Gabin Guelou

Pavan Ventrapati

Abdelhamid Kumar

Pierric Bourhim

Bernard Lemoine

Andrew R Raveau

Oleg Supka

Rabih Lebedev

Rahal Al

Marco Al Orabi

Koichiro Fornari

Suekuni

Gabin Guélou

Ventrapati Pavan Kumar

Abdelhamid Bourhim

Pierric Lemoine

Bernard Raveau

Andrew Supka

Oleg I Lebedev

Rabih Al

Rahal Al Orabi

Marco Fornari

Koichiro Suekuni

Emmanuel Guilmeau

Toppling the Transport Properties with Cationic Overstoichiometry in Thermoelectric Colusite: [Cu 26 Cr 2 Ge 6 ] 1+δ S 32

Keywords: Thermoelectric, sulfide, colusite, off-stoichiometry, conductive network, transport mechanism

come

Thermoelectricity (TE) has been considered as a potential source of energy for the past decades, in particular following the reports of materials with relatively high efficiency. 1 This efficiency can be characterized by the figure of merit, ZT = S²T/ρκ, which needs to be maximized by balancing conflicting electronic and thermal transport properties: a high Seebeck coefficient, S, a low electrical resistivity, ρ (to maximize the power factor PF = S²/ρ), and a low thermal conductivity, κ. Unfortunately, most performing materials involve difficult synthesis routes and/or toxic and expensive elements that critically limit their large-scale production. [START_REF] Biswas | Strained Endotaxial Nanostructures with High Thermoelectric Figure of Merit[END_REF][START_REF] Biswas | High-Performance Bulk Thermoelectrics with All-Scale Hierarchica l Architectures[END_REF][START_REF] Pei | Convergence of Electronic Bands for High Performance Bulk Thermoelectrics[END_REF] There is nonetheless a host of potential applications that can be reached if we can obtain good performances in bulk materials prepared from earth-abundant elements. [START_REF] Freer | Realising the Potential of Thermoelectric Technology: A Roadmap[END_REF] Among the promising groups of materials, ternary and quaternary copper sulfides have attracted considerable attention, [START_REF] Powell | Recent Developments in Earth-Abundant Copper-Sulfide Thermoelectr ic Materials[END_REF] with many reports of favorable raw material cost-to-efficiency ratio. Promising compositions include bornite Cu5FeS4, [START_REF] Qiu | Sulfide Bornite Thermoelectric Material: A Natural Mineral with Ultralow Thermal Conductivity[END_REF][START_REF] Guélou | Ball Milling as an Effective Route for the Preparation of Doped Bornite: Synthesis, Stability and Thermoelectric Properties[END_REF][START_REF] Kumar | Nontoxic and Abundant Copper Zinc Tin Sulfide Nanocrystals for Potential High-Temperature Thermoelectric Energy Harvesting[END_REF] germanite derivative Cu22Fe8Ge4S32, 10,11 stannoidite Cu8.5Fe2.5Sn2S12, 12 Cu2SnS3, 13 kesterite Cu2ZnSnS4, 14,15 Cu4Sn7S16, [START_REF] Bourgès | Low Thermal Conductivity in Ternary Cu4Sn7S16 Compound[END_REF] CuFeS2 [START_REF] Chen | Thermoelectr ic Enhancement of Ternary Copper Chalcogenide Nanocrystals by Magnetic Nickel Doping[END_REF] , tetrahedrites Cu12-xTxSb4S13 (T = Mn, Fe, Ni, Zn) [START_REF] Lu | High Performance Thermoelectricity in Earth-Abundant Compounds Based on Natural Mineral Tetrahedrites[END_REF][START_REF] Suekuni | High-Performance Thermoelectric Mineral Cu 12-x Ni x Sb 4 S 13 Tetrahedrite[END_REF][START_REF] Barbier | Structural Stability of the Synthetic Thermoelectr ic Ternary and Nickel-Substituted Tetrahedrite Phases[END_REF][START_REF] Barbier | A c c e p t e d m a n u s c r i p t[END_REF] and colusites Cu26T2M6S32 (T = V, Nb, Ta, Cr, Mo, W; M = Sn, Ge). [START_REF] Suekuni | High-Performance Thermoelectric Minerals: Colusites Cu 26 V 2 M 6 S 32 (M = Ge, Sn)[END_REF][START_REF] Suekuni | Tunable Electronic Properties and Low Therma l Conductivity in Synthetic Colusites Cu26-XZnxV2M6S 32 (x ≤ 4, M = Ge, Sn)[END_REF][START_REF] Bourgès | Structural Analysis and Thermoelectric Properties of Mechanically Alloyed Colusites[END_REF][START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Candolfi | Disorder-Driven Glasslike Thermal Conductivity in Colusite Cu26V2Sn6S32 Investigated by Mössbauer Spectroscopy and Inelastic Neutron Scattering. A c c e p t e d m a n u s c r i p t[END_REF][START_REF] Suekuni | Atomic-Scale Phonon Scatterers in Thermoelectric Colusites with a Tetrahedral Framework Structure[END_REF][START_REF] Lemoine | Therma l Stability of the Crystal Structure and Electronic Properties of the High Power Factor Thermoelectric Colusite Cu26Cr2Ge6S32[END_REF][START_REF] Kumar | High Power Factors of Thermoelectric Colusites Cu26T2Ge6S32 (T = Cr, Mo, W): Toward Functiona lization of the Conductive "Cu-S" Network[END_REF][START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] In previous reports, the peculiar behavior of the electrical transport properties in colusite, in particular in Cu26Cr2-xTxGe6S32 (T = Mo, W) [START_REF] Kumar | High Power Factors of Thermoelectric Colusites Cu26T2Ge6S32 (T = Cr, Mo, W): Toward Functiona lization of the Conductive "Cu-S" Network[END_REF][START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] and Cu26(V,Nb)2Sn6S32 [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Candolfi | Disorder-Driven Glasslike Thermal Conductivity in Colusite Cu26V2Sn6S32 Investigated by Mössbauer Spectroscopy and Inelastic Neutron Scattering. A c c e p t e d m a n u s c r i p t[END_REF][START_REF] Suekuni | Atomic-Scale Phonon Scatterers in Thermoelectric Colusites with a Tetrahedral Framework Structure[END_REF] was investigated in details and revealed the importance of cationic ordering/disordering on the electrical and thermal transport properties. In Cu26Cr2Ge6S32, exceptional transport properties (PF = 1.94 mW m -1 K -2 at 700 K) were recently reported and explained by the presence of interstitial Cr cations forming mixed tetrahedral-octahedral [CrS 4 ]Cu 6 complexes, which influence the geometry of the conductive "Cu 26 S 32 " framework. Our previous analysis of the colusite structure [START_REF] Kumar | High Power Factors of Thermoelectric Colusites Cu26T2Ge6S32 (T = Cr, Mo, W): Toward Functiona lization of the Conductive "Cu-S" Network[END_REF] showed that it derives from the sphalerite and consequently offers various tetrahedral cavities for hosting additional cations. Such structural properties should make possible the realization of cationic over A c c e p t e d m a n u s c r i p t stoichiometry with disorder and consequently has a great potential for changing its transport properties. The effect of deviation from ideal stoichiometry was shown in Cu26(V,Nb)2Sn6S32 [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF][START_REF] Candolfi | Disorder-Driven Glasslike Thermal Conductivity in Colusite Cu26V2Sn6S32 Investigated by Mössbauer Spectroscopy and Inelastic Neutron Scattering. A c c e p t e d m a n u s c r i p t[END_REF][START_REF] Suekuni | Atomic-Scale Phonon Scatterers in Thermoelectric Colusites with a Tetrahedral Framework Structure[END_REF] by sulfur volatilization using high-temperature sintering processes: cationic over-stoichiometr y associated with atomic-scale defects/disordered states, including interstitial defects, anti-site defects, and site splitting was achieved. In particular, a high concentration of antisite defects decreases the thermal conductivity drastically, and consequently increased the ZT significantly. [START_REF] Candolfi | Disorder-Driven Glasslike Thermal Conductivity in Colusite Cu26V2Sn6S32 Investigated by Mössbauer Spectroscopy and Inelastic Neutron Scattering. A c c e p t e d m a n u s c r i p t[END_REF] However, this sulfur volatilization process cannot be applied to Cu26Cr2Ge6S32 due to its lower thermal stability. [START_REF] Lemoine | Therma l Stability of the Crystal Structure and Electronic Properties of the High Power Factor Thermoelectric Colusite Cu26Cr2Ge6S32[END_REF] For this reason, we have investigated samples with global cationic compositio n larger than that corresponding to the ideal formula but working at lower temperature.

