Changjiang J Hu

Sarah Al Gharib

Yunlong L Wang

Pingping P Gan

Prof Qiuhao H J Li

Dr Sergey A Denisov

Sophie Le Caer

Dr Jacqueline Belloni

Jun Ma
email: junma@nuaa.edu.cn

Mehran Mostafavi

† Both

Ma

Dr S Al Gharib

Pr M Mostafavi

Radiolytic Approach for Efficient, Selective and Catalyst-free CO2 conversion at room temperature

The present study proposes a new approach for direct CO2 conversion using primary radicals from water irradiation. In order to ensure reduction of CO2 into CO2 -• by all the hydrated electrons, we use formate ions to scavenge simultaneously the parent oxidizing radicals H • and OH • producing the same transient CO2 -• radicals. Conditions are optimized to obtain the highest conversion yield of CO2. The goal is achieved under mild conditions of room temperature, neutral pH and 1 atm of CO2 pressure. All the available radicals are exploited for selectively converting CO2 into oxalate that is accompanied by H2 evolution. The mechanism presented accounts for the results and also sheds light on the data in the literature. The radiolytic approach is a mild and scalable route of direct CO2 capture at the source in industry and the products, oxalate salt and H2, can be easily separated.

Introduction

The excessive anthropometric carbon dioxide (CO2) emission, that is responsible for climate changes, has led to worldwide efforts for a drastic diminution of its atmosphere level. Beyond the progressive abandonment of fossil fuels and the storage and sequestration of CO2 surplus, the alternative strategy has been actively developed for efficient and cost-effective CO2 conversion into other eco-friendly and value-added chemicals. [START_REF] Snoeckx | [END_REF][2][3] Due to CO2 chemical stability, the existing methods of its conversion into less oxidized chemical species typically require a source of energy (i.e., heat, light, or electricity) to drive one-or multiply electron transfer reactions. For example, the reduction of CO2 by coal to form CO occurs at very high temperature (> 700 °C) due to a large positive reaction enthalpy (ΔH°298 K = 172 kJ mol -1 ). [4] The use for CO2 reduction electron-rich chemicals, e.g. CH4 and even H2, also consumes significant amounts of energy, even in the presence of a catalyst. [ 5 -7 ] The photocatalytic and electrocatalysis reduction of CO2 using renewable energy resources likely accomplish the carbon neutral energy cycle while synthesizing fuels. However, their slow kinetics together with side reactions lead to a low efficiency and diverse products. [8,9] The newly -developed catalysis covering various matter-scales exhibits superior performance for CO2 conversion, [10][11][12] but their ultimate practical viability is greatly hindered by several limiting factors such as high cost, lack of long-term stability, and upscale synthesis methodologies. [13] All these methods generally use multi-step processes based on complex chemistry, and essentially call for more direct and selective pathways.

The hydrated electron (eaq -) is the most powerful reducing species with a standard reduction potential of -2.9 VNHE that is larger than the reduction potential of CO2 /CO2 •-(-2.14 VNHE in water). [14] Therefore, eaq -can reduce aqueous CO2 at room temperature without any catalyst.

By taking advantage of the eaq -chemistry, the approaches of electrochemistry, [15] and more recently photoelectrochemistry [ 16 -18 ] have been developed to reduce CO2, but with a low Faradaic efficiency.

Herein, we exploit high-energy ionizing radiation as a promising method for aqueous CO2 reduction. [19] As the reducing radicals are directly formed in water at room temperature, this method is effective without requiring the use of any catalyst. However, up to now, the appropriated chemical conditions were not explored to enable maximizing CO2 conversion yield. Indeed, the fast back-oxidation by the symbiotic OH • radicals has to be eliminated which remains a challenge for the CO2 conversion efficiency. This work aims to reduce the aqueous CO2 into CO2 • , the key precursor of radiolytic products, by utilizing all primary radiation-induced water radicals. Among the few possible candidates to create the fully reducing conditions in irradiated aqueous CO2 solutions, formate ions were chosen as OH • and H • scavengers instead of methanol [20] or other organic chemicals, [21] because in this case, the reaction product is the same CO2 •-radical as in the reduction of CO2 by eaq -.

The oxidation reactions of formate by OH • and H • into CO2 •-and its second-order decay have already been extensively studied. [ 22 -28 ] However, this process is hindered by H + -catalysis leading to CO2 •-disproportionation back to formate and CO2. [ 29 ] Therefore, the present radiolytic experiments of CO2 reduction in presence of formate, under catalyst-free aqueous environment, were first performed at 1 atmosphere at various pH and then extended to higher CO2 pressure up to 30 atmospheres. Based on the stoichiometric radical yields values in water, we establish the reaction mechanism under various conditions in order to optimize the CO2 conversion.

Experimental methods

Materials

The gases CO2 (99.999%) and Ar (99.999%) were purchased from Air Liquide Industrial Gases Company or Nanjing Shangyuan Industrial gas plant. Sodium formate NaCOOH and formic acid HCOOH were purchased from Merck. Potassium oxalate NaC2O4 was obtained from Fluka, NaOH from Sigma Aldrich Chemistry and H2SO4 (98%) from Nanjing Chemical Reagent Co., Ltd). Aqueous solutions were prepared in deionized water (Millipore, resistivity 18.2 MΩ cm).

13 mL of the samples were saturated at 20 °C with CO2 through the septum. The hydration of CO2 into H2CO3 is weak: K ([H2CO3]/[CO2]) = 10 -2 . The pH was controlled by adequate addition of NaOH or HCO2H before the saturation by CO2.

-Irradiation and analysis

-radiolysis was carried out at room temperature under CO2 atmosphere using a panoramic 60 Co source. The dose rate, measured by the Fricke dosimeter, was 8 kGy h -1 (1 Gy = 1 J kg -1 ), and the doses were increased up to 60 kGy.

In the case of the experiments performed under pressure, a stainless-steel vessel was used at the Nanjing University. We found very small amount of CH4 at high pressure and one of the reasons of this production could be the reaction at the surface of stainless steel. However, as the CH4 amount is very low we neglect it. Moreover, the mechanism and the yield were deduced from the experiments obtained at one atmosphere in glass vessels where we did not find CH4.

Irradiation was carried out with a 60Co source. The dose rate in the vessel was determined to be 0.49 kGy h -1 .

Oxalate ions are quantified thanks to their optical absorption spectra. Measurements were performed using a Hewlett-Packard spectrophotometer). The absorbance is measured at 250 nm (ε = 28 mol -1 L cm -1 ) where the absorbance of formate anion is negligible (Figure SI-1). The sodium oxalate was also analyzed by high performance liquid chromatography (HPLC) (Figure SI-2), (Eclassical 3100 HPLC Dalian Elite Analytical Instruments) on a chromatographic column Shodex Sugar SH1011 (8.0 × 300 mm × 5.0 μm) and the eluent was an aqueous solution of H2SO4 (0.025 mol/L) at a flow rate of 1 mL/min and 50 °C with a UV absorption detection.

The gases produced (H2 and CO) were measured by a gas chromatography set-up (GC9790 Ⅱ co. Fuli instruments) equipped with a thermal conductivity detector (TCD) for H2 and a flame ionization detector (FID) for CO (Figures SI-3). To examine whether the CO arose from the ionization/excitation in the solution or in the headspace gaseous CO2(g), control radiolytic experiments of CO2(g) alone were performed (Figure SI-4). The contribution of the emitted CO from the gaseous decomposition is low (for example at 1 CO2 atm it is less than 2% of the total amount including the solution) (Figure SI-4), which is consistent with earlier radiolytic studies of gas-phase reactions that are often characterized by low product yields. The CO measurements were then systematically corrected from the CO formation arising from irradiated CO2(g) under the same conditions.

Results and Discussion

pH effects on radiolytic product yields at 1 atm CO2

The amounts of oxalate, CO and H2 are measured versus irradiation dose and pH in aqueous solutions under 1 atm of CO2 (or 3.8  10 -2 mol L -1 of CO2). First, preliminary experiments performed without formate on the radiolysis of CO2-saturated solutions showed that the oxalate yield is negligible, and the major product is H2. The radiolytic yield of H2, G(H2) ~ 0.4 ×10 -7 mol J -1 , is measured from the slope of the species amounts versus the absorbed dose (Figure SI-3).

In the presence of sodium formate (5  10 -2 mol L -1 ) as the OH • and H • radicals scavenger, the H2 radiolytic yield, G(H2) decreases from 2.75  10 -7 mol J -1 at pH 1 to 1.04 10 -7 mol J -1 at pH 7. The G(CO) value at 1 atm CO2 is very low at pH 7 and its maximum reaches 0.17 10 -7 mol J -1 at pH 2 and 3 (Figure 1 right).

The oxalate concentration increases at all pH linearly with the dose (Figure 1 left). The G(C2O4 2-) value increases from 0.05 ×10 -7 mol J -1 at pH 3 to 3.5 × 10 -7 mol J -1 at pH 7 following an opposite trend compared to H2 (Figure 1

(G(eaq -) + G(H) • + G(OH • )) = 3.6 10 -7 mol J -1 (see text)
Around pH 7, according to the mechanism described below, the CO2 consumption yield (G(-CO2)) corresponds to about the formate consumption yield (G(OH • )+G(H • )), or to the formation

yield of oxalate (1/2(G(OH • ) + G(H • ) + (G(eaq -)).
Therefore, the CO2 concentration after 60 kGy irradiation is estimated to be 1.3 × 10 -2 mol L -1 . That represents 66% of the initial concentration that have been converted. By comparison, the decrease of the formate concentration is only 17%. In addition, the yields of H2 and oxalate remain constant in this CO2 concentration range. We also showed that G(C2O4 2-) is independent on [HCO2 -] from 5 × 10 -2 to 10 -1 mol L -1 . Therefore, these observations at very high doses indicate that the radiolytic approach is effective for sustaining CO2 conversion in large amounts and that formate plays a significant role in the mechanism promoting the practicable and feasibility of this strategy.

CO2 pressure effects on the radiolytic yields

The increase of the CO2 pressure induces an increase of the CO2 aqueous concentration, and thus an acidification of the solution. The final pH obtained after having imposed different values of CO2 pressure can be calculated according to the various acid-base equilibria of the system for two initial pH (Figure SI-5). [ 30 ] Noteworthily, both HCO3 -and CO3 2-are present in solutions. However, the rate constants of the eaq -scavenging by these species (k < 10 6 M -1 s -1 ) are 3 orders of magnitude lower than that of reaction with CO2. Therefore, HCO3 -and CO3 2- are omitted in the reaction mechanism.

In the absence of formate, in CO2-saturated solutions, the substantial increase of the CO2 concentration by the pressurization leads to the rise of G(CO) from 0 to 0.18 ×10 -7 mol J -1 (30 atm, pH 5.3), but to a decrease of G(H2) from 0.4 to 0.1 × 10 -7 mol J -1 , respectively (Figure SI-3). The oxalate yield G(C2O4 2-) is negligible. In the presence of formate, for final pH ranging from 3.9 to 3.2 for a CO2 pressure varying from 1 to 30 atm, we observe a decrease of the H2 yield and a marked increase of the CO yield without any oxalate formation (Figure 2

left).

For final pH around 7, the effect of CO2 pressure on the yields of H2 and CO presents similar trend as at pH 3. As shown above (Figure 1 Right), G(C2O4 2-) measured at 1 atm of CO2 is the highest and equal to 3.5 × 10 -7 mol J -1 . However, when the CO2 pressure increases, G(C2O4 2 ) decreases down to a plateau value of 1.2 × 10 -7 mol J -1 at 30 atm (pH 5.5) (Figure 2 Right).

Obviously, in spite of a marked increase of the CO2 concentration, the increasing acidity caused by pressurization leads to a decrease of the oxalate yield.

It is worthy to note that CO2 pressure effects on decreasing G(C2O4 2-) and increasing G(CO) (Figure 2), are comparable to the pH effect observed at 1 atm pressure (Figure 1). However, the H2 formation yield does decrease to lower values at higher CO2 pressure (Figure 2) which is in contrast to the G(H2) increase when pH decreases at 1 CO2 atm (Figure 1).

Figure 2. Radiolytic formation yields of H2 and CO versus CO2 pressure from 1 to 30 atm (Left) in formic acid solution (pH 3). In these conditions, G(C2O4 2-) is negligible at any pressure. (Right) in solution containing initially 0.5 mol L -1 NaOH (final pH ranges from 7 to 5.5)

Mechanism Neutral medium, 1 atm CO2

The generation of various species from water radiolysis at pH 7 is expressed as:

H2O

eaq

(in brackets are the yields in 10 -7 mol J -1 unit). [31,[START_REF] Buxton | An overview of the radiation chemistry of liquids[END_REF] These yields correspond to a scavenging factor of 10 7 s -1 (or 100 ns after the radical production and their mutual reactions in spurs). Note that the direct effect of radiation on CO2 is negligible because of the low electron fraction (< 0.1) compared to water even at 30 atm pressure.

