Patrice Estellé
email: patrice.estelle@univ-rennes1.fr

Omid Mahian

Thierry F Maré

Hakan F Öztop

Natural convection of CNT water based nanofluids in a differentially heated square cavity

Keywords: CNT nanofluid, natural convection, cavity, volume fraction influence, Non-Newtonian effect Nomenclature

come

Natural convection of CNT water-based nanofluids in a

differentially heated square cavity

Introduction

Due to relevance in many engineering applications, heat transfer from natural convection in enclosures was largely investigated in literature. The last few years, studies mainly concern the use of nanofluids instead of pure conventional fluids like water, oil and ethylene glycol based fluids. In this field, the main results were reviewed in [START_REF] Haddad | A review on natural convective heat transfer of nanofluids[END_REF][START_REF] Öztop | A brief review of natural convection in enclosures under localized heating with and without nanofluids[END_REF] considering a large variety of configurations and enclosures and nanofluids nature as well. This evidences the potential of nanofluids as enhanced heat transfer fluids, enhancement being mostly dependent to nature of nanoparticles [START_REF] Öztop | Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids[END_REF]. It appears also that previous studies are mainly based on numerical works. Before performing a comprehensive numerical work to compute isotherm and streamline patterns within an enclosure, it could be first envisaged to use existing Nusselt number correlations for natural convection prediction, as recently suggested and demonstrated in [START_REF] Abouali | Computer simulations of natural convection of single phase nanofluids in simple enclosures: A critical review[END_REF]. This only requires the use of well selected theoretical models or preliminary evaluation of thermophysical properties of nanofluids. Some studies have been performed to show the uncertainties of theoretical models used evidencing the role of well representative correlations for heat transfer evaluation under natural convection [START_REF] Ho | Numerical simulation of natural convection of nanofluid in square enclosure: effects due to uncertainties of viscosity and thermal conductivity[END_REF][START_REF] Abouali | Numerical investigation of natural convection of Al2O3 nanofluid in vertical annuli[END_REF][START_REF] Hwang | Buoyancy-driven heat transfer of water-based Al2O3 nanofluids in a rectangular cavity[END_REF].

As already mentioned, a lot of studies have been conducted on various types of metallic and non-metallic oxide nanoparticles. However, whereas CNTs possess very high intrinsic thermal conductivity, there is no more report on natural convection of CNT nanofluids. Many works have been performed to evaluate the thermal conductivity of CNT nanofluids, as nicely reviewed in [START_REF] Murshed | Superior thermal features of carbon nanotubes-based nanofluids -A review[END_REF]. This is also shown in recent works [START_REF] Shanbedi | Experimental investigation of stability and thermophysical properties of carbon nanotubes suspension in the presence of different surfactants[END_REF][START_REF] Shamaeil | The effects of temperature and volume fraction on the thermal conductivity of functionalized DWCNTs/ethylene glycol nanofluid[END_REF]. However, there is still no reliable and universal model for thermal conductivity enhancement of CNT nanofluids, as shown in recent work [START_REF] Estellé | Thermal conductivity of CNT water based nanofluids: Experimental trends and models overview[END_REF]. In addition, reliable viscosity model for CNT nanofluids was only developed for high shear rate [START_REF] Halelfadl | Viscosity of carbon nanotubes waterbased nanofluids: Influence of concentration and temperature[END_REF][START_REF] Estellé | Viscosity measurements of multi-walled carbon nanotubes-based high temperature nanofluids[END_REF].

Natural convection of CNT nanofluids was experimentally investigated by Li et al. [START_REF] Li | Carbon-nanotube nanofluid thermophysical properties and heat transfer by natural convection[END_REF] with a closed rectangular vessel heated from the lower surface, the upper one being cooled by room temperature. It was shown that buoyancy induced convection and heat transfer capacity were reduced because of non-Newtonian nature of nanofluids compared to pure water. One can also mention a recent numerical study reported in [START_REF] Esfe | Natural convection in a trapezoidal enclosure filled with carbon nanotube-EG-water nanofluid[END_REF] considering a trapezoidal configuration filled with EG-water-CNT nanofluids with a vertical driven temperature difference and insulated side walls. The influence of inclination angles (15-75°), Rayleigh number (10 3 -10 6 ) and nanoparticles content (0.15-4.5% in vol.) were investigated. It was reported that buoyancy induced convection produces for Rayleigh numbers higher than 10 5 . In this range, average Nusselt number was found to decrease with increasing inclination angle with all tested nanoparticle contents. Higher Nusselt number was obtained at 30°, 0.15% in volume fraction and Ra=10 6 . This purpose of this note is to estimate the natural convection of CNT water based nanofluid stabilized with SDBS in a square enclosure from temperature dependant thermophysical properties previously measured. In this way, the available correlations for Nusselt number for natural convection of fluids in a square enclosure were used and the effects of non-Newtonian nature of nanofluids, following the volume fraction in nanotubes, were considered. Nanofluid temperature, nanoparticle volume fraction, temperature difference of heated vertical walls on Nusselt number were also presented and discussed.