We report herein on the effect of the cationic over-stoichiometry in the series [Cu26Cr2Ge6]1+δS32 on the carrier concentration and conduction mechanism together with the formation of structural defects. Our results, supported by first principles calculations, indicate that excess cations are likely to be interstitial Cu with the overall consequence of moderately improving the thermoelectric efficiency.

Samples with composition [Cu26Cr2Ge6]1+δS32 (δ = 0, 0.0159, 0.0240 and 0.0323), were prepared via mechanical alloying of the precursors followed by Spark Plasma Sintering (SPS). Mechanica lly alloyed powders show no obvious traces of secondary phases and overall poor crystallinity. After consolidation at 600°C using SPS, the crystallographic structure of colusite has been confirmed by Rietveld refinement of high-resolution Powder X-ray Diffraction (PXRD) data (Fig. S1-S2 and Table S1-S2 of the supporting information). Analysis of the patterns revealed traces (ca. S1-S2). Consequently, no obvious deviation from the structural model of pristine colusite can be established for the [Cu26Cr2Ge6]1+δS32 series using PXRD only. SEM/EDS analyses on polished surfaces confirmed the presence of Cu-S binary impurity in [Cu26Cr2Ge6]1+δS32 (δ = 0.0159 and 0.0323) and the otherwise good homogeneity in the cationic ratios for all samples (Fig. S3-S6, Table S3). It is unlikely that sulfur deficiency in the initia l composition would translate into sulfur vacancies in the crystallized phase. Rather, it is more sensible to consider the off-stoichiometric colusite as a cation rich phase where the many available interstitials are randomly occupied by a small fraction of excess cations. The presence of such partially filled interstitials, combined with the presence of disordered domains, has been demonstrated recently by Suekuni et al. [START_REF] Suekuni | Atomic-Scale Phonon Scatterers in Thermoelectric Colusites with a Tetrahedral Framework Structure[END_REF] in Cu26Nb2Sn6S32 and a more thorough investigation of the cationic repartition in high-performance Cu26V2Sn6S32 colusite is currently on-going. To support the experimental evidence, we have computed with first principles method the formatio n energy for sulfur vacancies as well as for Ge, Cr, and Cu interstitial. The results (Table 1) indicate that easy-to-form defects are Cu interstitials while S vacancies and other interstitials are less favored.

Table 1 Formation energy (in eV) of defects with respect to the formation energy of the pristine crystal. The values were computed assuming one defect per cell (comparable to  = 0.0323).

Effects associated to defect-defect interaction were investigated but found not significant. The two values associated with each defect correspond to different position for the vacancy or the interstitia l within the unit cell. In order to get insights in the level of disorder induced by cation over-stoichiometry at a local scale, transmission electron microscopy (TEM), including electron diffraction analysis (ED) and high-angular annular dark field scanning TEM (HAADF-STEM) were performed. When comparing HAADF-STEM micrographs taken on the pristine sample reported previously [START_REF] Kumar | High Power Factors of Thermoelectric Colusites Cu26T2Ge6S32 (T = Cr, Mo, W): Toward Functiona lization of the Conductive "Cu-S" Network[END_REF] and the over-stoichiometric [Cu26Cr2Ge6]1.024S32, we noticed the presence of ordered (see B region in Fig. 1) but also additional areas exhibiting significant levels of disorder (see A region in Fig. 1).

S vacancy Cu interstitial

Furthermore, in the ordered B domains, while no interstitial cations could be observed unequivocally, clear variations in the contrast, and particularly in brightness of atomic columns, evidence significant levels of cationic rearrangement, as opposed to pristine Cu26Cr2Ge6S32. These findings echo the observations made on Cu26V2Sn6S32 when sulfur deficiency was induced by high processing temperatures [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF] and confirm that temperature-induced cationic disorder could be somehow reproduced in the less thermally stable Cu26Cr2Ge6S32. A noticeable difference is found in the evolution of the lattice parameter, a, that remains unchanged in the present study, in opposition with the large increase observed in Cu26V2Sn6S32 with sintering temperature. Very recently, we demonstrated that the large reduction in thermal conductivity in Cu26V2Sn6S32, while it is temperature-induced, is caused by a higher concentration of anti-site defects rather than introducing vacancies or interstitial atoms. The rather unchanged lattice parameter here might suggest that the concentration of anti-site defects is lower than in thermally disordered Cu26V2Sn6S32. 26 W) solid solutions. [START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] Indeed, the presence of an excess cation dramatically changes the temperature behavior of the electrical resistivity, in particular with the apparition of an upturn from a semiconducting to a metallic T-dependence at ca. 400-500 K, Fig. 2(a). This effect was linked to the disorder caused by cation-size mismatch on the 2a site; [START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] however, in the present case, there is also a change in the charge carrier concentration from the filling of initially vacant interstitial sites by extra cations in the off-stoichiometric samples. As a consequence, the amplitude of the electrical resistivity and the Seebeck coefficient also changes significantly, as opposed to converging toward the same values as it was the case with isovalent substitution. [START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] Instead, both the Seebeck coefficient and the electrical resistivity increase with cation over-stoichiometry, resulting in a power factor that steadily decreases with increasing off-stoichiometry over the whole investigated temperature range (Fig. 2(e)). Nonetheless, the electrical performance remains high for a copperbased sulfide with a maximum power factor of ca. 1.9 mW m -1 K -2 and ca. 1.4 mW m -1 K -2 at 700 K for pristine Cr-Ge colusite and [Cu26Cr2Ge6]1.0323S32, respectively.

Such high performance in a thermoelectric sulfide is scarce and the transport mechanisms in colusite are well worth investigating. Hence, low-temperature Seebeck coefficient and electrical and thermal conductivities were measured down to 5 K for [Cu26Cr2Ge6]1+δS32 (δ = 0, 0.0159, 0.024) (Fig. S7). As expected from the presence of cation interstitials, the simultaneous increase of the Seebeck coefficient and the electrical resistivity in [Cu26Cr2Ge6]1+δS32 (δ = 0.0159, 0.024) compared to pristine compound is consistent with a reduced charge carrier concentration. This is Once again, this behavior resembles that of Cu26CrTGe6S32 (T = Mo, W) compounds with a nonetheless clear change in the temperature dependence of the Seebeck coefficient at 112 K, [START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] that may have been revealed by the lower carrier concentration in the present study. Indeed, in the case of the aforementioned solid solutions, the isovalent doping in samples had kept the lowtemperature Seebeck coefficient rather low, close to (or lower than) that of pristine Cr-Ge colusite. [START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] The low resulting Seebeck coefficient followed a nearly linear temperature dependence below 300 K that may have been occulting the transition. This further suggests that the observed change in conduction mechanisms is unrelated to microscopic features such as grain boundary, as we would not expect the Seebeck coefficient to exhibit such a marked transition. The peculiar behavior of the electrical resistivity between ca. 100 K and ca. 500 K can be associated (as it was demonstrated for the solid solutions) to chemical disorder, chemical localization, and/or variable range hopping (VRH) conductivity. [START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] Indeed, the temperature dependence of the Hall mobility, Fig. 3, switches over 100 K from typical acoustic phonon scattering ( T -3/2 ) for pristine colusite to an ionized impurity scattering type dependence ( T 3/2 ) for the off-stoichiometric compound.

