E A ; El-Maghrabi

H H ; Raynaud
email: raynaud@laplace.univ-tlse.fr

P ; Ali

H R ; Abd

Engy Ahmed Nada
email: engyahmednada@gmail.com

Heba Hassan El-Maghrabi
email: hebachem@yahoo.com

Patrice Raynaud

Hager Rabea Ali

Abd Saad

El-Wahab

Dina Yahea Sabry
email: d.y_sabry@yahoo.com

Yasser Mohamed Moustafa
email: ymoustafa12@yahoo.com

Amr Ahmed Nada

Enhanced Photocatalytic Activity of WS 2 /TiO 2 Nanofibers for Degradation of Phenol under Visible Light Irradiation

Keywords: nanofibers, tungsten disulfide, TiO 2, phenol photodegradation

Binary composite WS 2 /TiO 2 nanofibers (WTN) were elaborated by electrospinning technique. The photocatalytic efficiency of the binary nanofibers was changed via different ratios between WS 2 and TiO 2 . The structural, morphological and optical properties of the prepared nanofibers were evaluated by Raman spectroscopy, Fourier transform infrared spectroscopy (FT-IR), scanning electron microscopy (SEM) and UV-Vis diffuse reflectance spectroscopy (UV-vis/DRS), respectively. The prepared nanofibers showed a remarkable performance in photocatalytic efficiency of phenol compound degradation under visible light. WTN nanofibers showed superior photocatalytic activity (83%) and high stability of several cycles under visible light. Therefore, WS 2 /TiO 2 nanofibers have great prospects for the treatment of wastewater from toxic organic contamination due to their excellent photocatalytic performance reusability and recyclability.

Introduction

The development of industries throughout the world has led to increase water contamination which implies a serious threat to humans and environment [START_REF] Wang | Preparation of a novel sonocatalyst, Au/NiGa 2 O 4 -Au-Bi 2 O 3 nanocomposite, and application in sonocatalytic degradation of organic pollutants[END_REF][START_REF] Manavi | The development of aerobic granules from conventional activated sludge under anaerobic-aerobic cycles and their adaptation for treatment of dyeing wastewater[END_REF]. Water pollution is a matter of concern worldwide. Wastewater is caused by synthetic chemicals being released from a variety of anthropogenic industry sources. In particular, the petroleum refinery waste, which is one of the harmful sources for water, along with phenol compound which is one of these mischievous sources [START_REF] El-Maghrabi | Photocorrosion resistant Ag 2 CO 3 @Fe 2 O 3 /TiO 2 -NT nanocomposite for efficient visible light photocatalytic degradation activities[END_REF]. The different technologies for water treatment and purification have been extensively discussed in the literature as the design and the operation of affordable methods remains a challenge. The treatment methods include membrane separation, electrochemical method, chemical precipitation, adsorption, photocatalysis, etc. [START_REF] Pant | Synthesis and Characterization of ZnO-TiO 2 /Carbon Fiber Composite with Enhanced Photocatalytic Properties[END_REF]. It is pertinent to highlight that all treatment methods have their own technical and economic limitations for real-life applications. However, photocatalysis technique is one important method as it prevents to form the secondary pollutants [START_REF] Chandrabose | Removal and degradation of mixed dye pollutants by integrated adsorption-photocatalysis technique using 2-D MoS 2 /TiO 2 nanocomposite[END_REF].

Titanium dioxide (TiO 2 ) is one of the most significant photocatalysts, which is extensively utilized in various fields, such as electrochemistry and photocatalysis due to its several benefits such as low cost, nontoxicity, good chemical stability and excellent oxidizing power [START_REF] El-Maghrabi | Photocorrosion resistant Ag 2 CO 3 @Fe 2 O 3 /TiO 2 -NT nanocomposite for efficient visible light photocatalytic degradation activities[END_REF]. However, it has some drawbacks such as lack of activity in the abundant visible light energy because TiO 2 generally exhibits good photocatalytic activity in near ultraviolet (UV), due to its large bandgap energy (3.2 eV) [START_REF] Boningari | Novel one-step synthesis of nitrogen-doped TiO 2 by flame aerosol technique for visible-light photocatalysis: Effect of synthesis parameters and secondary nitrogen (N) source[END_REF]. To address these issues, it has been combined with other nanomaterials to enhance its photo-adsorption capacity under visible light such as CdS, NiO, ZnFe 2 O 4 , Ag 2 CO 3 , Fe 2 O 3 and Fe 3 O 4 [START_REF] Chen | BiSnSbO 6 -TiO 2 composites enhance LED light-driven photocatalytic antibacterial activity[END_REF][START_REF] Nur | A review on the development of elemental and codoped TiO 2 photocatalysts for enhanced dye degradation under UV-vis irradiation[END_REF]. The photocatalytic efficiency of TiO 2 -NFs can be improved by metals, non-metals and semiconductors [START_REF] Nada | Highly textured boron/nitrogen co-doped TiO 2 with honeycomb structure showing enhanced visible-light photoelectrocatalytic activity[END_REF][START_REF] Kawrani | Segregation of copper oxide on calcium copper titanate surface induced by Graphene Oxide for Water splitting applications[END_REF]. In addition, titanium dioxide is extensively utilized for environmental purification applications [START_REF] Nada | Functionalization of MCM-41 with titanium oxynitride deposited via PECVD for enhanced removal of methylene blue[END_REF] and tungsten disulfide (WS 2 ) is one of the important materials for TiO 2 enhancement.

Tungsten disulfide (WS 2 ) is one of the transition-metal dichalcogenide compounds which have inherent excellent photoactivity. The bandgap of WS 2 is approximately 1.9 eV when the thickness is reduced down to monolayers, which access the absorption from visible light and utilize solar energy sufficiently [START_REF] Yuan | Photoluminescence quenching and charge transfer in artificial heterostacks of monolayer transition metal dichalcogenides and few-layer black phosphorus[END_REF].

In this contribution, WS 2 /TiO 2 nanofibers were elaborated by combined 1D and 2D structures by electrospinning. The morphology, structure and optical properties of the nanocomposites were evaluated. The photocatalytic efficiency of WS 2 /TiO 2 nanofibers for the degradation of phenol, as an organic pollutant, was estimated under the visible light illumination.

Results and Discussion

Characterization of TWx Nanofibers

XRD spectra of the prepared WS 2 /TiO 2 nanofibers are shown in Figure 1. It is clear that the diffraction peaks of 25.5 004), (100), (100), ( 103) and (008), respectively (JCPDS No. 01-084-1285) of WS 2 [START_REF] Zhao | Enhanced catalytic activities of surfactant-assisted exfoliated WS2 nanodots for hydrogen evolution[END_REF]. The narrow and sharp peaks suggest that the product is highly crystalline. Where, the different weight ratios of TiO 2 to WS 2 (1:0, 0.9:0.1, 0.8:0.2, 0.7:0.3, 0.6:0.4 and 0.5:0.5) to obtain (TW0, TW10, TW20, TW30, TW40 and TW50, respectively). The WS 2 /TiO 2 nanofibers with different ratio between tungsten disulfide and titanium dioxide were observed with the use of Raman measurements. Moreover, Raman spectra of the prepared nanofibers are presented in Figure 2. In W sample, the bands at 126.28, 255.03, 320.11, 409.71, 701.09 and 799.35 cm -1 can be ascribed to the vibration bands of WS 2 nanosheets [START_REF] Voiry | Enhanced catalytic activity in strained chemically exfoliated WS 2 nanosheets for hydrogen evolution[END_REF][START_REF] Elías | Controlled synthesis and transfer of large-area WS 2 sheets: From single layer to few layers[END_REF]. For pure TiO 2 sample, the bands observed at 147.41, 198.17, 389.58, 502.31 and 626.37 cm -1 in the Raman spectrum can be assigned to E 1 g , E 2 g , B 1g (1), B 1g (2) + A 1g and E 3 g modes of anatase, respectively [START_REF] Xiang | Synergetic effect of MoS 2 and graphene as cocatalysts for enhanced photocatalytic H 2 production activity of TiO 2 nanoparticles[END_REF]. The combination of WS 2 with TiO 2 (in TW10, TW20, TW30, TW40 and TW50) resulted in a slight change in the anatase bands. In TW40 and TW50, the main band of anatase at 150 cm -1 is significantly changed due to the overlapping between the bands of TiO 2 and WS 2 and it means that there is a high conjugation between TiO 2 and WS 2 . The conjugation between WS 2 and TiO 2 can also be detected by the FTIR transmission spectra presented in Figure 2b. The band located around 3450 cm -1 is attributed to the O-H bonds for all nanofibers [START_REF] Golovnev | Calcium and strontium thiobarbiturates with discrete and polymeric structures[END_REF][START_REF] Smitha | Sol-gel route to synthesize titania-silica nano precursors for photoactive particulates and coatings[END_REF][START_REF] Golovnev | Hydrated and anhydrous cobalt (II) barbiturates: Crystal structures, spectroscopic and thermal properties[END_REF]. The overlap between the peaks of Ti-O and W-S was presented for all nanocomposite samples of TWx which confirmed the successful conjugation between TiO 2 and WS 2 .

