S p j /(ns) ≥ 0. In this case, coalition S obtains a zero profit, v β (S) = 0.

2. if V > δ s c S , then by (4) each member i ∈ S charges prices greater than their marginal cost, i.e. p i > c S , and by ( 5) the outsiders charge a zero average price, i.e. j∈N \S p j /(ns) = 0. In this case, coalition S obtains a positive profit v β (S) = s(Vδ s c S ) 2 /(4δ s ).

By solving successively the two minimization and maximization problems defined in (2), we can show that the α-characteristic function is well-defined too. The proof is similar to the one in Proposition 3.1, and so it is not detailed.

For general TU-games, Zhao (1999b) shows that the α and β-characteristic functions are equal when every strategy set is compact, every utility function is continuous, and the strong separability condition is satisfied. This condition requires that the utility function of any coalition S and each of its members' utility functions have the same minimizers. We argue that Zhao's result (1999b) holds for the specific class of Bertrand oligopoly TU-games with transferable technologies. First, we compactify the strategy sets by assuming that for every firm i ∈ N , X i = [0, p] where p is sufficiently large so that the maximization/minimization problems defined in ( 2) and ( 3) have interior solutions. Then, it is clear that every individual profit function π i is continuous. Finally, since the demand system is symmetric and cooperating firms in any coalition S operate at the same constant and identical marginal cost c S ∈ R + , it follows that Bertrand oligopoly TU-games satisfy the strong separability condition.

Corollary 3.3 Let (N, (X i , π i ) i∈N ) be a normal form Bertrand oligopoly game. Then, for every coalition S ∈ 2 N it holds that v α (S) = v β (S).

This result implies that outsiders' strategy profile p N \S that best punishes coalition S as a first mover (α-approach) also best punishes S as a second mover (β-approach).

Partial characterization of the core

In this section, we first provide an example in which the convexity does not hold for a large class of Bertrand oligopoly TU-games with transferable technologies. Then, we show that the convexity property is satisfied if firms' marginal costs are not too heterogeneous. Finally, even if the core can not be always fully characterized, we identify a subset of payoff vectors with a symmetric geometric structure easy to compute.

In the general framework of TU-games, the convexity property permits to characterize the core. However, this property may fail to hold in the present model as illustrated in the following example.

Example 4.1 Consider the Bertrand oligopoly situation (N, V, r, (c i ) i∈N ) where r = n, c 1 = c 2 = 0 and for all k ∈ N \{1, 2}, c k = 1. Fix any coalition S ∈ 2 N \{1,2} such that s = ⌈n/2⌉ and assume that V = δ s + 1. We want to prove that the convexity property fails to hold for coalition S. By point 2 of Remark 3.2 the worths of relevant coalitions to test (1) are given in the following table.

T S S ∪ {1} and S ∪ {2} S ∪ {1, 2} v β (T ) s 4(n -s + 1) (s + 1)(n -s + 2) 2 4(n -s) (s + 2)(n -s + 2) 2 4(n -s -1) Some elementary calculus show that v β (S ∪ {1, 2}) -v β (S ∪ {2}) < v β (S ∪ {1}) -v β (S)
for all n ≥ 6.

Although the convexity property does not always hold, the following result establishes that Bertrand oligopoly TU-games with transferable technologies are convex when firms' marginal costs are not too heterogeneous.

Theorem 4.2 Let v β ∈ G N o be a Bertrand oligopoly TU-game with transferable technologies corresponding to the Bertrand oligopoly situation (N, V, r,

(c i ) i∈N ) where V ≥ (1 + r)c n . Then v β is convex if rc 1 ≤ c n -c 1 ≤ V √ δ 1 + √ δ 3 1 √ δ 1 + 1 √ δ 3 - 2 √ δ 2 (6)
Proof: Before dealing with the convexity of v β , we establish three preliminary results. First, for any s ∈ {0, . . . , n} we deduce from δ s ≥ 1 that √ δ s ≤ δ s ≤ δ 0 = 1 + r and 1 = δ n ≤ √ δ s . These two results with the first inequality in (6