Additionally, the signal from Hall effect measurements at temperatures around the transition is disturbed by an anomalous contribution that is responsible for the few data points missing in Fig. 3. The relationship between this contribution and the changes in transport properties is undoubtedly complex and well worth further investigation that falls outside the scope of this study.

As with the solid solutions, the slope of dρ/dT changes sign around 100 K for the samples with extra cations, suggesting that the change in scattering mechanism also leads to an intermed iate state between an insulating and metallic regime.

As pointed out beforehand and in our previous work, [START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] the cation-size mismatch in the 2a position is responsible for a similar change of mechanism, only this time it is accompanied by a change in the charge carrier concentration. This suggests that cations occupying other initia lly empty interstitial sites, with chemical environment rather similar to that of the 2a position, and cationic disorder are extremely efficient in altering the conduction mechanism. The electrical transport properties of colusite, insured by the "Cu26S32" conductive framework, [START_REF] Kumar | High Power Factors of Thermoelectric Colusites Cu26T2Ge6S32 (T = Cr, Mo, W): Toward Functiona lization of the Conductive "Cu-S" Network[END_REF][START_REF] Kumar | Copper-Rich Thermoelectric Sulfides: Size Mismatch Effect and Chemical Disorder in the [TS4]Cu6 Complexes of Cu26T2Ge6S32 (T = Cr, Mo, W) Colusites[END_REF] can therefore be altered in a relatively similar fashion by either cation-size mismatch on the 2a site or cation over-stoichiometry. In other words, the cationic arrangement necessary to ensure a metallic In order to complete the investigation of the thermoelectric performances of [Cu26Cr2Ge6]1+δS32, the thermal conductivity was determined (Fig. 2(c)) and the lattice contribution was extracted using Wiedemann-Franz law (Fig. 2(d)). Overall, a clear diminution in the thermal conductivity was obtained from cation over-stoichiometry, in particular at the lower end of the investigated temperature range. A minimum at room temperature was determined for [Cu26Cr2Ge6]1+δS32 (δ ≥ 0.024) with ca. 1.5 W m K -1 , a nearly 100 % decrease from pristine Cu26Cr2Ge6S32. At 700 K, the total thermal conductivity is decreased to ca. 1.1 W m K -1 for [Cu26Cr2Ge6]1+δS32 (δ ≥ 0.024). The origin of this diminution is rather clear with no apparent change in the lattice contribution to the thermal conductivity (Fig. 2(d)). The diminution in the electrical resistivity is thus mainly responsible for the observed reduction. This is also true at low-temperature where the significa nt drop in thermal conductivity is mainly attributed to a decrease in the electronic contribution (Fig. S7c). This contrasts with the observations made on the V-Sn colusite, where order/disorder domains observed by HAADF-STEM, similar to those observed in [Cu26Cr2Ge6]1.024S32 (Fig. 1)

were initially held responsible for a large decrease in the lattice contribution to the thermal conductivity. [START_REF] Bourgès | High-Performance Thermoelectric Bulk Colusite by Process Controlled Structural Disordering[END_REF] This suggests that the disorder induced by cation over-stoichiometry, in contrast with high-temperature treatment, does not modify the distribution of cations enough so that the concentration of anti-site defects increases significantly, and thus does not shorten the acoustic phonon lifetimes. This is further supported by the absence of significant evolution in the unit cell parameter, a, in [Cu26Cr2Ge6]1+δS32. While some traces of secondary phase were found in some samples, they do not seem to have any impact of the lattice thermal conductivity at high temperature. Consequently, the calculated figure of merit is mildly improved at high temperature with a maximum ZT of 1.0 at 700 K for [Cu26Cr2Ge6]1.024S32, (Fig. 2f), the highest reported value in colusites to date. It is important to note that achieving ultra-low lattice thermal conductivity (κlat ≈ 0.2 W m -1 K -1 ) in Cr-Ge colusite, as we demonstrated was possible in Cu26V2Sn6S32, remains an open challenge and would lead to figure of merit values well over unity.

We have demonstrated that the presence of a slight excess of cations had an impact on the "Cu26S32" conductive framework by simultaneously altering the charge carrier concentration and the conduction mechanism in colusite, [Cu26Cr2Ge6]1+δS32. Many interstitial sites are available for the additional cations with a chemical environment similar to the 2a position, namely a particular tetrahedral-octahedral complex with competing ionic and metallic bonds. Additionally, this excess cation causes significant level of cationic rearrangement, as evidenced by HAADF-STEM.

However, the exact nature of the cations in the interstitial positions and their possible repartition is yet to be elucidated using powerful structural investigation tools such as synchrotron and/or neutron diffraction. Overall, along with a moderate improvement in the thermoelectric response, these findings open new ways for improvement via simultaneous charge carrier doping and transport mechanism modulation in copper sulfides.

Supporting Information

The following file is available free of charge.

  2 wt%) of a Cu8GeS6 (Pmn21, a = 7.04 Å, b = 6.97 Å, c = 9.87 Å) secondary phase only in the offstoichiometric samples [Cu26Cr2Ge6]1+δS32, where δ = 0.0159 and 0.0323. Rietveld refineme nts (Fig. S1-S2) indicate equivalent unit cell parameters and atomic coordinates for both pristine and

A c c e

  p t e d m a n u s c r i p t off-stoichiometry colusites (Table

A c c e p t e d m a n u s c r i p tFigure 1 .

 1 Figure 1. High resolution HAADF-STEM image of [Cu26Cr2Ge6]1.024S32 (cubic, P4 ̅ 3n) for the

A c c e

  p t e d m a n u s c r i p t

Figure 2 .

 2 Figure 2. Temperature dependence of the (a) electrical resistivity (), (b) Seebeck coefficient (S), (c) thermal conductivity (), (d) lattice thermal conductivity (L), (e) power factor (PF), and (f) figure of merit ZT in the [Cu26Cr2Ge6]1+δS32 series.

A c c e p t e d m a n u s c r i p t

  confirmed by low-temperature Hall effect measurements (inset in Fig.3) with a 3 times lower concentration at 5 K for the cation-rich colusites. The temperature dependence of the electrical resistivity is semiconducting for [Cu26Cr2Ge6]1+δS32 (δ = 0.0159, 0.024) down to ca. 100 K (Fig.S7(a)) as expected from the trend observed in Fig.2 (a). Below 100 K, the temperature dependence of both Seebeck coefficient and electrical resistivity changes radically for [Cu26Cr2Ge6]1+δS32 (δ = 0.0159, 0.024) while pristine Cu26Cr2Ge6S32 remains in a typical metallic regime (Fig.S7).

Figure 3 .

 3 Figure 3. Temperature dependence of the Hall mobility for Cu26Cr2Ge6S32, [Cu26Cr2Ge6]1.0159S32

A c c e p t e d m a n u s c r i p t

A c c e p t e d m a n u s c r i p t behavior and classical

  acoustic phonon scattering in pristine Cu26Cr2Ge6S32, is markedly upset by a slight cationic over-stoichiometry.