The reduction of CO2 occurs through the reaction with eaq -(reaction 2 in Table 1). This reaction has been found to be almost controlled by diffusion (k2 = 8.2  10 9 M -1 s -1 ), [20,21,[START_REF] Gordon | [END_REF] and the activation energy of k2 between 5 and 200 °C was measured to be 15.9 kJ mol -1 . [20] The value of the rate constant k2 warrants that all the solvated electrons reduce efficiently CO2. The second-order reaction of dimerization of radicals CO2 •-(reaction 3 in Table 1) into oxalate is in competition with the H + -catalysed disproportionation (reaction 4 in Table 1) which leads to the formation of the initial reactants. [22][23][24][START_REF] Keene | Pulse Radiolysis[END_REF][START_REF] Fojtik | [END_REF][27][28][29] In the absence of formate as an OH • scavenger, the oxalate yield G(C2O4 2-) = 0.02 × 10 -7 mol J -1 , is much less than G(e - aq)/2. This very low yield and inefficiency of the process are most likely ascribed to significant loss of CO2 •-caused by a secondary reaction with OH • radicals (reaction 5 in Table 1) faster than their dimerization (reaction (3)). However, the G(H2) value is close to the primary H2 yield G(H2) = 0.42 × 10 -7 mol J -1 .

Table 1. Reactions and corresponding rate constants involved in the mechanism of C2O4 2-, H2 and CO formation

Reactions

Rate constants (L mol -1 s -1 ) at 20 °C

- -• aq 2 2 e + CO CO  (2) 8.2  10 9 [20] -• -• 2- 2 2 2 4 CO + CO C O  (3) 1.4  10 9 [29] -• -• + -• - 2 2 2 2 2 2 CO + CO + H C-CO H CO O + HCO   (4) 1.4  10 9 [29] • -• - 2 2 OH + CO OH + CO  (5) • - -• 2 2 2 OH + HCO H O + CO  (6) 4.1  10 9 [22] • - -• 2 2 2 H + HCO H + CO  (7) 2.1  10 8 [31] - + • aq e + H H  (8) 2.3  10 10 [31] - 2 H + CO CO + OH   (9)   -• -• 2 2 2 2 CO + CO O C-CO (10)
    -• -• - 2 2 2 + 2 2 2 + O C-CO + O C-CO HCO + H 3 CO  (11)     -• -• 4 2- 2 2 2 2 2 2 O C-CO + O C-CO C O + 2 CO  (12) - -• 2 qf 2 CO + e CO  (13)     -• -• 2 2 2 2 + 2 - 3 +H O C-CO + O C-CO CO + HCO + 2 CO  (14)
In the presence of formate, at neutral pH, the H2 yield corresponds to the sum of the primary yield G(H2) = 0.42 × 10 -7 mol J -1 in reaction (1) and of the scavenging yield of H • radicals by formate (G(H • ) = 0.62 × 10 -7 mol J -1 ) in reaction (7). The experimental value G(H2) = 1.04 ×10 -7 mol J -1 (Figure 1 right) is in fair agreement with this sum G(H2) + G(H • ) (Figure 3 and Table 2). That means that in addition to primary radiolytic H2, all the H • radicals are exploited to produce H2.

G((C2O4 2-)) = ½ (G(eaq -) + G(H • ) + G(OH • ) = 3.5 ×10 -7 mol J -1 and G(H2) = (G(H • ) + G(H2))
= 1.04 ×10 -7 mol J -1 . At pH < 7, G(C2O4 2-) decreases because the H + -catalyzed disproportionation of CO2 •-is involved and becomes significant. At high CO2 pressure (in red), CO2 scavenges early the electrons before their hydration and the radicals CO2 •-react with H • radicals in spurs yielding CO. Simultaneously, the pH decrease favors the disproportionation, so that the oxalate yield becomes negligible. The arrow on the left indicates the time after the radiation-induced ionisation and excitation of the aqueous solution.

Table 2. Summary of product yields and predominant mechanisms in the radiolysis of CO2

solutions with sodium formate 5  10 -2 mol L -1 . dispr and dim are the disproportionation and the dimerization of the precursor radical CO2 -• , respectively. Primary yields of species arising from water radiolysis are in italic. 1.04

Irradiated

= G(H2) + G(H • )neutral 0.08 3.5 = ½ (G(eaq -)+ G(H) + G(OH • ))
Only dim

Part of CO2 •-produced in CO2-saturated sodium formate solutions is due to the e aq - scavenging by CO2 (reaction 2) and the other part is due to the scavenging of OH • and H • by formate (reactions 6 and 7). According to k6 and k7 values, and to the formate concentration, the scavenging half-time for reaction ( 6) is 4 ns and G4ns(OH • )  3.3 × 10 -7 mol J -1 . [34,35] Therefore, from reactions (2, 3, 5, 6 and 7), the maximum value of oxalate yield is G(C2O4

2-)calc = ½ (G(eaq -)+ G(H • ) + G(OH • )) = 3.6 × 10 -7 mol J -1 (dotted line in Figure 1 Right), with
G4ns(eaq -) = G4ns(OH • ))  3.3 × 10 -7 mol J -1 and G(H • )  0.62 × 10 -7 mol J -1 . The experimental yield obtained at pH 7: G(C2O4 2-) = 3.5 × 10 -7 mol J -1 is in excellent agreement with this maximum value. It is remarkable that under these mild conditions, the e - aq scavenging, which yields the conversion of CO2 at 1 atm into oxalate, is complete and that the disproportionation path (4) is inactive. Indeed, in formate and CO2 solutions, efficient reductive conditions of the aqueous solutions are fulfilled because, as an OH • and H • scavenger, formate inhibits the recombination reaction (5). [36,37] The scheme of the mechanism is summarized in Figure 3. The eaq -scavenging for selectively converting CO2 into oxalate is completely exploited. Moreover, the H • radical scavenging by formate yielding H2, a useful fuel, is also complete. The efficiency of the absorbed radiation energy is thus 70% for the oxalate formation (32% for the reducing conversion of CO2 and 38% for the formate oxidation), [ 38 ] plus 14.5% for the hydrogen formation.

Acidic medium

The yield of H2 increases in acid medium at CO2 atmospheric pressure (Figure 1) because eaq -are replaced by H • radicals via reaction (8) and more H • radicals are scavenged by formate (reaction (7)). Thus at pH 1 under 1 atm CO2, the yield would be G(H2)calc = G(H • )acidic + G(H2) = (3.42 + 0.42) × 10 -7 mol J -1 . However, the experimental value is lower: (G(H2)exp = 2.8 ×10 -7 mol J -1 ), showing that a small part of H • radicals are thus lost in another reaction, such as the CO2 •-radical reduction into CO (reaction ( 9)). This reaction may occur within the ionisation spurs before the homogenous dispersion of the radicals and explains the CO formation, although the experimental result is only G(CO) = 0.17 × 10 -7 mol J -1 .

At 1 atm of CO2, the oxalate yield G(C2O4 2-) decreases with the pH decrease (Figure 1). However, the simultaneous formation yield of the oxalate precursor CO2 •-(either formed by reactions 2 or 6 and 7) is pH-independent. In a previous mechanistic study [29] of CO2 •-arising from CO oxidation by OH • (without CO2), it was shown that at low pH the oxalate yield G(C2O4 2-) was strongly inhibited, and in contrast G(CO2) increased, owing to the disproportionation of CO2 •-into CO2 and formate (reaction 4). However, the rate constant of the second-order decay of CO2 •-(2k = 1.3  10 9 M -1 s -1 ) is also pH-independent. They concluded that the disproportionation (4) was catalysed by protons and favored by the acid form HCO2 • of the radical CO2 •-(pKa (HCO2 • /CO2 -• )  2.3). The protons would govern the disproportionation because of the presence of an intermediate head-to-tail adduct linking the carbon atom of one molecule to an oxygen atom of the other one. [29] Other values of CO2 •- pKa were proposed in the literature. [START_REF] Fojtik | [END_REF]27,39] However, the value recently determined by time resolved Raman spectroscopy measurements was pKa = 3.4. [40] We observe a more drastic influence of H + on the G(C2O4 2-) decrease than in ref. [29] ( Figure 1). The main difference here to that study is the saturation of the solution by CO2. The mechanism of the photoelectrochemical CO2 reduction proposed by Tryk et al. [16] involves the reaction of the radical CO2 •-with CO2 forming, at least partially, the transient dimer radical anion O2C-CO2 -• (reaction 10). Various structures and stabilities of this dimer radical anion have been ab initio calculated. [41] Here we assume also that the dimer radical anion O2C-CO2 -• is formed under CO2 atmosphere (reaction (10)). The fate of the dimer radical anion is not well known. [42] However, we propose that the dissociative disproportionation of O2C-CO2 -• radical (reaction 11) would be favored vs the dimerization (reaction 12). As it was recently suggested, the disproportionation of O2C-CO2 -• radical may also lead to CO formation (reaction 14). [43] Considering that the oxalate yield at a given pH is less in CO2-saturated than in Ar-saturated formate solutions of ref 29, the carbon-carbon bonding in the dimer radical O2C-CO2 -• structure possibly inhibits a new addition and reorganization into the oxalate in contrast to reaction (4).

Because the inflexion point of the oxalate yield increase is shifted to higher pH value than in conditions without CO2, the pKa (O2C-CO2H • /O2C-CO2 -• ) value is tentatively estimated at about 5.5 (Figure 1). As a result, the influence of protons in the H + -catalysed disproportionation is stronger for O2C-CO2 -• (reaction 11) than for CO2 -• (reaction 4). used and the efficiency is thus optimized. Furthermore, the radiation-induced reduction method is an achievable, promising and scalable route for industrial CO2 conversion. This method is promoted by the current availability of cost-effective industrial electron accelerators and 60 Co -γ sources. Though the aim of our work was not to give details on the economic aspect of this process and its possible valorization, it is known that the price of oxalate is higher than that of sacrificial formate. In addition, the valuable H2 is produced during this process. Several parameters intervene in the calculation of the total cost of the CO2 conversion. Eventually, our first estimation suggests that this simple process has a competitive energy cost compared to other chemical processes.

Conclusion

We demonstrated that the versatility of radiolytic pathways can be harnessed to direct the conversion of aqueous CO2, 1 atm / HCO2Na solutions at pH 7 at room temperature in order to produce value-added chemicals (oxalate, H2) with optimized yields. All the radiolytic radicals are scavenged so that the radiation energy is significantly exploited for converting CO2 into oxalate and H2. The method does not require the presence of any catalyst. The knowledge of radiolytic yield values of water radicals was exploited in order to unravel the reaction mechanisms of all the processes at stake in the systems. The mechanism sheds light also on other previous results. Moreover, we suggest that the ionizing radiation delivered by irradiation facilities may be directly and easily coupled with an industrial source of CO2. Therefore, the radiation-induced reducing radicals constitute a promising solution for the CO2 conversion.

Figure 1 .

 1 Figure 1. (Left) Representative measurements for the concentrations of oxalate (top), CO (middle) H2 (bottom), produced versus the irradiation dose at various pH. (Right) Yields ofH2, CO and oxalate in aqueous solution containing formate (5  10 -2 mol L -1 ) in the presence of CO2 at 1 atm (the concentration of CO2 in water is then 3.8  10 -2 mol L -1 ). Dotted line: Maximum oxalate yield, which is equal to ½ (G(eaq -) + G(H) • + G(OH • )) = 3.6 10 -7 mol J-1

Figure 3 .

 3 Figure 3. Scheme of the radiolysis of CO2-saturated solutions of sodium formate. At pH 7 and 1 atm (in black), the CO2 •-dimerization and the CO2 conversion into oxalate are optimized:

13 << 7 =

 137 G(H2) + G(H • )acidic0.

Acknowledgements

This work was supported by National Natural Science Foundation of China (Nos.11975122, 21906083), National Natural Science Foundation of Jiangsu Province (BK.2019030384) and Fundamental Research Funds for the Central Universities (N°. NE2020006).

Effects of high CO2 pressure

The pressure increase induces also a marked concentration increase of CO2 that scavenges rapidly in the radiation spurs the quasi-free electrons (eqf -) prior the hydration process (reaction (13)) as it was recently observed (Figure 3, red part). [30] In these conditions, the source of primary H • radicals (reacting with formate and producing H2, reaction 7) and of molecular hydrogen (reaction 1) progressively vanishes with the pressure increase. At pH = 3.9 to 3.2 for 1 to 30 atm respectively, we observe a decrease of the formation yield G(H2) to zero, and an increase of G(CO) (Figure 2). In contrast, the CO yield increases with the pressure because of the formation of CO via the reaction (9) between CO2 -• and the H • radicals issued from the dissociation of excited H2O* (Figure 3, red part). The OH • radicals are still scavenged by formate and yield CO2 -• , but their H + -catalysed disproportionation totally inhibits the formation of oxalate (Figure 3). Thus, the formation yield of oxalate is negligible under acidic conditions at any pressure.

