Joël Puibasset
email: puibasset@cnrs-orleans.fr

J.-M Pellenq

Water adsorption in disordered mesoporous silica (Vycor) at 300 K and 650 K: a Grand Canonical Monte Carlo simulation study of hysteresis

Keywords: BG_Keywords"). Grand Canonical Monte Carlo, water, mesoporous silica, adsorption, desorption, hysteresis, coexistence, grand potential, free energy

Word Style "BD_Abstract").

This numerical simulation paper focuses on the adsorption / desorption of water in disordered mesoporous silica glasses (Vycor-like). The numerical adsorbent was previously obtained by off lattice method, and was shown to reproduce quite well the micro-and meso-textural properties of real Vycor, as well as morphological (pore size distribution) and topological (pore interconnections) disorder. The water-water interactions are described by the SPC model while water-silica interactions are calculated in the framework of the PN-TrAZ model. The water adsorption/desorption isotherms and the configurational energies are calculated by the Grand Canonical Monte Carlo simulation method. The low pressure results compare well with experiments, showing the good transferability of the intermolecular potential. It is shown that if the hysteresis loop observed in the adsorption / desorption isotherm is considered as a true phase transition (which is actually still an open question in the case of disordered porous materials), then it is possible to calculate the grand potential by applying the thermodynamic integration scheme. The grand potential is shown to be multi-valued for low (subcritical) temperature, and continuous for high (supercritical) temperature. A coexistence point is found within the hysteresis loop, actually close to the vertical desorption line. Below the equilibrium chemical potential, the gas-like branch is stable whereas the liquid-like branch is metastable. The situation is reversed above the coexistence point.

INTRODUCTION

The adsorption of fluids in porous materials exhibits very interesting features. [1][START_REF] Gregg | Adsorption, Surface Area and Porosimetry[END_REF][START_REF] Rouquerol | Adsorption by Powders and Porous Solids[END_REF] For instance, fluids condensate in dense liquid-like state in mesoporous adsorbents. This well know phenomenon called " capillary condensation" has focused the attention of experimentalists as well as theoreticians for one century. Despite the numerous efforts to give a comprehensive analysis of the adsorption phenomenon, there is still open questions, as for instance whether the observed hysteresis loop in adsorption / desorption isotherms is related to true first-order phase transition. The development of molecular simulation allows to consider more and more realistic systems, with possible direct connections to industry or other applications. These realistic systems, as opposed to ideal models as cylindrical or slit pores, have to include the topological and the morphological disorder, as surface roughness, chemistry, large pore size distribution, interconnections between pores, etc.

In this molecular simulation study, we focus on the adsorption of water in mesoporous Vycor-like silica glass, which is of great interest due to large field of applications: geophysics, pharmaceutics, industry, or environment. The evoked processes, like water infiltration, de-pollution, phase separation, or catalysis, take advantage of the effect of confinement of water on nanometric scale. [START_REF] Dore | [END_REF][5][6] Most of the previous studies focused on the dynamical or structural properties of confined water. [7][8][9][10][11] However, the thermodynamic properties are also interesting, and actually sometimes essential: a molecular dynamics run should always be started from a thermodynamically representative (stable) initial configuration which reproduces, for instance, the correct density for the confined fluid. The Grand Canonical Monte Carlo method allows to simulate a real adsorption experiment where the temperature and the chemical potential of the fluid (or the pressure of the gas) are maintained constant. The adsorbent was previously constructed by off-lattice methods to mimic the micro-and meso-textural properties of real Vycor. The water adsorption isotherms at 300 K and 650 K are calculated in this numerical mesoporous glass sample, as well as the configurational energies, the isosteric heat of adsorption, the free energies, and the entropy. All these quantities are compared to experiments when possible, and differences are commented. The nature of the hysteresis loop observed at 300 K is also discussed.

NUMERICAL DETAILS

The atomistic Vycor-like mesoporous structure considered in this study has been generated by offlattice reconstruction algorithm. Portions of an initial cubic crystal of cristobalite (106.95 Å of edge) are cut out by applying the off-lattice functional (shown in Fig. 1) representing the gaussian field associated to the volume autocorrelation function of the studied porous structure. 12,13 It has been shown [14][15][16] that this numerical sample reproduces quite well micro-and meso-textural properties of real Vycor, and constitutes an improvement of the topological description of silica mesoporous glasses, because of the presence of disordered and interconnected pores. In order to have a realistic description of the surface chemistry, all silicon atoms in an incomplete tetrahedral environment are removed, and all oxygen dangling bonds are saturated with hydrogen atoms placed 1 Å from oxygen perpendicular to the surface.

The porosity of the sample is 0.28, the density 1.56 g/cm [START_REF] Rouquerol | Adsorption by Powders and Porous Solids[END_REF] , the specific surface area 210 m 2 /g, the mean pore diameter 36 Å (from chord distribution), and OH surface density 7 OH/nm 2 , all in agreement with experimental values: 0.28, 1.50 g/cm [START_REF] Rouquerol | Adsorption by Powders and Porous Solids[END_REF] , 200 m 2 /g, and around 7 OH/nm 2 if out-gassed around 400°C 15,16 , except for the pore size slightly smaller in our pseudo-vycor compared to real one (40 Å). It has been shown that this numerical sample reproduces quite well micro and meso textural properties of real vycor. 12,13 The adsorbed water is described with the SPC model [START_REF] Berendsen | Intermolecular Forces (B. Pullman[END_REF] , because it is a fast computable model well suited for very large systems (the completely saturated system contains more than 12000 water molecules). This model reproduces well the thermodynamic and structural properties around ambient temperature, like vapor pressure, enthalpy of vaporization, and radial distribution functions. [START_REF] Jorgensen | [END_REF] The interaction of water with Si, O and (surface) H Vycor species is assumed to remain weak, in the physisorption energy range. In this work, we have used a TrAZ form of the original PN-type potential function as reported for adsorption of rare gases and nitrogen in silicalite-1. 19 The PN-TrAZ potential function is based on the usual partition of the adsorption intermolecular energy restricted to two body terms only: it includes a dispersion interaction term, a repulsive short range contribution and an induction term. Three body dispersion (XYY) terms (where X is the adsorbate and Y an adsorbent species) can be obtained in principle from the same approach 20 but are neglected in the present study since they are expected to be significant for water molecules in contact with the silica surface (which is a small number of the total adsorbed amount). The choice of this particular model to describe the water/adsorbent potential was motivated by the good degree of parameter transferability. Indeed, in a previous study 21 , we found that using a set of potential parameters previously derived for adsorption in silica zeolite 19 augmented to take into account hydroxyl groups, the TrAZ model allows reproducing both low coverage experimental adsorption isotherm (amount adsorbed versus pressure at constant temperature) and isosteric heat curve with no further adjustment. In the TrAZ model, the adsorbatesurface energy (

A k i u ∈ ) of the k
th atom of molecule A at a position i in the simulation box, is given by

{ } 2 i E k H Si O j n ij kj n n n r b kj A k i r C f e A u ij kj α 2 1 , , 2 , 2 5 3 2 -       - = ∑ ∑ ∈ = - ∈ (1)
Where r ij is the distance between the matrix species j and the position i of the k th atom of a given molecule, f 2n are the so-called damping functions that depend on interspecies distance and on a repulsive parameter (see below) and finally E is the electric field created at adsorbate position i due to the inorganic species ; all other symbols are physical parameters that depend on electronic properties of interacting species only. The sums runs over all (j) atomic sites in the matrix that are oxygen, silicon and hydrogen atoms. The first term in the sum is a Born-Mayer term representing a two-body form of the short range repulsive energy due to finite compressibility of electron clouds when approaching the adsorbate at very short distance of the pore surface. There is one such a term per pair of interacting species. The repulsive parameters (A kj and b kj ) are obtained from mixing rules of like-atoms pairs (see below). The second term in the above equation is a multipolar expansion series of the dispersion interaction in the spirit of the quantum mechanical perturbation theory applied to intermolecular forces. [START_REF] Stone | The Theory of Intermolecular Forces[END_REF] It has been shown that two (and three body) dispersion C 2n coefficients for isolated or in-condensed phase species can be obtained from the knowledge of the dipole polarizability and the effective number of polarizable electrons N eff of all interacting species 19,20 which are closely related to partial charges that can be obtained from ab initio calculations. The f 2n terms in the above equation are damping functions of the form:

( ) ij r kj b n 2 0 m m ij kj n 2 e . ! m r b 1 f - = ∑         - = (2)
The role of these damping functions is to avoid divergence of the dispersion interaction at short distance where the wave functions of the two species overlap (i.e. when the interacting species are at contact). [START_REF] Tang | [END_REF] They allow taking into account the possible electronic exchange which has a non zero probability at short distance even for two close shell structures. For each pair of interacting species, they are parametrized with the single b kj repulsive parameter. The damped dispersion multipolar expansion can be seen as a convenient way to make the perturbation theory valid at short inter-atomic separations.

The last term in equation ( 1) is the induction interaction as written in the context of the quantum mechanical perturbation theory applied to intermolecular forces. [START_REF] Stone | The Theory of Intermolecular Forces[END_REF] It represents an attractive energy arising from the coupling of the polarisable electronic cloud of the adsorbate atom of polarizability α k at position i with the electric field E i induced by the charges carried by framework species (O, Si and H) that result from the bonding process within the inorganic matrix itself. In the case of water interacting with vycor, one has to parametrize six different adsorbate/adsorbent-species potentials; all atomic parameters and coefficients are given in Tables 1 and2. Note that repulsive parameters for like-pairs are taken from a previous work on the simulation of rare gases in silicalite 19 using Bohm and Ahlrichs 24 combination rules based on ab initio results at the Hartree-Fock level approximation which only describes the repulsive energy term in the case of non charged and polar species. This type of potential function based on the PN-TrAZ parameterization method was used in various studies of molecular and covalent fluids at interfaces from open surfaces 21,25,26 to microporous zeolites [27][28][29][30][31][32] and more recently in the case of mesoporous vycor-like materials 14,21,26,[33][START_REF] Puibasset | 6th Int. Symp. on the Characterization of Porous Solids[END_REF][35][36] .