A c c e p t e d m a n u s c r i p t

A c c e p t e d m a n u s c r i p t

  Crucial Role of Selenium for Sulphur Substitution in the Structural Transitions and Thermoelectric Properties of Cu5FeS4 Bornite. Dalt. Trans. 2017, 46 (7), 2174-2183. https://doi.org/10.1039/c6dt04204j. (10) Pavan Kumar, V.; Paradis-Fortin, L.; Lemoine, P.; Caignaert, V.; Raveau, B.; Malaman, B.; Le Caër, G.; Cordier, S.; Guilmeau, E. Designing a Thermoelectric Copper-Rich Sulfide from a Natural Mineral: Synthetic Germanite Cu22Fe8Ge4S32. Inorg. Chem. 2017, 56 (21), 13376-13381. https://doi.org/10.1021/acs.inorgchem.7b02128. (11) Pavan Kumar, V.; Paradis-Fortin, L.; Lemoine, P.; Le Caer, G.; Malaman, B.; Boullay, P.; Raveau, B.; Guélou, G.; Guilmeau, E. Crossover from Germanite to Renierite-Type Structures in Cu22-XZnxFe8Ge4S32 Thermoelectric Sulfides. ACS Appl. Energy Mater. 13) Shen, Y.; Li, C.; Huang, R.; Tian, R.; Ye, Y.; Pan, L.; Koumoto, K.; Zhang, R.; Wan, C.; Wang, Y. Eco-Friendly p-Type Cu2SnS3 Thermoelectric Material: Crystal Structure and Liu, M. L.; Huang, F. Q.; Chen, L. D.; Chen, I. W. A Wide-Band-Gap p -Type Thermoelectric Material Based on Quaternary Chalcogenides of Cu2ZnSnQ4 (Q=S,Se).

	m a n u s c r i p t m a n u s c r i p t
	Supplementary_Information.pdf: Including the experimental section (p. S-1 to S-5); PXRD patterns and corresponding Rietveld refinements; elemental mappings; low-temperature electrical resistivity, Seebeck coefficient and thermal conductivity measurements. A c c e p t e d Transport Properties. Sci. Rep. 2016, 6, 32501. A c c e p t e d https://doi.org/https://doi.org/10.1038/srep32501.
	Corresponding Author (14)
	* Emmanuel.guilmeau@ensicaen.fr

2019, 2 (10), 7679-7689. https://doi.org/https://doi.org/10.1021/acsami.0c00094. (12) Pavan Kumar, V.; Barbier, T.; Caignaert, V.; Raveau, B.; Daou, R.; Malaman, B.; Caër, G. Le; Lemoine, P.; Guilmeau, E. Copper Hyper-Stoichiometry: The Key for the Optimizatio n of Thermoelectric Properties in Stannoidite Cu8+xFe3-XSn2S12. J. Phys. Chem. C 2017, 121 (30), 16454-16461. https://doi.org/10.1021/acs.jpcc.7b02068. (Appl. Phys. Lett. 2009, 94 (20), 202103. https://doi.org/https://doi.org/10.1063/1.3130718.

Acknowledgement

The authors would like to thank Christelle Bilot and Jerôme Lecourt for technical support and the financial support of the French Agence Nationale de la Recherche (ANR-15-CE05-0027), FEDER and Region Normandy. K.S. thanks the International Joint Research Program for Innovative Energy Technology funded by the Ministry of Economy, Trade and Industry (METI), Japan. M.F. and A.R.S. acknowledge collaboration with the AFLOW Consortium (http://www.aflow.org) under the sponsorship of DOD-ONR (Grants N000141310635 and N000141512266).

Author Contributions

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

Funding Sources

ANR-15-CE05-0027 DOD-ONR (Grants N000141310635 and N000141512266).


Douadia Bougherara

Sandrine Costa

Gilles Grolleau

Lisette Ibanez

Dealing with the aversion to the sucker's payoff in public goods games

Keywords: Experiments, Public good, Sucker's payoff, Assurance. JEL codes: C72, C91, H41

A usual explanation to low levels of contribution to public goods is the fear of getting the sucker's payoff (cooperation by the participant and defection by the other players).

In order to disentangle the effect of this fear from other motives, we design a public good game where people have an insurance against getting the sucker's payoff. We show that contributions to the public good under this 'protective' design are significantly higher and interact with expectations on other individuals' contribution to the public good. Some policy implications and extensions are suggested.

Dealing with the aversion to the sucker's payoff in public goods games 1. Introduction and motivation

Social dilemmas involve a conflict between a cooperative strategy, benefiting the group but potentially costly to the individual, and a defection strategy, detrimental to the group but benefiting the individual. A common social dilemma is the prisoner's dilemma involving four types of payoffs as indicated in figure 1 where T>R>P>S [START_REF] Rapoport | A note on the 'index of cooperation' for prisoner's dilemma[END_REF]. Mutual cooperation brings a reward (R) payoff to each player; mutual defection leads to a punishment (P) payoff; when one player cooperates (resp. defects) while the other defects (resp. cooperates), he gets the sucker (S) payoff (resp. the temptation payoff). Low levels of contribution are frequently reported in games with a prisoner's dilemma structure. Two main drivers are greed and fear (Ahn et al., 2001). Out of greed, one player may defect to benefit from the good at the expense of the other player. Out of fear, individuals may prefer to defect to avoid the sucker payoff. The greed motivation is the classical opportunistic free-riding behavior. The fear motivation is an aversion to the sucker payoff.

Even if the outcomes seem similar between greed and fear motivation -that is reduced contributions to public goods-the drivers are different. Indeed, greed leads to free riding behavior and is an opportunistic behavior where the individual seeks to consume more than his fair share of a public resource while defection out of fear is a priori non-opportunistic and results from the uncertainty concerning others' behaviors.

The aversion to the sucker's payoff has been notably introduced in the analysis of public goods with threshold effects. In this case, the agent does not contribute for the production of a public good because he fears that the good will not be produced because too many other players will defect. Given that the production of the public good requires a minimum level of contributions, if the contributions are insufficient, the good will not be produced and the individual will feel he squandered his contribution1 [START_REF] Sen | Isolation, Assurance and the Social Rate of Discount[END_REF][START_REF] Runge | Institutions and the Free Rider : the Assurance Problem in Collective Action[END_REF][START_REF] Schmidtz | The Limits of Government, An Essay on the Public Goods Argument[END_REF][START_REF] Wiener | A Framework for Promoting Cooperation[END_REF]. For example, a driver can renounce purchasing an environmentally friendly car because he is convinced that his isolated contribution is too weak to induce a perceptible improvement in air quality, except if he is convinced that a sufficient number of other drivers will also contribute by purchasing an environmentally friendly car 2 . While the aversion to the sucker's payoff is exacerbated in the case of public goods with threshold effects, we contend that it remains an impediment to higher contributions, even when there is no threshold effect.

In a survey, [START_REF] Rapoport | Prisoner's Dilemma: A Study in Conflict and Cooperation[END_REF] showed that cooperation rates in prisoner's dilemmas increase when the 'sucker' payoff decreases. The 'strong [emphasis added] desire to avoid being a sucker' is supported by an empirical regularity that 'when a manipulation (…) has the effect of increasing the likelihood that the group's goal will be achieved, subjects are more likely to cooperate' [START_REF] Wiener | A Framework for Promoting Cooperation[END_REF]; see also [START_REF] Taschian | Measuring attitudes Toward Energy Conservation: Cynicism, Belief in Material Growth, and Faith in Technology[END_REF].

Using experimental games, [START_REF] Fehr | Cooperation and Punishment in Public Goods Experiments[END_REF] demonstrate that people are willing to punish free-riding -even if it is costly for them-in order to avoid getting the sucker's outcome.

Aversion to the sucker's payoff stems from many behavioral assumptions. [START_REF] Wilkinson-Ryan | The Sucker Norm[END_REF] discusses what it is in the sucker payoff that deters from cooperation. The authors argue that to be a sucker, three conditions must be satisfied. First, the individual must "either give more than he gets or get less than he deserves". That points to fairness considerations.