Thermophysical properties of nanofluids

The nanofluids considered here have been previously investigated in [START_REF] Estellé | Thermal conductivity of CNT water based nanofluids: Experimental trends and models overview[END_REF][START_REF] Halelfadl | Viscosity of carbon nanotubes waterbased nanofluids: Influence of concentration and temperature[END_REF][START_REF] Halelfadl | Optimization of thermal performance and pressure drop of a rectangular microchannel heat sink using aqueous carbon nanotubes based nanofluid[END_REF][START_REF] Halelfadl | Efficiency of carbon nanotubes water based nanofluids as coolants[END_REF].

They consist of MWCNTs (carbon purity 90%; nanotubes density 1800 kg/m 3 ) dispersed by using ultrasonic processing in de-ionized water and stabilized by sodium dodecyl benzene sulfonate (SDBS) as surfactant. The average size of the nanotubes is 1.5 µm in length and 9.2 nm in diameter, respectively. Different nanofluid samples were prepared following the procedure described in Ref. [START_REF] Halelfadl | Viscosity of carbon nanotubes waterbased nanofluids: Influence of concentration and temperature[END_REF] with volume fraction varying from 0.418% to 0.0055%. The rheological properties of these nanofluids, and corresponding base fluids, were previously measured in Ref. [START_REF] Halelfadl | Viscosity of carbon nanotubes waterbased nanofluids: Influence of concentration and temperature[END_REF] for temperature range between 20 and 40°C. For these temperatures, a Newtonian behavior was reported for base fluids with a decrease of viscosity values with surfactant content and temperature decrease, as seen in Table 1. It should be noted that surfactant content is related to nanotubes loadings, with a constant weight ratio of 2. Table 2 indicates the behavior of nanofluids (Newtonian or non-Newtonian) depends on volume fraction and temperature of suspensions. The rheological behaviors can be described by the following relationships between shear stress τ and shear rate γ& .

For Newtonian fluid,

γ µ τ & = (1)
With µ the viscosity.

For non-Newtonian fluid,

n γ η τ & = ( 2
)
where η is the consistency and n is the flow index behavior.

By means of curve fitting of previous experimental results, the rheological parameters of equations ( 1) and ( 2), namely viscosity, consistency and flow index were obtained as reported in Tables 1 and2 as a function of surfactant content, temperatures, and nanoparticle loadings.

The thermal conductivity of nanofluids and base fluids considered here was also previously measured in Ref. [START_REF] Estellé | Thermal conductivity of CNT water based nanofluids: Experimental trends and models overview[END_REF], showing that thermal conductivity of nanofluids enhances with both volume fraction and temperature. As shown in Ref. [START_REF] Estellé | Thermal conductivity of CNT water based nanofluids: Experimental trends and models overview[END_REF], the lack of reliability of existing thermal conductivity models for CNT nanofluids with our results evidences the need to use experimental data rather than theoretical correlations.

The density of nanofluids was also measured in a recent study [START_REF] Halelfadl | Efficiency of carbon nanotubes water based nanofluids as coolants[END_REF]. These data were here used and additional measurements were also performed following the same procedure to cover the entire range of volume fraction presently investigated.

In absence of experimental data, the heat capacity of the nanofluids was calculated from the following theoretical correlation [START_REF] O'hanley | Measurement and model validation of nanofluid specific heat capacity with Differential Scanning Calorimetry[END_REF], neglecting the heat capacity of surfactant [START_REF] Halelfadl | Efficiency of carbon nanotubes water based nanofluids as coolants[END_REF].

( ) ( )( ) ( ) bf np bf p np p nf p C C C ρ φ φρ ρ φ ρ φ - + - + = 1 1 , (3)
In the same way, the volumetric thermal expansion coefficient β of nanofluids was analytically evaluated from the following equation [START_REF] Khanafer | Buoyency-driven heat transfer enhancement in a twodimensional enclosure utilizing nanofluids[END_REF] as

( ) ( )( ) ( ) bf np bf np nf ρ φ φρ ρβ φ ρβ φ β - + - + = 1 1 (4)
Here, within the range of tested temperatures the thermal expansion coefficient for the CNT nanotubes was taken to 2 × 10 -5 K -1 from Ref. [START_REF] Deng | Coefficient of thermal expansion of carbon nanotubes measured by Raman spectroscopy[END_REF]. It should be noted that the effect of temperature on heat capacity of water was obviously considered. Equation (3) induces the decrease of volumetric thermal expansion coefficient of nanofluid with increasing volume fraction. The thermal expansion coefficient of nanofluids depends also on temperature and decreases with temperature rise.

Problem description

A differentially heated square cavity filled with CNT nanofluid is considered in this study. The schematic of the problem is shown in Fig. 1. On the right side of the enclosure, the temperature is considered as hot and denoted Th. On the left side the temperature is cold and denoted Tc. The mean temperature of the hot and cold walls corresponds to the nanofluid temperature. Since the nanofluid thermophysical properties are temperature dependent for the investigated temperatures 20, 30 and 40°C, this implies that non-dimensional numbers also vary with temperature.