The surface area results showed that S BET of TW0, TW10, TW20, TW30, TW40 and TW50 are 17, 25, 42, 61, 73 and 78 m 2 /g. The surface area of (1D) TiO 2 was enhanced by interacting with the (2D) structure of WS 2 . It is well known that TiO 2 with large surface area has excellent photocatalytic activity as a result of providing more active adsorption sites and speed transport channels for large organic molecules which were detected in TiO 2 with WS 2 .

The morphology of fabricated nanofibers was observed using scanning electron microscope as presented in Figure 3a-f. The T, TW10, TW20, TW30, TW40 and TW50 nanofibers have 97 ± 8, 120 ± 5, 188 ± 4, 210 ± 8, 240 ± 7 and 290 ± 9 nm average diameter, respectively. The increase in the diameter of the WS 2 -enriched nanofibers was attributed to the viscosity of electro-spun solution coming from WS 2 , which contributed to the successful incorporation between WS 2 and TiO 2 nanofibers. These findings are consistent with those of Raman measurements. EDX mappings for TW40 are presented in Figure 3g-l. The distribution of WS 2 is uniform with TiO 2 nanofibers. AAS demonstrated that the total Ti:W atomic ratios were approximately 0.902:0.098, 0.801:0.199, 0.704:0.296, 0.602:0.398 and 0.505:0.495 in TW10, TW20, TW30, TW40 and TW50, respectively. Indicating that TiO 2 and WS 2 were uniformly distributed throughout the network structure according to the preparation designed. Their uniform distribution should contribute to their photocatalytic activity as presented. The optical characteristics of the fabricated nanofibers were determined using UV-Vis diffuse reflectance (UVDR), as presented in Figure 4. The UVDR-spectra of the nanofibers were examined in the range of 200-800 nm using UV-Vis optical spectroscopy. The optical band gaps (Eg) were obtained by the following equation:

αhv = A(hv -Eg) (n/2) ( 1
)
where α is the absorption coefficient, v is the frequency of light and n is the constant of proportionality (n equals 4 for the indirect transition in the prepared nanofibers) [START_REF] Atuchin | Exploration on anion ordering, optical properties and electronic structure in K 3 WO 3 F 3 elpasolite[END_REF][START_REF] Ji | Comparative investigations of the crystal structure and photoluminescence property of eulytite-type Ba 3 Eu(PO 4 ) 3 and Sr 3 Eu(PO 4 ) 3[END_REF][START_REF] Atuchin | Exploration of structural, thermal, vibrational and spectroscopic properties of new noncentrosymmetric double borate Rb 3 NdB 6 O 12[END_REF]. The prepared nanofibers had a red shift due to joining of WS 2 with TiO 2 . As per the following sequence: 3.18, 2.87, 2.72, 2.68, 2.57 and 2.45 eV for TW0, TW10, TW20, TW30, TW40 and TW50, respectively. This recorded shift was due to the conjugation between WS 2 and TiO 2 , also this conjugation was detected in the Raman and FTIR [START_REF] Voiry | Enhanced catalytic activity in strained chemically exfoliated WS 2 nanosheets for hydrogen evolution[END_REF]. The TW40 sample has reflectance lower than other nanofibers due to the high absorption of light. In contrast, TW50 had a greater amount of WS 2 , which led to the aggregation of WS 2 and reduced light absorbed, as seen in the SEM image.

Photocatalytic Degradation of Phenol under Visible Light by TWx Nanofibers

The adsorption test was evaluated in the dark for 1 h before illumination. Then the photocatalytic activity of phenol degradation was detected every hour during 4 h. The photodegradation of phenol reached 13, 47, 57, 74, 83 and 66% for TW0, TW10, TW20, TW30, TW40 and TW50, respectively, as presented in Figure 5a. The TW0 sample (TiO 2 nanofibers) provided low activity for photodegradation of phenol ~13% due to large band gap of 3.18 eV. In contrast, the combination of WS 2 with TiO 2 enhanced the catalytic activity of phenol degradation, reaching 83% for TW40. This can be attributed to the band gap decrease followed by active absorption of visible light, as obtained by UV-Vis diffuse reflectance spectra. In addition, the photocatalytic active sites were increased by the incorporation of 1D WS 2 with 2D TiO 2 nanofibers. The kinetic of phenol photodegradation by prepared nanofibers is presented in Figure 5b. The phenol degradation follows the first-order kinetics model of Langmuir-Hinshelwood which is presented in Equation ( 3) [START_REF] Elshypany | Magnetic ZnO Crystal Nanoparticle Growth on Reduced Graphene Oxide for Enhanced Photocatalytic Performance under Visible Light Irradiation[END_REF]. The apparent order rate constant (k) for TW0, TW10, TW20, TW30, TW40 and TW50 is 2.235 × 10 -4 , 0.0017, 0.0024, 0.0043, 0.0057 and 0.0028 min -1 , respectively. Rate constants grow in this order: TW40 > TW30 > TW50 > TW20 > TW10 > TW0. The TW40 sample has the maximum rate constant (0.0057) that is attributed to the high interaction between WS 2 and TiO 2 , as presented in SEM and Raman, besides the synergistic effect between WS 2 and TiO 2 nanofibers. In order to determine the role of hydroxyl radicals ( • OH), holes (h + ) and superoxide anions ( • O 2-) in the efficiency of photodegradation of phenol, we used different free radical trapping agents with TW40 nanofibers, as presented in Figure 5c. The scavengers were Tert-butyl alcohol (TBA), disodium ethylenediaminetetraacetic acid (Na 2 -EDTA) and p-benzoquinone (BQ) to trap free radicals of hydroxyl radicals ( • OH), holes (h + ) and superoxide radicals ( • O 2-), respectively [START_REF] Elshypany | Elaboration of Fe 3 O 4 /ZnO nanocomposite with highly performance photocatalytic activity for degradation methylene blue under visible light irradiation[END_REF][START_REF] Nada | Photoelectrocatalysis of paracetamol on Pd-ZnO/N-doped carbon nanofibers electrode[END_REF]. The photodegradation of phenol varied according to different sacrificial agents. Predominantly, the photocatalytic degradation in presence of the TBA (5 mM) reduced to 36%. Therefore, the photogenerated • OH radical is the important free radical component during the phenol photodegradation process [START_REF] Elshypany | Elaboration of Fe 3 O 4 /ZnO nanocomposite with highly performance photocatalytic activity for degradation methylene blue under visible light irradiation[END_REF][START_REF] Nada | Photoelectrocatalysis of paracetamol on Pd-ZnO/N-doped carbon nanofibers electrode[END_REF]. Moreover, TW40 has high photodegradation stability up to five cycles, as presented in Figure 5d. The phenol photodegradation remained at the level of 80% after 5 cycles. The results obtained for the TW40 sample indicate a high photocatalytic degradation of phenol with a narrow bandgap, as well as an excellent stability of photocatalytic activity. We can note also that the crystallinity of TW40 nanofibers (Figure S1) and the morphology (Figure S2) were stable after five cycles as confirmed by XRD and SEM, respectively.