Minimal image convention is adopted to calculate all interactions. Long range mean-field correction to dispersion terms cannot be precisely calculated because the density is not uniform. Actually, the size of the box is large enough (107 Å) so that large distance contributions are negligible. The electrostatic contribution is evaluated by summing on neutral subgroups of atoms of highest symmetry (tetrahedral silicon SiO 4 and surface hydroxyls OH). Implementation of Ewald summation procedure 37,38 has proven to be of little improvement, probably due to the large box size, absence of isolated charges, and negligible total dipolar moment of the substrate (due to isotropy in normal orientation on Vycor surface). Calculations show that the largest part of the substrate contribution to electrostatics comes from the surface dipoles (hydroxyls), the tetrahedral SiO 4 giving rise to weak quadrupolar term only.

Considering the polarization contributions (last term of equation 1), one has to take into account the fact that the SPC model already contains some polarization effects (treated as effective contributions in the bulk phase parameterization). It is then impossible to simply add the surface polarization contribution (the local electric field from surface and water molecules have to be added instead). However, since the surface contribution term was shown to be small, it was decided to neglect this contribution.

In order to save computing time, the water/substrate interactions have been calculated in advance and saved on a grid. This is possible because the substrate is kept rigid in this study. The grid mesh has been chosen equal to 1 Å. Due to memory constraints it could not be made thinner. However, close to the silica surface, this mesh is not fine enough for small dipolar molecules like water. This is why the contribution of the substrate has been divided in two parts, inside and outside a sphere of 6.5 Å in radius, centered on the grid cell to be evaluated. Only the contribution from outside is saved on the grid, the rest being calculated directly during the course of the simulation. A neutrality constraint has been added so as to avoid fast oscillations of electrostatic contributions due to isolated charges coming in and out of the 6.5 Å radius sphere: for instance O and H atoms from the same surface hydroxyl group are considered together. The choice of the radius is a compromise between CPU time saving and grid smoothness. Three different grids are necessary: two dispersion-repulsion grids for both O and H atoms of water, plus one for electrostatic potential. Since there are eight corners for each cubic cell, and since the 6.5 Å radius sphere has to be centered on the cell, the total number of grid used is 8 time 3, and interpolation in each cell is made between 8 grids. All grids are defined everywhere in the simulation box due to the short distance cutoff. The result obtained with this method for the whole interaction is always within 4% of the value obtained by a completely direct calculation with all the atoms of the simulation box. Our method is then very accurate, and very fast (few tens of atoms taken into account instead of the 60 000 species that are present in the silica texture of our vycor numerical sample). This procedure is well suited to study adsorption properties of polar molecules and allows considering large systems.

The water-water interaction calculations become excessively slow at low temperature when the substrate is almost completely filled (more than 10000 water molecules). It was then decided to implement a hierarchy method 39 . The simulation box is divided in a 3-dimensional array of 5 Å wide cells. The total interaction energy of a water molecule with the others is the sum of the individual contributions of the molecules contained in the neighboring cells (within a distance of 15 Å) and the multipole representation for the other cells. The uncertainty introduced by this method does not exceed few percent. Let us first focus on the adsorption isotherm obtained at 300 K and for pressure lower than the water saturating vapor pressure. The Fig. 3 shows our numerical results (circles), as well as two experimental data found in the literature for comparison. The first one (solid line) is a gravimetric measurement by T.

ADSORPTION / DESORPTION ISOTHERMS AND CONFIGURATIONAL ENERGIES adsorption/desorption isotherms

Takei, et al. [START_REF] Takei | [END_REF] in a 7930 Vycor sample characterized by nitrogen adsorption (pore size around 4.72 nm in diameter). The other one (dashed line) has been obtained by Markova, et al. 42 with a double twin microcalorimeter, in a 7930 Vycor glass (pore size between 4 and 7 nm). Since the geometric characteristics of the experimental and numerical Vycor are not exactly identical, the choice of the normalization slightly influences the comparison. The best normalization to compare low coverage adsorption isotherms would be to divide it by the geometric surface area of the sample. However, due to the uncertainties in its experimental determination, this normalization has been discarded, and two other are proposed. In the upper figure, the adsorbed water mass has been normalized to the mass of the Vycor sample, and plotted as a function of the pressure, related to the chemical potential by assuming that the water vapor is ideal, which is a good approximation up to the saturating vapor pressure 43 . The pressure is actually normalized to the saturating vapor pressure P 0 = 0.044 bar at 300 K for the SPC model 43,44 . In the lower figure, the amount of adsorbed water has been normalized to its maximum (full saturation) for each case. This normalization seems to be less sensitive to morphological details of the sample.

However, it is very sensitive to the choice of the saturating coverage. In our case, since the denser the liquid, the less effective the insertion of a new molecule by GCMC, the complete saturation might not be reached due to the possible incompleteness of the convergence of the simulation run. The first normalization is then probably more reliable. The calculated isotherm (circles) shows a rapid increase at low pressure (below 10% of the saturating pressure) characteristic of high adsorbate-substrate interaction, as expected for hydrophilic silica surfaces. The agreement of simulation results, which fall in between the experimental determinations up to the mono-layer coverage (around P/P O = 0.6) in both normalization cases, validates the good transferability of the potential model used in this study.

Let us now consider the whole adsorption / desorption isotherms at 300 K and 650 K. The results are now normalized to the surface area of the substrate exposed to water vapor (in µmol / m 2 ), and plotted in Fig. 4 as a function of the chemical potential of water imposed by the GCMC simulation. It is important to note that this chemical potential contains only both the ideal translational and excess contributions:

the ideal rotational contribution -kT(-4.09+3/2lnT), with T in Kelvin, is not taken into account. [START_REF] Frenkel | Understanding Molecular Simulation[END_REF] As can be seen, the high temperature (650 K) isotherm is reversible, whereas the low temperature (300 K) case presents a large hysteresis. The ambient temperature adsorption / desorption curve is of type IV in the IUPAC classification. The steep rise around µ=-45.5 kJ/mol is associated to the capillary condensation of water in the mesoporosity of the vycor glass. The rather large pore size distribution explains why this " condensation" branch is not vertical, in agreement with experimental observations. As can be seen, the desorption branch does not follow the adsorption branch. It is probably due to the energetic barrier necessary to nucleate a gas bubble into the adsorbed liquid-like water prior to the desorption of water.

However, once a low enough chemical potential is reached to nucleate the gas-like phase, the desorption proceeds in one step (vertical desorption branch). These simulation results are in qualitative agreement with experimental data, with an asymmetric adsorption / desorption hysteresis characteristic of disordered and interconnected pores. 1,[START_REF] Takei | [END_REF]47 However, in our simulations the capillary condensation occurs for a chemical potential higher than expected. The hysteresis loop extends beyond the bulk saturating chemical potential. An energetic study of the surface of Vycor during the adsorption process allows to propose explanations.

The energetic characteristics of the adsorbent has been measured by evaluating the distribution in energy of interaction of a water molecule with the substrate in the vicinity of the surface. From a technical point of view, the position and orientation of the water molecule are generated randomly within a fixed volume containing the adsorbent surface. The water/substrate interaction energy is calculated. After a large sampling in positions and orientations is acquired, one can extract the distribution of the interaction energy with the substrate. In other words, it corresponds, in the six dimensional positional and orientational coordinate space of one particle, to the volume density for a given value of interaction energy. This distribution has been calculated in three cases. Firstly, in our numerical Vycor sample, where the position of the water molecule is chosen arbitrarily in the simulation box. Secondly, the same energetic distribution has been calculated for the same Vycor sample hydrated by one monolayer of water and considered as the whole " substrate" . Finally, the same calculation has been carried out for half a space filled with bulky water previously equilibrated at 300 K by canonical Monte Carlo run. This allows to compare the energetic characteristics of the surface of dry Vycor, hydrated Vycor, and water liquid/gas interface. The results obtained are given in Fig. 5. As can be seen, the three curves follow the same general feature which can be described as follows. Firstly, the long tail in the positive values of energies (up to infinity), corresponds to positions chosen very close to the surface or inside the substrate (repulsive interaction). Secondly, the large peak around zero (weak interaction) corresponds mainly to contributions from regions of space far from the substrate, which is not of our interest. Finally, the negative energy part of the distribution is the most important since it concerns the attractive sites created by the substrate. The question of the normalization has to be raised.

The total surface below the curve corresponds to the volume of the six dimensional space defined by the positional and orientational coordinates of the test-water molecule. Its value depends on the total volume of the sampling box, and is not an intrinsic surface property. Then it cannot be used for normalization.

Furthermore, the relative amplitudes of the three portions of the curve, as previously described, depend on the relative spatial extension occupied by the substrate and the voids in the simulation box, and are again not surface properties. Then the absolute amplitudes cannot be normalized or compared. The two quantities which are characteristic of the surface are the lower limit in the energy distribution (most energetic site) and the slope of the curve in the negative region, which gives the relative abundance of the sites according to their energy. As can be seen, for dry Vycor, the site population decreases at a rate of a factor 3.66 for a decrease in energy of 10 kJ/mol, down to a low energy limit of -80 kJ/mol which is the absolute minimum in the energy landscape of our numerical mesoporous Vycor glass. The surface energy distribution of the water liquid/gas interface decreases by a factor 4.7 every 10 kJ/mol, down to -50 kJ/mol. The third curve shows that the site population of a water monolayer adsorbed on the Vycor surface decreases by a factor 11.5 every 10 kJ/mol down to an absolute minimum of about -40 kJ/mol.

As shown by the Fig. 5, the dry Vycor surface is much more attractive than the bulk liquid water flat surface, and explains the hydrophilic properties of the mesoporous glass. On the other hand, the water monolayer adsorbed on Vycor presents much less attractive sites than the flat liquid/gas water interface.