In formate solutions initially added with 0.5 mol L -1 NaOH at increasing CO2 pressure (final pH = 7 to 5.4 for 1 to 30 atm, respectively), the oxalate yield decreases from G(C2O4 2-) = 3.5×10 -7 mol J -1 at 1 atm to a plateau at 1.3 ×10 -7 mol J -1 at pressures > 10 atm (Figure 2). However, even if the part of CO2 -• radicals, arising from the fast (reactions 2) and ultrafast (reaction 13) CO2 reduction, increases versus the part arising from formate oxidation by OH • scavenging (reaction 6), the total yield G(CO2 -• ) does not change. Above 10 atm (pH 5.4), CO2 -• radicals undergo only partly the dimerization into oxalate (reactions 3 and 12) because of the competition with the H + -catalysed disproportionation (reaction ( 11)) (Figure 3 and Table 2)). In fact, the value of the yield G(C2O4 2-) = 1.3 × 10 -7 mol J -1 at pressures > 10 atm (Figure 2) is close to the value at the same pH 5 at 1 atm (Figure 1). The effect of pH is thus more decisive than that of CO2 concentration in the conversion into oxalate.

The mechanism proposed in the present study (Figure 3 and Table 2) could be also of interest for previous results. The eaq -chemistry contributing to scalable CO2 reduction has recently attracted increasing attention for example in photoelectrocatalytic [17] or plasma [15] processes where the analogy with the initial reduction of CO2 by eaq -in radiolysis was underlined.

However, the main feature in these studies is that the symbiotic oxidizing species which are unavoidably formed together with eaq -, are responsible of back-reactions and low overall efficiencies of the reduction. In contrast, in the present radiolytic approach the oxidizing species are scavenged by formate, which is chosen to contribute moreover to a selective production of oxalate and H2. The mechanism in Figure 3 and Table 2 explains how the radiation radicals are


Philippe Bich
email: bich@univ-paris1.fr

Julien Fixary
email: julien.fixary@univ-paris1.fr

Structure and oddness theorems for pairwise stable networks

Keywords: Weighted Networks, Pairwise Stable Networks Correspondence, Generic Oddness. JEL Classification: C72, D85

We determine the topological structure of the graph of pairwise stable weighted networks. As an application, we obtain that for large classes of polynomial payoff functions, there exists generically an odd number of pairwise stable networks. This improves the results in Bich and Morhaim ([5]) or in Herings and Zhan ([14]), and can be applied to many existing models, as for example to the public good provision model of Bramoullé and Kranton ([8]), the information transmission model of ), the two-way flow model of Bala and Goyal ([2]), or Zenou-Ballester's key-player model ([3]).

Introduction

Pairwise stability concept, proposed in Jackson and Wolinsky ( [START_REF] Jackson | A strategic model of social and economic networks[END_REF]), plays a central role in strategic network formation models. First introduced for unweighted networks, it was then extended for weighted networks (i.e. networks for which relationships are measured by real numbers between 0 and 1), and recently, it was proved that a pairwise stable weighted network exists for large classes of payoff functions (see Bich and Morhaim [START_REF] Bich | On the existence of Pairwise stable weighted networks[END_REF][START_REF] Bich | On the existence of pairwise stable weighted networks[END_REF]).

Roughly, a network is pairwise stable if "no two agents could gain from linking and no single agent could gain by severing one of his or her link" (see [START_REF] Jackson | A strategic model of social and economic networks[END_REF]). A frequently mentioned issue in network formation literature is the too large number of pairwise stable networks, which can be a problem because it prevents the use of comparative statics analysis. Thus, it seems quite important to study the question of the number of pairwise stable networks.

In game theory, this question has been intensively studied for Nash equilibria: for example, several papers have proved generic finiteness and oddness of the set of mixed Nash equilibria of finite games (e.g. [START_REF] Blume | The algebraic geometry of perfect and sequential equilibrium[END_REF][START_REF] Govindan | Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes[END_REF][START_REF] Harsanyi | Oddness of the number of equilibrium points: A new proof[END_REF][START_REF] Mas-Colell | Generic finiteness of equilibrium payoffs for bimatrix games[END_REF][START_REF] Govindan | On the generic finiteness of equilibrium outcome distributions in game forms[END_REF][START_REF] Pimienta | Generic Determinacy of Nash Equilibrium in Network Formation Games[END_REF][START_REF] Herings | A differentiable homotopy to compute nash equilibria of n-person games[END_REF][START_REF] Wilson | Computing equilibria of n-person games[END_REF]).

The aim of this paper is twofold:

1. First, we characterize the topological structure of the graph of pairwise stable networks P, formally the set of all pairs (v, g), where v = (v 1 , ..., v n ) is a profile of payoffs functions of n agents involved in the network, and g is a pairwise stable network associated to v. Under mild assumptions, we prove that P is homeomorphic to the set of profiles of payoff functions. This result is similar to Kohlberg-Mertens theorem ( [START_REF] Kohlberg | On the strategic stability of equilibria[END_REF]), but for pairwise stable networks, instead of Nash equilibria. 12. Second, we prove that for large classes of polynomial payoff functions, there exists generically an odd number of pairwise stable networks. 2 There exist some analogous results for Nash equilibria ( [START_REF] Govindan | Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes[END_REF][START_REF] Wilson | Computing equilibria of n-person games[END_REF][START_REF] Harsanyi | Oddness of the number of equilibrium points: A new proof[END_REF][START_REF] Blume | The algebraic geometry of perfect and sequential equilibrium[END_REF][START_REF] Mas-Colell | Generic finiteness of equilibrium payoffs for bimatrix games[END_REF][START_REF] Govindan | On the generic finiteness of equilibrium outcome distributions in game forms[END_REF]), but, to our knowledge, they apply only to affine payoff functions.

To get an idea of our two main results, first consider the following example, introduced in Bich-Morhaim ( [START_REF] Bich | On the existence of pairwise stable weighted networks[END_REF]): three researchers are labeled in N = {1, 2, 3}. For every i ∈ N and every j = i, researchers i and j write a paper together, and we call x ij ∈ [0, 1] the (normalized) amount of time spent together by i and j on their common project. Assume u 1 (x 12 , x 13 , x 23 ) = x 12 + x 13 -c 12 x 2 12 -c 13 x 2 13 : the linear term captures that u 1 is increasing with respect to the total amount of work of 1, and the quadratic term measures the cost of working on the different projects. The payoff functions are assumed to be similar for researchers 2 and 3: u 2 (x 12 , x 13 , x 23 ) = x 12 + x 23 -c 21 x 2 12 -c 23 x 2 23 and u 3 (x 12 , x 13 , x 23 ) = x 13 + x 23 -c 31 x 2 13 -c 32 x 2 23 . The optimal amounts of time for i are xij = min{ 1 2c ij , 1} for all j = i, and it is easy to see that the unique pairwise stable weighted network is xij (c) = min{1, 1 2 max{c ij ,c ji } }, where c = (c 12 , c 13 , c 23 ) ∈ (0, +∞) 3 and ij is equal to {1, 2}, {1, 3} or {2, 3}.

Each profile of payoff functions can be represented by c ∈ (0, +∞) 3 , thus, denoting ḡ(c) the pairwise stable network associated to c, the graph of pairwise stable networks can be written P = {(c, ḡ(c)) : c ∈ (0, +∞) 3 }, and it is clearly homeomorphic to the set (0, +∞) 3 of profile of payoff functions, the homeomorphism being the projection on the first factor. Our first main result (Structure Theorem 2.1) is a generalization of this example.

To illustrate our second main result, suppose now that the payoffs of the three researchers are equal to u(x 12 , x 13 , x 23 ) = -x 12 x 13 x 23 -c(x 12 +x 13 +x 23 ), for some parameter c ≥ 0. For every c > 0, (0, 0, 0) is the unique pairwise stable network. But for c = 0, there is an infinite number of pairwise stable networks (since any network for which two links have weights equal to zero is pairwise stable). Thus, there is a finite number of pairwise stable networks for all parameters c ≥ 0 except for c = 0. Our second result (Oddness Theorem 3.3) is a generalization of this example: we prove that for large classes of profiles of payoff functions, there is generically an odd number of pairwise stable networks (which means intuitively that the situations for which the number of pairwise stable networks is even or infinite are exceptional).

We describe several standard models in network formation literature to which our Oddness Theorem can be applied: the public good provision model of Bramoullé and Kranton ([8]), the information transmission model of ), the two-way flow model of Bala and Goyal ( [START_REF] Bala | A noncooperative model of network formation[END_REF]), or Zenou-Ballester's key-player model ( [START_REF] Ballester | Who's who in networks. wanted: The key player[END_REF]).

Our paper is organized as follows: in Section 2, we first recall notations and basic facts about pairwise stability (Subsection 2.1), we define the graph of pairwise stable networks (Subsection 2.2) and we state our first main structure theorem (Subsection 2.3). In Section 3, after some reminders about real algebraic geometry and algebraic topology (Subsection 3.1), we state generic finitness and oddness of the set of pairwise stable networks (Subsection 3.2). Then, several examples are given (Subsection 3.3).

2 Structure of the graph of pairwise stable networks

Reminders about pairwise stability

Throughout this paper, 3 we fix some integer n ≥ 2, and we define N = [ [1, n]], called the set of agents. The set L = {{i, j} : (i, j) ∈ N 2 , i = j} denotes the set of links on N and G = [0, 1] L the set of (weighted) networks on N . We now define a society, which is a n-tuple of payoff functions (one for each agent) defined on the set of all possible networks on N .

Definition 2.1. A society is an element v = (v 1 , . . . , v n ) ∈ F(G, R) n , where each mapping v i is called the payoff function of agent i ∈ N .
Notations. Every link {i, j} ∈ L is denoted ij. For every network g and every ij ∈ L, the element g {i,j} is simply denoted g ij . For every ij ∈ L, we denote

L -ij = L -ij (i.e L -ij = {kl ∈ L : kl = ij}) and G -ij = [0, 1] L -ij . For every ij ∈ L, every network g -ij ∈ G -ij and every w ∈ [0, 1], g = (w, g -ij )
∈ G is the network defined by g kl = g kl for every kl = ij and g ij = w. For every network g ∈ G and every ij ∈ L, g -ij ∈ G -ij is defined as g -ij = (g kl ) kl =ij .

We now recall the seminal concept of pairwise stability, introduced by Jackson and Wolinsky ( [START_REF] Jackson | A strategic model of social and economic networks[END_REF]) for unweighted societies, 4 and extended by Bich and Morhaim for weighted societies (see [START_REF] Bich | On the existence of Pairwise stable weighted networks[END_REF], [START_REF] Bich | On the existence of pairwise stable weighted networks[END_REF]). Definition 2.2. A network g ∈ G is pairwise stable with respect to a society v = (v 1 , . . . , v n ) if for every ij ∈ L, the two following conditions hold:

1. For every w ∈ [0, g ij ), v i (w, g -ij ) ≤ v i (g) and v j (w, g -ij ) ≤ v j (g). 2. For every w ∈ (g ij , 1], v i (w, g -ij ) ≤ v i (g) or v j (w, g -ij ) ≤ v j (g).
The set of pairwise stable networks of the society v is denoted P(v). 3 Let us recall some mathematical definitions or notations used in this paper: for every sets X and Y , F(X, Y ) denotes the set of functions from X to Y . For every finite set X, card(X) (or sometimes | X |) denotes the cardinal of X. A correspondence Φ from a set X to another set Y is a mapping from X to the set of all subsets of Y , and we denote it by Φ : X Y . For every g ∈ G, the Euclidean norm g of g is defined by g = ij∈L g 2 ij . A mapping f from a topological space X to another topological space Y is proper if for every compact subset K of Y , f -1 (K) is compact in X. For every topological space X and Y , a homotopy from a continuous function f : X → Y to another continuous function g : X → Y is a continuous function H : [0, 1] × X → Y such that H(0, .) = f (.) and H(1, .) = g(.). We say that: (1) f and g are homotopic if there exists a homotopy H from f to g; (2) f and g are properly homotopic if there exists a homotopy H from f to g such that H -1 (K) is compact for every compact subset K of Y . A function f : G → R is said to be polynomial if f is a polynomial function of the weights g ij of g. 4 By definition, the set of unweighted networks is G u = {0, 1} L and an unweighted society is a n-tuple

v = (v 1 , . . . , v n ) ∈ F(G u , R) n .