Materials and Methods

Chemical

Titanium (IV) isopropoxide 97%, WS2 powder, n-butyl lithium solution, hexane, absolute ethanol (99%), acetic acid (98%), polyvinylpyrrolidone (PVP; Mw = 1,300,000 g/mol) and phenol were purchased from Sigma Aldrich (St. Louis, MO, USA). All purchased compounds were used as received, without further purification.

Preparation of WS 2 /TiO 2 Nanofibers

Tungsten disulfide (WS 2 ) with TiO 2 nanofibers was prepared using the electrospinning technique by different ratio between WS 2 and TiO 2 . First, the solution (a) was prepared by 0.3 g of PVP dissolved in 3 mL of ethanol and 2 mL of acetic acid while stirring for 30 min, then 1.5 mL of TTIP was added with stirring for 30 min. The solution (b) was prepared with different weight ratios of WS 2 in 2 mL of ethanol and sonicated for 2 h to obtain (TW0, TW10, TW20, TW30, TW40 and TW50 for TiO 2 : WS 2 is 1:0, 0.9:0.1, 0.8:0.2, 0.7:0.3, 0.6:0.4 and 0.5:0.5, respectively). After that, solution (b) was mixed to solution (a) with stirring for 30 min. The solution was added in syringe with a stainless-steel nozzle with a diameter of 0.7 mm under flow rate 1 mL/h and voltage power (1.5 kV/cm) between the nozzle and the rotating collector. The collected nanofibers were calcined at 400 • C for 3 h.

Characterization

The nanofiber vibrational properties were observed using a dispersive Raman microscope with a laser wavelength of 532 nm and a power of 10 mW (Senterra, Bruker, Bremen, Germany). The functional groups of the prepared composite nanofibers were recognized using FTIR spectrometer (Perkin Elmer, Waltham, MA, USA) with the use of standard KBr pellet technique. The surface area of the nanofibers was detected from nitrogen adsorption-desorption isotherms at liquid nitrogen temperature using Micromeritics ASAP 2010 equipment (degassing conditions: 200 • C/12 h). The titanium and tungsten elemental concentrations in the as-prepared nanofibers were determined using atomic absorption spectroscopy (AAS) (Analyst 400, PerkinElmer, Bremen, Germany). The morphology of nanofibers was detected by SEM (Quanta-250 FEG, FEI, Netherlands) methods. The UV-Vis-DR spectra were measured by a Jasco V-570 (Jasco Inc., Easton, MD, USA).

Photocatalytic Activity Measurement

The photocatalytic activity of the WS 2 /TiO 2 nanofibers was detected by tracing the photodegradation of phenol. The photocatalyst efficiency was detected by analyzing the decrease of phenol concentration during exposure to visible light radiation. The reaction temperature was fixed at 25 ± 0.2 • C by circulating water over the photoreactor. The photocatalytic performance was detected by using 10 mg of photocatalyst in 200 mL of phenol solution (50 ppm) under a 500 W linear halogen lamp (visible light radiation). The solution mixture was stirred in the dark for 1 h to obtain the adsorption/desorption equilibrium of phenol [START_REF] Nasr | Enhanced visible-light photocatalytic performance of electrospun rGO/TiO 2 composite nanofibers[END_REF]. Then, the mixture was exposed to visible light for 4 h. Following, the sample was collected from reactor every 1 h and the catalyst was eliminated by a centrifuge to detect the organic pollutants' concentration in the solution by high-performance liquid chromatography (HPLC) equipped with a photodiode array detector (Agilent 1200 series, Agilent, Colorado Springs, CO, USA).

The degradation efficiency was measured by the following equation:

degradation efficiency % = (((C 0 -C))/(C 0 )) * 100 [START_REF] Manavi | The development of aerobic granules from conventional activated sludge under anaerobic-aerobic cycles and their adaptation for treatment of dyeing wastewater[END_REF] where C 0 and C are the initial and final phenol concentration, respectively [START_REF] Sugi | Photocatalytic Degradation efficiency of CuXZn1-XO Composite[END_REF].

The kinetics of phenol photocatalytic degradation were calculated as follows (where k (min -1 ) is the apparent rate constant): ln (C t /C 0 ) = -kt

(3)

Conclusions

In brief, we have successfully prepared WS 2 /TiO 2 nanofibers with different ratios between WS 2 (1D) and TiO 2 (2D). Structural and morphological analyses of the prepared nanofibers confirmed the incorporation between WS 2 and TiO 2 . Significantly, the photocatalytic efficiency of TW40 for phenol degradation was as high as 83%. The remarkable photocatalytic performance of TW40 (1D/2D structure) was attributed to the synergism effect between WS 2 and TiO 2 to improve electron transfer. In addition, the linkage between WS 2 and TiO 2 enhanced the photocatalytic stability of the nanofibers. The trapping test for the photogenerated free radicals and holes of TW40 demonstrated that the photogenerated hydroxyl radicals ( • OH) have a significant role in the photodegradation of phenol as an organic pollutant. Therefore, WS 2 /TiO 2 nanofibers act as promising materials for photo-degradation of organic pollutants in water.

Figure 1 .

 1 Figure 1. XRD for W, TW0, TW10, TW20, TW30, TW40 and TW50 nanofibers of TiO

Figure 1 .

 1 Figure1. XRD for W, TW0, TW10, TW20, TW30, TW40 and TW50 nanofibers of TiO 2 :WS 2 (0:1, 1:0, 0.9:0.1, 0.8:0.2, 0.7:0.3, 0.6:0.4 and 0.5:0.5, respectively).

Figure 2 .

 2 Figure 2. (a) Raman spectra and (b) FTIR for TW0, TW10, TW20, TW30, TW40 and TW50 nanofibers.

Figure 3 .

 3 Figure 3. SEM images for (a) TW0, (b) TW10, (c) TW20, (d) TW30, (e) TW40 and (f) TW50 nanofibers

Figure 3 .

 3 Figure 3. SEM images for (a) TW0, (b) TW10, (c) TW20, (d) TW30, (e) TW40 and (f) TW50 nanofibers; (g,h) Images of SEM/EDX elemental mapping of TW40 (i) oxygen, (j) titanium, (k) tungsten and (l) sulfur in TW40.

Figure 4 .Figure 4 .

 44 Figure 4. UV-Vis diffuse reflectance spectra of TW0, TW10, TW20, TW30, TW40 and TW50 nano-

Figure 5 .

 5 Figure 5. (a) Photocatalytic degradation of phenol for prepared nanofibers under visible light irradiation; (b) the ln(C/C0) changing versus time during visible light irradiation for prepared nanofibers; (c) Trapping test of the photogenerated free radicals and holes with TW40 nanofibers; (d) The stability cycles of phenol photodegradation using TW40 nanofibers under visible light irradiation for 4 h.

  • , 38.1 • , 48.1 • , 54.1 • , 55.3 • and 62.9 • are correspondent to the standard characteristic peaks position of (101), (004), (200), (105), (211) and (204) crystal planes of anatase TiO 2 , respectively (JCPDS No. 01-084-1285). In addition, the peaks at 28.98 • , 32.81 • , 33.64 • , 39.64 • and 58.48 • can be well indexed to the (002), (

Funding: This research received no external funding.

Conflicts of Interest:

The authors declare no conflict of interest.

Supplementary Materials:

The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/inorganics10040054/s1, Figure S1: XRD of TW40 nanofibers before and after photocatalytic reaction.;


Bertrand Crettez

Rabia Nessah

Tarik Tazdaït

ON THE STRONG HYBRID SOLUTION OF AN N-PERSON GAME Second Revised Version

Keywords: N -Person Game, coalition structure, hybrid solution, strong hybrid solution

We propose a new notion of coalitional equilibrium, the strong hybrid solution, which is a refinement of Zhao's hybrid solution. It is well suited to study situations where people cooperate within coalitions but where coalitions compete with one another. In the strong hybrid solution, as opposed to the hybrid solution, the strategy profile assigned to each coalition is strongly Pareto optimal. We show that there exists a strong hybrid solution whenever preferences are partially quasi-transferable.