This hydrophobicity of the water/Vycor monolayer for the next incoming water molecules explains why capillary condensation extends beyond the saturating chemical potential. The strong distortion of the tetrahedral structure of water in the vicinity of the silica surface 7,[9][10][11]26 may explain this relative hydrophobicity.

configurational energies and isosteric heat of adsorption

The Fig. 6 gives the molar configurational energy (in kJ per mol of water molecules) at 300 and 650 K as a function of the chemical potential. The 650 K curve is reversible. For very low coverage, the configurational energy per particle decreases with an increase in coverage. The first molecules adsorb on the most energetic sites, which explains the low initial value for the configurational energy. The subsequent molecules probably adsorb on sites almost as energetic as the first ones, and close enough to allow the molecules to establish hydrogen bonds with the first ones. The strength of the interaction with the silica maintain the molecules close to each other, despite the high temperature. It is the onset of such hydrogen bonds between water molecules which could explain the decrease in energy. Then, the most energetic sites are finally occupied. The next molecules have to adsorb on less and less energetic sites.

Furthermore, at this high temperature (higher than bulk supercritical temperature) the hydrogen bonds are destabilized (gas-like structure), the interaction with the substrate becoming too weak at large coverage to confine the molecules in a liquid-like structure. This results in an increase of the configurational energy per molecule. Finally, as the Vycor fills up, the water density increases, and the structure is more and more liquid-like. This enable more and more hydrogen bonds to be established, which makes the configurational energy go through a maximum and finally decrease again to reach a plateau for high chemical potential. At 300 K, the configurational energy per molecule shows a hysteresis loop, which is a consequence of the hysteresis in adsorption isotherms. For very low chemical potential, the configurational energy decreases down to a plateau at -75 kJ/mol, which corresponds to the most energetic sites in the porous glass. This very hydrophilic character of the Vycor explains why the amount of adsorbed water increases very rapidly for low relative pressure (P/P O < 0.1, or µ < -50.8 kJ/mol) (see Fig. 3). On the other hand, for the highest chemical potential values, the configurational energy per particle reaches a maximum (around -50 kJ/mol) close to the bulk water cohesive energy. The fact that the energetic characteristics of water saturating the Vycor tend to the bulk properties shows that the substrate interaction does not extend beyond few layers. This relatively short range interaction is not an artefact in the potential calculation since no distance cut-off (except the minimal image convention) has been used for electrostatic calculations. On contrary, it is thought that the dipolar character of the hydroxyl groups attached to the surface induces actually relatively short range interactions in a polar fluid like water. For intermediate chemical potential values, the configurational energy increases with chemical potential. The water-substrate interaction is significantly stronger than the water-water interaction. As the quantity of adsorbed fluid increases, the configurational energy per particle increases too, ranging from water-substrate intensity to water-water intensity.

Contrary to the previous case, the configurational energy does not go through a maximum. This is essentially because at low (sub-critical) temperature the fluid adsorbed on the surface adopts approximately a local liquid-like structure whatever the coverage.

The configurational energy is not a directly accessible quantity, from an experimental point of view.

The measurable quantity is the isosteric heat of adsorption, which measures the heat released by the adsorption of a given amount of fluid, as a function of the total amount of fluid previously adsorbed in the substrate. Numerically, this quantity is deduced from cross-fluctuations in energy (U) and adsorbed quantity (N) through the formula:

40 T R N - N N U - N U - q 2 2 st + = (3)
The results are given in Fig. 7 for both temperatures, as a function of coverage in g/g for comparison with experimental data. Generally, the low chemical potential value of the isosteric heat of adsorption gives information on the strength of the adsorbate/adsorbent interaction. The low coverage isosteric heat of adsorption at 650 K and 300 K are close to 55 kJ/mol and 80 kJ/mol respectively. The numerical results at 300 K are in good agreement with experimental values. 42 At 300 K, the curve shows that beyond one monolayer coverage (around 0.06 g/g), the isosteric heat reaches 44 kJ/mol, that is to say the latent heat of vaporization of water (horizontal line). The strong influence of the silica surface has then disappeared in the second and subsequent layers, which shows again that the surface interactions are actually relatively short range in a polar fluid.

GRAND POTENTIAL, HELMHOLTZ FREE ENERGY AND ENTROPY CALCULATION grand potential calculation

The hysteresis obtained by GCMC could be seen as the envelope of a van der Waals loop. In such an analysis the system should have a stable equilibrium within the loop. From a general point of view, such an issue can be discussed if the system is completely characterized from a thermodynamic point of view, that is to say if one calculates the appropriate thermodynamic potential. More precisely, since our simulations are performed in the Grand Canonical Ensemble, the thermodynamic potential to be considered is the grand potential

N S T E µ - - = Ω
, where E is the total internal energy, T the temperature, S the entropy, µ the chemical potential, and N the number of particles in the system. However, this quantity is not a direct output of the GCMC simulation. A way to circumvent this problem was proposed by Peterson and Gubbins 48 . In their method, they take advantage of two differential equations to calculate the grand potential along reversible paths, from a reference point up to both the vapor and liquid branches. The two differential equations link the grand potential Ω to the number of particles and the total energy of the system which are given by the GCMC simulation. The first equation is the Gibbs adsorption isotherm, obtained by differentiating Ω along constant temperature paths:

N V T - =       ∂ Ω ∂ , µ (4)
where N is the number of particles in the simulation box of volume V. The second equation is obtained along paths of constant chemical potential µ:

( ) ( ) µ µ N E T T V - =       ∂ Ω ∂ , / 1 / ( 5
)
where E is the total energy of the system, which is the sum of the configurational energy U and the kinetic energy taken equal to 1/2 kT per degree of freedom (3 for position and 3 for rotation if the rotational contribution is taken into account). This set of equations allows to calculate the difference in the reduced grand potential between any two states of the system. More precisely, the integration has to be performed along a reversible path: the supercritical adsorption isotherm (650 K) is then used to connect the gas-like and liquid-like phases. Additional points have been simulated in the liquid-like phase to increase the precision in the integration along constant µ paths.

To get absolute values for Ω, one has to know at least one point. For µ low enough so that the adsorption isotherm is proportional to the activity

( 3 Λ µ β e
), and follows essentially the ideal gas approximation, one can perform analytically the integration of equation 4 from minus infinity up to the value of the chemical potential of the first GCMC point to get the ideal gas value for the grand potential:

T k N B id - = Ω (6)
This ideal gas approximation has been used for each adsorption isotherm. As previously mentioned one has to take care of the degrees of rotational freedom. Since the chemical potential entering the GCMC method does not contain the ideal rotational contribution, the integration procedure gives the translational ideal gas plus the excess contributions only, without the ideal rotational contribution.

However, it is straightforward to show that the ideal rotational contribution to the grand potential is exactly zero for all temperatures. Thus, the calculations actually give the complete grand potential.

The result for the grand potential obtained along all isotherms is shown in Fig. 8. Since the grand potential is an extensive quantity, very small values are expected for one simulation box. In order to obtain more natural values, it was decided to consider the grand potential divided by the volume of the simulation box. Inspection of Fig. 8 shows differences between the two branches: at 300 K the grand potential is multi-valued, whereas it is a smooth function at 650 K. This behavior is typical of a first order phase transition with a critical point between 300 K and 650 K. The intersection of the two branches at 300 K for µ = -45.63 gives the equilibrium or coexistence point between the gas-like and the liquid-like " phases" . As can be noted, this coexistence point is found close to the vertical desorption line (µ = -46.1 kJ/mol). The lowest portions of the branches, or absolute minimum of the grand potential, give the stable phase for a given chemical potential, the other branches corresponding to metastable states. For µ<-45.63 kJ/mol, the gas phase is the stable one, whereas the liquid-like phase is metastable (-46.1 < µ < -45.63) or unstable (µ <-46.1 kJ/mol). On the other hand, for µ>-45.63 kJ/mol the liquidlike phase is the stable one whereas the gas-like phase is metastable up to µ = -44.3 kJ/mol and instable above.

The discussion of the real nature of this " transition" is probably one of the most difficult. [49][50][51][52] The previous analysis would tend to argue in favor of a first order phase transition. However, this conclusion actually relies on the hypothesis that the observed adsorption / desorption branches correspond to welldefined thermodynamic phases. It is definitely not established in this work that the observed hysteresis is the envelope of a van der Waals loop. However, within such a framework, it was possible to calculate a coexistence point.

Helmholtz free energy and entropy calculation

The knowledge of the grand potential enable to calculate all thermodynamic functions of interest. For instance, the Helmholtz free energy and the entropy which are conceptually very important quantities, are not so commonly calculated in simulation works. The molecular Helmholtz free energy f (in kJ per mole of water molecules) is calculated according to the formula

µ + Ω = = N N F f (7)
and plotted as a function of the amount of adsorbed water in Fig. 9. Here again, the ideal rotational contribution to molecular free energy -kT(-4.09+3/2lnT), with T in Kelvin, is not taken into account. As can be seen, the free energy increases with coverage, as expected since the grand potential is negative and

2 , /N N f T V Ω - =       ∂ ∂ (8)
Furthermore, the molecular free energy diverges for zero coverage, following the ideal gas behavior.

The free energy is a continuous (reversible) function of the coverage for supercritical temperature (650 K), and presents a discontinuity for subcritical temperature (300 K).

The entropy of the system is conceptually very interesting since it measures the number of accessible states for a given state of the system. However, this quantity is rarely directly calculable, except by means of very sophisticated methods. Nevertheless, it can be deduced from the molecular Helmholtz free energy f and the molecular configurational energy u, according to the formula:

T f u R s - + = 2 3 ( 8
)
where R is the ideal gas constant, arising from the kinetic part of the internal energy of water molecules.

The rotational ideal gas contribution (3/2-4.09+3/2lnT)R (J mol -1 K -1 ) is included. The Fig. 10 shows the entropy as a function of the chemical potential µ. For supercritical temperature, the entropy decreases with chemical potential, due to excluded volume effects: the larger the density, the lower the accessible volume for each water molecule. The 300 K case presents interesting features. For low chemical potential, the entropy decreases with coverage for the same reason as for supercritical temperature.