Graph of pairwise stable networks

The payoff functions considered in this paper satisfy some concavity assumption: more precisely, for every i ∈ N , let For every i ∈ N , the set F i is endowed with the norm5

F i = v i ∈ C 0 (G, R) : ∀j = i, ∀g -ij ∈ G -ij , g ij → v i (g ij , g -ij ) is concave and C 1 on [0, 1] and F = i∈N F i . Here, a function f : [0, 1] → R is said to be C 1 if it is continuous on [0, 1], C 1 on ]0, 1[, f -(0)
v i i = max{max{ v i ∞ , ∂ ij v i ∞ } : j = i}
and F with the product norm. Here,

∂ ij v i denotes the function from G to R defined by ∀g = (g ij , g -ij ) ∈ G, ∂ ij v i (g) = ∂v i ∂g ij (g ij , g -ij ).
Definition 2.3. The pairwise stable networks correspondence (or PSN correspondence) is the correspondence

P : F G v → P(v)
and the graph of the PSN correspondence (sometimes called simply the graph of pairwise stable networks) is defined by

P := Gr(P ) = {(v, g) ∈ F × G : g ∈ P(v)}.
We endow P with the following product norm (v, g) = max{max{ v i i : i = 1, . . . , n}, g } where g denotes the Euclidean norm of g. We denote the projection from P to F by π, i.e. π(v, g) = v for every (v, g) ∈ F × G.

The structure theorem

In this section, we determine the topological structure of P, the graph of pairwise stable networks: Theorem 2.1. (Structure Theorem) The projection mapping π : P → F is properly homotopic to some homeomorphism η : P → F.

The proof is given in Appendix 4.1. For a similar result applied to mixed Nash equilibria of games, see Kohlberg-Mertens' structure theorem ( [START_REF] Kohlberg | On the strategic stability of equilibria[END_REF]), and also a recent generalization due to Predtetchinsky ( [START_REF] Predtetchinski | A general structure theorem for the nash equilibrium correspondence[END_REF]).

Actually, Corollary 2.1 proves that Theorem 2.1 remains true if we replace F by a subset of F satisfying some additional stability assumption: for every i ∈ N, let

A i = {g ∈ G → j =i α ij g ij + c : (α ij ) j =i ∈ R n-1 , c ∈ R},
that is to say, v i ∈ A i if v i is affine and does not depend on g -ij , for every j = i. In particular, A i ⊂ F i . For every subset U of F, we define

P U := {(v, g) ∈ P : v ∈ U}
and we consider π | P U the restriction of π to P U , that is to say

∀(v, g) ∈ P U , π | P U (v, g) = v.
Similarly, η | P U denotes the restriction of η (the mapping defined in the above theorem) to P U .

Corollary 2.1. For every i ∈ N , consider U i ⊂ F i such that for every v i ∈ U i and every

A i ∈ A i , v i + A i ∈ U i , and let U = i∈N U i . Then, η | P U : P U → U is a homeomorphism which is properly homotopic to π | P U : P U → U.
The proof is given in Appendix 4.3.

3 Generic finiteness and oddness of the graph of pairwise stable networks

Reminders about real algebraic geometry and algebraic topology

Let us first recall the definition of a semi-algebraic subset of R m and of a semi-algebraic mapping (see Bochnak [START_REF] Bochnak | Real Algebraic Geometry[END_REF]). Definition 3.1. A semi-algebraic subset of R m is a subset of the form

s i=1 r i j=1 {x ∈ R m : f i,j i,j 0}
where i,j denotes either < or = and f i,j ∈ R[X 1 , . . . , X m ] for every i = 1, . . . , s and every j = 1, . . . , r i . Thus, by definition, semi-algebraic sets are closed under finite unions and finite intersections. Actually, these sets are also closed under projections, a consequence of Tarski-Seidenberg principle:

Proposition 3.1. If S is a semi-algebraic subset of R m+p and Π is the canonical projection from R m+p to R m , then Π(S) is a semi-algebraic subset of R m .
From this result, we can also define semi-algebraic sets using quantified variables (or m-tuples of variables) which range in R, R m or more generally in a semi-algebraic set (for example, see Bochnak [START_REF] Bochnak | Real Algebraic Geometry[END_REF], p. 28). Definition 3.2. A first-order formula of the language of ordered fields with parameters in R is a formula written with a finite number of conjunctions, disjunctions, negations, and universal or existential quantifiers on variables in semi-algebraic sets, starting from atomic formulas which are formulas of the kind f (x 1 , . . . , x m ) = 0 or g(x 1 , . . . , x m ) > 0, where f and g are polynomials with coefficients in R. Proposition 3.2. Let Φ(x 1 , . . . , x m ) be a first-order formula of the language of ordered fields with parameters in R. Then, {x ∈ R m : Φ(x)} is a semi-algebraic set.

For example, if f is a polynomial with four variables and if S 1 denotes the unit circle of R 2 , then the set {(x, y) ∈ R 2 : ∀(z, t) ∈ S 1 , f (x, y, z, t) ≥ 0} is semi-algebraic (as S 1 is itself a semi-algebraic set). Definition 3.3. Let S be a semi-algebraic subset of R m and T be a semi-algebraic subset of R p . A mapping f : S → T is semi-algebraic if its graph

Gr(f ) = {(x, f (x)) : x ∈ S} is a semi-algebraic subset of R m × R p .
A semi-algebraic homeomorphism is a homeomorphism which is semi-algebraic (in that case, f -1 is also semi-algebraic). We now define the dimension of a semi-algebraic set (see Bochnak [START_REF] Bochnak | Real Algebraic Geometry[END_REF], Theorem 2.3.6. p. 33 and Corollary 2.8.9., p. 53). Definition 3.4. For every semi-algebraic subset S of R m , there exists an increasing sequence of non-negative integers

d 0 ≤ d 1 ≤ • • • ≤ d k such that S = k i=0 S i ,
the union being disjoint, where S i is semi-algebraically homeomorphic to ]0, 1[ d i for every i = 0, . . . , k (where, by convention, ]0, 1[ 0 is a point). The dimension of S is defined as dim(S) := max{d 0 , d 1 , . . . , d k } (and does not depend on the decomposition of S).

Then, we have the following properties: Proposition 3.3. Let S, S 1 and S 2 be semi-algebraic subsets of R m :

1. If f : S → R p is semi-algebraic and is a bijection from S to f (S), then dim(S) = dim(f (S)).

2. If S 1 ⊂ S 2 , then dim(S 1 ) ≤ dim(S 2 ).

3. dim(S 1 × S 2 ) = dim(S 1 ) + dim(S 2 ).

Definition 3.5. Let S be a semi-algebraic set and S 0 be a semi-algebraic subset of S. We say that S 0 is a generic semi-algebraic subset of S if S 0 is open in S and if dim(S -S 0 ) < dim(S).

Theorem 3.1. Let S and T be two semi-algebraic sets such that dim(S) ≤ dim(T ) and f : S → T be a surjective continuous semi-algebraic mapping. Then, there exists a generic semi-algebraic subset T 0 of T such that for every t 0 ∈ T 0 :

• f -1 (t 0 ) is a (nonempty) finite set,

• there exists an open neighborhood

V t 0 of t 0 such that f -1 (V t 0 ) is a finite union of pairwise disjoint open sets (V k t 0 ) k∈K (where K is a finite set) such that for every k ∈ K, f | V k t 0 is a homeomorphism between V k t 0 and V t 0 .
Proof. To prove this theorem, we use the following theorem (see Bochnak [START_REF] Bochnak | Real Algebraic Geometry[END_REF], p. 224):

Theorem 3.2. Let S and T be two semi-algebraic sets and f : S → T be a continuous semi-algebraic mapping. Then, there exists a generic semi-algebraic subset T 0 of T such that f has a semi-algebraic trivialization6 over each semi-algebraically connected component7 of T 0 .

Now, let us consider t 0 ∈ T 0 and the connected component C 0 of T 0 which contains t 0 . We apply the theorem above to C 0 : thus, there exists a semi-algebraic trivialization of f over C 0 with fiber K. In particular, C 0 × K and f -1 (C 0 ) are semi-algebrically homeomorphic, thus we obtain that

dim(C 0 × K) = dim(C 0 ) + dim(K) = dim(f -1 (C 0 )) with dim(f -1 (C 0 )) ≤ dim(S) and dim(C 0 ) = dim(T ), since f -1 (C 0 ) ⊂ S and T 0 is open in T (thus C 0 is also open in T ). Now, f -1 (t 0 ) is also semi-algebraically homeomorphic to K (indeed, K is semi-algebraically homeomorphic to {t 0 } × K and θ | {t 0 }×K is a semi-algebraic homeomorphism from {t 0 } × K to f -1 (t 0 )). In particular, dim(f -1 (t 0 )) = dim(K), thus from the above equality, we obtain that dim(f -1 (t 0 )) = dim(f -1 (C 0 )) -dim(C 0 ) ≤ dim(S) -dim(T ).
But, by assumption, dim(S) ≤ dim(T ), and in particular we obtain that dim(f -1 (t 0 )) = 0. Finally, as f -1 (t 0 ) is a 0-dimensional semi-algebraic set, it is a finite (nonempty) set (see Bochnak [START_REF] Bochnak | Real Algebraic Geometry[END_REF], Theorem 2.3.6.) with the same cardinal as K. Then, defining V t 0 an open neighborhood of t 0 included in C 0 , and

V k t 0 = θ(V t 0 × {k}) for every k ∈ K, we easily get that: 1. The sets V k t 0 , k ∈ K, are open (because V t 0 × {k} is open in C 0 × K and θ is a homeomorphism) and disjoint. 2. For every k ∈ K, f | V k t 0 is a homeomorphism between V k t 0 and V t 0 , because f | V k t 0 = Π • θ -1 | V k t 0
(where Π is the projection from V t 0 × {k} to V t 0 ) and because both Π and

θ | V k t 0
are homeomorphisms.

The oddness theorem

In this section, we prove that for some large classes of polynomial payoff functions (which we call regular ), we obtain the generic oddness of the set of pairwise stable networks. From now, we denote by R[g] the set of polynomial functions from G to R, and we denote by R δ [g] the vector space of polynomial functions whose degree is less or equal to δ ∈ N. We consider on R δ [g] the topology induced by the norm defined in Section 2.2 (actually, from the equivalence of norms in finite dimensional spaces, any norm would give the same topology).

For every agent i ∈ N , we consider U i a subset of R δ i [g] for some fixed integer δ i ∈ N. In particular, any element v i ∈ U i can be expressed as

v i (g) = k∈N L α i k g k where g ∈ G, k = (k ij ) ij∈L ∈ N L is a multi-index (with α i k = 0 if deg(k) := ij∈L k ij > δ i
) and where g k := ij∈L g k ij ij is called a monomial. We also recall that for every monomial g k = 0, the degree of g k is by definition equal to deg(k), and that the degree of v i is equal to max{deg(k) : α i k = 0} (which is equal, by convention, to -∞ when v i = 0). For example, if we consider three agents and if v 1 (g 12 , g 13 , g 23 ) = -g 2 12 g 23 +3g 12 g 5 13 g 4 23g 2 13 g 23 , then we can write

v 1 (g) = α 1 k 1 g k 1 + α 1 k 2 g k 2 + α 1 k 3 g k 3 with k 1 = (2, 0, 3), k 2 = (1, 5, 4
) and k 3 = (0, 2, 1) and with α 1

k 1 = -1, α 1 k 2 = 3 and α 1 k 3 = -1 (in particular, deg(v i ) = 10
). The mapping which associates to every payoff function v i ∈ R δ i [g] its coefficients (with respect to some predefined order of the monomials in R

[g]) is an isomorphism from R δ i [g] to R m i for some integer m i . Let us consider its restriction ϕ i : U i → R m i . In particular, for every i ∈ N , ϕ i is a homeomorphism from U i to ϕ i (U i ).
In the following, we note m = i∈N m i ,

S i := ϕ i (U i ), S = i∈N S i and ϕ := × i∈N ϕ i : U → R m . Moreover, for every agent i ∈ N , we consider the subset A i of R[g] of affine functions in (g ij ) j =i , that is the set of functions A i ∈ A i of the following form A i (g) = j =i α ij g ij + c
for every g ∈ G, where c ∈ R and α ij ∈ R for every j = i. Definition 3.6. The set U := i∈N U i is said to be regular if for every i ∈ N , the two following conditions are fulfilled:

1. (Stability assumption) For every v i ∈ U i and every

A i ∈ A i , v i + A i ∈ U i .

(Semi-algebraicity assumption)

S i = ϕ i (U i ) (the set of coefficients of polynomials in U i ) is a semi-algebraic set.
In the following theorem, proved in Appendix 4.4, for every x ∈ S, we denote by v x := ϕ -1 (x) the family of polynomials in U associated to x. Theorem 3.3. (Oddness Theorem) For every regular subset U ⊂ F, there exists a generic semi-algebraic subset G of S such that for every x ∈ G, the society v x has an odd number of pairwise stable networks.