For a square cavity with differentially heated vertical walls filled with Newtonian fluid, the correlation for average Nusselt number proposed by Turan et al. [START_REF] Turan | Laminar natural convection of powerlaw fluids in a square enclosure with differentially heated side walls subjected to constant temperatures[END_REF], expressed by equation ( 5), can be used. This equation appears as an improvement of the Berkovskie-Polevikov equation [START_REF] Berkovsky | Numerical study of problems on high-intensive free convection[END_REF] in particular for high Prandtl numbers.

in the above, c1, c2 and c3 are given by the following relationships for n ≤ 1 c1 = 1.343; c2 = 0.065; c3 = 0.036 [START_REF] Shamaeil | The effects of temperature and volume fraction on the thermal conductivity of functionalized DWCNTs/ethylene glycol nanofluid[END_REF] also, in this case Ra and Pr numbers are respectively expressed as

n n n p n k TL C g Ra η βρ 1 2 1 + + ∆ = (11) n n p L C k 2 2 2 Pr - -         = ρ ρ η (12)
It should be noted that equations [START_REF] Murshed | Superior thermal features of carbon nanotubes-based nanofluids -A review[END_REF], [START_REF] Estellé | Thermal conductivity of CNT water based nanofluids: Experimental trends and models overview[END_REF] and ( 12) reduce to equations ( 5), ( 6) and [START_REF] Hwang | Buoyancy-driven heat transfer of water-based Al2O3 nanofluids in a rectangular cavity[END_REF], respectively, when n is taken to 1, i.e. for Newtonian fluids. The correlations given by equations ( 8) to ( 12) are for 10 4 ≤Ra≤10 6 and 0.6<n≤1 [START_REF] Turan | Laminar natural convection of powerlaw fluids in a square enclosure with differentially heated side walls subjected to constant temperatures[END_REF].

A square cell dimension of 25x25mm² was here considered. In addition, the ∆T=Th -Tc values were tested and selected to check previous conditions related to Ra and Pr numbers for both base fluids and nanofluids. In addition, as previous equations were based on a Boussinesq approximation, the driving temperature difference should remain small for this to be valid.

Consequently, ∆T=Th -Tc starts from 0.1 up to 4, 2 and 1.5 for nanofluids average temperature of 20, 30 and 40°C respectively. This also implies that Ra number of nanofluids cannot be obtained for some nanoparticle volume fractions following the difference in wall temperature. As shown in Table 2, flow index n is well within the validity domain of equations whatever the nanoparticle loadings and average temperature considered.

Results and discussion

The results are presented in two subsections. First, the variation of Nusselt number for base fluids is presented. Next, the effects of various parameters such as temperature and concentration are investigated on the magnitude of Nusselt number of nanofluids. First, it is observed from Fig. 2 that, at both fixed average temperature and temperature difference between hot and cold walls, Nusselt numbers of base fluids are quite constant. It should be recalled that for the range of surfactant content, viscosity of base fluids increase with surfactant content (see Table 1) while, inversely, thermal conductivity decreases [START_REF] Shanbedi | Experimental investigation of stability and thermophysical properties of carbon nanotubes suspension in the presence of different surfactants[END_REF][START_REF] Shamaeil | The effects of temperature and volume fraction on the thermal conductivity of functionalized DWCNTs/ethylene glycol nanofluid[END_REF].

Base fluids

Based on the use of equation ( 6), this evidences a weak dependence of Nusselt number on Prandtl number. Obviously, when surfactant content is fixed, Fig. 2 shows the increase of Nusselt numbers of base fluids with respect to the increase in temperature difference between hot and cold walls. This is observed for all average temperatures of base fluids. Finally, with a fixed value of 0.1°C for temperature difference between hot and cold walls, one can see in Fig. 2 that Nusselt numbers of base fluids increase with average temperature of base fluids. Similar trends were obtained when the temperature difference between hot and cold walls is changed (not reported here).

Nanofluids

The effect of nanoparticle content, average temperature of nanofluids and driving temperature difference between hot and cold walls on Nusselt number of nanofluids are pointed out in Fig. 3. First, Fig. 3 shows that Nusselt number of nanofluids increases with the driving temperature between vertical walls. Moreover, it is also observed that Nusselt number decreases when the average temperature of nanofluids is increased. This is particularly true when the temperature rises from 30 to 40°C. In addition, Nusselt number is lowered by nanoparticle content increase. This shows the competition between thermal conductivity which increases with both temperature and nanoparticle content while viscosity decreases with temperature but largely increases with nanoparticle volume fraction. So, this balance drives the heat transfer enhancement in the enclosure from natural convection. So, it is evidenced that heat transfer of CNT nanofluids from natural convection in enclosure is rather sensitive to their rheological properties and in particular their non-Newtonian nature for volume fraction higher than 0.055%. Actually, shear-thinning behavior penalizes the onset and sustaining of natural convection in particular for low temperature difference between vertical walls.