However, it should be noted that the absolute value of entropy for equivalent coverage is lower for the low temperature case. This expected result is simply the consequence of confinement of water molecules close to the silica surface at low temperature. More interestingly, the entropy per particle steeply increases around the capillary condensation, and then continues to slowly decrease along the liquid branch. In terms of number of accessible states, it means that during capillary condensation the accessible volume in phase space increases. This can be interpreted by considering that at low temperature, the molecules are confined close to the surface up to the monolayer. Their entropy then slowly decreases with chemical potential (increase in density). Above this coverage, the newly added molecules are free to move in a much larger space since the interaction with the Vycor hydrated by one monolayer of water is significantly weaker, as shown previously in Fig. 5. This explains the steep increase in entropy. However, as the pores are completely filled, the entropy decreases again with coverage due to excluded volume effects.

CONCLUSION

The adsorption of water in disordered mesoporous silica glass (Vycor-like) has been analysed by GCMC simulation method at two temperatures: 300 K and 650 K. The results show a hysteresis loop in the adsorption/desorption data at low temperature, while the high temperature curve is reversible. The configurational energy has also been calculated, showing hysteresis at low temperature, and reversible but non monotonic behavior at high temperature. The amount adsorbed at low pressure, as well as the isosteric heat of adsorption are in good agreement with published experimental data, which validates the good transferability of the interaction model used. However, close to the saturating pressure the amount of water adsorbed in simulation in significantly lower than expected. The analysis of the distribution in energy of the attractive sites of the system shows that the simulated monolayer of water adsorbed on

Vycor is less hydrophilic than the bulk water liquid/gas interface. This effect is related to the strong distortion in the structure of the adsorbed monolayer. This explains why the capillary condensation fails to occur for pressures lower than the saturating vapor pressure of bulk water. Even if it is expected that the hydrated Vycor be less hydrophilic than the dry Vycor, the simulation results are certainly exaggerated. It is thought that allowing mobility of the surface hydroxyl groups (the whole substrate is kept rigid during this exploratory study) could enable the first water monolayer to recover partially the bulk water structure, and therefore some hydrophilicity.

The hysteresis loop observed in the adsorption / desorption is analyzed within the hypothesis of a true phase transition: each branch is considered as a real " phase" with thermodynamic quantities associated, like coverage, and configurational energy. The thermodynamic integration scheme can be applied, which allows to calculate the grand potential, the Helmholtz free energy and the entropy for instance.

The grand potential is shown to be multi-valued for low (subcritical) temperature, and continuous for high (supercritical) temperature. A coexistence point is found within the hysteresis loop, actually close to the desorption vertical line. Below the equilibrium chemical potential, the gas-like branch is stable whereas the liquid-like branch is metastable. The situation is inverted above the coexistence point.

However, as pointed out, this first order transition behavior relies on the hypothesis that the observed

For a given

  temperature, starting from low chemical potential and an empty initial configuration, and then increasing the chemical potential step by step, the Grand Canonical Monte Carlo (GCMC) technique[START_REF] Nicholson | Computer simulation and the statistical mechanics of adsorption[END_REF] allows to calculate the water adsorption isotherm in the mesoporous silica sample. An equal number of trial for translation, rotation, and creation or destruction of molecules has been chosen for each Monte Carlo (MC) step at fixed chemical potential. The configurations are considered for visual examination and statistical calculations every block of 10 4 Monte Carlo steps (4.10 4 MC trials) along the Markov chain, so that they are de-correlated. The Fig.2shows a typical equilibration run at a relative pressure of 0.236 from an initial configuration at relative pressure 0.075. The number of particles is drawn as a function of the number of blocks of 10 4 MC steps. As can be seen equilibration is reached after approximately 3.7 10 4 blocks, that is to say around 8.10 5 trials per water molecule. To acquire data, another 10 6 trials per molecule were run, where the number of particles fluctuates around a constant value. Actually, The higher the coverage, the larger these numbers to compensate the decrease in insertion acceptance level. After the complete saturation of the Vycor sample is reached for high chemical potential µ, one can start decreasing µ to acquire the desorption branch (hysteresis).

  FIGURE CAPTIONS (Word Style " VA_Figure_Caption" ).

FIGURE 1 : 3 - 2 :

 132 FIGURE 1:

FIGURE 3 :

 3 FIGURE 3:

FIGURE 4 :

 4 FIGURE 4:

FIGURE 5 :

 5 FIGURE 5:

  FIGURE 6:

Figure 1 Figure 2 number of blocks of 10

 1210 Figure 1

Figure 6 chemical

 6 Figure 6

Table 1 :

 1 PN-TrAZ parameters for water and vycor species

	a 0 =0.529177Å, 1E h =3.1578 10 5 K

Table 2 :

 2 Dispersion and repulsion parameters obtained in the framework of the PN-TrAZ model.

						water		vycor
	Atoms			Ow	Hw	Si	O	H
	q (e)			-0.82	0.41	+2	-1	+0.5
	A (E h )			247.7	1.338	6163.4	1543.5	1.338
	b (a 0 -1 )			2.075	2.11	2.395	2.19	2.11
	Polarizabilities (a 0	3 )	7.56	2.655	2.36	8.03	2.25
	N eff			4.476	0.414	1.52	4.656	0.324
	Water	Silica
	species	species	C 6 (E h a 0	6 )	C 8 (E h a 0 8 )	C 10 (E h a 0 10 )	A(E h )	b (a 0 -1 )
	H	O	8.3157	151.22	2668.6	45.444	2.149
	H	Si	2.4874	32.919	-	90.811	2.243
	H	H	1.7340	22.983	-	1.338	2.11
	O	O	34.850	735.319	13540.4	618.3	2.131
	O	Si	10.513	172.72	-	1235.5	2.223
	O	H	6.4843	116.27	-	18.2	2.092

Configurational energy during water adsorption / desorption in Vycor, versus chemical potential at 300 K and 650 K. Lines are guides to the eyes.


Francis Bloch
email: francis.bloch@polytechnique.edu

Geoffroy De Clippel
email: declippel@brown.edu

CORES OF COMBINED GAMES

Keywords: Jeux coopératifs, coeur, additivité, négociation multidimensionnelle Cooperative games, Core, Additivity, Issue Linkage, Multi Issue Bargaining Classification JEL: C70, C71

Cet article analyse le coeur de jeux combinés, obtenus en additionnant deux jeux sous forme de fonction caractéristique. Nous montrons que l'ensemble des jeux à utilité transférable peut être partitionné en classes d'équivalences, de telle sorte que le coeur du jeu combiné est égal à la somme des coeurs des jeux composants si et seulement si les deux jeux composants appartiennent à la même classe d'équivalence. Par ailleurs, pour des jeux non balancés, la relation binaire associant deux jeux non balancés dont la combinaison a un coeur vide est non transitive. Toutefois, nous identifions une classe de jeux non balancés telles que, chaque jeu, combiné avec tout autre jeu non balancé, a un coeur vide.

Introduction

The broad subject of the paper is the study of bargaining and cooperation when multiple issues are at stake. We have two complementary objectives in mind:

1. Identify conditions under which negotiating over different issues separately is equivalent to negotiating over these issues simultaneously;

2. Identify situations in which combining issues reduce conflict in bargaining.

We use games in coalitional form, a classical model to study cooperation, to tackle these two questions. The coalitional function specifies for each coalition the surplus to be shared should its members cooperate. The simplicity of this reduced-form approach, making no direct reference to the underlying social or economic alternatives, comes at a cost. Indeed, relating the cooperative opportunities associated to different issues to the cooperative opportunities of the combined issues is possible in this framework only if the different issues are independent (e.g. bargaining over the allocation of different goods with quasi-linear preferences that are additively separable). In such cases, the coalitional function associated to the combined issues is simply the sum of the coalitional functions associated to each issue taken separately. Spillovers are certainly an important characteristic of multi-issue bargaining, and further analysis of non-welfarist models is needed to understand their implication. We argue in the present paper that a good understanding of multi-issue bargaining is important even in the absence of such spillovers.

Multi-issue bargaining was of central importance to Professor Shapley when studying values for games in coalitional form, as illustrated by his motivation for the additivity axiom: "The third axiom ("law of aggregation") states that when two independent games are combined, their values must be added player by player " (Shapley, 1953, page 309).

Put differently, additivity implies that the outcome of multi-issue negotiations does not depend on the agenda chosen by the negotiators. Whether issues are discussed separately or "packaged" in different ways does not affect the result of the negotiation. In Professor Shapley's view, this agenda independence is a natural requirement to impose on a solution concept.

However, the Shapley value is the only solution concept for which additivity is posited as an axiom. Other solution concepts, whether they are based on alternative axiomatizations, like the Nash bargaining solution, or more positive considerations, like the core, do not satisfy this property of agenda independence. In this paper, we focus attention on the core primarily because of its importance in economic theory. Other solution concepts are briefly discussed in Section 5.

It is well known that the core is superadditive (see for example, [START_REF] Peleg | On the Reduced Game Property and its Converse[END_REF] axiomatization of the core), so that the core of the combination of two games is always larger than the sum of the core of the two components. Intuitively, by combining two negotiation processes, and forcing players to make coalitional objections on the issues simultaneously, it is easier to sustain an imputation than when players can make separate objections on the two issues. Hence, the specific question we tackle in this paper is the following: For which pairs of games is the core of the combination of the two games exactly equal to the sum of the core of the component games? This offers a formal statement to the first objective listed at the beginning of the paper.