3.3 Some applications of Theorem 3.3

Polynomial payoff functions concave with respect to agents' weights

For every i ∈ N , let us consider the set U i of polynomial functions with a degree less or equal to some fixed integer δ i and which are concave in g ij for each j = i. The following corollary states that societies with payoff functions in U i have generically an odd number of pairwise stable networks. Corollary 3.1. There exists a generic semi-algebraic subset G of S such that for every x ∈ G, v x has an odd number of pairwise stable networks.

Proof. We show that U = i∈N U i is a regular set:

1. Stability assumption is satisfied, since the sum of a concave function and an affine function is a concave function.

2. Semi-algebraicity assumption is satisfied: indeed, for every i ∈ N and every g ∈ G, any payoff function in U i can be written v i (g) = k∈N L α i k g k . Then we can define the polynomial function

P i : (α i := (α i k ) k∈N L , g) → k∈N L α i k g k ,

and finally

S i = {α i : ∀g ∈ G, ∀j = i, ∂ 2 P i (α i , g ij , g -ij ) ∂g 2 ij ≤ 0}
which is a semi-algebraic set, from G being semi-algebraic and from Proposition 3.2 (see Section 3.1).

Finally, from Theorem 3.3, we obtain Corollary 3.1.

Linear costs of link formation

For every i ∈ N , let vi be a polynomial function in F i (i.e. concave in g ij for each j = i), and let L = {(i, j) : (i, j) ∈ N 2 , i = j} denotes the set of directed links on N . Let x ∈ R L , denoted x = (x i,j ) (i,j)∈L , and consider the society parameterized by x, v x (.) = (v x i (.)) i∈N where for every i ∈ N , v x i (g) := vi (g)j =i x i,j g ij for every g ∈ G. Here, x i,j ∈ R can be interpreted as a marginal cost of maintaining the weight g ij of link ij (at least when x i,j ≥ 0). The following corollary states that societies with such payoff functions have generically an odd number of pairwise stable networks. Corollary 3.2. There exists a generic semi-algebraic subset G of R L such that for every x ∈ G, v x has an odd number of pairwise stable networks.

Proof. Define U i = {g → vi (g)j =i x i,j g ij : ∀j = i, x i,j ∈ R}. We can easily show that U = i∈N U i is a regular set: stability assumption is obviously satisfied, and for every i ∈ N , the set S i is a finite product of singletons (corresponding to coefficients of v) and of copies of R, thus is a semi-algebraic set (i.e. semi-algebraicity assumption is satisfied). Finally, from Theorem 3.3, there exists a generic semi-algebraic subset G of S = Π i∈N ϕ i (U i ) such that for every x ∈ G , ϕ -1 (x ) has an odd number of pairwise stable networks. Now, since the mapping f : x ∈ R L → ϕ(v x ) is a semi-algebraic homeomorphism from R L to S, we get that for every x ∈ G = f -1 (G ), v x has an odd number of pairwise stable networks, and G is a generic subset of R L .

Quadratic costs of link formation

Consider the same example as above, but where the payoff functions are parameterized by x = (x i,j ) (i,j)∈L ∈ (0, +∞) L , where L = {(i, j) : (i, j) ∈ N 2 , i = j} denotes the set of directed links on N , and y = (y i,j ) (i,j)∈L ∈ R L as follows: v x,y i (g) := vi (g)j =i x i,j g 2 ijj =i y i,j g ij for every g ∈ G. Here, the cost of maintaining the weight g ij of link ij is assumed to be quadratic. Again, we get that societies with such payoff functions have generically an odd number of pairwise stable networks: Corollary 3.3. There exists a generic semi-algebraic subset G of (0, +∞) L × R L such that for every (x, y) ∈ G, v x,y has an odd number of pairwise stable networks.

Proof. It is easy to check that U = i∈N U i is a regular set, where

U i = {g → vi (g) - j =i x i,j g 2 ij -j =i y i,j g ij : ∀j = i, x i,j ∈ (0, +∞), y i,j ∈ R}.

A public good provision model

Consider the following modification of a model introduced by Bramoullé and Kranton ( [START_REF] Bramoullé | Public goods in networks[END_REF]): consider n agents, each of them being characterized by some level of effort e i ∈ [0, +∞) (it could be the amount of time a consumer spends researching a new product). The marginal cost of effort of each individual is z ∈ R. Agents interact in some network g, and g ij measures the benefit for agent i (resp. j) of agent's i (resp. agent's j) efforts.

For every x = (x i,j ) (i,j)∈L ∈ R L , consider a society v x (.) = (v x i (.)) i∈N parameterized by x, and defined by: for every i ∈ N , v x i (g) := b(e i + j =i g ij e j ) -ze ij =i x i,j g ij , where b : R → R is a concave polynomial benefit function and for every j = i, x i,j ∈ R is the marginal cost of maintaining the weight g ij of the link ij.

In their paper, Bramoullé and Kranton are interested by the optimal levels of efforts e i (i = 1, ..., n), which are here exogenous, and they consider exogenous values of the weights g ij ∈ {0, 1}, which are here endogenous, and belong to [0, 1].

Corollary 3.4.

There exists a generic semi-algebraic subset G of R L such that for every x ∈ G, v x has an odd number of pairwise stable networks.

Proof. The proof is similar to the linear cost example, remarking that b(e i + j =i g ij e j )ze i is a polynomial function in F i (i.e. concave in g ij for each j = i).

Information transmission

We now apply Theorem 3.3 to a weighted version of an information transmission model due to Calvó-Armengol [START_REF] Calvó | Pairwise-stability and nash equilibria in network formation[END_REF] (see also [START_REF] Bich | On the existence of pairwise stable weighted networks[END_REF]). We consider n agents: if agent i and agent j are in a full relationship (g ij = 1), some information can be transmitted from one agent to another agent with some probability p ij . We assume that if the relationship is weighted

(g ij ∈ [0, 1]), the probability of transmission is p ij g ij . The payoff of agent i ∈ N is defined by v x i (g) = 1 - j∈N -{i} (1 -p ij g ij ) - j∈N -{i} x i,j g ij ,
for every x = (x i,j ) (i,j)∈L ∈ R L , and where g is a weighted network on N . The first term corresponds to the probability that the message is transmitted to agent i, and the second term to the cost of maintaining his links (i.e. x i,j ∈ R is the marginal cost of maintaining the weight g ij of the link ij between agent i and agent j). Denoting v x (.) = (v x i (.)) i∈N , another application of the linear cost example treated above is the following corollary: Corollary 3.5. There exists a generic semi-algebraic subset G of R L such that for every x ∈ G, v x has an odd number of pairwise stable networks.

A two-way flow model

We now adapt a model of Bala and Goyal [START_REF] Bala | A noncooperative model of network formation[END_REF] to weighted networks: we consider n agents, and for every agent i and agent j = i, a path from i to j is a finite sequence x 0 = i, x 1 , . . . , x k = j of distinct elements of N . Let P j be the (finite) set of all paths from i to j, and n i (g) be the sum on all paths in P j (j = i) of the product of weights along these paths. We can interpret n i (g) as the benefit that agent i receives from his links.

For every agent i, define v x (.) = (v x i (.)) i∈N and

v x i (g) = n i (g) - j∈N -{i} x i,j g ij ,
where g ∈ G and where x i,j ∈ R is the marginal cost of maintaining the weight g ij of the link ij between agent i and agent j (at least when x i,j > 0). Denoting x = (x i,j ) (i,j)∈L ∈ R L , we get from the linear cost example above: Corollary 3.6. There exists a generic semi-algebraic subset G of R L such that for every x ∈ G, v x has an odd number of pairwise stable networks.

Zenou-Ballester's model

We consider the following modification of a model introduced by Zenou and Ballester ([3]): n agents interact in some (endogenous) network g. Each agent i has chosen a (exogenous) level of effort e i ∈ [0, +∞). The (strictly concave) payoff function of each agent i has the following form

v x i (g) = α i e i + 1 2 σ ii e 2 i + j =i g ij (x i,j + e i e j ),
where ((α i ) i∈N , (σ ii ) i∈N ) ∈ R n ×(-∞, 0) n are fixed parameters, and where x = (x ij ) (i,j)∈L ∈ R L . As in the previous corollaries, denoting v x (.) = (v x i (.)) i∈N , we get: Corollary 3.7. There exists a generic semi-algebraic subset G of R L such that for every x ∈ G, v x has an odd number of pairwise stable networks. We first construct the homeomorphism η from P to F (see Step 1-4 below), and we prove that η is properly homotopic to π in Step 5. Let us fix some element g 0 ∈ G, and consider

η : P → F (v, g) → v g
where for every i ∈ N and every γ ∈ G,

v g i (γ) = v i (γ) + j =i ∂ ij v i (g ij , g -ij ) -∂ ij v i (g ij , g 0 -ij ) (γ ij -g ij ) + j =i g ij γ ij , (1)
that is v g i and v i are equal up to an affine mapping. Moreover, to define the inverse of η, we consider ρ :

F → F × G v → (ṽ, g v )
where for every i ∈ N , ṽi (γ) = v i (γ) -

j =i ∂ ij v i (g v ij , g v -ij ) -∂ ij v i (g v ij , g 0 -ij ) (γ ij -g v ij ) - j =i g v ij γ ij (2)
and where for every link ij ∈ L, g v ij = min{w v i,j , w v j,i }, with w v i,j ∈ [0, 1] be the unique solution of the strictly concave maximization problem max

w∈[0,1] v i (w, g 0 -ij ) - w 2 2
and w v j,i ∈ [0, 1] be the unique solution of the strictly concave maximization problem max

w∈[0,1] v j (w, g 0 -ij ) - w 2 2 .
In the proof, we will use the following first order necessary and sufficient condition for a maximum:

Lemma 4.1. A C 1 concave function f : [0, 1] → R has a maximum at x ∈ [0, 1]
if and only if one of the three following conditions is true:

1. f (x) = 0 and x ∈ (0, 1). vi (γ) -v i (γ) = j =i (∂ ij ṽi (g v ij , g v -ij ) -∂ ij ṽi (g v ij , g 0 -ij )(γ ij -g v ij )) - j =i (∂ ij v i (g v ij , g v -ij ) -∂ ij v i (g v ij , g 0 -ij )(γ ij -g v ij )) = j =i (∂ ij (ṽ i -v i )(g v ij , g v -ij ) -∂ ij (ṽ i -v i )(g v ij , g 0 -ij ))(γ ij -g v ij ).
But (ṽ i -v i )(γ) is an affine mapping of γ, and in particular, for each j = i, ∂ ij (ṽ iv i )(g ij , g -ij ) does not depend on g -ij , thus the last term is nul, which finishes Step 2.

Step 3: we show that ρ • η = id P .

For every (v, g) ∈ P, we have (ρ • η)(v, g) = ρ(v g ) = ( ṽg , g ), where g = g v g , that is for every ij ∈ L, g ij = min{w i,j , w j,i }, with w i,j ∈ [0, 1] be the unique solution of the strictly concave maximization problem max

w∈[0,1] v g i (w, g 0 -ij ) - w 2 2 ,
and w j,i ∈ [0, 1] be the unique solution of the strictly concave maximization problem max

w∈[0,1] v g j (w, g 0 -ij ) - w 2 2 .
We have to show that ṽg = v and that g = g .

First, we show that g = g .

Let us consider an arbitrary link ij ∈ L. Recall that since (v, g) ∈ P, g is pairwise stable with respect to v. The following lemma (cf. Bich and Morhaim [START_REF] Bich | On the existence of Pairwise stable weighted networks[END_REF], [START_REF] Bich | On the existence of pairwise stable weighted networks[END_REF]) establishes the relationship between a weight of a link between agents i and j at some pairwise stable network and the optimal weights for agent i and agent j: Lemma 4.2. Let g = (g ij ) ij∈L be some pairwise stable network with respect to a society v = (v i ) i∈N . Moreover, for every ij ∈ L, let

I i = [a i , b i ] = arg max w∈[0,1] v i (w, g -ij ) and I j = [a j , b j ] = arg max w∈[0,1] v j (w, g -ij )
. 8 Then we have:

g ij ∈ [min{a i , a j }, min{b i , b j }].
To prove that g ij = g ij , without any loss of generality (up to a permutation of i and j if necessary), we can assume a i ≤ a j . In particular, from the above lemma, we get g ij ∈ [a i , b i ] in all cases. Thus, g ij ∈ I i , and as a maximizer of v i (., g -ij ), it has to satisfy the necessary (and sufficient) condition of Lemma 4.1 for this function. Now, the derivative of the function q i : w → q i (w

) := v g i (w, g 0 -ij ) -w 2 2 at g ij is ∂ ij v g i (g ij , g 0 -ij ) -g ij , which is also equal to ∂ ij v i (g ij , g -ij ) (computing ∂ ij v g i (g ij , g 0 -ij
) thanks to Equation ( 1)). Thus, q i has also to verify the necessary (and sufficient) condition of Lemma 4.1 at w = g ij , and by strict concavity of q i , g ij has to be the unique maximum of this function, i.e. g ij = w i,j . To finish, we only have to prove that g ij = w i,j , or equivalently that w i,j ≤ w j,i . Since this inequality is true when w i,j = 0 or when w j,i = 1, we can now assume w i,j > 0 and w j,i < 1. The same computation as above gives that the derivative of q j : w → q j (w) := v g j (w,

g 0 -ij ) -w 2 2 at g ij is ∂ ij v j (g ij , g -ij )
. Thus, since w j,i < 1 is the unique maximizer of q j , we get q j (w j,i ) ≤ 0 from Lemma 4.1. Also, since v j (., g -ij ) is concave and since w i,j = g ij ≤ b j (because g ij ≤ min{b i , b j }), we get that ∂ ij v j (w i,j , g -ij ) ≥ 0, and in particular, q j (w i,j ) ≥ 0.