Finally, the influence of nanoparticle content, average temperature of nanofluids and driving temperature difference between hot and cold walls on relative Nusselt number, defined as the ratio of Nusselt number of nanofluids to the one of base fluids is displayed in Fig. 4. Similar trends that the ones reported in figure 3 are observed as Nusselt number of base fluid is quite constant with surfactant increase which is related to nanoparticle increase also. Consequently, the best enhancement in heat transfer from natural convection due to presence of nanoparticles is achieved at lower temperature and within the range of 0.0055% to 0.055% in volume fraction with a weak influence of temperature difference between hot and cold walls.

Conclusions

The prediction of natural convection of CNT-water based nanofluids within a square enclosure partially heated was performed. The Nusselt number in the enclosure was evaluated from available correlations depending on thermophysical properties of nanofluids. In contrast to main previous investigations, where theoretical predictions for thermal conductivity and viscosity were used, experimental data were here considered, avoiding uncertainties related to unappropriated models. Moreover, the effect of average temperature and volume fraction of nanofluids, driving temperature between hot and cold walls and role of surfactant was investigated. The influence of non-Newtonian nature of some nanofluids depending on nanoparticle content and temperature was also discussed.

The results mainly evidence that Nusselt number of nanofluids is lowered by nanoparticle content increase related to Non-Newtonian behaviour of nanofluids and temperature increase, inversely to thermal conductivity. An optimum in natural convection due to presence of nanoparticles was reported at both low temperature and nanoparticle content.

Nu

Temperature difference between hot and cold walls /°C Tav=20°C; 0.287% Tav=30°C; 0.287% Tav=40°C; 0.287% Tav=20°C; 0.0055% Tav=30°C; 0.0055% Tav=40°C; 0.0055%

ρ 1 β

 1 density, kg m -3 k thermal conductivity, W m -1 K -1 µ viscosity, Pa s γ& shear rate, s -1 τ shear stress, Pa η consistency (Pa s n ) n flow index behavior (-) Cp specific heat, J kg -1 K -thermal expansion coefficient (K -1 ) φ nanoparticle volume fraction Nu Nusselt number Ra Rayleigh number Pr Prandtl number L Length and height of the enclosure, m g gravity, m s -

  of non-Newtonian fluids, the average Nusselt number can be expressed in terms of algebraic function of Ra, Pr, and flow index behavior "n"[START_REF] Turan | Laminar natural convection of powerlaw fluids in a square enclosure with differentially heated side walls subjected to constant temperatures[END_REF] as

Figure 2

 2 Figure 2 shows the variations of Nusselt number for deionized water and the mixture of water and SDBS as surfactant with different volume fractions where the temperature difference between hot and cold walls and the average temperature of base fluids change.

Figure 1 .Figure 2 .

 12 Figure 1. Schematic of the considered problem

Figure 3 .Figure 4 .

 34 Figure 3. Nusselt number of CNT nanofluids in function of volume fraction, average nanofluid temperature and temperature difference between hot and cold walls

Figure 1 .

 1 Figure 1. Schematic of the considered problem

Figure 2 .

 2 Figure 2. Influence of surfactant content, temperature difference between hot and cold walls and average temperature of base fluids on the Nusselt number of base fluids in square cavity.

Figure 3 .Figure 4 .

 34 Figure 3. Nusselt number of CNT nanofluids in function of volume fraction, average nanofluid temperature and temperature difference between hot and cold walls

Table Captions Table 1 .

 Captions1 Rheological properties of base fluids in function of temperature and SDBS volume fraction.

Table 2 .

 2 Rheological properties of nanofluids in function of temperature and nanoparticles volume fraction.


Zefeng Chen

Xianran He

Fei Ding

Hang Li

Yongmin Zhang

Yangguanggao

Zhou Cao

Chengzhong Su

Chiron Approaches to Anti-tumor Natural Product Fuzanin D

Keywords: Synthesis, Chiral pool, Anti-tumor, Cytotoxicity, Olefination Scheme 2 Retrosynthetic analysis of Fuzanin D

Pyridine-containing natural products which mainly originate from animals, plants, eukaryotes and prokaryotes fascinate chemical community due to their various biological activities, 1 such as cytotoxic, 2 antimicrobial, 3 antifungal, 4 anti-HIV, 5 neurotoxic, 6 anti-inflammatoryactivities. 7 Fuzanin D, a disubstituted pyridine-containing natural product, incorporated a long chain with a E,E-conjugated diene moiety, continuous hydroxyl groups, and one methyl group, was isolated from the culture supernatant of Kitasatospora sp. IFM10917 by Ishibashi and the coworkers in 2009. 8 The biological activities determination of Fuzanin D in vitro displayed inhibitory activity against DLD-1 cells with IC50 value of 41.2 μM. Besides, it also displayed moderate inhibition of Wntsignal transcription at 25 mM.