Our main result (Proposition 1) shows that the core of the sum of two games v and w is equal to the sum of the cores of v and w if and only if the extreme points of the cores of v and w are defined by the same set of coalitional constraints (see Proposition 1). Because the latter property defines an equivalence relation among games, we conclude that the set of all balanced transferable utility games can be partitioned into equivalence classes such that the core of the combination of two games is equal to the sum of the cores of the components if and only if the two games belong to the same class. One of these equivalence classes (where the extreme points are determined by any increasing sequence of coalitions) is the set of convex games introduced by [START_REF] Shapley | Cores of Convex Games[END_REF]. Hence, the combination of two convex games does not result in an expansion of the set of core allocations. By contrast, whenever two games v and w are taken from two different equivalence classes, the core of the combined game is strictly greater than the sum of the core of its components. When v and w are close, a simple continuity argument shows that the difference between the core of v + w and the sum of the cores of v and w is small. In other cases, the difference can be extremely large, as the dimension of the core of v + w may exceed the dimension of the sum of the cores (for example, even when the cores of v and w are singletons, the core of v + w may be a set of full dimension in the set of imputations.)

The core of two games with an empty core may be non-empty (Example 1). In such cases, bargaining over each component would lead to an impasse or to partial cooperation, but efficiency can be recovered (on both components) by combining the issues. This illustrates the relevance of the second objective introduced in the first paragraph. Formally, we would like to characterize pairs of games with an empty core whose sum has a non-empty core. Unfortunately, our characterization of the set of games for which the core is additive does not carry over to games with empty cores. The binary relation associating two games v and w whose combination has an empty core is not transitive. This is easily understood: for two games v and w to be such that the combined game v + w has an empty core, it is sufficient that one of the balanced 1 collections of coalitions has a worth exceeding the worth of the grand coalition in both games v and w. Now consider a triple of games v, w, z. The worth of the balanced collection C may exceed the worth of the grand coalition in both v and w and the worth of the balanced collection D may exceed the worth of the grand coalition in both w and z. However, v and z may very well not share any balanced collection whose worth exceeds the grand coalition, and be such that the core of v +z is nonempty. Put differently, for a game to be unbalanced, one only requires one of the balanced collection to have a greater worth than the grand coalition, so that the set of games with empty cores is not defined by a set of linear inequalities, and is in fact typically not convex. In spite of this, we can identify a convex subset of the class of unbalanced games which has the following property: for any game in that class, the combination of this game with any other game with empty core also has an empty core (Proposition 2). Intuitively, this subset contains those games which are hardest to "balance" with other games, and may create the more difficulties in negotiations.

To the best of our knowledge, the only previous studies of the additivity of the core in the cooperative game theoretic literature are due to [START_REF] Tijs | Additive Stable Solutions on Perfect Cones of Cooperative Games[END_REF]. They identify three subclasses of games on which the core is additive (including the class of convex games). Our results complement and extend their analysis by showing that in fact the entire set of balanced games can be partitioned into subclasses on which the core correspondence is additive. The literature on noncooperative games has paid more attention to simultaneous, multi-issue bargaining. In a two-player setting, [START_REF] Fershtman | The Importance of the Agenda in Bargaining[END_REF] and [START_REF] Busch | Bargaining Frictions, Bargaining Procedures and Implied Costs in Multiple Issue Bargaining[END_REF] extend [START_REF] Rubinstein | Perfect Equilibrium in a Bargaining Model[END_REF]'s alternating offers game to a multi-issue setting, where players bargain over each issue in a predefined sequence. They show that the equilibria of this multi-issue bargaining differ considerably from the single-issue model. In later contributions to this literature, [START_REF] Bac | Issue-by-issue Negotiation: The Role of Information in Time Preference[END_REF], [START_REF] Inderst | Multi-issue Bargaining with Endogenous Agenda[END_REF] and [START_REF] In | Agenda Restrictions in Multi-Issue Bargaining[END_REF] allow players to endogenously choose on which issue to bargain, and show that players have an incentive to manipulate strategically the agenda. Issue linkage has also been studied in noncooperative games representing international negotiations across countries. It has long been argued that combining negotiations over different dimensions (trade, protection of the environment) may have beneficial effects (see for example [START_REF] Carraro | Policy Coordination for Sustainability: Commitments, Transfers and Linked Negotiations[END_REF]. [START_REF] Conconi | Issue Linkage and Issue Tie-In in Multilateral Negociations[END_REF] propose a model of issue linkage and evaluate this argument using a parameterized model of international trade and environmental negotiations. Issue linkage also appears implicitly in the literature on mergers in Industrial Organization (e.g. [START_REF] Perry | Oligopoly and the Incentive for Horizontal Merger[END_REF] and [START_REF] Farrell | Horizontal Mergers: An Equilibrium Analysis[END_REF]). In order to be profitable, a merger must involve two dimensionsboth a cost and a market dimensions -and result in cost synergies as well as market concentration.

The rest of the paper is organized as follows. In the next Section, we recall the standard definitions of coalitional games and the core. In Section 3, we analyze the combination of games with nonempty cores. We first provide intuition by analyzing symmetric games with three players, then provide our general result on the partitioning of the set of balanced games. We illustrate this result by computing exactly this partition in four-player symmetric games. Section 4 contains our results for games with empty cores. We provide an example to show that the set cannot be partitioned, and discuss how four-player games with empty cores can or cannot be combined to obtain a nonempty core. We finally identify the class of noncompensable games, which, combined with any other game with an empty core, still retain an empty core. The additivity of other cooperative solution concepts is briefly discussed in Section 5. Section 6 contains the proof, and in particular a key lemma on the addition of convex polyhedra defined by systems of linear inequalities.

Preliminaries

Let N be a set of players. A cooperative game is described by a coalitional function v which assigns to every nonempty subset S of N a real number, v(S), called the worth of the coalition. Game will be assumed to be superadditive: v(S ∪ T ) ≥ v(S) + v(T ), for any two disjoint coalitions S and T . We denote the set of all such n-player games by Γ(n). A game is convex if the players' marginal contributions are non decreasing:

v(S ∪ {i}) -v(S) ≤ v(T ∪ {i}) -v(T ), for each pair (S, T ) of coalitions such that S ⊆ T .
An imputation is a vector x ∈ N that is feasible, efficient, and individually rational: i∈N x i = v(N ) and x i ≥ v({i}), for each i ∈ N . The core of a cooperative game v is the set of payoff vectors x ∈ N that are feasible when all the players cooperate, and which cannot be improved upon by any coalition:

i∈N x i ≤ v(N ) and i∈S x i ≥ v(S) for each coalition S. Let A be the (2 n -1) × n matrix encoding coalitional membership: A S,i = 1 if i ∈ S and A S,i = 0 if i ∈ S,
for each coalition S and each player i. Then,

C(v) = {x ∈ N | i∈N x i = v(N ), Ax ≥ v}.
This rewriting highlights the fact that the core is a bounded convex polyhedron defined by a system of linear inequalities. As any such set, the core is characterized by its set of extreme points -points which cannot be obtained as convex combinations of other points in the set. Equivalently, a payoff vector x is an extreme point of the core of v if there exists a collection (S k ) n k=1 of coalitions such that i∈S k x i = v(S k ), for each k, and these n equations are linearly independent.

The system of linear inequalities defining the core may be inconsistent, in which case the core is empty. [START_REF] Bondareva | Some Applications of Linear Programming Methods to the Theory of Cooperative Games[END_REF] and [START_REF] Shapley | On Balanced Sets and Cores[END_REF] proposed a characterization of games with nonempty core based on balanced collections of coalitions. A collection (S k ) K k=1 of coalitions is balanced if there exists a collection (δ k ) K k=1 of real numbers between 0 and 1 (called balancing weigts) such that

S k |i∈S k δ k = 1, for each i ∈ N . A game v is balanced if and only if k δ k v(S k ) ≤ v(N ), for each balanced collection (S k ) K
k=1 of coalitions and each collection (δ k ) K k=1 of balancing weights. The core of a game v is nonempty if and only if the game v is balanced. The set of all balanced superadditive n-player games is denoted β(n).

3 Combining games with a nonempty core

Three-player symmetric games

In this Section, we suppose that the two component games v and w are balanced. In order to gain intuition, we first consider three-player symmetric normalized games. For these games v({i}) = 0, v(N ) = 1 and v(S) = v 2 ∈ (0, 1) for any coalition S with two players. It is easy to see that the core of game v is empty if (1, 0, 0) (0, 1, 0) (0, 0, 1) When the game is balanced but not convex, the core has the shape of a triangle in the simplex, and has three extreme points given by (2v 2 -1, 1 -v 2 , 1 -v 2 ) (and the permutations). When the game is convex, the core has the shape of an hexagon, characterized by six extreme points given by (0, v 2 , 1 -v 2 ) (and the permutations). Clearly, by summing up two balanced, nonconvex games v and w, one obtains a new balanced nonconvex game v + w. This combined game again has a core shaped like a triangle, and all the extreme points of C(v + w) are be decomposed as sums of extreme points of C(v) and C(w). Similarly, by summing up two convex games v and w, one obtains a new convex game v +w. The core of v +w is shaped like an hexagon and all the extreme points of v + w can be decomposed as sums of extreme points of C(v) and C(w). However, if one combines a convex game with a nonconvex balanced game, the core of the combined game cannot be equal to the sum of the cores of the games. To see this, let us combine a nonconvex game -v 2 ∈ ( 1 2 , 2 3 ] -with a strictly convex game -w 2 ∈ [0, 1 2 ). Observe that, for any point in C(v), x i > 2v 2 -1. Hence, for any points in C(v) + C(w), x i > 2v 2 -1. However, the core of v + w is either shaped as a triangle or as an hexagon. In the latter case, it contains extreme points on the boundary, so that C(v + w) is a strict superset of C(v) + C(w). In the former case, the core of v + w contains an extreme point for which a player's payoff equals

v 2 > 2 3 ,
t t t t t t a) v 2 = 0.4 b) v 2 = 0.6
x i = 2(v 2 + w 2 ) -2 = (2v 2 -1) + (2w 2 -1) < 2v 2 -1,
where the last inequality is obtained because w 2 < 1/2. Yet that player's minimal payoff is 2v 2 -1 in C(v) and 0 in C(w). Hence, again C(v + w) is a strict superset of C(v) + C(w).