Second, "the victim must have evinced some kind of trust for his eventual antagonist, and then had that trust betrayed". That is linked to trust and betrayal. And finally, "each sucker has to some extent contributed to his own state" that is "it involves self-blame". Indeed, the dimensions involved in the sucker's payoff aversion are manifold. Obviously, we will not be able to deal with them all but we focus on the aversion to others contributing an overall amount of less than 75% of the individual's contribution (see next section).

In this article, we study an insurance mechanism to increase cooperation in the face of sucker's payoff aversion. Since sucker's payoff aversion stems from the uncertainty on other players' behavior rather than from opportunistic behavior, we investigate the impact of an insurance mechanism where players are insured against large losses in case they are the only contributors in the group. Rather than advocating for insurance schemes or contracts from a theoretical viewpoint (e.g., [START_REF] Schmidtz | The Limits of Government, An Essay on the Public Goods Argument[END_REF], we question their effectiveness to improve the funding of public goods. In order to disentangle the effect of the sucker's payoff aversion from other factors on the level of contributions, we design a public good game where participants are partially insured against defection by other players. The contribution level to public goods when a partial insurance mechanism is implemented has not been investigated in the literature. In other words, our paper answers to the following question: does the provision of an insurance mechanism lead to higher levels of contribution to public goods and to what extent? We report two main results. First, we corroborate that aversion to the sucker's payoff matters in overall contribution to public goods. The implementation of an insurance mechanism has a positive impact on the individual's contribution. Second, the insurance mechanism also affects the individual's expectations regarding the contributions of other participants. As all other agents also benefit from the same insurance mechanism, their incentive to defect is equally reduced. This effect simultaneously (i) reinforces the positive effect of the insurance mechanism at the individual's level as the probability to end up with the sucker's payoff is reduced, ceteris paribus but (ii) also decreases the overall individual's contribution because he expects that given that other players will contribute more, he can contribute less. Ultimately, the overall effect of implementing an insurance mechanism on the individual's contribution remains positive.

The remainder of the paper is organized as follows. Section 2 describes the experiment and stipulates the theoretical predictions. Section 3 presents and discusses the results. Section 4 concludes and provides some policy implications.

Experimental Design and Implementation

In this section we present the experimental design and the theoretical predictions given our treatments and our choice of parameters for the experiment.

Basic design

We use two treatments namely the Reference treatment that corresponds to a standard public good game and the Insurance treatment where we provide subjects with an insurance against the risk of getting the sucker's payoff. In the Reference treatment, subjects are endowed with 20 tokens they allocate between a private investment which earns one euro per token and a public investment which earns 0.4€ per token as in any standard public good experiment. Given other players' contribution i c -, player i chooses the level of contribution i c that maximizes the following payoff function:

1 ( , ) 20 0.4 20 0.6 0.4

n i i i k i i k u c c c c c c - - = = -+ = - + ∑
In the Reference treatment, the Nash equilibrium is to contribute nothing and the social optimum to contribute all the endowment. The reason for low contributions may lie in greed (leading to free riding behavior) but also in fear (the aversion to the sucker's payoff). To distinguish these effects, we design a second treatment.

In the Insurance treatment, subjects have the same payoff function as in the Reference treatment except that another payoff function (alternative payoff) substitutes to the standard payoff if the other players in the group is too low. Given other players' contribution i c -, player i chooses the level of contribution i c that maximizes the following payoff function:

( , ) [ (20 0.6 0.4 );(20 0.3 ) ]

i i i i i v c c Max c c c - - = - + -
The Nash equilibrium of this game is still to contribute nothing and the social optimum to contribute all. However, the worst payoff for player i that is to be the only one to contribute ("sucker's payoff") is now relatively better, ( , 0) 20 0.

3 i i v c c = - (in the Reference treatment, this worst payoff was i i c c u 6 . 0 20 ) 0 , ( - =
). Note that the insurance mechanism insured against the case where the others give an overall contribution of less than 75% of the individual's contribution. The individual is indeed indifferent to the insurance mechanism when ), then the Insurance treatment provides higher payoffs than the Reference treatment. However, if player i is a relatively small contributor ( 4 3

3 4 i i c c -= .

Predictions

i i c c - ≤
), the Insurance treatment is equivalent to the Reference treatment. In a pure 'homo economicus' model, the

Insurance mechanism should play no role even when 0 1 5 i c - < < . Non contribution remains the dominant strategy. However, with other models of behavior where human beings are not 'pure egoists' (e.g., [START_REF] Croson | Theories of Commitment, Altruism and Reciprocity: Evidence from Linear Public Goods Games[END_REF], the Insurance mechanism will play a role. What behavior can we expect? By providing an insurance against the sucker payoff to all participants, the Insurance mechanism leads the individual to anticipate that others will contribute more. This anticipation can exert an influence in two opposite ways. On the one hand, if the individual exhibits reciprocal preferences, he will contribute more to match the higher contributions of other participants. On the other hand, if the individual exhibits altruistic preference that can be crowded out by expectations that other participants will contribute at higher levels, he will decrease his own contribution. In sum, in addition to the direct effect of the Insurance mechanism on the individual i , there is also an indeterminate indirect effect through the individual' expectations on the contribution levels of other participants.

Experimental results

We first present the sample and the sessions, then some summary statistics and finally the econometric results.

Sample and sessions

The experiment has been performed at the ENGREF (Nancy, France) and gathered a sample of 64 students. Subjects were randomly distributed among groups of four players. In each session, there were 4 groups. There were two sessions per treatment. The statistical analysis does not take into account the panel structure of the data. We take it into account in the econometric analysis.

Econometric results

Our data displays a panel structure and we are interested in time-invariant variables such as the treatment. The use of a random effect model that includes dummy variables for groups shows there is no individual specific effect. Thus, we use an ordinary least square model.

0 1 2 3 4 1 i -i k ki k c Insurance E(c ) Period Group e α α α α α ≠ = + + + + + ∑
The dependent variable is an individual i's contribution to the public good ( i c ).

Independent variables are the treatment dummy variable (Insurance) equal to one if the treatment is the Insurance treatment, an individual i 's expectations on what the other three individuals in his group will contribute in the same period t (

-i E(c ) ), the period number (Period), and an indicator variable for each group minus one (Group).

Individual i 's expectations on others' behavior is unobservable. Thus, we used three proxies for the variable [START_REF] Timothy | Environmental Labeling and Incomplete Consumer Information in Laboratory Markets[END_REF]Gangadarhan, 2002 or in Croson, 2007). We consider that player i updates his beliefs on others' behavior on a period by period basis. In the actual computation method, we simply use the actual contribution of other players in the group as a proxy for individual i 's expectations. In the myopic computation method, player i takes account only of the last period without considering the preceding periods. In the nonmyopic computation method, player i updates his beliefs in period (N+1) by a weighted mean where the behavior of others in period (N-1) is projected on periods 1 to (N-2). The three computation methods yield the same results. In the article, we display only the actual computation method. Table 3 presents summary statistics for the dependent and independent variables. The econometric results for all individuals are presented in Table 4. In line with the predictions in section 2.2., we have introduced an interaction effect between the treatment and the expectations. According to the analysis performed by [START_REF] Croson | Theories of Commitment, Altruism and Reciprocity: Evidence from Linear Public Goods Games[END_REF], this negative correlation associated with positive levels of contributions reveals altruism on the part of participants. There is a crowding out effect. When participants expect high contributions from others in the group, they will decrease their contribution to the public good. Given such behavioral patterns, we predicted an increased negative effect of expectations in the Insurance treatment. However, the interaction effect between the Insurance treatment and the expectations is positive, although small. When the treatment has an insurance device against the sucker payoff, higher expectations will lead to higher contributions.