INTRODUCTION

To study cooperation in strategic non-transferable utility games, one can rely on the concepts of coalitional equilibria. Those concepts differ in the way outsiders are assumed to react to the formation of a deviating coalition. The three traditional concepts of coalitional equilibria, all introduced in [START_REF] Aumann | Acceptable points in general cooperative n-person games[END_REF], include the α-core, the β-core [START_REF] Aumann | The core of a cooperative game without side payments[END_REF][START_REF] Aumann | Acceptable points in general cooperative n-person games[END_REF]) and the strong Nash equilibrium [START_REF] Aumann | Acceptable points in general cooperative n-person games[END_REF]). An action profile is in the α-core if no coalition can guarantee a higher payoff for each of its members by choosing another action, independently of the actions * Université Panthéon-Assas, Paris II, CRED, EA 7321, 21 Rue Valette -75005 Paris, France. Contact: bertrand.crettez@u-paris2.fr.

† (i) IESEG School of Management, 3 rue de la Digue 59000 Lille, France and (ii) LEM-CNRS 9221, 3 rue de la Digue, 59000 Lille, France. Contact: r.nessah@ieseg.fr ‡ CIRED, CNRS, EHESS, Ecole des Ponts, Nogent sur Marne, France. Contact: tazdait@centre-cired.fr § We thank four anonymous reviewers as well as Emel Filiz-Ozbay Co-Editor-in-Chief, for helpful comments on a previous version of this work.

1 taken by the other players. On the other hand, an action profile is in the β-core if no deviating coalition can guarantee a higher payoff for each of its members, because coalition outsiders have a dominant blocking strategy. In a strong Nash equilibrium, no coalition can guarantee a higher payoff for each of its members assuming that coalition outsiders stick to their equilibrium strategies.

Scarf [1971] gave existence results of α-core in games with nontransferable utilities, which were later extended for local α and β-cores by [START_REF] Yano | A local theory of cooperative games[END_REF]. [START_REF] Kajii | A generalization of Scarf's theorem: An α-core existence theorem without transitivity or completeness[END_REF] showed the non-emptiness of the α-core without assuming that agents' preference relations are transitive or complete. Zhao [1999aZhao [ , 1999b] ] obtained general existence results for α and β-cores in games with transferable utility. More recently, [START_REF] Uyanik | On the nonemptiness of the α-core of discontinuous games: Transferable and nontransferable utilities[END_REF] addressed the existence of α-core in discontinuous games.

In addition, Yang [2017a[START_REF] Yang | Some generalizations of Kajii's theorem to games with infinitely many players[END_REF] generalized Kajii's theorem to games with infinitely many players, Yang [2017b] considered the essential stability of α-core and [START_REF] Yang | Hadamard well-posedness of the α-core[END_REF] tackled the Hadamard well-posedness of the α-core.1 Martins-da-Rocha and [START_REF] Yannelis | Non-emptiness of the alpha-core[END_REF] established the existence of the α-core in games where agents' preferences are non-ordered and with infinite dimensional strategy spaces. [START_REF] Askoura | The ex-ante α-core for normal form games with uncertainty[END_REF] analyzed the ex-ante α-core in games with uncertainty while [START_REF] Noguchi | Alpha cores of games with nonatomic asymmetric information[END_REF] focused on the α-cores in games with asymmetric information.

Yet, the coalitional equilibrium concepts initially defined by Aumann are probably too demanding. They require that each conceivable coalition should not break away from a prealably agreed strategy profile. As a consequence, the class of games for which the α-core and the β-core exist is not large. In that connection, it is worthy to note that in many "real-world" situations not all the coalitions are relevant (for instance, it is doubtful that certain countries can ever cooperate, so different are their views on the issue at hand, be it for international trade, environmental protection and so on). To put it another way, some coalition can form more easily than others. [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF] is a major attempt to study coalition games with restricted formation of coalitions.

Specifically, he introduces the hybrid solution, a solution concept pertaining to the case where there is cooperation inside coalitions, but competition between coalitions. The main issue here is the internal stability of the coalitions belonging to a given coalition structure.

Consider for instance Catalogna in Spain, Corsica in France, or Scotland in Great Britain. In each of these cases, a region (that is, a subcoalition of a given coalition) contemplates breaking away from an existing coalition (a Member Sate of the European Union). But to the best of our knowledge, none of these regions endeavors to join, or merge, with another member of the European Union (that is, another coalition).

In relation to the remark above, the hybrid solution is a strategy profile that is stable against deviations by any subcoalition of a coalition belonging to a given partition of the players. More precisely, any subcoalition of a given coalition S can be blocked by some decisions taken by the outsiders in the coalition, assuming that all the other players (in the other coalitions) stick to their strategies. Here, blocking means that a subcoalition cannot choose a strategy such that the payoff obtained by any of its members is strictly greater than what he obtains with the hybrid solution, when the outsiders in the coalition minimize his gain. Zhao (ibid) showed the existence of the hybrid solution and Zhao [1999aZhao [ , 1999b] ] extended the notions of TU α and β cores to the hybrid solution.2 Existence of a version of the hybrid solution for games with non-ordered preferences, infinite dimensional action spaces (namely Hausdorff topological vector spaces) and infinitely many players was studied in [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF]. In their approach, however, the definition of blocking is slightly different from Zhao's. It is actually similar to the definition used in the α-core.3

This paper introduces a new coalitional equilibrium concept, namely the strong hybrid solution, that applies for any given coalition structure C. This definition furthers the approach of [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF] in the sense that it introduces yet another definition of blocking. Here, we propose to say that an alternative strategy chosen by a subcoalition is blocked by the outsiders in the coalition if it cannot guarantee a payoff at least as great as the reference one for each of its members, and strictly higher than the reference one for at least one of its members, independently of the actions taken by all the other players (like in the hybrid solution, we assume that the players outside the coalition stick to their strategies). To wit, because coalition members are assumed to cooperate, there is no reason why they would not collectively choose an alternative strategy profile that does no good nor bad for all of them, and strictly benefits to at least one member. With this definition, one can show that the set of strong hybrid solution is a subset of the Yang and Yuan's hybrid solutions, which are themselves a subset of Zhao's hybrid solutions. Hence the assertion that the notion of strong hybrid solution is a refinement of the hybrid solution. Our main contribution is to provide an existence result for the strong hybrid solution. This existence result is obtained under the assumption that preferences are partially quasi-transferable. 4The remainder of the paper is organized as follows. Section 2 sets out formal definitions, establishes some properties for the strong hybrid solution and compare it with alternative ones.

Section 3 establishes our existence result for the strong hybrid solution. Section 4 studies the condition of partially quasi-transferable preferences in two economic models. Section 5 provides some concluding remarks.

DEFINITIONS AND PROPERTIES

In this section, we first present our setting. Then, we state some definitions and properties for different notions of coalitional equilibria (including ours).

PRELIMINARIES

Consider an n-person game G = I, X i , u i where I = {1, ..., n} is the set of players, X i (a nonempty set in R l(i) , l(i) ≥ 1) is player i's strategy set and u i :

X → R is player i's payoff function, where X = i∈I X i .
For each nonempty coalition S of players, and for any subset K of S, define -S = {i ∈ I : i / ∈ S} and S -K = {j ∈ S : j / ∈ K}. If S is a singleton {i}, we denote by -i all the players but i.

We also denote by X S = i∈S X i the set of strategies of coalition S's players. A coalition structure C = {P 1 , ..., P m } is a partition of I. That is, i=1,...,m P i = I and P i ∩ P j = ∅, for each i = j.

A coalition structure C being given, we can express any strategy profile x = (x 1 , ..., x n ) ∈ X as

x = (x P 1 , x P 2 , .., x Pm ), where x P i ∈ X P i .

We say that a game G is compact, convex, and continuous if for all i ∈ I, X i is compact, convex, and u i is continuous on X.