A general result

The study of three-player symmetric games thus shows that the set of balanced games can be partitioned into two subclasses on the basis of the extreme points of the core. The core of the combined game is equal to the sum of the cores of the component games if and only if the two component games belong to the same subclass. Our main result shows that this intuition can be generalized to any n-player transferable utility game. We will prove this statement as a corollary to a general result on convex polyhedra.

In the lemma from the Appendix, we define an equivalence relation between two bounded convex polyhedra

P (A, b) = {x ∈ N |Ax ≥ b} and P (A, b ) = {x ∈
N |Ax ≥ b} if the extreme points of the two polyhedra are defined by the same constraints. We first consider the generic case where every extreme points is characterized by exactly N equalities. In that case, when P ). This is proven by induction on the number of players in the games. For n = 1, the polyhedra are subsets of the line, and the inclusion is verified. For higher values of n, we pick an extreme point of P (A, b + b ) and show that it can be decomposed as the sum of two vectors in P (A, b) and P (A, b ). This is done by isolating one of the players, i, redefining an n -1 dimensional polyhedron by using one of the binding constraints to define x i as a function of x -i and applying the induction hypothesis to the lower dimensional polyhedron.

In the nongeneric case where some extreme points of the polyhedron P (A, b) or P (A, b ) satisfy strictly more than N equalities, the argument must be adapted. In Then C(v) + C(w) = C(v + w) if and only if there exist two sequences of games v k and w k converging to v and w such that, for all k, the games v k and w k belong to the same equivalence class in the quotient set β(n)/R.

Four-player symmetric games

We illustrate the partition of the set of balanced games into equivalence classes by considering normalized four-player symmetric games -N = {1, 2, 3, 4}, v(N ) = 1 and v({i}) = 0, for each i ∈ N . Let v 2 denote the value of two-player coalitions and v 3 the value of three-player coalitions. Superadditivity requires that 2 depicts the subsets of games where the extreme points of the cores are defined by the same constraints. The computations underlying Figure 2 are given in Section 6.2. The set of four-player symmetric games can be partitioned into eight regions, labeled from A to H. In each region, the extreme points of the core are characterized by the same constraints. Three regions stand out. Region A corresponds to the class of convex games. The work of [START_REF] Shapley | Cores of Convex Games[END_REF] and [START_REF] Ichiishi | Super-modularity: Applications to convex games and to the greedy algorithm for LP[END_REF] imply that a game is convex if and only if the extreme points of the core coincide with the vectors of marginal contribution. Convex games form an equivalence class of the relation R, so that the core of the sum of any two convex games is equal to the sum of the cores.

v 2 ∈ [0, 1/2] and v 3 ∈ [v 2 , 1]. Figure
(See also [START_REF] Tijs | Additive Stable Solutions on Perfect Cones of Cooperative Games[END_REF] on that point.) Region G corresponds to games where the extreme points of the core are characterized by constraints involving only three-player coalitions, or the dual imputation set. This is the class of games K d introduced by Driessen and Tijs (1983) -and for which [START_REF] Tijs | Additive Stable Solutions on Perfect Cones of Cooperative Games[END_REF] also note that the core is additive. Finally, region H (for which v 3 > 3 4 ) corresponds to games with empty cores.

Notice that any game in the interior of a region is a generic game, where extreme points are defined by exactly N equalities. Games at the boundary between two regions are nongeneric, and can be approached by sequences of generic games belonging to the interior of the games in the two regions. Hence, combining a game on the boundary with a game belonging to any of the two regions does not increase the size of the set of stable agreements.

While the core correspondence is additive inside each region of Figure 2, it is never additive when we combine two games belonging to different regions. Notice however that, generically, when the two component games are close, the difference between the core of the combined game and the sum of the cores of the component games cannot be too large. To see this, define the game v ∨w which assigns to each coalition S the worth max{v(S), w(S)}. Generically, when v and w are sufficiently close, the game v ∨ w will belong to the same equivalence class as one of the component games. Suppose without loss of generality that v∨wRv. Then, C(v∨w)+C(v) = C(v∨w+v). Recall that, on the set of games balanced games β(n), the core correspondence is continuous (See e.g. Peleg and Sudholter (2003), Theorem 9.2.4). Hence, when v and w are close enough, the differences between C(v∨w)\C(w) and C(v∨w+v)\C(v+w) cannot be too large, so that the difference C(v + w) \ (C(v) + C(w)) is bounded and converges to zero when v converges to w.

However, Figure 2 also shows that there are special cases where the difference between the core of the combined game and the sum of the cores of the component games can be extremely large. In fact, it is possible to combine two component games where the core collapses to a single point, and obtain a full dimensional core. For example, pick two games v and w such that v 2 = 1 2 and w 3 = 3 4 . For each of these games, the core is a single point ( 14 , 1 4 , 1 4 , 1 4 ). However, the sum of the two games can belong to any of the regions A,B,D,E,F or G, where the core is a full-dimensional set.

Combining games with an empty core 4.1 Examples

We now consider the combination of unbalanced games. First, as the next example shows, the combination of two games with empty cores may very well possess a nonempty core. In this sense, it is worthwhile to combine games, or to link negotiations which would otherwise result in an impasse.

Example 1 Let n = 4. Let v(123) = 2 3 , v(S) = 1 2 if |S| = 2, 3 and 4 ∈ S, v(N ) = 1, v(S) = 0 otherwise. Let w(S) = 0 if |S| < 3, w(S) = 4 5 if |S| = 3 and w(N ) = 1.
In Example 1, game v has an empty core, because the worth of the grand coalition is smaller than the sum of worths in the balanced collection C ∞ = {{123}, {14}, {24}, {34}}. Game w is unbalanced with respect to the collection C ∈ = {{123}, {124}, {134}, {234}}. However, it is easy to check that the imputation ( 12 , 1 2 , 1 2 , 1 2 ) belongs to the core of v + w.

Example 1 illustrates a crucial difference between the conditions characterizing a nonempty core (the equations defining the extreme points of the core), and the conditions guaranteeing emptiness of the core (the inequalities for balanced collections). Whereas a nonempty core is characterized by all the equations determining extreme points, emptiness of the core is guaranteed as long as one of the balanced collections has a worth greater than the grand coalition. As the next example shows, this difference implies that the binary relation linking two unbalanced games whose combination is also unbalanced may not be transitive.

Example 2 Let n = 5. Let v(S) = 0 if |S| < 4, v(S) = 5 6 if |S| = 4, v(N ) = 1. Let w(S) = 0 if |S| = 2, w(S) = 3 4 if |S| = 3, 4 and w(N ) = 1. Let z(S) = 0 if |S| = 2, z(S) = 3 4 if S = 3, z(S) = 5 6 if S = 4 and z(N ) = 1.
In example 2, v is unbalanced because of the collection of coalitions of size 4, w because of the collection of coalitions of size 3, and z because of both collections of coalitions of size 4 and 3. Hence, the combined games v + z (respectively w + z ) is unbalanced, because the worth of the grand coalition is smaller than the sum of worths of the balanced collection of coalitions of size 4 (respectively 3). However, the core of the combined game v + w is nonempty, and contains for example the symmetric allocation ( 25 , 2 5 , 2 5 , 2 5 , 2 5 ). Put differently, the difficulty is that the set of games with empty cores (for which only one of the balanced collections may pose problems) need not be convex, whereas classes of games with nonempty cores whose extreme points satisfy the same constraints are always convex. This is illustrated in Figure 3, which graphs unbalanced symmetric games with 5 players, as a function of the values v 3 and v 4 of the three and four player coalitions. In region A, the core is empty because any imputation is blocked by a four-player coalition; in region B the blocking coalitions are coalitions of size 3. In region C, both coalitions of sizes 3 and 4 are blocking. From Figure 3, it is clear that combinations of games of regions A and B may have nonempty cores. On the other hand, any combination of symmetric games involving one component in region C has an empty core.

Noncompensable games

Figure 3 suggests that there exist unbalanced games which, combined with any other unbalanced game, produce an empty core. We will term these games noncompensable games. Formally, a game v is noncompensable if, for any game w such that C(w) = ∅, C(v + w) = ∅. Clearly, the class of noncompensable games contains those games for which imputations are blocked by all proper minimal balanced collections. 2For example, if n = 3, the only proper minimal balanced collection is the collection C = {{1, 2}, {1, 3}, {2, 3}. Hence any game such that: v(12) + v(13) + v(23) > 2v(123) is unbalanced and noncompensable. If n = 4, the proper minimal balanced collections are (up to a permutation)

C = {{1, 2, 3}, {1, 4}, {2, 4}, {3, 4}}, C = {{1, 2, 3}, {1, 2, 4}, {1, 3}, {1, 4}}, C = {{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}}.
Hence, a symmetric four-player game is noncompensable if and only if 2v 3 + 3v 2 > 1 and v 3 > 3 4 . Notice in particular that this set is smaller than region H in Figure 2. In fact, some four-player symmetric games (for which v 3 > 3 4 but 2v 3 + 3v 2 ≤ 1) cannot be compensated by another unbalanced symmetric game, but can be compensated by nonsymmetric games (see Example 1).

We now prove that the set of noncompensable games is exactly equal to the set of games for which imputations are blocked by all minimal proper coalitions.

Proposition 2 A game v is noncompensable if and only if, for all proper minimal balanced collections of coalitions C = {S 1 , ...S j , ..., S J }, j δ j v(S j ) ≥ v(N ).

Additivity of other cooperative solution concepts

In this paper, we characterize the classes of cooperative games on which the core is additive. In this concluding section, we briefly comment on the generalization of our results to other cooperative solution concepts, and discuss the existing literature on additivity axioms in cooperative game theory.