Recently, the Rao's group has achieved the syntheses of Fuzanins C, D and their derivatives as well as preformed biological evaluation on anticancer in vitro (Scheme 1). 9 On the basis of synthesis, the original absolute configuration of Fuzanin D was suggested to be reassigned. In the synthesis of Fuzanin D, Sharpless epoxidation was utilized to set the chiral centers, and Julia olefination was utilized to construct carbon carbon double bond. Herein, we would like to describe an alternative synthesis of Fuzanin D using L-arabinose or ethyl Llactate as chiral pool.

Scheme 1Structures of Fuzanin C and Fuzanin D.

As shown in Scheme 2, synthesis of Fuzanin D (1) could be achieved by Wittig olefination between building block 2 and 3-methylpicolinaldehyde3 followed by removal of cyclohexyli-dene group. Compound 2 was prepared from commercially available L-arabinose through several conventional manipulations, while compound 3 could be easily obtained by selective oxidation of 2,3-dimethyl pyridine according to a literature procedure. [START_REF] Dunn | Organic Preparations and Procedures International[END_REF] Synthesis of building block 2 was commenced with a known compound 3,4-O-isopropylidene-L-arabopyranose4 (Scheme 3). 11 Oxidative cleavage of compound 4 with NaIO4 followed by treatment of the crude product with Na2CO3 gave protected L-erythrose5. 12 Without further purification, compound 5 was subjected to Wittig olefination 13 with in-situ generated methyl Wittig reagent in toluene at reflux to provide enol 6 in 52% yield for 2 steps. To note, ring-opening reaction by Wittig reagent gave a low yield of 6 when using THF as solvent at 0 °C to room temperature. Iodination of enol 6 using standard Appel reaction condition 14 gave compound 7 in a high yield (90%). Reduction of iodide 7 was carried out smoothly with an excess amount of LiAlH4 (10 eq) to give terminal methyl-containing compound 8 in 82% yield. In consideration of low boiling point of compound 8, isopropylidene group was replaced by bulky cyclohexylidene group through a 2-step conversion in a yield of 88%. Olefin cross-metathesis (OCM) between compound 9 and allyl bromide under a catalytic amount of Grubbs II exclusively generated trans olefin 2 (J = 15.6 Hz) in 81% yield.

Paper 2017-01-23 page 2 of 4 mixture of diastereoisomers was obtained from ethyl L-lactate over 3 steps superior to a literature procedure 15 when using DCM as solvent instead of Et2O. After that, compound 10 and allyl bromide was subjected to the similar OCM as described for 2 to give trans olefin 11 in 75% yield. Compound 11 was converted into compound 2 in a high yield (89%) through unblocking of silyl ether under acidic condition followed by protection with cyclohexylidene group. With building block 2 and 3-methylpicolinaldehyde 3 in hand, assembly of them became our next goal. As shown in Scheme 5, treatment of compound 2 with PPh3 in CH3CN at reflux gave the corresponding alkyltriphenylphosphonium salt 12. Without purification, the alkyltriphenylphosphonium salt 12 was deprotonated by n-BuLi at -78 °C in THF to provide in-situ Wittig reagent, which reacted with 3-methylpicolinaldehyde 3 to give an inseparable E,Z-mixture in a ratio of 1.7:1 according to H-NMR analysis. Fortunately, the E,Z-mixture could be purified at the subsequent deprotection step. Thus, the natural product Fuzanin D (1) was obtained in 42% yield for 3 steps. The value of optical rotation and spectral data were in good accordance with the previously reported literature. 9

Scheme 5 Synthesis of Fuzanin D (1). Reagents and conditions: (a) PPh3, CH3CN, 80°C; (b) (i) n-BuLi (2.5M), THF, then 3-methylpicolinaldehyde 3, -78 °C; (ii) PTSA, MeOH, 42% yield for 3 steps.

In conclusion, synthesis of Fuzanin D (1) has been achieved from L-arabinose, ethyl L-lactate in 9 steps, 6 steps, respectively. Highlight of the protocol was using natural chiral pool to set stereogenic centers, and adopting Wittig olefination and OCM as crucial steps. Our scheme was flexible and facile, and could be applied to synthesize homologs or derivatives of Fuzanin D, which would be reported in due course.

The experimental section has no title; please leave this line here.

All reagents were commercially available and used directly without further purification unless otherwise stated. Column chromatography was carried out by using silica gel (100-200 mesh). Routine monitoring of reactions was used silica gel 60 F254 TLC plates. 1 H and 13 CNMR spectra were recorded with Bruker Avance III spectrometer at 400 MHz, 100 MHz respectively, relative to Me4Si (δ = 0 ppm) as internal standard. HRMS were measured with a Bruker micro-TOFQ II mass spectrometer. IR spectra were recorded on neat sampleswith a NICOLET iS 10 infrared spectrometer.Optical rotations were recorded with a AUTOPOL IV automatic polarimeter.