For each C in the coalition structure C, we denote by

∆ C the |C| -1 simplex of R |C| , that is ∆ C =    λ ∈ R |C| + : |C| i=1 λ i = 1    .
Let E be a vector space and Z be a convex set in

E. A function f : Z → R is concave on Z if, for any z 1 , z 2 in Z and for any θ ∈ [0, 1], θf (z 1 ) + (1 -θ)f (z 2 ) ≤ f (θz 1 + (1 -θ)z 2 ). It is convex on Z if -f is concave on Z. A function f : Z → R is quasiconcave on Z if, for any z 1 , z 2 in Z and for any θ ∈ [0, 1], min {f (z 1 ), f (z 2 )} ≤ f (θz 1 + (1 -θ)z 2 ). It is quasiconvex on Z if -f is quasiconcave on Z. A function f : Z → R is strictly-quasiconcave on Z if, for any z 1 , z 2 in Z
with z 1 = z 2 and for any θ ∈ (0, 1), min {f

(z 1 ), f (z 2 )} < f (θz 1 + (1 -θ)z 2 ). A correspondence C : Z E is said to be closed, or has closed graph, if its graph Gr(C) = {(x, y) ∈ Z × E : y ∈ C(x)} is a closed subset in Z × E.
(For more details see [START_REF] Aliprantis | Infinite dimensional analysis: A Hitch-hiker's Guide[END_REF]).

For any two vectors a, b ∈ R n , we write a b if a i ≥ b i for all i = 1, ..., n, a b if a i > b i for all i = 1, ..., n and a b if a b and a = b.

COALITIONAL EQUILIBRIA

We first recall the definitions of the α and β-cores of an n-person game introduced in Aumann [1961,1959]. A strategy profile x is in the α-core if for every coalition K and for each deviation

x K of K, players in -K can choose a strategy profile x -K such that the payoff obtained by any member of K at (x K , x -K ) cannot be higher than its value when all the players stick to x. A strategy x is in the β-core if for every coalition K, players in -K can choose a strategy x -K such that for each deviation x K of K, the payoff obtained by any member of K at (x K , x -K ) cannot be higher than its value when all the players stick to x K .

We next recall the formal definition of the hybrid solution.

DEFINITION 2.1 [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF]) Given a coalition structure C, a feasible strategy x ∈ X is a hybrid-solution if, for every coalition S in C and for any subcoalition K of S there is

no strategy profile x K in X K such that u K (x K , x -S ) u K (x), where u K (x K , x -S ) = {inf x S-K u i (x K , x S-K , x -S ), i ∈ K}.
A slightly different definition of the hybrid solution appears as follows in [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF].

DEFINITION 2.2 [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF]) Given a coalition structure C, a feasible strategy x ∈ X is a hybrid solution if, for every coalition S in C and for any subcoalition K of S there is no

x K in X K such that u K (x K , x S-K , x -S ) u K (x), for all x S-K in X S-K .
The next property states a link between the two definitions above of the hybrid solution.

Property 2.1 Any hybrid solution in the sense of [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF] is a hybrid solution in the sense of [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF].

This result stems from the fact that for the hybrid solution in the sense of [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF],

there is no x K such that u K (x K , x -S ) u K (x).
We next introduce our notion of strong hybrid solution.

DEFINITION 2.3 Given a coalition structure C, a strategy profile x in X is a strong hybrid solution of game G if for every coalition S in C and for any subcoalition K of S, there does

not exist x K in X K such that for each x S-K in X S-K u K (x K , x S-K , x -S ) u K (x), with u K (.) = (u i (.), i ∈ K).
That is, x ∈ X is a strong hybrid solution if, given any coalition S in C and any subcoalition K of S, for each deviation x K of K, players in S -K have a strategy x S-K such that at

(x K , x S-K , x -S
), the payoffs of players in K cannot be higher than what they would obtain by playing x K , when the players of -K play x -K . This notion of coalitional equilibrium is stable against the deviations of each subcoalition K of each coalition S in C.

The hybrid solution and the strong hybrid solution rely on different definitions of a blocking coalition. In a strong hybrid solution an alternative strategy of a subcoalition is blocked by the outsiders in the coalition if it cannot guarantee a payoff at least as great as the reference one for each of its members, and strictly higher than the reference one for at least one of its members, independently of the actions taken by outsiders (the players outside the coalition sticking to their strategies).

By contrast, with the hybrid solution an alternative strategy of a subcoalition is blocked by the outsiders in the coalition if it cannot guarantee a payoff strictly higher than the reference one for all its members, independently of the actions taken by outsiders (the players outside the coalition sticking to their strategies).

In our view, the strong hybrid solution is more consistent with the assumption that coalition members cooperate whereas coalitions compete with one another. To wit, because coalition members are assumed to cooperate, there is no reason why they would not collectively choose an alternative strategy profile that does no good nor bad for all of them, and strictly benefits to at least one member.

From definition 2.3, it is immediate to obtain the two following properties.

Property 2.2 The strong hybrid solution is individually rational, i.e., for each deviation x i of every player i in coalition S, there exists a punishment strategy x S-i of players in S -i such that player i cannot be better off when players in -S play x -S .

Property 2.3 The strong hybrid solution is such that for every coalition S in C, the strategy profile

x S is a solution to the following multiobjective optimization problem X S , {u i (., x -S ), } i∈S . 5

In particular, this implies that x S is Pareto-efficient in X S .

The next property states that the strong hybrid solution is a refinement of the hybrid solutions of [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF] and [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF]).

Property 2.4 Any strong hybrid solution is also a hybrid solution (both in the senses of [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF] and [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF]).

That is because, every strong hybrid solution is a hybrid solution in the sense of [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF], and then from property 2.1, is also a hybrid solution in the sense of [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF].

The next example illustrates the differences between the notions of coalitional equilibria defined above.

EXAMPLE 2.1 Consider a three-player game G such that

I = {1, 2, 3}, X i = [0, 1], i = 1, 2, 3 and          u 1 (x) = 1 -(x 1 -x 2 ) 2 + x 3 +    3 2 x 1 if x 1 ≥ 1 2 , -3 2 x 1 + 3 2 if x 1 ≤ 1 2 u 2 (x) = x 2 + x 3 , u 3 (x) = x 3 .
Let C = {{1, 2}, {3}}. The only strong hybrid solution is given by (x 1 , x 2 , x 3 ) = (1, 1, 1). The set of hybrid solutions (both in the senses of [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF] and [START_REF] Yang | Some generalizations of Zhao's theorem: Hybrid solutions and weak hybrid solutions for games with nonordered preferences[END_REF]) is

x = (x 1 , x 2 , x 3 ) ∈ X : x 1 ∈ [ 7 -2 √ 6 4 , 1], x 2 = x 3 = 1 .
Observe that the unique strong hybrid solution Pareto-dominates all the hybrid solutions (except, of course, when the later coincides with the former). Since by assumption players 1 and 2 cooperate, it seems relevant that they will grasp any opportunity to increase their payoffs, even if only one player's payoff actually increases.

We now relate the strong hybrid solution to other solutions concepts. Recall that a strategy profile

x ∈ X is a Nash equilibrium of the game G if,

u i (y i , x -i ) ≤ u i (x)
, for all i ∈ I and for all y i ∈ X i .

5 Due to its connection with mathematical programming, multiobjective programming is more appealing than alternative terms such as vector maximization or multicriteria decision making. More importantly, [START_REF] Borm | Pareto Equilibria in Multiobjective Games[END_REF] first used the term multiobjective games (MOG). However, several later authors had used different terms such as Multicriteria Games, which were unnecessary because it not only caused confusion in the field but also showed a bit disrespect for the original contributors.

The next result property is easily obtained.

Property 2.5 If the coalition structure C is such that C = {{1}, {2}, ..., {n}}, then the set of strong hybrid solutions and Nash equilibria coincide.

We next consider a refinement of the Nash equilibria.

DEFINITION 2.4 [START_REF] Aumann | Acceptable points in general cooperative n-person games[END_REF]) A strategy profile x in X is a strong Nash equilibrium of game G if, for every coalition S, there is no y S in X S such that u S (y S , x -S ) u S (x).

That is, a strategy profile is a strong equilibrium if no coalition can profitably deviate from this profile. A straightforward implication is that any strong equilibrium is both Pareto efficient and stable with regard to the deviation of any coalition. We then have:

Property 2.6 Any strong equilibrium is also a strong hybrid solution.

We consider yet another coalition equilibrium notion, which adopts the concept of absolute optimal solution in multiobjective programming [START_REF] Zhao | Results on the absolute solutions of multiobjective mathematical programming[END_REF]) and in games [START_REF] Nessah | Absolute optimal solution for a compact and convex game[END_REF]).