We first note that, whenever a solution is defined by a system of linear inequalities, a direct application of the Lemma from Section 6.1 shows that the set of cooperative games can be partitioned into equivalence classes where the solution is additive. For example, [START_REF] Laussel | Conflict and Cooperation; The Structure of Equilibrium Payoffs in Common Agency Games[END_REF] analyze the Pareto frontier of sets

U (v) = {(u 1 , ..., u n )|u i ≥ 0, i∈S u i ≤ v(N ) -v(N \ S)
} for a given cooperative game v. From our analysis, it is clear that the convex polyhedron corresponding to the sum of two games v and w is equal to the sum of the convex polyhedra, U (v + w) = U (v) + U (w) if and only if the extreme points of U (v) and U (w) are defined by the same coalitions. 3 On the other hand, , the Lemma does not apply if the solution concept is not a unique polyhedron but a finite union of polyhedra, like the M i 1 bargaining set [START_REF] Davis | Existence of Stable Configurations for Cooperative Games[END_REF], [START_REF] Maschler | The Inequalities that Determine the Bargaining Set M i 1[END_REF], or the kernel [START_REF] Davis | The Kernel of a Cooperative Game[END_REF] and [START_REF] Maschler | A Characterization, Existence Proof and Dimension Bounds for the Kernel of a Game[END_REF]). Suppose for illustration that a solution can be written as the union of two polyhedra:

S(v) = A(v) ∪ B(v).
Even if we consider two games v and w with the same binding coalitions in the two polyhedra A and B, so that A(v + w) = A(v) + A(w) and B(v + w) = B(v) + B(w), there is no guarantee that S(v + w) = S(v) + S(w). In fact, it is easy to check that (A(v + w) ∪ B(v + w)) ⊆ (A(v) ∪ B(v)) + (A(w) ∪ B(w)), with strict inclusion for generic games.

We next consider solutions defined as unique points rather than convex polyhedra. Of course, the Shapley value satisfies additivity. [START_REF] Peters | A Note on Additive Utility and Bargaining[END_REF] and (1986) provides an axiomatic characterization of solutions to Nash's bargaining problem which satisfy additivity and variants of superadditivity. [START_REF] Charnes | On sasymptotic behavior of some nuclei of n-person games and the piecewise linearity of the nucleolus[END_REF] and [START_REF] Kohlberg | On the Nucleolus of a Characteristic Function Games[END_REF] prove that the nucleolus is piecewise linear in the following sense. For any imputation x, and any coalition S, compute the excess function e(x, S) = v(S) -x(S), and order the coalitions, by decreasing values of the excess, to obtain an array of coalitions b(x, v) = (b 1 (x, v), ..., b 2 n -1 (x, v). Partition then the set of coalitional games in such a way that v and w belong to the same equivalence class if and only if, at the nucleolus of the two games, ν(v) and ν(w), the array of coalitions satisfy b(ν(v), v) = b(ν(w), w). Then, for any two v and w in the same equivalence class, ν(v + w) = ν(v) + ν(w).

Finally, we would like to emphasize that, in our opinion, the study of the additivity of the core is only a first step in a research program on multi-issue cooperation. In the future, we hope to extend the analysis by studying noncooperative models of sequential coalitional bargaining on multiple issues. Since b ∼ b , we must have:

Appendix 6.1 A useful lemma

P (A, b ) = [ b k A k , b l A l ],
and hence

P (A, b) + P (A, b ) = [ b k + b k A k , b l + b l A l ].
The desired conclusion follows from the fact that P 4 Let i ∈ N be such that A m,i = 0.5 Observe that

(A, b + b ) ⊆ [ b k +b k A k , b l +b l A l ]. Let N ≥ 2 be
P (A, b) ∩ {x ∈ N |A m x = b m } = {x ∈ N |x -i ∈ P ( Ā, b), x i = b m -A m,-i x -i A m,i } P (A, b ) ∩ {x ∈ N |A m x = b m } = {x ∈ N |x -i ∈ P ( Ā, b ), x i = b m -A m,-i x -i A m,i } P (A, b+b )∩{x ∈ N |A m x = b m +b m } = {x ∈ N |x -i ∈ P ( Ā, b+ b ), x i = b m + b m -A m,-i x -i A m,i }
where Ā, b and b are defined as follows:

bk = b k - A k,i b i A m,i b k = b k - A k,i b i A m,i Āk,j = A k,j - A m,j A m,i for
e i = bm+b m -A m,-i e -i A m,i . Taking x i = bm-A m,-i x -i A m,i and x i = b m -A m,-i x -i A m,i
, we obtain that e = x + x , where x ∈ P (A, b) and x ∈ P (A, b ), as desired.

Let us finally drop the assumption that each inequality appearing in Ax ≥ b is binding at some extreme point of P (A, b).

Equivalence classes of four-player symmetric games

We characterize (up to a permutation) the different categories of extreme points, and the conditions on the games for which those extreme points belong to the core. Each extreme point is characterized by a set of 3 coalitions for which the inequalities: x(S) = v(T ) are binding. By superadditivity, we can restrict attention to coalitions which have a nonempty intersection -if two coalitions S and T with S ∩ T = ∅ are used, this must imply that v(S ∪ T ) = v(S) + v(T ), a nongeneric condition. Furthermore, we only have to consider collections of coalitions for which the conditions are independent. This leaves us with the following possible extreme points: E1 Coalitions 1,12,123. Extreme point (0, v 2 , v 3 -v 2 , 1 -v 3 ). For this extreme point to be in the core, we must have:

v 3 ≥ 2v 2 , 1 ≥ 2v 3 -v 2 .
E2 Coalitions 1,12,13. Extreme point (0, v 2 , v 2 , 1 -2v 2 ). For this extreme point to be in the core, we must have: v 2 ≤ 1 3 , 2v 2 ≥ v 3 We compute:

k δ k w(T k ) = k|T k =S j δ k α i∈T k x i + k|T k ⊃S j δ k α i∈T k (x i -) = i∈N ( k|T k i,T k =S j δ k αx i + k|T k i,T k ⊃S j δ k α(x i -) = α i∈N ( k|T k i,T k ⊇S j δ k x i - k|T k i,T k ⊃S j δ k < α i∈N x i .
The last inequality is obtained because k|T k i,T k ⊇S j δ k ≤ 1 and can only be equal to 1 for all agents i if T k ⊇ S j for all sets T k in the collection C . However, because C = C, there must exist some set T k ⊃ S j , which implies that k|T k i,T k ⊃S j δ k = 0 for some player i. Armed with this result, we can define, for any proper minimal balanced collection C = C, the positive quantity ∆(w, C ) = j δ j w(S j )k δ k w(T k ) α .

Notice also that ∆(w, C ) does not depend on α. Now, consider the combined game v + w. For the collection C, , we have that for any collection C , v(N ) + w(N ) > k δ k (v(T k ) + w(T k )), so that the core of v + w is nonempty.

Figure 1

 1 Figure 1 Core of symmetric three-player games

  (A, b) + P (A, b ) = P (A, b + b ), any extreme point of P (A, b + b ) can be decomposed as the sum of two elements of P (A, b) and P (A, b ). These vectors have to be extreme points of the polyhedra P (A, b) and P (A, b ), and furthermore neither P (A, b) nor P (A, b ) can possess additional extreme points. This shows that, whenever P (A, b) + P (A, b ) = P (A, b + b ), the extreme points of P (A, b) and P (A, b ) must be defined by the same constraints. To prove the converse statement, we need to show that, when extreme points are defined by the same constraints, P (A, b + b ) ⊂ P (A, b) + P (A, b

Proposition 1

 1 that case, P (A, b) + P (A, b ) = P (A, b + b ) if and only if one can construct sequences of generic polyhedra b k and b k converging to b and b such that b k and b k are equivalent for all k. Applying this lemma to the core of cooperative games, we obtain the following result. Consider the equivalence relation R on β(n), where vRw if and only if the extreme points of C(v) and C(w) are defined by the same constraints.

Figure 2

 2 Figure 2 Equivalence classes of four-player symmetric games

Figure 3

 3 Figure 3 Five-player symmetric games with empty cores

For

  each positive integers M and N , let A M,N be the set of couples (A, b), where A is an (M xN )-matrix and b is an N -vector such that P (A, b) = {x ∈ N |Ax ≥ b} is non-empty and bounded. For each extreme point of P (A, b), let M e (A, b) be the set of binding constraints at e, i.e. M e (A, b) = {m ∈ {1, . . . , M }|A m e = b m }. Two vectors b and b are equivalent (given A), b ∼ b , if there exists a bijection f between the set of extreme points of P (A, b) and the set of extreme points of P (A, b ) such that M e (A, b) = M f (e) (A, b ), for each extreme point e of P (A, b). Lemma 1 Let (A, b) and (A, b ) be two elements of A M,N , for some integers M and N . The two following properties are equivalent: 1. P (A, b + b ) = P (A, b) + P (A, b ).