-i E(c ) (as in

Conclusion

We examined the effect of the aversion to the sucker's payoff on contribution to public goods, using experimental games. Our results confirm that the aversion to the sucker's payoff plays a significant role in explaining contribution to public goods. Implementing an insurance mechanism plays a direct positive role on the individual's contribution and a positive indirect role through the individual's expectations on other's contribution. When the expected cooperation rate is relatively high, the insurance scheme reinforces the positive role of expectations. A clear implication from our study is that public goods contribution schemes can increase the size of individual contributions thanks to refunding mechanisms in the event that the provision point is not reached. If people perceive their contribution as pivotal, that is, their contribution will 'make a difference' 3 , they are more likely to contribute more.

Our study has limitations that give room for several extensions. For example, our insurance mechanism was partial and we do not investigate how different levels of insurance (from no insurance to full insurance) can impact on overall contribution to public goods with respect to the anticipated cooperation rate. An additional extension relates to the effect of heterogeneous agents (e.g. big and small contributors to public goods) on the functioning of insurance schemes. Moreover, in real life, insurance mechanisms can correspond to various devices that are likely to impact differently on contributions (e.g., [START_REF] List | The Effects of Seed money and refunds on Charitable Giving: Experimental Evidence from a university Capital Campaign[END_REF]. We contend that people may, regardless from the end-outcome, extract 'procedural' utility from the way the insurance scheme is functioning [START_REF] Benz | Introducing Procedural Utility: Not Only What but also How Matters[END_REF]. For instance, the common knowledge of the presence of a sufficient portion of individuals willing to contribute to the public goods, regardless of others' contributions in the population can provide a natural 'insurance mechanism' preventing to some extent the aversion to the sucker's payoff in a different way when compared to a formal contract reimbursing people in case of insufficient overall contributions. This natural insurance mechanism may explain why ecolabelling schemes performs much better in some countries (e.g., Germany) compared to other countries (e.g. France).

Documents de

  Figure 1. Typical prisoner's dilemma

Figure 2 Figure 2 .

 22 Figure 2 displays (i) in plain lines, player i 's payoff as a function of his own contribution and depending on the contribution of the three other players and (ii) in dotted lines, the player i 's alternative payoff as a function of his own contribution.

Figure 3 :

 3 Figure 3: Box plot of average group contribution as a function of the treatment

Table 1 . Organized sessions

 1 Reference treatment and 27.863 (standard deviation: 15.532) for the Insurance treatment.Figure3gives a box plot representation of the average group contribution over the periods and reveals a higher median for the groups in the Insurance treatment. A two-sample Wilcoxon rank-sum test is performed to test for a difference of distribution of group contributions between the two treatments. The results suggests group contributions were higher in the Insurance treatment at a 1% significance level (z= -6.258).

	Session Treatment Number of	Number of
			groups	participants
	1	Reference	4	16
	2	Reference	4	16
	3	Insurance	4	16
	4	Insurance	4	16
	3.2. Sample statistics
	The average group contribution is 22.625 tokens (standard deviation: 18.226) for the

Table 3 . Description and statistics of variables used in the regression analysis

 3

	Variable	Description	#Obs.	Mean
				(SD)
	i c	Individual i's contribution to the public good	1280	6.311
				(6.732)
	Insurance	Dummy (=1 if Insurance treatment and 0	1280	0.500
		otherwise)		(0.500)
	E(c ) -i	Individual i's expectations on what the other three individuals in his group will contribute	1280	18.933 (13.856)

Table 4 . OLS regression of individual i's contribution to the public good for all individuals

 4 From table 4, we see that the Period has always a negative effect on individual contributions. It is a common result in experimental data. The data analysis shows a positive effect of the principal effect of Insurance treatment. The alternative payoff provides participants with an insurance against the risk of getting the sucker's payoff. Individuals are averse to the sucker's payoff. The principal effect of expectations is negative, although small.

		Coefficient	SD	P>|t|
	Insurance	2.160	0.949	0.023
	Expectations	-0.136	0.028	0.000
	Insurance X Expectations	0.068	0.032	0.034
	Period	-0.401	0.033	0.000
	Constant	6.905	0.776	0.000
	(Dummies for group not reported here)
	Nb obs.		1280
	Adj-R2		0.2701

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[START_REF] Sen | Isolation, Assurance and the Social Rate of Discount[END_REF] defines this problem as the assurance problem.

Another example can be related to the effects of seed money and refunds which increase significantly the contribution level of charitable giving (e.g.,[START_REF] List | The Effects of Seed money and refunds on Charitable Giving: Experimental Evidence from a university Capital Campaign[END_REF].

La liste intégrale des Documents de Travail du LAMETA parus depuis 1997 est disponible sur le site internet : http://www.lameta.univ-montp1.fr

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Marcel Boyer
email: marcel.boyer@umontreal.ca

-Topic -M. Boyer -Competition law and health crisis

HAL is a

economic institutions and organizations by sorcerers' apprentices and fake intellectuals. The latent discourse against markets, competition, economic freedom, and globalization, will likely amplify in the coming months.

3. The ability to trade, especially with strangers, is a distinguishing characteristic of humans. This ability is unique to us and far exceeds the simple reciprocity observed in other animals, where it is typically limited to individuals of the same clan or family-frequently involving goods of the same undiversified kind received or consumed within a relatively short period of time.

The emergence of economics 4. The emergence of economics as a positive science, which studies the evolving incremental accumulation of trade, coordination, and incentivization mechanisms that has allowed a truly collective intelligence to flourish within mankind, has led to a normative science of ways and means, tools and instructions to foster the further development, improvement, and resilience of our collective intelligence.

5. The development of civilization follows a fundamental guiding principle: the ongoing quest-in a sometimes orderly and stepwise and sometimes random and haphazard progression-for mechanisms of exchange, trade and specialization, coordination, communication, and incentivization that are increasingly efficient and effective.

6. This quest relies on factors such as urban density, open and integrated transportation systems, specialization of tasks, reduced environmental footprint, and social institutions that generate mutual trust between strangers. Each one of us, focused on our specialized task, is literally depending on a vast number of strangers: simply counting the people who worked for my well-being this hour would take many days.

The collective intelligence 7. The collective intelligence that coordinates the work of billions of cells (all of us) within modern society seen as a social body, through competitive markets, market-like institutions, and competitive international trade, needs to, and does continuously, bolster this body's resilience to the inevitable shocks and spells of chaos and dysfunction: pandemics, wars and military escapades, and economic crises. As a result, our world has become more cooperative, safer, more resilient, and more innovative. The marginal deviant behaviour reported in the news and to be eradicated remains mostly insignificant. 8. To paraphrase Matt Ridley, let's just say that sometimes we have no choice but to play Russian roulette, remaining optimistic while venturing into the unknown. It takes time for Mother Nature to get the live round into the proper chamber, combining high contagion with asymptomatic carriers and a significant mortality rate, but sometimes she pulls it off, leaving us with the major pandemics and catastrophes of the past and today's COVID-19-black swan events. 9. So far, we have made it through every one of them, though sometimes with considerable losses and scars that persist for decades.

10. There are those who complain that we should have seen it coming and been better prepared on the basis of the precautionary principle. However, when confronted with many risks, especially those that are unknown (the unknown unknowns, the ones we don't know we don't know) 2 or poorly understood and of very low probability, a surfeit of caution may result in needless and costly paralysis and stagnation. However, once the triggering event has been observed, it would be a serious error to allow the effects to unfold exponentially. That is where we are at with COVID-19.

The financial crisis and recession of 2007-2010 11. The financial crisis and recession of 2007-2010 was caused by the gaming of poorly designed incentive mechanisms and a collapse of interbank confidence because of free-riding and financial fraud. The solution was a massive injection of financial liquidity to prevent a total collapse and targeted investments of public funds in major financial and manufacturing institutions. This government largesse was offered on draconian and costly terms. Banks were charged high interest rates to motivate them to quickly restore their balance sheets by raising new equity. And failing private firms were acquired, once bankrupt, by the government and later resold. 3 This was strong medicine. 4 The current situation is very different 12. The current situation is very different! At its core, the COVID-19 crisis does not find its roots in distorted incentives or a generalized loss of confidence, but rather in a lack of medical knowledge compounded by social proximity, which, having previously improved our public health and resilience, 2 D. Rumsfeld (2002), the former US Secretary of Defense: "[T]here are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns -the ones we don't know we don't know. And In spite of the government heavy-handed interventions, the market discipline eventually prevailed and, contrary to entrenched popular beliefs, responsibilities and liabilities were broadly imposed and implemented.

has now mutated into an engine of dissemination of the virus. We are living through a pandemic and a global economic stress that will be remembered for generations.