Finally, q j (w j,i ) ≤ q j (w i,j ), thus in particular w i,j ≤ w j,i because q j is strictly decreasing from strict concavity of q j . This proves that g = g .

Second, we show that ṽg = v.

The proof is very similar to the proof of Step 2 (i.e. the proof that η • ρ = id F ): for every i ∈ N , by definition of ρ, since ρ(v g ) = ( ṽg , g ) = ( ṽg , g), we have:

ṽg i (γ) = v g i (γ) - j =i ∂ ij v g i (g ij , g -ij ) -∂ ij v g i (g ij , g 0 -ij ) (γ ij -g ij ) - j =i g ij γ ij (5)
and by definition of η

v g i (γ) = v i (γ) + j =i ∂ ij v i (g ij , g -ij ) -∂ ij v i (g ij , g 0 -ij ) (γ ij -g ij ) + j =i g ij γ ij . (6)
Summing the two above equations, we get

ṽg i (γ) -v i (γ) = j =i (∂ ij (v i -v g i )(g ij , g -ij ) -∂ ij (v i -v g i )(g ij , g 0 -ij ))(γ ij -g ij ).
But (v i -v g i )(γ) is an affine mapping of γ, and in particular, for each

j = i, ∂ ij (v i - v g i )(g ij , g -ij )
does not depend on g -ij , thus the last term is nul, which finishes the proof that ṽg = v and the proof of Step 3.

Step 4: we show that η and ρ are continuous mappings.

Recall that for every (v, g) ∈ P, η(v, g) = v g , where for every i ∈ N and every γ ∈ G,

v g i (γ) = v i (γ) + j =i ∂ ij v i (g ij , g -ij ) -∂ ij v i (g ij , g 0 -ij ) (γ ij -g ij ) + j =i g ij γ ij .
To prove the continuity of η, notice that:

1. The mapping (v, g) ∈ P → v i ∈ F i is continuous, as a projection, since F is endowed with the product norm (see Section 2.2).

2. The mapping (v, g)

∈ P → γ → j =i ∂ ij v i (g ij , g -ij ) -∂ ij v i (g ij , g 0 -ij ) (γ ij -g ij ) (from P to F i )
is continuous: first, to prove the continuity of the mapping

F ij : (v, g) ∈ P → F ij (v, g) : γ → ∂ ij v i (g ij , g -ij )γ ij ,
consider (v, ḡ) ∈ P and for every (v, g) ∈ P, compute

F ij (v, g)-F ij (v, ḡ) i = max{max{ F ij (v, g)-F ij (v, ḡ) ∞ , ∂ ik F ij (v, g)-∂ ik F ij (v, ḡ) ∞ } : k = i} Now, for every γ ∈ G, F ij (v, g)(γ) -F ij (v, ḡ)(γ) = ∂ ij v i (g ij , g -ij )γ ij -∂ ij vi (ḡ ij , ḡ-ij )γ ij = ∂ ij (v i -vi )(g ij , g -ij )γ ij -(∂ ij vi (ḡ ij , ḡ-ij ) -∂ ij vi (g ij , g -ij ))γ ij ,
which implies that

F ij (v, g) -F ij (v, ḡ) ∞ ≤| ∂ ij (v i -vi )(g ij , g -ij ) | + | (∂ ij vi (ḡ ij , ḡ-ij ) -∂ ij vi (g ij , g -ij )) | ≤ v i -vi i + | (∂ ij vi (ḡ ij , ḡ-ij ) -∂ ij vi (g ij , g -ij )) |,
which proves that lim

(v,g)→(v,ḡ) F ij (v, g) -F ij (v, ḡ) ∞ = 0.
Moreover,

∂ ij F ij (v, g)(γ) -∂ ij F ij (v, ḡ)(γ) = ∂ ij v i (g ij , g -ij ) -∂ ij vi (ḡ ij , ḡ-ij )
which gives by a same argument as above

∂ ij F ij (v, g)(γ) -∂ ij F ij (v, ḡ)(γ) ∞ ≤ v i -vi i + | (∂ ij vi (ḡ ij , ḡ-ij ) -∂ ij vi (g ij , g -ij )) |, which proves that lim (v,g)→(v,ḡ) ∂ ij F ij (v, g)(γ) -∂ ij F ij (v, ḡ)(γ) ∞ = 0.
This proves finally the continuity of F ij . We prove similarly the continuity of the mappings (v, g)

∈ P → γ → ∂ ij v i (g ij , g -ij ) -∂ ij v i (g ij , g 0 -ij
) g ij and of the mappings (v, g) ∈ P → γ → g ij γ ij for every j = i. Summing all these mappings, we finally get the continuity of η. Now, to prove the continuity of ρ, we recall that for every v ∈ F, ρ(v) = (ṽ, g v ) where for every i ∈ N and for every γ ∈ G,

ṽi (γ) = v i (γ) - j =i ∂ ij v i (g v ij , g v -ij ) -∂ ij v i (g v ij , g 0 -ij ) (γ ij -g v ij ) - j =i g v ij γ ij (7)
and where for every ij ∈ L, g v ij = min{w v i,j , w v j,i }, with w v i,j be the unique maximum of the strictly concave function q i (w) := v i (w, g 0 -ij ) -w 2 2 on [0, 1] and w v j,i be the unique maximum of the strictly concave function q j (w

) := v j (w, g 0 -ij ) -w 2 2 on [0, 1].
Actually, the continuity of ρ can be proved similary as the continuity of η. The only additional point to be proved is that the mapping v → g v is continuous, which is a consequence of the continuity of the mappings v → w v i,j and v → w v j,i (from Berge theorem). 9

Step 5: the mapping

H : [0, 1] × P → F, (t, (v, g)) → H(t, (v, g)) = (1 -t)π(v, g) + tη(v, g)
is a proper homotopy between π and η.

For every i ∈ N , let us define the following mapping

G i : [0, 1] × F × G → F i by G i (t, (v, g))(γ) = v i (γ)+t j =i ∂ ij v i (g ij , g -ij )-∂ ij v i (g ij , g 0 -ij ) (γ ij -g ij )+ j =i g ij γ ij (8)
for every (t, (v, g)) ∈ [0, 1] × F × G and for every γ ∈ G, and let

G : [0, 1] × F × G → F defined by G = (G i ) i∈N . It is easy to see that the restriction of G to [0, 1] × P is equal to H. Let us define ψ : [0, 1] × F × G → [0, 1] × F × G by ψ(t, (v, g)) = (t, G(t, (v, g)), g)
for every (t, (v, g)) ∈ [0, 1] × F × G. The mapping ψ is invertible: for every t ∈ [0, 1], ψ -1 (t, (v, g)) = (t, G(t, (v, g)), g) where G = ( Gi ) i∈N and where for every i ∈ N , Gi :

[0, 1] × F × G → F i is defined by Gi (t, (v, g))(γ) = v i (γ) -t j =i ∂ ij v i (g ij , g -ij ) -∂ ij v i (g ij , g 0 -ij ) (γ ij -g ij ) + j =i g ij γ ij
(similar to Step 2 and Step 3). Moreover, ψ and ψ -1 are continuous (similar to Step 4). Thus, for every compact subset K of F,

H -1 (K) = {(t, (v, g)) ∈ [0, 1] × P : H(t, (v, g)) ∈ K} = {(t, (v, g)) ∈ [0, 1] × P : ψ(t, (v, g)) ∈ [0, 1] × K × G} = ψ -1 ([0, 1] × K × G) ∩ ([0, 1] × P).
9 Indeed, Berge Theorem states that if we consider a continuous correspondence Φ : X Y with nonempty compact values and if we consider a continuous mapping f : X × Y → R, then the argmax correspondence of the maximization problem max y∈Φ(x) f (x, y) has a closed graph and nonempty values. In particular, if we take Y = [0, 1], X = F, Φ(x) = [0, 1] for every x ∈ X and finally f :

(v, w) ∈ F × [0, 1] → f (v, w) = v i (w, g 0 -ij ) -w 2 /2
(which is continuous), then we get that the (single-valued) correspondence v → w v i,j has a closed graph, or equivalently that the mapping v → w v i,j is continuous.

But ψ -1 is a continuous mapping and [0

, 1] × K × G is a compact subset of [0, 1] × F × G, thus ψ -1 ([0, 1] × K × G) is compact in [0, 1] × F × G. Moreover, [0, 1] × P is closed in [0, 1]×F ×G (see lemma below). Thus, finally H -1 (K) = ψ -1 ([0, 1]×K ×G)∩([0, 1]×P)
is compact as a closed subset of a compact set, which proves that H is a proper mapping.

Lemma 4.3. The set P is a closed subet of F × G.

Proof. Consider a sequence (v k , g k ) in P which converges to (v, g) ∈ F × G. To prove that (v, g) ∈ P, suppose by contradiction that (v, g) / ∈ P:

1. Suppose first that there exists some ij ∈ L and some w < g ij such that v i (w, g -ij ) > v i (g). Since v i is continuous, this implies that there exists k ≥ 0 such that

v k i (w, g k -ij ) > v k i (g k ) (indeed, from the topology defined on F, v k i (g k ) converges to v i (g) since | v k i (g k )-v i (g) |≤| v k i (g k )-v i (g k ) | + | v i (g k )-v i (g) |≤ v k i -v i ∞ + | v i (g k )-v i (g) |≤ v k i -v i i + | v i (g k ) -v i (g) |, and similarly v k i (w, g k -ij ) converges to v i (w, g -ij )
). But this contradicts that g k is pairwise stable with respect to v k .

2. Now, suppose that there exists some ij ∈ L and some w > g ij such that v i (w, g -ij ) > v i (g) and v j (w, g -ij ) > v j (g). Since v i and v j are continuous, this implies that there

exists k ≥ 0 such that v k i (w, g k -ij ) > v i (g k ) and v k j (w, g k -ij
) > v j (g k ) (same proof as above), which again contradicts that g k is pairwise stable with respect to v k .

Topological degree of a continuous mapping

In this subsection, we gather all the properties of topological degree which are used in this paper. To every continuous mapping f : S m → S m , one can associate an integer deg(f ) ∈ Z, called the degree of f (or topological degree, or Brouwer degree) which satisfies the following properties:

(i) (Degree of identity). deg(id S m ) = 1 (see Dold [START_REF] Albrecht | Lectures on algebraic topology. Die Grundlehren der mathematischen Wissenschaften[END_REF], Proposition 4.2., (i), p. 62). (ii) (Homotopy invariance). If f : S m → S m and g : S m → S m are homotopic, then deg(f )=deg(g) (see Dold [START_REF] Albrecht | Lectures on algebraic topology. Die Grundlehren der mathematischen Wissenschaften[END_REF], Proposition 4.2., (iii), p. 62 or Hatcher [START_REF] Hatcher | Algebraic topology[END_REF], p. 134). (iii) (Surjectivity). For every continuous function f : S m → S m such that deg(f ) = 0, f is surjective (see Hatcher [START_REF] Hatcher | Algebraic topology[END_REF], p. 134). (iv) (Homeomorphism). If f : S m → S m is a homeomorphism, then deg(f ) ∈ {-1, 1} (see Hatcher [START_REF] Hatcher | Algebraic topology[END_REF], p. 135, in the proof of Proposition 2.29).