DEFINITION 2.5 [START_REF] Nessah | Absolute optimal solution for a compact and convex game[END_REF]) Given a coalition structure C, a feasible strategy

x ∈ X is an absolute optimal solution relatively to C (or C-absolute optimal solution) of game G if, for every coalition S in C, we have u j (y S , x -S ) ≤ u j (x), for each j in S and y S in X S .

Hence, a strategy profile x is an absolute optimal solution relatively to C if no player in any coalition S in C, can be better off when players in S deviate from their C-absolute optimal strategy profile x S . This equilibrium is stable against deviations from any player in S.

Property 2.7 Any absolute optimal solution relatively to C is also a strong hybrid solution.

EXISTENCE RESULT

This section establishes a general existence result for a strong hybrid solution. This result relies on the specific assumption that preferences are partially quasi-transferable as well as on the arguments used in the proof of Theorem 3 in [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF].

To explain this specific assumption we shall need some notations and definitions. Firstly, without loss of generality, we assume that min x∈X u i (x) > 0 for each i ∈ I. Secondly, recall that for each

x ∈ X, S ⊆ C, u S (x S , x -C ) is defined as follows

u S (x S , x -C ) = inf y C-S ∈X C-S u i (x S , y C-S , x -C ), i ∈ S .
DEFINITION 3.1 (Partially quasi-transferable preferences) Given a coalition structure C, a game G satisfies the assumption of partially quasi-transferable preferences if, for every x ∈ X,

C ∈ C, x C ∈ X C , γ ∈ ∆ C , every S ⊆ C, t > 0 such that u S (x S , x -C ) u S (x C , x -C ) tγ S ,
there exist a sub-coalition S ⊆ C and a strategy x S ∈ X S such that

u S (x S , x -C ) tγ S .
This condition actually states that if the payoff of each member i of coalition S is such that

u i (x S , x -C ) ≥ u i (x C , x -C
) ≥ tγ i (with a strict inequality for at least one i in the first inequation), it is possible to find a subcoalition S whose members are able to secure a payoff satisfying the condition:

u i (x S , x -C ) > tγ i .
Definition 3.1 is connected with the notion of quasi-transferability of preferences that we recall next.

DEFINITION 3.2 We say that a game satisfies the quasi-transferable preferences property if for each coalition S, whenever x S i y S for all i ∈ S, and there is j in S such that x S j y S , then there is xS in X s such that xS i y S for all i in S.

This definition is adapted from a [START_REF] Rader | Theory of general economic equilibrium[END_REF] (who introduced it in a general equilibrium context)

and was also used in, e.g, [START_REF] Diamantaras | On the set of Pareto efficient allocations in economies with public goods[END_REF]. In relation to the above definition,

the assumption of partially quasi-transferable preferences refers to a transformation of the initial preferences, and only concerns a subset of agents (hence the term partially) of the initial coalition (notice that the agents in the subcoalition are assumed to do better despite the fact that their former teammates in S \ S joins people in C \ S do lower their payoffs).

We shall illustrate definition 3.1 in the following example.

EXAMPLE 3.1 Let us consider the following four players contribution game in which each player can either contribute to a collective action or retract his contribution. Let I = {1, 2, 3, 4} be the set of players, X i = [0, 1] be the strategy space of player's i, and player i's payoff function be given by

u i (x) = -ix i + j =i
x j , i = 1, 2, 3, 4.

Let the coalition structure C = {{1, 2}, {3, 4}}. This game satisfies the assumption of partially quasi-transferable preferences. Indeed, let

x ∈ X, C = {i, j} ∈ C, x C ∈ X C , γ ∈ ∆ C , t > 0, S ⊆ C such that u S (x S , x -C ) u S (x C , x -C ) tγ S .
We distinguish the following cases:

1) If S = {h} with h = i, j. Then obviously we have

u h (x h , x -C ) > u h (x C , x -C ) tγ h . 2) If S = C, then    u i (x i , x j , x -C ) > u i (x i , x j , x -C ) tγ i , u j (x i , x j , x -C ) ≥ u j (x i , x j , x -C ) tγ j . This implies that    -ix i + x j > -ix i + x j , x i -jx j ≥ x i -jx j .
We can show that both x i < x i and x j < x j . Since -ix i + x j > -ix i + x j , then

x i < x i + x j -x j i .
Let us choose > 0 such that

x i + < x i + x j -x j i .
Therefore there exists a strategy x S ∈ X S (defined by xi = x i + and xj = x j ) such that

u S (x S , x -C ) tγ S .
Our existence result is as follows.

THEOREM 3.1 A coalition structure C being given, assume that the game G is continuous, compact and convex and the functions u i are quasiconcave in x C , for each C ∈ C, i ∈ C. Then if the game satisfies the partially quasi-transferable preferences assumption, it has a strong hybrid solution.

The proof of this theorem uses the lemmata 3.1 and 3.2 established in [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF]. Let us introduce some notations used in the statements of these results. Let C ∈ C and S any subcoalition in

C (S ⊆ C). Let t S : ∆ C × X -C → R be the function defined by t S (α, y) = max t>0, x S ∈X S t    u S (x S , y) -tα S 0, ce C-S -tα C-S 0, ( Problem 1)
where

α C = (α S , α C-S ) ∈ ∆ C , e C = (1, . . . , 1) ∈ R |C| and c ∈ R + is such that u i (x) < c for all i ∈ N and x ∈ X. Finally, let t : ∆ C × X -C → R be defined by t C (α, y) = max S⊆C t S (α, y).
The first of the two lemmas drawn from [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF] is presented next.

LEMMA 3.1 [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF] When the game G is convex, continuous, and the function u i are quasiconcave in x C , for each C ∈ C, for all i, the following two properties hold.

1) The function f

C : ∆ C × X -C → R n defined by f C (α, y) = t C (α, y)α is continuous in (α, y).
2) The correspondence

θ C : ∆ C × X -C C defined by θ C (α, y) = S ⊆ C : t S (α, y) = t C (α, y)
is nonempty and has a closed graph.

Now for each S ⊆ C in C, let h S : C → ∆ C be the function defined by h S (i) = 1 |S| if i ∈ S and 0 otherwise. Define also the function g C : ∆ C × ∆ C → ∆ C by g C (α, β) = α i + max 0, β i -1 n 1 + n j=1 max 0, β j -1 n , i = 1, . . . , |C| and the correspondence σ C : ∆ C × X -C ∆ C by σ C (α, y) = co {h S : S ∈ θ(α, y)} .
Furthermore. We say that a collection B = {S 1 , . . . , S |B| } of subsets of C is balanced if there exists y S j ∈ R ++ for each S j ∈ B, such that for all x C in X C , j,x C ∈S j y S j = 1.

LEMMA 3.2 [START_REF] Zhao | The hybrid solutions of an n-person game[END_REF]) When the game G is convex, continuous, and the function u i are quasiconcave in x S for each S ∈ C and C ∈ C, for all i, then the correspondence π

C : ∆ C × ∆ C × X -C ∆ C × ∆ C defined by π C (α, β, y) = {g C (α, β)} × σ C (α, y)
satisfies the two following properties.

1) It has closed bounded nonempty convex values and a closed graph.

2) For each fixed y ∈ X -C , the correspondence π C (α, β) = π C (α, β, y) has a fixed point (α * , β * ) and at this fixed point θ C (α * , y) is a balanced set.

In addition to the two above lemmata, our proof of Theorem 3.1 will make use of a third lemma which we shall prove next. This lemma refers to the notion of strong core solution which we state beforehand:

DEFINITION 3.3 Let a coalition C in C be given. A strategy profile x C is said to be a strong core solution of the subgame G(x -C ) = S, X S , u S (x S , x -C ) if, for any subcoalition S of C, there does not exist x S in X S such that u S (x S , x -C ) u S (x).

We shall also say that the game G(x -C ) = S, X S , u S (x S , x -C ) is balanced if for all y in R |C| and any balanced collection of coalitions θ such that for each S in θ, there exists x S ∈ X S satisfying u S (x S , x -C ) ≥ y S , then there exists

x C ∈ X C such that u C (x C , x -C ) ≥ y C .
LEMMA 3.3 Assume that the game G is continous, compact and convex. Assume also that the functions u i are quasiconcave in x S , for each S ∈ C and C ∈ C, and such that min X u i (x) > 0 for all i. Suppose further that preferences are partially quasi-transferable. Then the correspon-

dence δ C : ∆ C × X -C X C defined by δ C (α, y) = {x C ∈ X C : u C (x C , y) -t C (α, y)α 0}
satisfies the next two properties.