13. The actions taken by most governments thus seem appropriate, at least as of March 31: temporarily reduce social proximity to curb the spread of the virus, accept that this will induce an expected significant but temporary economic recession, cushion the harm to individuals and businesses resulting from this recession by public support programmes, invest massive resources as of immediately in the search for a vaccine, and accept an explosion of public debt way above the limits governments were trying to abide by.

14. These public policies come with significant costs and risks. Reducing social proximity risks to translate into national proximity rules against international trade and institutions, a major factor of economic development and growth, and civilization. Accepting that these policies will trigger a deep recession if not an outright depression is playing with the fire of inequalities as recessions are tilted against the poor, individuals and countries. Cushioning the harm to individuals and businesses through generous public programmes risks developing a sense of irresponsible dependence and reliance on pyromaniac firefighting governments. Investing massive resources in pharmaceutical research on COVID-19 may grasp resources from other important research, pharmaceutical or otherwise. Finally, accepting an explosion of public debt will require strong discipline to pay it back through either savings, taxation, and inflation or reduced lower-quality public services.

15. The ratchet effect of Leviathanesque governments and public sector bureaucracy is the direst possible consequence of the above policies. While hoping that the measures are adequate, we must not neglect to prepare for the economic recovery when the time is right. The first and foremost challenge will be to avoid throwing out the baby with the bathwater as fear may remain present. We must be careful not to damage the engines of growth and civilization that has generated enormous benefits to mankind over the last decades or centuries: the quest for efficiency and effectiveness through exchange, trade and specialization; and the capacity and willingness to identify and adopt inventions and innovations, whether technological, social or organizational. Such capacity and willingness are rooted in individual attitudes towards change, which itself depends on incentive mechanisms and financial and insurance instruments, which individuals, organizations, and firms can rely on to manage change and global risks as well as fears.

16. Shrinking reliance on competitive markets and competitivelike institutions and a general tendency to turn in upon ourselves and return to outmoded concepts of buying local or national represent the direst risks of current public policies. Insofar as the USA behaves in an irresponsible "Me First" way at the international level by forcing private firms to terminate contracts with foreign firms and governments under the Defense Production Act (1950),5 other national governments are likely to follow in a movement toward a bad but stable Nash equilibrium, with disastrous effects for all. 17. More than ever, we must secure strong international health and agro-food clusters. These two clusters could be the first sectors to experience increasing government intervention, interference and regulation, under the flag of health and food sovereignty, for the benefit of some niche stakeholders and countries but at the expense of mankind as a whole. Stressing national clusters would impact negatively the situation in developing countries, thereby increasing the risk of new pandemics in developed countries. More than ever, we as humans are all in the same boat, but the boat is too big for some to see it as such.

18. The current fear-based policies will make further development of a one-world vision of globalization and international trade very difficult. One can expect that this development will be stopped and even recede. It is important to recall and stress that all countries can and will benefit from such trade, independently of their absolute competitiveness. This statement is arguably the most important finding in modern economic theory. It is the foundation of free-trade policies against protectionism, that is, the foundation of policies favouring social well-being, poverty eradication, wealth creation, and social and economic growth against the specific private interests of lobby groups, whatever the grandiloquence of such interest groups.

19. Paul A. Samuelson, the 1970 Nobel laureate in economic sciences, once answered a challenge from mathematician Stanislaw Ulam to name one proposition in all of the social sciences which is both true and non-trivial. 6 His answer: "Comparative advantage. That it is logically true need not be argued before a mathematician; that it is not trivial is attested by the thousands of important and intelligent men who have never been able to grasp the doctrine for themselves or to believe it after it was explained to them."

The huge Increase in public debt 20. The huge increase in public debt may send the signal that debt is good and apparently cheap. If governments could implement such a recourse to debt to fight COVID-19, why not use debt also to overcome climate change, to invest more in our health and education sectors at all levels, to meet our significant needs in infrastructures, etc. Remember the warning of the dismal science: Needs are infinite, but resources are limited. The ensuing drop in fiscal discipline, rather than moving us towards a more cooperative world, is the seed of forthcoming aggressive political battles and social disruption.

21. This brings us a dangerous fallacy, which is often repeated both in the private and public sector. In its simplest form, the fallacy is as follows: since the private sector cost of capital (cost of borrowing or raising equity finance) is higher than the public sector cost of capital (cost of borrowing), then the cost of a public sector business must necessarily be less than the cost a private sector firm would incur for producing, distributing and delivering the same goods and services. Yes, governments can borrow at lower rates than private organizations but an important cost of government borrowing is hidden from the casual observation of published rates or yields.

22.

A government can borrow at lower rates because it has the right and power to raise if necessary additional taxes to reimburse its debt holders, that is, if its activities and projects fail to deliver the expected returns. A private sector firm has no such right or power and this justifies the requirement by lenders of a higher interest rate. However, from the point of view of the citizens who are the ultimate risk bearers as customers and taxpayers, the right and power of the government to literally withdraw money from their bank accounts to cover financial distress situations does have a price: it is the option value today of the government right to require and obtain from them additional funds to cover what may turn out to be ex post socially non-profitable projects. The cost of the explosion of public debt is much higher than it looks, not for governments, not for lenders, but for citizens and taxpayers.

23.

In other words, if the citizens were to grant a private organization the right and power to "tax" them if it ends up in financial distress, then this organization would be able to raise capital at the same conditions as the government. Hence, the claim that the governmental sector can produce at lower costs because the government can raise money at lower interest rates is a subtle but clear fallacy.

The reforms of social democratic societies 24. The reforms of social democratic societies' "capitalism for the people." 7 Pre-COVID-19, there was significant pressure being exerted on social democratic societies and their institutions to adapt to a more competitive environment in their political, economic, social and cultural spheres. To maintain their social security programme that has conditioned their economic performances and characterized their high quality of life, social democratic societies were under pressure to become more efficient and effective in delivering those programmes as well as other public and social goods and services, efficient in reaching the goals and objectives set and effective in doing it at the lowest possible cost.

25.

There is a real danger that the COVID-19 crisis will obscure the urgent renewal of the legal, political and social interfaces between the public and private sectors in making 7 To borrow the title from L. Zingales (2012), A Capitalism for the People, Basic Books.

democratic societies strong and innovative ones, societies in which justice, equity and entrepreneurship are valued principles.

26. The COVID-19 has in no way reduced the creeping inefficiency in the production, distribution and delivery of public and social goods and services. This inefficiency has many roots, but the most important ones find their sources in two subsets of factors. The first subset revolves around the omnipresent confusion between goals/objectives and ways/means. This confusion is the source of numerous fruitless debates. It is time to clarify the goals and objectives and to ensure that the most efficient, least costly and least risky ways and means are used and harnessed to reach the goals and objectives to be pursued.

27.

The second subset proceeds from the capture of large segments and portions of the production, distribution and delivery processes of the public and social goods and services by well-organized, entrenched and highly protected interest groups and lobbies. Those groups have, over the years, become capable of imposing hurdles of many kinds, which have impaired sustainable performance and productivity gains. COVID-19 may have entrenched even more their power.

28.