) (δ 0 c 1 ≤ δ n c n ) implies that ∀s ∈ {1, . . . , n -2}, δ s c 1 ≤ δ s+2 c n (7)
Second, for any s ∈ {0, . . . , n -1} we have δ s -δ s+1 = r/n. From the strict monotonicity and strict convexity of function f : R ++ -→ R ++ which assigns to every x ∈ R ++ the positive real number f (x) = 1/ √ x, it follows that ∀s ∈ {1, . . . , n -2},

1 √ δ 1 + 1 √ δ 3 - 2 √ δ 2 ≤ 1 √ δ s + 1 δ s+2 - 2 δ s+1 (8)
Third, for any s ∈ {1, . . . , n} it holds that δ s < 1 + r. Hence we deduce from V ≥ (1 + r)c n that for every coalition S ∈ 2 N , we have V > δ s c S . Thus, the worth of every coalition S ∈ 2 N is given by point 2 of Remark 3.2.

We want to prove that (1) holds in v β ∈ G N o . First assume that S = ∅ and consider any firms i, j ∈ N such that c j ≤ c i without loss of generality. It follows from δ 2 < δ 1 and c j ≤ c i that v β ({i}) + v β ({j}) < v β ({i, j}). Then assume that S = ∅ and define the TU-game ṽβ ∈ G N which assigns to every coalition S ∈ 2 N , the worth ṽβ (S) = (4/s)v β (S) = (Vδ s c S ) 2 /δ s with ṽβ (∅) = 0. For every coalition S ∈ 2 N , it follows from the monotonicity4 of ṽβ that inequality in (1) is passed from ṽβ to v β so that we focus on the TU-game ṽβ . Consider any firms i, j ∈ N , take any coalition S ∈ 2 N \{i,j} and without loss of generality assume that c j ≤ c i . For any

α 1 , α 2 ∈ R + and any a, b ∈ R it holds that α 1 a 2 -α 2 b 2 = ( √ α 1 a+ √ α 2 b)( √ α 1 a- √ α 2 b).
In the TU-game ṽβ the marginal contribution of firm i to coalition S is then given by Although the core can not be always fully characterized, a subset of payoff vectors with a symmetric geometric structure can be identified. Given a Bertrand oligopoly situation (N, V, r, (c i ) i∈N ), let w β ∈ G N o be the Bertrand oligopoly TU-games with the most efficient technology corresponding to the Bertrand oligopoly situation (N, V, r, (ĉ i ) i∈N ) where for all i ∈ N , ĉi = c 1 . Since firms operate at the constant and identical marginal cost c 1 , it holds that w β is convex [START_REF] Lardon | Convexity of Bertrand oligopoly TU-games with differentiated products[END_REF].

Theorem 4.3 Let v β ∈ G N
o be a Bertrand oligopoly TU-game with transferable technologies corresponding to the Bertrand oligopoly situation (N, V, r,

(c i ) i∈N ). Then it holds that conv({m σ (w β ) : σ ∈ Π N }) ⊆ C(v β ),
where w β ∈ G N o is the associated Bertrand oligopoly TU-game with the most efficient technology.

Proof: We know from the convexity of w β that conv({m σ (w β ) : σ ∈ Π N }) = C(w β ). It remains to prove that C(w β ) ⊆ C(v β ). For both TU-games v β and w β the worths of any coalition S are given by v β (S) = π S (p S , p N \S ) and w β (S) = π S (p S,w β , p N \S,w β ). We distinguish two cases:

1. take any coalition S ∈ 2 N such that 1 ∈ S. Then, by ( 4) and ( 5) it holds that for all i ∈ S, p i = p i,w β and j∈N \S p j = j∈N \S p j,w β which implies that v β (S) = w β (S). In particular, we have v β (N ) = w β (N ).