2,3-O-isopropylidene-L-erythrose (5)

To a solution of 3,4-O-isopropylidene-L-arabopyranose4 (10.4 g, 54.71mmol) in MeOH (100mL) was added NaIO4 solution (16 g dissolved in 90 mL water) by dropping funnel, and the mixture was allowed to stir at room temperature. After the reaction completed (monitoring by TLC), the solution was basified (pH 9) by the addition of Na2CO3 powder, and stirred for another 1h. The undissolved white solid was filtered and the filtrate was extracted with DCM (3 × 150 mL). The combined organic layer was evaporated under reduced pressure. The residue was purified by silica gel column chromatography (petroleum ether-EtOAc, 3 : 1) to give compound 5 as a syrup. The spectral data agreed with the reported literature. 12 ((4S,5R)-2,2-dimethyl-5-vinyl-1,3-dioxolan-4-yl)methanol ( 6)

To a suspension of methyl triphenyl phosphoniumbromide (15 g, 42.13 mmol) in dried toluene (100 mL) was added dropwise n-BuLi solution (2.5 M in hexane, 19.3 mL, 48.25 mmol) at 0 °Cunder a nitrogen atmosphere. The formed yellow suspension was stirred for 45 min at the same temperature, then compound 5(3 g, 18.74 mmol) in 10 mL of toluene solution was added dropwise. After the addition completed, the ice bath was replaced by an oil bath. The mixture was heated at reflux, andquenched with saturated aqueous NH4Cl (100 mL) until TLCindicated a complete consumption of the starting material.The organic layer was removed and the water phase wasextracted with EtOAc (3 × 100 mL), and the combined organiclayer was concentrated in vacuo. Column chromatography of the residue on silica gel (petroleum ether-EtOAc, 3 : 1) generated compound 6 as a yellow oil (1.54 g, 52% yield for 2 steps).

[α]D 25 -39.2 (c 0.35, CHCl3). IR (neat):1637, 1376, 1215, 1164, 1039 cm -1 .

1 HNMR (400MHz, CDCl3): δ = 5.91-5.83 (m, 2H), 5.39 (d, 1H, J = 17.2 Hz), 5.28 (d, 1H, J = 14.0 Hz), 4.65 (t, 1H, J = 6.8 Hz), 4.26 (q, 1H, J = 6.0 Hz), 3.58 (d, 1H, J = 5.6 Hz), 1.51 (s, 3H), 1.39 (s, 3H) ppm. 13 C NMR (100MHz, CDCl3): δ =133.11, 119.15, 109.02, 78.43, 78.39, 62.21, 27.93, 25.37 To a solution of compound 6 (410 mg, 2.59 mmol) in THF (10 mL) was added I2 (1.18 g, 4.65 mmol), PPh3 (1.22g, 4.65 mmol), imidazole (530 mg, 8.54 mmol) in sequence. The mixture was heated at reflux till the completion of the reaction monitoring by TLC. Saturated Na2S2O3 solution (10 mL) was added to quench the reaction, the organic layer was seperated, extracted with EtOAc (3 × 10 mL), and the combined organiclayer was concentrated. The residue was purified by silica gel column chromatography (petroleum ether-EtOAc, 50 : 1) to furnish compound 7 (625 mg, 90% yield) as a yellow oil.

[α]D 25 -1.45 (c 0.76, CHCl3) Treatment of compound 7 (600 mg, 2.24 mmol) in MeOH (10 mL) with p-toluenesulfonic acid (PTSA) (20 mg, 0.105 mmol), the mixture was allowed to stir at room temperature until the complete consumption of the starting material monitoring with TLC. The solution was concentrated in vacuo, and the residue was dissolved in CH3CN (20 mL). Cyclohexanone (550 mg, 5.6 mmol) was added subsequently. The resulting solution was stirred for additional 2h at room temperature,neutralized with Et3N and concentrated. Column chromatography of the residue on silica gel (petroleum ether-EtOAc, 50 : 1) provided compound 8 (607 mg, 88% yield) as a colorless oil.

[α]D 25 -1.0 (c 1.08, CHCl3).

IR (neat): 2932, 2854, 1448, 1280, 1108, 928 cm -1 .

1 HNMR (400MHz, CDCl3): δ = 5.87-5.79 (m, 1H), 5.41 (d, 1H, J = 16.8 Hz), 5.31 (d, 1H, J = 10.4 Hz), 4.61 (t, 1H, J = 6.0 Hz), 4.41 (q, 1H, J = 6.4 Hz), 3.14 (dd, 1H, J = 10.0, 7.6 Hz), 3.03 (dd, 1H, J = 10.0, 6.4 Hz), 1.69-1.64 (m, 4H), 1.62-1.56 (m, 4H), 1.39-1.38 (m, 2H) ppm.