To every continuous mapping f : S m → S m and every y ∈ S m , one can also associate an integer deg(f ) |y ∈ Z called the local degree of f at y which satisfies the following properties:

(v) (Local homeomorphism). If f is a homeomorphism from some neighborhood of y to some neighborhood of f (y), then deg(f ) |y ∈ {-1, 1} (see Hatcher [START_REF] Hatcher | Algebraic topology[END_REF], p. 135-136 or Dold [START_REF] Albrecht | Lectures on algebraic topology. Die Grundlehren der mathematischen Wissenschaften[END_REF], Example 5.4., p. 67). (vi) (Additivity of degree modulo 2). For every f : S m → S m continuous and for every z ∈ S m , if f -1 (z) = {y 1 , . . . , y m }, then

deg(f ) = m k=1 deg(f ) |y k [2].
(see Hatcher [START_REF] Hatcher | Algebraic topology[END_REF], Proposition 2.30., p. 135-136 or Dold [START_REF] Albrecht | Lectures on algebraic topology. Die Grundlehren der mathematischen Wissenschaften[END_REF], Proposition 5.8., p. 68). In particular, from the formula above, if for every k = 1, . . . , m, f is a homeomorphism from some neighborhood of y k to some neighborhood of f (y k ) and if deg(f ) = 1, then m is odd.

Proof of Corollary 2.1

The idea is to follow the proof of Theorem 2.1 by restricting η to P U : the only important point is to remark that η(P U ) ⊂ U, which is true because for every i ∈ N , (η

| P U ) i (v, g)(γ) = v i (γ) + j =i ∂ ij v i (g ij , g -ij ) -∂ ij v i (g ij , g 0 -ij ) (γ ij -g ij ) + j =i g ij γ ij
where η | P U = ((η | P U ) i ) i∈N , and because the mapping

γ → j =i ∂ ij v i (g ij , g -ij ) -∂ ij v i (g ij , g 0 -ij ) (γ ij -g ij ) + j =i g ij γ ij
belongs to A i . Thus, we can use the stability assumption in Corollary 2.1.

Proof of Theorem 3.3

Consider a regular set U = i∈N U i of profiles of payoff functions such that for every i ∈ N , U i ⊂ F i (equivalently, each v i ∈ U i is polynomial and concave in g ij for each j = i). From Theorem 2.1, there exists a homeomorphism η (defined in Equation ( 1)) from P, the graph of pairwise stable networks, to F. Recall that π : P → F denotes the projection from P to F, and that for every profile of polynomials payoff functions v ∈ U, ϕ(v) denotes the vector of its coefficients.

Since U is regular, Corollary 2.1 implies that η | P U , the restriction of η to P U (the graph of pairwise stable networks restricted to U) is a homeomorphism from P U to U.

In the first part of the proof (from Step 1 to Step 8), we assume that S = ϕ(U), the set of coefficients of profiles of polynomials in U, is equal to R m . Actually, the proof below is similar if we only assume that S is homeomorphic to R p , for some integer p ≤ m. The general case where S is simply semi-algebraic is considered at the end of the proof (see Step 9).

Step 1: the set

M := {(ϕ(v), g) ∈ R m × G : (v, g) ∈ P U } is semi-algebraic and the mapping η : M → R m , (x, g) → ϕ(η | P U (ϕ -1 (x), g)) is a semi-algebraic homeomorphism. In particular, dim(M) = m.
First, we show that M is semi-algebraic. From the concavity assumption of payoff functions in U, we can remark that the condition (x, g) ∈ M, that is the condition "g is a pairwise stable network of ϕ -1 (x)", is equivalent to the following conditions (where we recall that

v x = (v x i ) i∈N = ϕ -1 (x)): for every ij ∈ L, 1. either g ij ∈ (0, 1), ∂ ij v x i (g ij , g -ij ) = 0 and ∂ ij v x j (g ij , g -ij ) ≥ 0, 2. or g ij ∈ (0, 1), ∂ ij v x j (g ij , g -ij ) = 0 and ∂ ij v x i (g ij , g -ij ) ≥ 0, 3. or g ij = 0 and [∂ ij v x i (g ij , g -ij ) ≤ 0 or ∂ ij v x j (g ij , g -ij ) ≤ 0], 4. or g ij = 1 and [∂ ij v x i (g ij , g -ij ) ≥ 0 and ∂ ij v x j (g ij , g -ij ) ≥ 0]
. These conditions involve a finite number of equalities and of inequalities with semialgebraic mappings, thus M is a semi-algebraic set, as announced.

Second, we show that the mapping η is semi-algebraic. From its definition and the definition of η (in Equation 1), this is equivalent to say that

(x, g) → ϕ γ → v x i (γ) + j =i ∂ ij v x i (g ij , g -ij ) -∂ ij v x i (g ij , g 0 -ij ) (γ ij -g ij ) + j =i g ij γ ij
is a semi-algebraic mapping. We can see directly that this is the case, since each coefficient of the polynomial function

γ → v x i (γ) + j =i ∂ ij v x i (g ij , g -ij ) -∂ ij v x i (g ij , g 0 -ij ) (γ ij -g ij ) + j =i g ij γ ij
is a polynomial function of (x, g). Now, the following diagram summarizes the situation:

P P U M F U R m π η π | P U η | P U φ π η ϕ
where π | P U and η | P U are the restrictions of η and π to P U and where π and η are the 'transportations" of π | P U and η | P U from M to R m by the homeomorphisms ϕ and φ :

P U → M, (v, g) → (ϕ(v), g). With this notation, π = ϕ • π | P U • φ-1 (which is the usual projection from M to R m ) and η = ϕ • η | P U • φ-1 .
We can remark that η is a semialgebraic homeomorphism from M to R m , since both η | P U and φ are homeomorphisms. This implies (see [START_REF] Bochnak | Real Algebraic Geometry[END_REF], Theorem 2.8.8.) that the dimensions of M and of R m coincide, thus that dim(M) = m. This ends the proof of Step 1.

Step 2: extensions of π and η to the compactifications of M and R m . The network g ∈ G is pairwise stable in the society v x if and only if (x, g) ∈ π-1 (x). Thus, to finish the proof of Theorem 3.3, we need informations on the cardinal of π-1 (x), which can be done by using topological degree. The topological degree we use (see Section 4.2) is applied to functions defined on spheres (which avoids boundary issues). But π is not defined on a sphere (but on M): that is why we try to extend π on the compactification of M, which will be proved to be homeomorphic to some m-dimensional sphere. We recall that every noncompact locally compact Hausdorff space (X, τ X ) admits a (Alexandroff one-point) compactification X * . Formally, X * = X ∪{∞} (where ∞ is any element which does not belong to X) and X * is endowed with the following topology:

O ⊂ X * is open in X * if (i) either ∞ / ∈ O and O is open in X, (ii) or ∞ ∈
O and the complement of O is compact for the topology τ X . Also, let us recall that if X and Y are noncompact locally compact Hausdorff space and f : X → Y is a continuous proper mapping, then it can be extended (uniquely) to a continuous mapping f * :

X * → Y * for which f (∞) = ∞.
Since M is homeomorphic to R m (the homeomorphism being η), M is a noncompact locally compact Hausdorff space and thus, it admits a compactification M * . From Corollary 2.1 and from stability assumption in Definition 3.6, π | P U and η | P U are proper mappings. In particular, since ϕ and φ are homeomorphisms, π and η are also proper mappings. Thus, these mappings admit (unique) continuous extensions π * : M * → (R m ) * and η * : M * → (R m ) * (we can notice that η * is a homeomorphism). Moreover, we recall that (R m ) * is also homeomorphic to S m , the m-sphere (where the inverse of the stereographic projection, denoted s : (R m ) * → S m , is such a homeomorphism).

The following diagram summarizes the situation:

M M * S m R m (R m ) * S m π η π * η * s•η * π η s where π = s • π * • (s • η * ) -1 = s • π * • (η * ) -1 • s -1 η = s • η * • (s • η * ) -1 = s • id (R m ) * • s -1 = id S m
and where s • η * is a homeomorphism from M * to S m .

Step 3: π * (resp. π) and η * (resp. η) are homotopic.

We recall that from Corollary 2.1, there exists some proper homotopy H | P U between π | P U and η | P U . Using this homotopy, we can deduce an homotopy between π and η. Indeed, the map

H : [0, 1] × M → R m (t, (x, g)) → ϕ(H | P U (t, ( φ-1 (x, g)))
is continuous (by composition) and we obtain that H(0, x, g) = φ(H | P U (0, φ-1 (x, g))) = φ(η| P U ( φ-1 (x, g))) = η(x, g)

• First, L is compact in M: indeed, if (x k , g k ) is sequence in L , by definition, for every integer k, there exists t k ∈ [0, 1] ∩ [t -, t + ] such that (t k , (x k , g k )) ∈ H-1 (K) which is compact since K is compact and H is a proper. Thus, (t k , (x k , g k )) admits some subsequence converging to some ( t, (x, ḡ)) ∈ H-1 (K), and finally (x, ḡ)

∈ L since t ∈ [0, 1] ∩ [t -, t + ].
• Second, let us prove there exists > 0 such that V = (]t-, t+ [∩[0, 1])×(L c ∪{∞}) contains (t, x, g): this is always true if (x, g) = ∞. If this is false when (x, g) = ∞, then in particular (choosing = 1 k for every integer k ≥ 1), we get that (x, g) ∈ L 1 k for every k ≥ 1. This means that there exists a sequence t k converging to t such that for every k ≥ 1, H(t k , (x, g)) ∈ K. Passing to the limit, from the continuity of H and the compactness of K (which is closed for the considered topology), we get that H(t, (x, g)) ∈ K, which contradicts that (t, (x, g))

∈ ( H * ) -1 (O) = ([0, 1] × {∞}) ∪ H-1 (K) c (since this implies in particular (t, (x, g)) ∈ H-1 (K) c , that is H(t, (x, g)) / ∈ K). • Third, let us prove that V ⊂ ( H * ) -1 (O). If (t , (x , g )) ∈ V and (x , g ) = ∞, then we obtain directly that (t , (x , g )) ∈ ( H * ) -1 (O). Moreover, if (x , g ) = ∞ and (t , (x , g )) ∈ (]t -, t + [∩[0, 1]) × L c , then by definition of L , for every t ∈ [0, 1] ∩ [t -, t + ] we have (t , (x , g )) / ∈ H-1 (K)
, and in particular, H(t , (x , g )) / ∈ K, which implies that (t , (x , g )) ∈ ( H * ) -1 (O) and ends the proof.

Step 4: the degree of π is equal to 1, and π is surjective (which implies that for every v ∈ U, there exists a pairwise stable network).

From the previous step, π : S m → S m and η : S m → S m are homotopic, thus from homotopy invariance of topological degree (see Appendix 4.2, Property (ii)), we have deg(π) =deg(η). But from Step 2, we know that η = id S m , thus deg(η) = 1 (see Appendix 4.2, Property (i)). In particular, deg(π) = 1, which implies that π is a surjective mapping (see Appendix 4.2, Property (iii)).

This also implies that π is surjective: indeed, for every x ∈ R m , s(x) ∈ S m , thus there exists y ∈ S m such that π(y) = s(x), which gives x = (s -1 • π)(y), thus x = π • (s • η) -1 (y) (by definition of π). Remark that for every v ∈ U, we also have that there exists y ∈ S m such that v = (π | P U • φ-1 • (s • η) -1 )(y), which proves that π | P U is also surjective: equivalently, for every v ∈ U, there exists a pairwise stable network.

In particular, we recover (but for a smaller class of payoff functions) the result of Bich and Morhaim [START_REF] Bich | On the existence of pairwise stable weighted networks[END_REF], which proves the existence of a pairwise stable network for every continuous society v = (v 1 , . . . , v n ) such that v i is quasiconcave with respect to g ij , for every link ij.

Step 5: there exists a generic semi-algebraic subset G of R m such that for every x ∈ G, π-1 (x) is nonempty and finite.

Indeed, π : M → R m is a surjective semi-algebraic continuous mapping (from Step 4), and dim(M) = m (from Step 1). Thus from Theorem 3.1 (see Section 3.1), we obtain the existence of a generic semi-algebraic subset G (now fixed) of R m such that for every x ∈ G, π-1 (x) is nonempty and finite.

Step 6: for every x ∈ G, the set of pairwise stable networks of v x = ϕ -1 (x) is nonempty and finite (we call K x its cardinal).

Indeed, from Step 5, we know that for every x ∈ G, π-1 (x) is nonempty and finite. However, since φ is a homeomorphism, we obtain that φ-1 (π

-1 (x)) = π -1 | P U (ϕ -1 (x)) = π -1 | P U (v x
) is also nonempty and finite (with the same cardinal as π-1 (x)). But π -1

| P U (v x ) = {(v x , g) : g is a pairwise stable network of v x }.
Step 7: for every x ∈ G, there exists Vx an open neighborhood (in S m ) of s(x) such that π-1 ( Vx ) is a union of pairwise disjoint open sets ( V k x ) k∈K (where K is a finite set of cardinal K x ) and such that for every k ∈ K, π| V k x is a homeomorphism between V k x and Vx .