2. take any coalition S ∈ 2 N such that 1 ∈ S. We distinguish three cases:

-if V ≤ δ s c 1 then by point 1 of Remark 3.2, it holds that p i = c S and p i,w β = c 1 which implies that v β (S) = w β (S) = 0.

-if δ s c 1 < V ≤ δ s c S then by points 1 and 2 of Remark 3.2, it holds that p i = c S and p i,w β > c 1 which implies that v β (S) = 0 < w β (S).

-if δ s c S < V then by point 2 of Remark 3.2, we deduce from δ s c 1 ≤ δ s c S that v β (S) ≤ w β (S).

In all cases, for every coalition S ∈ 2 N \{N } we have v β (S) ≤ w β (S) and v β (N ) = w β (N )5 which implies that C(w β ) ⊆ C(v β ).

The following example shows that C(w β ) may be a large subset of C(v β ) and is easy to compute due to its symmetric geometric structure.

Example 4.4 Consider the Bertrand oligopoly situation (N, V, r, (c i ) i∈N ) where N = {1, 2, 3}, V = 10, r = 3, c 1 = 1, c 2 = 3 and c 3 = 5. For every coalition S ∈ 2 N , the worths v β (S) and w β (S) are given in the following We see that the core C(w β ) is a large subset of the core C(v β ). Moreover, since w β is symmetric6 by definition, the set of marginal vectors {m σ (w β ) : σ ∈ Π N } is easy to compute.

Concluding remarks

A natural question would be whether the results in this article remain valid for convex cost functions. Insofar as the convexity property does not hold in this more general case, it seems difficult to provide an appealing sufficient condition which guarantees such a property is verified.

Other approaches as the γ-approach discussed in the introduction can be considered. The computation of the worth of any coalition then requires to consider a competition setting in which cooperating firms face external ones acting individually. Unlike the α and β-approaches in which only the coalition profit function is considered, outsiders' profit functions must be taken into account. This interesting but difficult subject is left for further research.

  table. The core C(v β ) contains all payoff vectors x ∈ R 3 + such that i∈N x i = 60.75, 4.08 ≤ x 1 ≤ 56.75, 0.08 ≤ x 2 ≤ 44.75 and x 3 ≤ 44.75. The included core C(w β ) contains all payoff vectors x ∈ R 3 + such that i∈N x i = 60.75 and for all i ∈ {1, 2, 3}, 4.08 ≤ x i ≤ 44.75. The 2-simplex below represents these two core configurations.

	S v β (S) 4.08 0.08 {1} {2} {3} {1, 2} {1, 3} {2, 3} {1, 2, 3} 0 16 16 4 60.75
	w β (S) 4.08 4.08 4.08	16	16	16	60.75

Zhao shows that the core is non-empty for general TU-games.

When r approaches zero, products become unrelated, and when r approaches infinity, products become perfect substitutes.

A TU-game v ∈ G N is monotonic if ∀S ∈ 2 N , ∀T ∈ 2 N : S ⊆ T, v(S) ≤ v(T ).

Since v β (∅) = w β (∅) = 0, this proves that w β is an efficient convex cover of v β[START_REF] Rulnick | Convex Covers of Symmetric Games[END_REF].

A TU-game v ∈ G N is symmetric if there exists a function f : N -→ R such that for every coalition S ∈ 2 N , v(S) = f (s).

where

We want to show that A i (S)

Then, we show that B i (S) ≤ B i (S ∪ {j}) which is equivalent with c j ≤ c i to prove the following inequality

By (7), the second inequality of ( 6) and ( 8) it follows that

which concludes the proof.

Note that condition (6) does not depend on all firms' marginal costs in the industry but only those of the less and most efficient firms. Moreover, for any heterogeneity level of the marginal costs c nc 1 satisfying (1 + r)c 1 ≤ c n , the second inequality in (6) ensures that the convexity property holds if the intercept of demand is sufficiently large.