13 C NMR (100MHz, CDCl3): δ = 132. 69, 119.33, 109.76, 78.78, 78.03, 38.09, 35.05, 25.09, 24.08, 23.75, 4.13

(2S,3R)-2-methyl-3-vinyl-1,4-dioxaspiro[4.5]decane (9)

To a solution of compound 8 (200 mg, 0.649 mmol) in dry THF (8 mL) was added LiAlH4 (246 mg, 6.49 mmol) powder under a nitrogen atmosphere. The suspension was stirred at room temperature for 2h, then quenched by the addition of saturated solution of K/Na tartrate (10 mL) cautiously, extracted with EtOAc (3 × 10 mL), and the combined organiclayer was concentrated. The residue was purified by silica gel column chromatography (petroleum ether-EtOAc, 50 : 1) to furnish compound 9 (96.9 mg, 82% yield) as a colorless oil.

[α]D 25 +12.3 (c 0.3, CHCl3).

IR (neat): 2927, 2854, 1448, 1366, 1112, 926 cm -1 . 13 C NMR (100MHz, CDCl3): δ = 134. 91, 118.10, 108.78, 79.74, 73.66, 38.18, 35.13, 25.27, 24.19, 23.88, 16.31

(2R,3S)-2-((E)-3-bromoprop-1-en-1-yl)-3-methyl-1,4dioxaspiro[4.5]decane (2)

Method A: To a solution of compound 9 (615 mg, 3.376 mmol) and allyl bromide (840 mg, 6.94 mmol) in DCM (10 mL) was added Grubbs II catalyst (85.9 mg, 0.101 mmol). The mixture was allowed to stir at reflux until the completion of the reaction, and concentrated in vacuo. Column chromatography of the residue on silica gel (petroleum ether-EtOAc, 50 : 1) provided compound 2 (749 mg, 81% yield) as a colorless oil.

Method B:

To a solution of compound 11 (900 mg, 2.92 mmol) in MeOH (10 mL) was added p-toluenesulfonic acid (PTSA) (25 mg, 0.131 mmol), the mixture was allowed to stir at room temperature until the completion of the reaction monitoring with TLC. The solution was concentrated under reduced pressure, and the residue was dissolved in CH3CN (20 mL). After that, cyclohexanone (0.76 g, 7.74 mmol) was added. The resulting solution was stirred for another 2h at room temperature, neutralized by the addition of Et3N and concentrated.

Column chromatography of the residue on silica gel (petroleum ether-EtOAc, 50 : 1) gave compound2 (0.71 g, 89% yield) as a colorless oil.

[α]D 25 +6.1 (c 0.41, CHCl3).

IR (neat): 2929, 1608, 1416, 1243, 1025 cm -1 .

1 HNMR (400MHz, CDCl3): δ = 5.94 (dt, 1H, J = 15.2, 7.2 Hz), 5.71 (dd, 1H, J = 15.2, 7.2 Hz), 4.49 (t, 1H, J = 6.4 Hz), 4.32 (dt, 1H, J = 12.8, 6.4 Hz), 3.94 (d, 2H, J = 7.6 Hz), 1.68-1.61 (m, 4H), 1.60-1.52 (m, 4H), 1.39-1.38 (m, 2H), 1.12 (d, 3H, J = 6.4 Hz) ppm.

13 C NMR (100MHz, CDCl3): δ = 131. 93, 129.61, 109.05, 78.02, 73.82, 38.12, 35.03, 31.68, 25.22, 24.17, 23.85, 16.24 66, 116.51, 76.73, 71.37, 25.85, 18.11, 17.69, -4.39, -4.80 ppm.

(2S,3R,E)-6-bromo-2-((tert-butyldimethylsilyl)oxy)hex-4-en-3-ol (11)

To a solution of compound 10 (2.5 g, 11.56 mmol) and allyl bromide (3.5 g, 28.93 mmol) in DCM (10 mL) was added Grubbs II catalyst (310 mg, 0.365 mmol). The mixture was stirred for 5 h at reflux, and concentrated in vacuo. Column chromatography of the residue on silica gel (petroleum ether-EtOAc, 50 : 1) provided compound 11 (2.67 g, 75% yield) as a yellow oil.

[α]D 25 +10.

Fuzanin D (1)

PPh3 (630 mg, 2.40 mmol) was added to a solution of compound 2 (600 mg, 2.19 mmol) in CH3CN (20 mL), the mixture was heated at reflux under a nitrogen atmosphere till a complete consumption of the starting material. The solution was concentrated in vacuo to furnish compound 12 as a white powder, which was directly subjected to the coming Wittig olefination without further purification. To a suspension of compound 12 (587 mg, 1.09 mmol) in dried THF (10 mL) was added n-BuLi (2.5 M in THF, 0.5 mL, 1.25 mmol) via a syringe at -78 °C under a nitrogen atmosphere. After stirring for 45 min at the same temperature, 3-methylpicolinaldehyde3 (110 mg, 0.91 mmol) dissolved in 5 mL of dried THF was added. The mixture was allowed to warm to room temperature gradually, and quenched with saturated aqueous NH4Cl (8 mL) till the completion of the reaction. The organic layer was separated, and the water phase was extracted with EtOAc (3× 10 mL), the combined organic layer was dried over anhydrous Na2SO4, concentrated in vacuo. The residue was dissolved in MeOH (10 mL), then PTSA (10 mg, 0.05 mmol) was added to the above solution. The reaction was quenched by the addition of Et3N till the reaction finished, concentrated. Column chromatography of the residue on silica gel (petroleum ether-EtOAc, 2 : 1) furnished Fuzanin D (100 mg, 42% yield for 3 steps) as an amorphous solid.