2 .Lemma 2 F 1 α

 221 There exist two sequences (b k ) k∈N and (b k ) k∈N in N such that (b k ) k∈N converges to b, (b k ) k∈N converges to b , and b k ∼ b k for each k ∈ N. The proof requires another lemma. Let α be a strictly positive real number, and let (A, b) and (A, b ) be two elements of A M,N , for some integers M and N . If P (A, b + b ) = P (A, b) + P (A, b ), then P (A, αb + b ) = P (A, αb) + P (A, b ).Proof of Lemma 2: It is always true that P (A, αb) + P (A, b ) ⊆ P (A, αb + b ). So we have to prove the other inclusion. We first assume that α > 1. Let x be an element of P (A, αb + b ). Consider the correspondence F :P (A, b ) → 2 P (A,b ) defined as follows: (y ) = {z ∈ P (A, b )|(∃z ∈ P (A, b)) : z + z = x -y α + y }, for each y ∈ P (A, b ). Observe that A( x-y α + y ) ≥ b + b (the total coefficient of y , α-, is positive because α > 1). Hence F is non-empty valued. It is easy to check that it is also convex-valued, and has a closed graph. Kakutani's fixed point theorem implies that there exists y in P (A, b ) such that y ∈ F (y ). Hence x-y α ∈ P (A, b), and x = (x -y ) + y ∈ P (A, αb) + P (A, b ). Suppose now that α < 1. We have:P (A, αb)+P (A, b ) = αP (A, b)+αP (A, b α ) = α[P (A, b) + P (A, b α )] = αP (A, b + b α ) = P (A, αb + b ).The penultimate equality follows from the previous paragraph. The other equalities are straightforward.Proof of Lemma 1: (1 ⇒ 2) For each k ∈ N, let b k = k k+1 b + 1 k+1 b and b k = 1 k+1 b + k k+1 b .Notice that if e k is an extreme point of P (A, b k ), then there exists a unique extreme point x of P (A, b) and a unique extreme pointx of P (A, b ) such that e k = k k+1 x + 1 k+1 x . In addition, M e k (A, b k ) = M x (A, b) ∩ M x (A, b ). Indeed, if e k is anextreme point of P (A, b k ), then there exists a set L of N independent lines such that A L e k = b k L . By Lemma 2, there exist x ∈ P (A, b) and x ∈ P (A, b ) such that e k = k k+1 x + 1 k+1 x . It must be that A L x = b L and A L x = b L . So x and x are the unique vectors in P (A, b) and P (A, b ) whose weighted sum coincides with e k . It must also be that x and x are extreme points of P (A, b) and P (A, b ), respectively. Finally, A m e k = b k m if and only if A m x = b m and A m x = b m (the necessary condition follows from the fact that x ∈ P (A, b) and x ∈ P (A, b )). Conversely, observe that if there exists an extreme point x of P (A, b) and an extreme point x of P (A, b ) such that M x (A, b) ∩ M x (A, b ) contains N independent lines, then k k+1 x + 1 k+1 x is an extreme point of P (A, b k ). A similar argument holds to show that 1 k+1 x + k k+1 x is an extreme point of P (A, b k ). For each extreme point e k of P (A, b k ), let f (e k ) be the vector 1 k+1 x + k k+1 x , where x is the unique extreme point of P (A, b) and x is the unique extreme point of P (A, b ) such that e k = k k+1 x + 1 k+1 x . The previous paragraph implies that f (e k ) is an extreme point of P (A, b k ). It also implies that f is a bijection, and that M e k (A, b k ) = M f (e k ) (A, b k ), for each extreme point e k of P (A, b k ). We thus have established Condition 2, since (b k ) k∈N converges to b, and (b k ) k∈N converges to b . (2 ⇒ 1) It is always true that P (A, b)+P (A, b ) ⊆ P (A, b+b ). So we have to prove the other inclusion. Consider the correspondence φ associating to any vector b the nonempty bounded convex polyhedron P (A, b). Because λP (A, b)+(1-λ)P (A, b ) ⊆ P (A, λb + (1 -λ)b ), the graph of φ is convex, and by Corollary 9.2.3 in Peleg and Sudhölter (2003), the correspondence φ is lower hemi continuous. Because P (A, b) is defined by a set of continuous, linear inequalities, the correspondence φ is clearly upper hemi continuous, and hence fully continuous. Now take a point x in P (A, b+b ). Because φ is lower hemi continuous, there exist sequences b k and b k converging to b and b such that x ∈ P (A, b k +b k ). Furthermore, because the polyhedron P (A, b + b ) is only determined by the sum b + b , we are free to choose two sequences b k and b k such that b k ∼ b k . Suppose that we have proven that P (A, b + b ) = P (A, b) + P (A, b ) for each pair (b, b ) of N -vector such that b ∼ b . Then, x ∈ P (A, b k ) + P (A, b k ) for all k. Because the correspondence φ is upper hemi continuous, this implies that x ∈ P (A, b) + P (A, b ). The preceding argument shows that in order to finish the proof of Lemma 1, it is sufficient to show that P (A, b + b ) = P (A, b) + P (A, b ) for each pair (b, b ) of N -vector such that b ∼ b . We proceed by induction on N . Suppose first that N = 1. Then P (A, b) = [

  such that the desired inclusion holds for all N ≤ N -1. We prove now that P (A, b + b ) ⊆ P (A, b) + P (A, b ), for all (A, b) and (A , b) in A M,N such that b ∼ b . Assume first that (A, b) is such that for each m ∈ {1, . . . , M }, there exists an extreme point e of P (A, b) such that A m e = b m . Let e be an extreme point of P (A, b + b ). We will be done with this part of the proof after showing that e ∈ P (A, b) + P (A, b ). Let m be such that A m e = b m + b m .

  all k ∈ {1, . . . , M } \ {m} and all j ∈ {1, . . . , N } \ {i}. Notice that the first two sets above must be non-empty because m ∈ M e (A, b) for some extreme point e of P (A, b) (and hence m ∈ M f (e) (A, b ), since b ∼ b ). P ( Ā, b) and P ( Ā, b ) are thus non-empty, and belong to A M -1,N . Take an extreme point ē in P ( Ā, b). Then e = (ē, e i = bm-A,m,-iē -i A m,i ) is an extreme point of P (A, b) such that m ∈ M e (A, b). Because b ∼ b , there exists an extreme point e of P (A, b ) such that m ∈ M e (A, b ) and all other constraints are defined by the same equalities as ē. It is easy to check that the N -1 vector ē such that e i = b m -A,m,-iē -i A m,i is an extreme point of P ( Ā, b ), so that b ∼ b . The induction hypothesis implies that P ( Ā, b + b ) = P ( Ā, b) + P ( Ā, b ). Hence there exists x -i ∈ P ( Ā, b) and x -i ∈ P ( Ā, b ) such that e -i = x -i + x -i and

  Let z ∈ P (A, b + b ), and let L ⊆ {1, . . . , M } be the set of inequalities that are binding at some extreme point of P (A, b) (or P (A, b ), since b ∼ b ). In particular, we have z ∈ P (A L , b L + b L ). By our previous argument, there exists x ∈ P (A L , b L ) and x ∈ P (A L , b L ) such that z = x + x . Let m ∈ {1, . . . , M } \ L. Notice that P (A -m , b -m ) ∩ {y ∈ N |A m y = b m } = ∅ (as otherwise the extreme points of this convex polyhedron would be extreme points of P (A, b), contradicting the fact that m ∈ L). Since P (A -m , b -m ) is convex and has a nonempty intersection with {y ∈ N |A m y ≥ b m }, we conclude that P (A -m , b -m ) ⊆ {y ∈ N |A m y > b m }. Iterating the argument, we conclude that A L x ≥ b L implies that Ax ≥ b. Similarly, A L x ≥ b L implies that Ax ≥ b . Hence z ∈ P (A, b) + P (A, b ).

  that C(w) = ∅. Next, consider any other proper minimal balanced collection, C = {T 1 , ..., T k , ..., T K }.

j

  δ j (v(S j ) + w(S j )) < v(N ) + α i∈N x i . Now, for any α, we can pick a value γ < min{x i } α such that γ < v(N )j δ j v(S j ) and hence j δ j (v(S j ) + w(S j )) ≤ v(N ) + w(N ).Next, consider any other proper minimal collection C . Then,k δ k (v(T k ) + w(T k )) = k δ k v(T k ) + j δ j v(S j ) -αDelta(w, C ). Hence, v(N ) + w(N )k δ k (v(T k ) + w(T k )) = w(N )j δ j v(S j ) + α∆(w, C ) +v(N )k δ k v(T k ) = γ + α∆(w, C ) + v(N )k δ k v(T k ). Hence, if one picks α > max C ( k δ k v(T k )-v(N )-γ) ∆(w,C )

  and the game is convex if v 2 ≤ 1 2 . Figures 1a and 1b illustrate the shape of the core of three-player symmetric games, when v 2 ≤ 1 2 and 1 2 ≤ v 2 ≤ 2 3 .

		(0, 0, 1)
		t			t
		t t			t t
		t			t
		t		t
	t		t	t	t
			t	t
	t t			t t t	t t
	t			t	t
	t			t	t
	t		t	t
	t	t t	t	t t	t	t t
	(1, 0, 0)			(0, 1, 0)

As in[START_REF] Bondareva | Some Applications of Linear Programming Methods to the Theory of Cooperative Games[END_REF] and[START_REF] Shapley | On Balanced Sets and Cores[END_REF] -the reader is reminded of the formal definition in Section

Recall that a balanced collection is minimal if any subcollection is unbalanced and proper if no two sets in the collection are disjoint. It is easy to verify that these are the only balanced collections where the inequality j δjv(Sj) ≤ v(N ) needs to be checked.

We are grateful to Hideo Konishi for pointing this reference to us.

The vector e is characterized by n equations of this type. The difficulty in this part of the proof stems from the fact that we are not sure a priori that these equations applied to b and b lead to an element of P (A, b) and P (A, b ), respectively.

We assume without loss of generality that A does not have a line with only zero entries. Such lines are redundant and can be deleted anyway.

E3 Coalitions 1,123,124. Extreme point (0, 2v 3 -1, 1 -v 3 , 1 -v 3 ). For this extreme point to be in the core, we must have: 1 2 leqv 3 ≤ 2 3 and 2v 3 ≥ v 2 + 1. E4 Coalitions 12,13,123. Extreme point (2v

. For this extreme point to be in the core, we must have:

). For this extreme point to be in the core we must have:

). For this extreme point to be in the core we must have:

. For this extreme point to be in the core, we must have: 3 4 ≥ v 3 ≥ 2 3 and 2v 3 ≥ 1 + v 2 .

Proof of Proposition 2

⇐ This implication is obvious. If imputations are blocked for all proper minimal balanced collections, clearly the game is non compensable. ⇒ Suppose that there exists one proper minimal balanced collection C = {S 1 , ..., S j , ..., S J } for which j δ j v(S j ) < v(N ). Construct then a game w as follows. Fix a vector

w(T ) = 0 ifT S j for anyj

We first check that the game w is superadditive. Because the collection C is proper, S j ∩ S k = ∅ for all j, k. As the game w only puts positive worth on supersets of sets in C, we conclude that if S ∩ T = ∅, either w(S) = 0 or w(T ) = 0. This implies that, in order to check that w is superadditive, we only need to verify that w is monotonic. Given that < min{x i } n and γ < min{x i } α , monotonicity is always satisfied. Next note that