1) It has closed bounded nonempty convex values and a closed graph.

2) If θ C (α, y) is a balanced family and G(y) is a balanced game for each y, then each x C ∈ δ C (α, y) is a strong core solution of G(y).

PROOF. The first assertion is then a direct consequence of the continuity of the functions u i , t C (α, y)α and the quasiconcavity of

u i in x C .
Let us prove the second assertion. As θ C (α, y) is a balanced family and G(y) is a balanced game for each y, we can show as in the proof of lemma 3 in Zhao that t C (α, y) = tC (α, y).

Now let x C ∈ δ C (α, y). If x C is not a strong core solution of G(y), then there is a coalition S ⊆ C and x S ∈ X S such that u S (x S , y) u S (x C , y). (3.1) Since x C ∈ δ C (α, y) and t C (α, y) = t C (α, y), then (3.1) becomes u S (x S , y) t C (α, y)α S .
But by the choice of the constant c and since

x C ∈ δ C (α, y), then ce C-S t C (α, y)α C-S . that    u S (x S , y) tα S ce C-S tα C-S .
Hence, t and x S is feasible for the Problem 1. Then t ≤ t C (α, y) which contradicts the relation

t > t C (α, y).
Proof of Theorem 3.1.

Let Ω be the set defined by

Ω =   i=1,...,r (∆ C i × ∆ C i )   × X
where r is the cardinal of C.

Let Ψ : Ω → 2 Ω be the correspondence defined by

Ψ (α 1 , β 1 ), . . . , (α r , β r ) , x =   i=1,...,r π C i (α i , β i , x -C i )   ×   i=1,...,r δ C i (α i , x -C i )   .
By Lemmas 3.1-3.3, the correspondence Ψ is defined on a compact convex set and has nonempty closed convex values and a closed graph. Then by Kakutani fixed point Theorem, Ψ has a fixed point (α i , β i ), for each i = 1, . . . , r and there is a vector x = (x C 1 , . . . , x Cr ) such that

(α i , β i ) ∈ π C i (α i , β i , x -C i ) and x C i ∈ δ C i (α i , x -C i ). Since u i is continuous on X, quasiconcave in x S
and the game G(x -C i ) is balanced [START_REF] Scarf | On the existence of a cooperative solution for a general class of N -person games[END_REF], therefore by Lemma 3.3, x is a strong hybrid solution.

Examples

In this section we study the condition of partially quasi-transferable preferences in two examples of games having a strong hybrid solution. The first example includes three versions of the garbage game introduced by Shapley and Shubik [1969]. The second example is a coalitional game with environmental externalities.

Some versions of the garbage game

• A garbage game with side-payments Consider the version of Shapley and Shubik [1969] garbage game studied in [START_REF] Laffont | Cours de théorie microéconomique[END_REF], in which four neighbors decide on where to dump their garbage. Assume that each neighbor has an endowment of a transferable good equal to one and a quantity of garbage also equal to one. Let z ji ≥ 0 be the quantity of good transfered by neighbor j to neighbor i, and x ji ≥ 0, the quantity of garbage that neighbor j locates in the garden of neighbor i.

For each i, 0 ≤ 4 j=1 z ij ≤ 1, 0 ≤ 4 j=1 x ij ≤ 1, 0 ≤ z ij .
Suppose that all agents i have the same payoff function

u i (x ji , z j i) = 4 j=1 z ji - 4 i=1 x ji , (4.1)
and assume that there are only two coalitions. The first coalition includes neighbors 1, 2, 3 while the second only includes neighbor 4.

Then, one can check that the strategy profile z ii = 1 for all i, x i4 = 1, for all i = 1, 2, 3,

x 41 = 1/3, x 44 = 0 is both a hybrid and a strong hybrid solution. Notice that this game is continous, compact and convex and the functions u i are quasi-concave. Let us check that indivudals' preferences are partially quasi-transferable. To do this we shall study the two coalitions in turn.

Case C = {4}.

The property is clearly satisfied for this coalition.

Case C = {1, 2, 3}.

Let us examine all the subcoalitions S included in {1, 2, 3} (as well as the coalition {1, 2, 3} itself).

• S = {1, 2, 3}.

Suppose that the following inequalities hold

4 j=1 z ji - 4 j=1 x ji ≥ 3 j=1 zji + z 4i - 3 j=1 xji -x 4i ≥ γ i t, i = 1, 2, 3. (4.2)
and that at at least one of the left-hand sides inequalities hold strictly. One can check that it is always possible to rearrange the transfers among individuals in such a way that all the left-hand sides inequalities become strict.

• S = {1, 2}.

j=1,j =3 z j1 - 4 j=1,j =3 x j1 -1 ≥ 3 j=1 zj1 + z 41 - 3 j=1 xj1 -x 41 ≥ γ 1 t (4.3) 4 j=1,j =3 z j2 - 4 j=1,j =3 x j2 -1 ≥ 3 j=1 zj2 + z 42 - 3 j=1 xj2 -x 42 ≥ γ 2 t. (4.4)
Again, we can always design some transfers of good ensuring that the left-hand sides inequalities are both strict.

The argument is similar for the other two-person subcoalitions. Moreover, one can check that all one-person S coalitions satisfy the partially quasi-transferable preferences property as well.

Therefore preferences are partially quasi-transferable.

• A garbage game with bounded payoff for one player

We now pay focus on a variant of the garbage game considered above. Assume that the payoff function of all neighbors but neighbor 2 is given by

u i (x ji , z ji ) = 3 ×   4 j=1 z ji   - 4 j=1
x ji , i = 1, 2, 4. (4.5) Furthermore, assume that the payoff function of neighbor 2 reads

u i (x ji , z ji ) = min    1 2 , 4 j=1 z j2    . (4.6)
That is, neighbor 2's payoff is bounded above and does not depend on the quantity of garbage located in his garden. Thus, preferences are not partially quasi-transferable (since that property is not satisfied for the subcoalition {2}). Yet the game is continous, compact and convex and the functions u i are quasi-concave.

Consider the strategy profile:

z ii = 1 (for i = 1, 3, 4), z 21 = 1/2, x i4 = 1, x 4i = 1/3, x 44 = 0.
One can check that profile is a strong hybrid solution. Therefore, the assumption that preferences are partially quasi-transferable is not necessary for the existence of a strong hybrid solution.

Also notice that some hybrid solutions are not strong. To see this, let us pay attention to the strategy profile: z ii = 1, x i4 = 1, x 4i = 1/3, x 44 = 0 and concentrate on the coalition {1, 2, 3}. Consider the subcoalition {1, 2}. One can see that it is impossible to increase the payoff of neighbor 2 since it is equal to its maximum value. The same remark applies for coalition {2, 3}. Now consider coalition {1, 3}. Both neighbor cannot jointly increase their payoffs since if they deviate, player 2 can split his garbage equally between them. Hence the strategy profile is a hybrid solution. Yet, it is not a strong hybrid solution. For instance, let us concentrate on coalition {1, 2}. Suppose that neighbor 2 gives a quantity of good 1/2 to neighbor 1. Then his payoff remains equal to 1/2.

Assume that to punish the deviation by neighbors 1 and 2, neighbor 3 transfers all his garbage in the garbage of neighbor 1. The latter's payoff is then equal to 25/6, whereas the payoff associated with the candidate equilibrium strategy profile equals 8/3. Thus in the deviation considered above, neighbor 2's payoff is unchanged, but neighbor 1's increases.

• A garbage game with altruism

There are four players and two coalitions as before. In this version of the garbage games, we assume that players 1, 3 and 4 have the same preferences, which are given by the payoff function

3 × 4 j=1 z ji -4 i=1 x ji , i = 1, 3, 4.
Preferences of player 2 are given by: 4 j=1 z j2 + z 21 . Thus, player 2 is altruistic: his payoff increases when he transfers some good to player 1.