Let x ∈ G, and let C x be the connected component of G containing x. From Step 6, we know that π-1 (x) is nonempty and finite, so we obtain that (π * ) -1 (x) = (π) -1 (x) and π-1 (s(x)) are also non empty and finite, with the same cardinal as π-1 (x) (as s is a homeomorphism), i.e. of cardinal K x . Moreover, from Theorem 3.1 (see Section 3.1), there exists an open neighborhood V x of x such that π-1 (V x ) is a union of pairwise disjoint open sets (V k x ) k∈K (where K is a finite set) such that for every k ∈ K, π| V k

x is a homeomorphism between V k x and V x . Denote by Vx the image by the homeomorphism s of V x and for every k ∈ K, denote by V k

x the image by the homeomorphism s • η * of V k x . Since s is a homeomorphism, Vx is an open neighborhood of s(x). Also, since s • η * is a homeomorphism, π-1 ( Vx ) = k∈K V k x , and ( V k x ) k∈K is a family of pairwise disjoint open sets such that for every k ∈ K, π| V k x is a homeomorphism between V k x and Vx .

Step 8: for every x ∈ G, there is an odd number of pairwise stable networks of the society v x = ϕ -1 (x). We want to prove that for every x ∈ G, the integer K x (introduced in Step 6) is odd. From Step 7, since π-1 (s(x)) is a finite subset of k∈K V k x (with cardinal equal to K x =card(K)) and since for every k ∈ K, π| V k x is a homeomorphism between V k

x and Vx , we obtain that π-1 (s(x)) = {y k i.e. that K x is odd.

x : k ∈ K} for some y k x ∈ V k x , k ∈ K.
Step 9: the general case.

We now return to the general case. We recall that for every i ∈ N , ϕ i : U i → R m i is the topological embedding which associates to any polynomial function in U i its coefficients in R m i , R δ i [g] is the vector space of polynomial functions with a degree less or equal to δ i ∈ N, and ϕ := × i∈N ϕ i : U → R m .

For i ∈ N fixed, we denote by K -i the linear subspace of R δ i [g] generated by all the monomials in R δ i [g], except the ones in A i , i.e.

K -i := span({g k : k ∈ N |L| , deg(k) ≤ δ i } -{g ij : j = i}).
Then, denoting also (by abuse of notation) ϕ i , the mapping from R δ i [g] to R m i such that for every P ∈ R δ i [g], ϕ i (P ) is the m i -tuple of coefficients of P , we can define π -i : R m i → ϕ i (K -i ) to be the projection from the set R m i of coefficients of polynomial functions in R δ i [g] to the set ϕ i (K -i ) of coefficients of polynomial functions in K -i . Since S i = ϕ i (U i ) is semialgebraic (from semi-algebraicity assumption in Definition 3.6), from Tarski-Seidenberg's theorem (see Section 3.1), each set π -i (S i ) is semi-algebraic. In particular, from a wellknown decomposition result for semi-algebraic sets (see Definition 3.4 in Section 3.1),

π -i (S i ) = κ i k=1 T i k ,
the union being disjoint, where κ i ∈ N and where for every k ∈ {1, . . . , κ i }, T i k ⊂ π -i (S i ) ⊂ ϕ i (K -i ) is semi-algebraic and homeomorphic to ]0, 1[ t i k (for some t i k ∈ N). Thus,

π -1 (S 1 ) × • • • × π -n (S n ) = κ 1 k=1 T 1 k × • • • × κn k=1
T n k =

(k 1 ,...,kn)∈Λ

(T 1 k 1 × • • • × T n kn ) (9)
where Λ := i∈N {1, . . . , κ i }. In particular, for every k = (k 1 , . . . , k n ) ∈ Λ, the set

U k := ϕ -1 1 (T 1 k 1 ) × • • • × ϕ -1 n (T n kn )
is homeomorphic to ]0, 1[ t k , for some t k ∈ N. For every k = (k 1 , . . . , k n ) ∈ Λ, we now consider the set

V k := U k + n i=1 A i = n i=1 (ϕ -1 i (T i k i ) + A i )
and we define V k i i := ϕ -1 i (T i k i ) + A i , for every i ∈ N . We can notice that for every k = (k 1 , . . . , k n ) ∈ Λ and every i ∈ N :

1. Stability assumption (in Definition 3.6) is satisfied for V k i i .

2. Semi-algebraicity assumption (in Definition 3.6) is satisfied for V k i i because ϕ i (V k i i ) = T i k i + ϕ i (A i ) is a semi-algebraic set, since T i k i and ϕ i (A i ) are both semi-algebraic sets. 10 3.

V k i i is included in F i because T i k i ⊂ π -i (S i ) = π -i (ϕ i (U i )), which implies that ϕ -1 i (T i k i ) ⊂ (ϕ -1 i • π -i • ϕ i )(U i ) ⊂ U i .
4. V k i i is homeomorphic 11 to ϕ -1 i (T i k i ) × A i , thus homeomorphic to (0, 1) t i k i , for some t i k i ∈ N. Thus, from the previous steps, we know that for every k = (k 1 , . . . , k n ) ∈ Λ, there exists a generic semi-algebraic subset G k of S k := ϕ(V k ) such that for every x ∈ G k , the society v x = ϕ -1 (x) has an odd number of pairwise stable networks. This implies that for every x ∈ G := k∈Λ G k , the society v x = ϕ -1 (x) has an odd number of pairwise stable networks.

To finish, we can notice that G is generic in S because:

1. The (S k ) k∈Λ forms a covering of S = ϕ(U); indeed,

k∈Λ S k = k∈Λ ϕ(V k ) = k∈Λ ϕ( n i=1 (ϕ -1 i (T i k i ) + A i )) = k∈Λ n i=1 (ϕ i (ϕ -1 i (T i k i ) + A i )) = k∈Λ ( n i=1 (T i k i + ϕ i (A i ))) = k∈Λ ( n i=1 T i k i ) + ( n i=1 ϕ i (A i )) = π -1 (S 1 ) × • • • × π -n (S n ) + ( n i=1 ϕ i (A i )) (from Equation (9)) = (π -1 (S 1 ) + ϕ 1 (A 1 )) × • • • × (π -n (S n ) + ϕ n (A n )) = (π -1 (ϕ 1 (U 1 )) + ϕ 1 (A 1 )) × • • • × (π -n (ϕ n (U n )) + ϕ n (A n )) = ϕ 1 (U 1 ) × • • • × ϕ n (U n ) = ϕ(U).
2. From the previous step, the complement of G in S is included in k∈Λ (S k -G k ): indeed, if x ∈ S and x / ∈ G, then since (S k ) k∈Λ forms a covering of S, there exists l ∈ Λ such that x ∈ S l . But x / ∈ G l , thus finally

x ∈ S l -G l ⊂ k∈Λ (S k -G k )
whose dimension is strictly less than dim(S), because dim(S k -G k ) <dim(S k ) ≤ dim(S). 10 The sum of two semi-algebraic subsets S 1 and S 2 of some Euclidean space is a semi-algebraic set: S 1 + S 2 = f (S 1 × S 2 ), where f : (x, y) → x + y is a polynomial function and where S 1 × S 2 is a semi-algebraic set. Moreover, ϕ i (A i ) is a subspace of R mi , thus is semi-algebraic.

11 Indeed, the mapping f : ϕ -1 i (T i ki ) × A i → ϕ -1 i (T i ki ) + A i , (x, y) → x + y is such a homeomorphism, since the vector subspace generated by ϕ -1 i (T i ki ) and the vector subspace A i are in direct sum.

  and f + (1) (the left and right derivative of f at 0 and 1) exist, and if lim x→0 f (x) = f -(0) and lim x→1 f (x) = f + (1).

  Also, from Property (v) of local degree (see Appendix 4.2) we obtain that deg(π) | y k x is equal to 1 or -1 for every k ∈ K. Now, from additivity of the topological degree with respect to local degrees (see Appendix 4.2, Property (vi)) we obtain (modulo 2) that 1 = deg(π) = k∈K deg(π) | y k x = k∈K 1 = card(K) = K x [2]

Kohlberg-Mertens theorem states that the graph of the mixed Nash equilibrium correspondence, if defined on the space of finite games, is homeomorphic to the space of finite games. Predtetchinsky ([START_REF] Predtetchinski | A general structure theorem for the nash equilibrium correspondence[END_REF]) generalizes this result for more general payoff functions than those involved in mixed extensions of finite games.

A recent work ofHerings and Zhan ([14]) provides the same result for affine payoff functions, using homotopy methods.1 Documents de travail du Centre d'Economie de la Sorbonne -2021.[START_REF] Jackson | A strategic model of social and economic networks[END_REF]

Documents de travail du Centre d'Economie de la Sorbonne -2021.[START_REF] Jackson | A strategic model of social and economic networks[END_REF]

This norm is defined on the vector space{v i ∈ C 0 (G, R) : ∀j = i, ∀g -ij ∈ G -ij , v i (•, g -ij ) is C 1 on [0, 1]}.4 Documents de travail du Centre d'Economie de la Sorbonne -2021.16

We recall that a semi-algebraic trivialization of f over a semi-algebraically connected component C of T 0 with fiber K (where K is a semi-algebraic set) is a semi-algebraic homeomorphism θ : C × K → f -1 (C) such that f (θ(c, x)) = c for every (c, x) ∈ C × K. Remark that when f -1 (C) = ∅ this is automatically true, taking the empty mapping θ = ∅.

Let us recall that in R m , a semi-algebraic set is semi-algebraically connected if and only if it is connected (see Bochnak[START_REF] Bochnak | Real Algebraic Geometry[END_REF], Theorem 2.4.5., p. 35).

These sets are closed nonempty intervals, since the functions maximized are concave and continuous on a compact set. 15 Documents de travail du Centre d'Economie de la Sorbonne -2021.16

2. f (x) ≥ 0 and x = 1.

3. f (x) ≤ 0 and x = 0.

Step 1: we show that ρ(F) ⊂ P.

Let v ∈ F and ρ(v) = (ṽ, g v ). We have to prove that the network g v is pairwise stable with respect to ṽ. For that, we fix a link ij ∈ L. Since g v ij = min{w v i,j , w v j,i }, we can assume, without any loss of generality (permutting w v i,j and w v j,i if necessary), that g v ij = w v i,j , which corresponds to the case where w v i,j ≤ w v j,i . First, we prove that g v ij = w v i,j maximizes the payoff function ṽi (., g -ij ) of agent i: from Lemma 4.1, we have to prove that ṽi (., g v -ij ) satisfies one of the three conditions of the lemma at g v ij = w v i,j . But an easy computation gives at every w ∈ [0, 1]:

But this last quantity is also the derivative with respect to w of the concave function

on [0, 1], and writing the first order and necessary condition given by Lemma 4.1 applied to this function, we also get (still from Lemma 4.1) that g v ij = w v i,j maximizes ṽi (., g v -ij ). In particular, this implies that Condition 1. of pairwise stability is fulfilled for agent i at g v ij , and that Condition 2. of pairwise stability is also fulfilled. It remains to show that Condition 1. of pairwise stability is satisfied for agent j at g v ij . It is clearly the case if g v ij = 0, thus we can assume g v ij > 0, and finally the proof of Step 1 will be finished if we prove that ṽj (.,

) ≥ 0, or, after a computation similar to this in Equation (3), it is equivalent to

But by definition, the concave function w → v j (w, g 0 -ij ) -w 2 2 reaches its maximum at w j,i : hence, from Lemma 4.1 and concavity, its derivative at g v ij ≤ w j,i , which is precisely

has to be nonnegative. This ends the proof that g v is pairwise stable with respect to ṽ. Finally, we have proved that ρ(F) ⊂ P, and we now consider, by abuse of notation, that ρ is a mapping from F to P.

Step 2: we show that η • ρ = id F .

For every v ∈ F, let us define v := (η • ρ)(v) = η(ṽ, g v ). We have to prove that v = v. By definition of η, we have, for every i ∈ N :

and summing Equation ( 4) and Equation ( 2), we get by definition of η, and similarly that

by definition of π. We can notice that, as H | P U is a proper map, H is also proper. Now, let us define the following map:

and let us show that H * is an homotopy between π * and η * . First, we can remark that

This homotopy can be "transported", using the homeomorphisms s • η * and s, to an homotopy Ĥ between π and η, which is defined in the following way:

Now, we only have to prove that H * is a continuous map: let O ⊂ (R m ) * be an open subset in (R m ) * and let us prove that ( H * )

To prove that this set is open in [0, 1] × M * , let us consider an element (t, (x, g)) ∈ ( H * ) -1 (O) and let us find an open neighborhood of (t, (x, g)) which is included in ( H * ) -1 (O). We will prove that there exists > 0 such that

is the neigborhood we are looking for, where L := {(x , g ) ∈ M : ∃t ∈ [0, 1] ∩ [t -, t + ] : (t , (x , g )) ∈ H-1 (K)} = {(x , g ) ∈ M : ∃t ∈ [0, 1] ∩ [t -, t + ] : H(t , (x , g )) ∈ K}.