It is always time to reaffirm the preponderance of goals and objectives and give all citizens the right to displace inefficient providers of public and social goods and services. Many observers claim that our health, education, infrastructure, and environment systems are failing to provide citizens with the level of services that they could and should provide, in spite of increasingly important resources being invested. How could and did we end up with so many problems in the production and distribution of public and social goods and services, considering that we have become richer and richer almost every year over the last half-century and more? 29. If social democratic societies really aim to improve or simply maintain their broadly defined social protection and security programmes (including universal access to highquality education, training and health services, unemployment benefits, environmental protection and restoration policies, extensive water and sanitation services, recreational and cultural activities, etc.), they will have to run those programmes and deliver the associated public and social goods and services they are supposed to provide in a much more efficient and effective way. Competition is key.

Competition is key 30. Competition is the key to the emergence of more efficient and effective ways and means by which the public and social goods and services will or should be provided in the future. The increasing economic pressures arising from both the globalization of markets and the internationalization of cultures will lead to a reduction, shrinkage or even abandonment of those publicly provided programmes one way or anotherperhaps not officially, but certainly in practice with lower-quality goods and services and/or slower delivery, increased uncertainty, and lower dependability. This reduced quality of public and social goods and services will occur not because we cannot afford them anymore but because these goods and services are produced with increasing inefficiency and ineffectiveness.

31. The current political debate on the failure of the health system, education system, and infrastructure system to deliver the goods and services they are supposed to produce and distribute is centred on government budget allocation. Different groups call for more money, sometimes under the more acceptable pseudonym of "reinvestment," for health, education, infrastructure, environment, etc. These calls will become louder, now that we have learned that we could increase debt through the ceiling.

32.

The fundamental problem is one of organizational or systemic efficiency and effectiveness, not one of money or budget per se. Social democratic societies have become welleducated, highly skilled societies with significant entrepreneurial capabilities. They have also reached a high level of tolerance for diversity, not regarding fundamental principles and objectives (justice, equity, inclusion, efficiency, effectiveness), but regarding the different ways and means by which those principles and objectives may be achieved or met in practice in different contexts.

33.

Misunderstanding the role of competition and the reality of uncertainty and risk can lead to years of suboptimal and even wasteful allocation of resources, human, natural and technological. Human behaviour can be explained from two major fears: the fear of competition and the fear of uncertainty, insecurity and risk. In the current fight against COVID-19, both fears may reinforce each other to move us toward a big brother nationalist society and be powerful engines of stagnation and negative growth.

34.

But they can also be powerful engines of growth and opportunities to increase the well-being of all. Negating or misunderstanding the role of competition and improperly assessing the importance of uncertainty, insecurity and risk are the two most important roadblocks towards an improved social democratic society. rationally if there is no other action leading to preferred consequences to those of the action chosen." Rationality is an amoral concept that sees saints, criminals, and, of course, ordinary citizens, as rational people: rationality can serve the betterment of society as well as its enslavement. When properly understood, rationality presents the greatest advantage of allowing predictions of human behaviour and, in particular, changes in behaviour due to altered incentives.

37.

No economist would pretend that everyone is rational in the above sense in all circumstances and at all times. The notion of rationality must be understood in a broad sense, including constrained and bounded rationality as well as imperfect and incomplete information rationality. Moreover, the oft-assumed selfishness of the individual incorporates interests and opinions of others insofar as they are part of the individual's preferences.

38.

Behaviour is a function of preferences and incentives. It is difficult to change preferences, but incentives can be used to lead individuals towards contributing not only to their own well-being but also to the well-being of all.

39. The rationality of the individuals leads quite naturally to the second postulate: incentives are a powerful tool that favours efficiency in reaching the objectives of the open social democratic model and project. The importance of properly understanding and designing incentive mechanisms can be illustrated by the agricultural crisis of 1959-1961 in continental China. Autonomous agricultural collectivization began around 1952 and was immediately a clear success: the agricultural production increased in an impressive way between 1952 and 1958. In contexts where information can be manipulated, production cooperatives can be extremely profitable if certain organizational requirements, mainly those that allow for the proper handling of coordination and motivation through adequate mechanisms, are met. It appears that the organizational structure of the Chinese agricultural cooperatives met these requirements in the first few years.

40. The number of cooperatives had grown to more than 735,000 in 1957 with 119 million households as members for an average 160 households per cooperative. Members of a cooperative had the option of withdrawing their labour or physical capital in order to join another cooperative project if they believed that the productivity or their share of the benefits was insufficient in the former cooperative. 41. Following the success of the first cooperatives, the Chinese government decided in 1958-1959 to extend the collectivization project to the whole agricultural production. The government cooperatives gathered an average 5,000 households and the right of withdrawal was abolished to simplify the administration of the system. Compensation was also changed from a distribution based on points of merit to a system primarily based on the members' needs, independently of productivity. The mutual observation of the comrades' effort provided was possible when there were 160 households in the cooperative but impossible with 5,000 households. Abolishing the right of withdrawal made the complementary threat from more productive members totally void. By 1961, grain production plummeted more than 30% below the levels reached in 1958.

42. Justin Yifu Lin (1990), 9 an economist of the University of Beijing at that time, attributes most of the fall in production to the modified organization of the cooperatives. The modification significantly reduced the possibility of effective coordination and efficient incentives for effort and resulted in a famine that caused an estimated 30 million deaths! 43. Such misunderstood role of asymmetric information leading to opportunism, free-riding and hold-up behaviour can have disastrous effects. Efficient contracting for the production or distribution of public and social goods and services must include incentive-compatible mechanisms that are intended to optimally reduce the impact of such potential sources of inefficiency.

44.

The third basic principle is that competition generates efficiency, growth, and consequently well-being. This postulate is quite often subject to ill-informed and biased criticism. The following is a typical criticism: "Competition is not the way to create a strong community. If you compete with your neighbour, there will be a winner and a loser. We do not want losers."

45. The absence of competition generates only losers, besides the bureaucratic central-planning illuminated leaders who claim to know better than the citizens themselves what is good for them. Proper, open and transparent competition pushes everyone upwards. Modern history hardly leaves any space for doubt regarding relevancy and truth in the statement that such proper competition generates a win-win society where markets and solidarity are reconciled for the benefit of all.

46. Complementary competitive mechanisms, such as benchmarking, competitive tendering, public-private partnerships, and competitive answers to NIMBY could be particularly efficient as needed transition mechanisms in the public sector.

47.

Only proper, open and transparent competitive mechanisms, making an optimal use of new information and communication technologies, can guarantee the emergence of a society where the interests of the citizens prevail, where choices of production, consumption and investments, public and private, are made on the basis of the best information available, best competencies available, and best development prospects. 48. Let us hope that the march towards more cooperation and competition, more innovation, and more globalization, that is, the march towards a more civilized one-world vision survives the current handling of the COVID-19 crisis. There are significant risks that it will not, at least for a long time. 

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  35. To (re)build an open social democratic model and project after the COVID-19 crisis is behind us, three principles should be relied on: the rationality of individuals; the efficiency of incentive mechanisms; the efficiency of competitive mechanisms.36. Rational behaviour can be characterized as the pursuit of a coherent set of objectives and the use of appropriate means to reach them. Frank Hahn (1978) 8 proposes the following definition: "Given a set of possible actions, the agent chooses 8 F. H.Hahn (1978), On Non-Walrasian Equilibria, Review of Economic Studies 45, 1-16.

  .Lin (1990),Collectivization and China's Agricultural Crisis in 1959-1961, Journal of Political Economy 98, 1228-1252. See also J. Y. Lin and D. T.Yang (2000), Food Availability, Entitlements and the Chinese Famine of1959-61, The Economic Journal 110, 136-158.

Such international trade restrictions could be termed "overreaching/illegal use of the Act."

In Understanding theWTO, World Trade Organization (2007). https://www.wto.org/english/thewto_e/whatis_e/tif_e/utw_chap1_e.pdf.