Let us check that preferences are partially quasi-transferable. We consider the two coalitions in turn.

Case C = {4}.

The property is clearly satisfied for this coalition.

Case C = {1, 2, 3}.

Let us consider all the subcoalitions S included in {1, 2, 3} (and the coalition {1, 2, 3} itself).

• S = {1, 2, 3}.

Suppose that the following inequalities hold

3   4 j=1 z j1   - 4 j=1 x j1 ≥ 3   3 j=1 zj1 + z 41   - 3 j=1 xj1 -x 41 ≥ γ 1 t (4.7) 4 j=1 z j2 + z 21 ≥ 3 j=1 zj2 + z 42 + z21 ≥ γ 2 t (4.8) 3   4 j=1 z j3   - 4 j=1 x j3 ≥ 3   3 j=1 zj3 + z 43   - 3 j=1 xj3 -x 43 ≥ γ 3 t, (4.9)
and that at least one of the left-hand sides inequalities hold strictly. Let us focus one, say player i, for whom the inequality is strict. Clearly, is always possible to rearrange the transfers among individuals in such a way that all the left-hand sides inequalities become strict.

• S = {1, 2}

Finally, consider the following strategy profile: z ii = 1 for all i = 2, z 22 = 0, z 21 = 1, x i4 = 1 for all i = 1, 2, 3, z 4i = 1/3 for all i = 1, 2, 3. This profile is a strong hybrid solution (it is impossible to increase the welfare of any deviating coalition).

Notice that there is a set of hybrid solutions indexed by the transfer of good by player 2 to player 1. For instance, consider the solutions obtained as above but with z 22 = 1 -z 21 , 0 ≤ z 21 ≤ 1.

Then there is a strong hybrid solution strictly included in the set of hybrid solutions (set z 21 = 1).

The last two versions of the garbage game illustrate the fact that the condition of partially quasitransferable preferences is not necessary for the existence of a strong hybrid solution, nor it is necessary for a strong hybrid solution to be strictly included in the set of hybrid solutions.

A coalitional game with environmental externalities

Consider a coalitional game with environmental externalities where there are n agents (e.g., countries, firms). Assume that these agents produce and consume the same good and that production is a polluting activity. Let y i denote the production of agent i and let x i be the emission of pollutants resulting from this production. Also assume that y i = g i (x i ) and that u i (x) = g i (x i ) -v i i∈I x i , where g i : R + → R + is an increasing function such that g i (0) = 0 and v i : R + → R + is increasing on R + . Thus agent i's payoff increases with the consumption of the produced good (g i (x i ))

and decreases with global pollution ( i∈I x i ). Further, assume that X i = {x i : 0 ≤ x i ≤ x 0 i } (x 0 i is agent i's maximum emission level), and that both g i and v i are continuous. Now, suppose that there are x ∈ X, x S ∈ X S , C ∈ C, x C ∈ X C , γ ∈ ∆ C , S ⊆ C, t > 0 such that u S (x S , x -C ) u S (x C , x -C ) tγ S . This implies that there is an agent i in S such that

g i (x i ) -v i   j∈S x j + j∈C\S x 0 j + j∈-C x j   > tγ i . (4.16)
Here, the agents j ∈ C \ S set their emission levels equal to x 0 j in order to punish the deviation of coalition S. Necessarily, x i > 0 (because v(.) takes non-negative values, and tγ i is non-negative).

Then by continuity of g i (.) there exists > 0 such that x i -> 0, and such that (4.18) This proves that preferences are partially quasi-transferable.

g i (x i -) -v i   x i -+ j∈S,j =i

CONCLUSION

We have proposed a notion of strong hybrid solution, which is a refinement of Zhao's hybrid solution. Like this solution, the strong hybrid solution is well suited to study situations where people cooperate within coalitions but where coalitions compete with one another. In the strong hybrid solution, as opposed to Zhao's solution, the strategy profile assigned to each coalition is strongly-Pareto optimal. To put it another way, all possible gains of cooperation at the coalition level have been exhausted.

We have also given existence results for the strong hybrid solution in games with compact and convex strategy spaces.

We would like to point out that the concept of hybrid solutions, as well as that of strong hybrid solution introduced in this paper can be used to study the existence of the consensus equilibria introduced by [START_REF] Di | The consensus equilibria of mining gap games related to the stability of Blockchain Ecosystems[END_REF] for the mining gap games related to the stability of Blockchain Ecosystems, which plays a key role for the development of Ecosystems. The strong hybrid solution may also be applied to study coalitional game with altruism (see, e.g., the multiobjective games analyzed in [START_REF] Lejano | Modeling the commons as a game with vector payoffs[END_REF]).

There are at least two issues that can be studied in further research. The first one is the existence of the strong hybrid solution in games with discontinuous payoffs. Another interesting issue is the existence of this solution with general strategy spaces, and with an arbitrarily large number of players.

  i . (4.17) Now set S = S and xk = x k , k = i, xi = x i -. Since v k (.) is decreasing it holds that for every agent k in S:g k (x k ) -v k

Roughly said, a strategy profile in the α-core of a game is essential if whenever we slightly modify the game, the α-core of the modified game has elements which are close to the strategy profile. A game is Hadamard well-posed if its α-core has a unique element and if this element changes continuously with the conditions defining the game.

[START_REF] Zhao | The hybrid equilibria and core selection in exchange economies with externalities[END_REF] also applies the hybrid solution concept to an exchange economy with externalities.[START_REF] Zhao | TU oligopoly games and industrial cooperation, in Handbook of Game Theory and Industrial Organization[END_REF] reviews different solution concepts for TU games in Industrial Organization.

This definition implies that there is no subcoalition which can choose a strategy such that the payoff obtained by any of its members is no lower (as opposed to not strictly greater as in[START_REF] Zhao | The hybrid solutions of an n-person game[END_REF]) than what he obtains with the hybrid solution, when the outsiders in the coalition minimize his gain.

Following Rader [1972], preferences are said to be quasi-transferable if whenever a strategy profile x is at least as good as another one x for a group of players, and strictly better for at leat one agent, then there is a third profile x which strictly better than x for all the agents. Our notion of partially quasi-transferable preferences builds on Rader's definition.

(4.11)

We can reason as before. When one of the left-hand side inequality is strict, one can make a transfer that striclty benefits the other player of the deviating coalition.

The same reasoning used above applies.

• S = {1, 3}.

Assume that the following two inequalities hold.

and that at least one of them holds strictly. Again, we can always design some transfers of good ensuring the the left-hand sides inequalities are strict.

Finally, all the one-person S coalitions satisfy the partially quasi-transferable preferences property.

Therefore preferences are partially quasi-transferable. The game satisfies all our assumptions.

Therefore there exists a strong hybrid solution.

Now consider the following strategy profile: z ii = 1 for all i, x i4 = 1 for all i = 1, 2, 3, z 4i = 1/3 for all i = 1, 2, 3. This profile is a hybrid solution (it is impossible to (weakly) increase the welfare of any deviating coalition). Yet, it is not a strong hybrid solution. To see this, consider coalition {1, 2}. Clearly if player 2 transfers a quantity 1/2 to player 1, his welfare is unchanged while that of player 1 strictly increases (even when player 3 minimizes the deviants' payoffs).


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Polymerization-Induced Self-Assembly (PISA) for in situ drug encapsulation or drug conjugation in cancer application

  CAUTIONThis manuscript has been accepted after peer review and corresponds to a version prior to editing, proofreading, and formal publication. The final version available on the editor website may be different from this version as a result of the above editing process. Readers should consider obtaining the final version from the journal website if they want to ensure full accuracy of information. The corresponding author can also be contacted in case of need.

Hien Phan, Mélissande Cossutta, Claire Houppe, Clémence Le Coeur, Sylvain Prevost, Ilaria Cascone, José Courty, Jacques Penelle, Benoit Couturaud To cite this version: Hien Phan, Mélissande Cossutta, Claire Houppe, Clémence Le Coeur, Sylvain Prevost, et al.. Polymerization-Induced Self-Assembly (PISA) for in situ drug encapsulation or drug conjugation in cancer application. Journal of Colloid and Interface Science, 2022, 618, pp.173-184. 10.1016/j.jcis.2022.03.044. hal-03875308