[α]D CHCl3).

IR (neat): 3227, 2921, 1614, 1583, 1449, 1417, 1372, 1297, 1072, 989 cm -1 . 13 C NMR (100MHz, CDCl3): δ =153. 15, 146.72, 138.71, 134.82, 133.73, 132.49, 131.10, 127.90, 122.22, 76.03, 70.40, 18.85, 17.72

  ppm. HRMS (ESI):m/z[M -H] -calcd for C8H13O3: 157.0865; found: 157.0842.

1

  HNMR (400MHz, CDCl3): δ = 5.84-5.75 (m,1H), 5.30 (d, 1H, J = 17.2 Hz), 5.22 (d, 1H, J = 10.0 Hz), 4.47 (t, 1H, J = 6.8 Hz), 4.35-4.29 (m, 1H), 1.68-1.64 (m, 4H), 1.62-1.58 (m, 4H), 1.40-1.39 (m, 2H), 1.14 (d, 3H, J = 6.4 Hz) ppm.

1

  HNMR (400MHz, CDCl3): δ = 8.40 (d, 1H, J = 3.6 Hz), 7.44-7.37 (m, 2H), 7.04 (dd, 1H, J = 7.6, 4.4 Hz), 6.80 (d, 1H, J = 14.8 Hz), 6.55 (dd, 1H, J = 15.2, 10.8 Hz),6.00 (dd, 1H, J = 15.6, 6.8 Hz), 4.24-4.21 (m, 1H), 3.93- 3.91 (m, 1H), 2.36 (s, 3H), 1.16 (d, 3H, J = 6.4 Hz) ppm.

d InstitutParisien de ChimieMoléculaire, CNRS UMR 8232, Université Pierre et Marie Curie-Paris 6, 4 Place Jussieu, 75005Paris, France e-mail_gaoyangguang@126.com; yongmin.zhang@upmc.fr.

HNMR (400MHz, CDCl3): δ=5.99-5.91 (m, 1H), 5.74 (dd, 1H, J = 15.2, 6.0 Hz), 4.05-4.03 (m, 1H),3.95 (d,

2H, J = 7.6 Hz),3.87-3.81 (m, 1H), 2.33 (d, 1H, 4.0 Hz), 0.89 (s, 9H), 0.07 (s, 6H) ppm.13 C NMR (100MHz, CDCl3): δ=133.59, 128.30, 75.26, 71.25, 32.15, 25.84, 18.09, 17.88, -4.37, -4.81 ppm.

Acknowledgment

We are grateful tothe financial support from Natural Science Foundation of Hubei Province of China (No. 2015CFB333) for Youth Fund and Hubei Chenguang Talented Youth Development Foundation, China.

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Stéphane Le Roux

Infinite subgame perfect equilibrium in the Hausdorff difference hierarchy

Keywords: infinite multi-player games in extensive form, subgame perfection, Borel hierarchy, preference characterization, Pareto-optimality

Subgame perfect equilibria are specific Nash equilibria in perfect information games in extensive form. They are important because they relate to the rationality of the players. They always exist in infinite games with continuous real-valued payoffs, but may fail to exist even in simple games with slightly discontinuous payoffs. This article considers only games whose outcome functions are measurable in the Hausdorff difference hierarchy of the open sets (i.e. ∆ 0 2 when in the Baire space), and it characterizes the families of linear preferences such that every game using these preferences has a subgame perfect equilibrium: the preferences without infinite ascending chains (of course), and such that for all players a and b and outcomes x, y, z we have ¬(z <a y <a x ∧ x < b z < b y). Moreover at each node of the game, the equilibrium constructed for the proof is Pareto-optimal among all the outcomes occurring in the subgame. Additional results for non-linear preferences are presented.

Introduction

Game theory is the theory of competitive interactions between agents having different interests. Until the late 1960's an agent would usually represent a human or group of humans, when game theory was mainly meant for economics and political science. Then game theory was also applied to evolutionary biology [START_REF] Smith | The logic of animal conflicts[END_REF] and to theoretical computer science [START_REF] Büchi | Solving sequential conditions by finite-state strategies[END_REF], especially to system verification and system synthesis (against given specifications). Classically, the verification or synthesis problem is represented as a game with two players: the system trying to win the game by meeting the specifications, and the environment trying to win the game by preventing the system from doing so. The two players play this game in turn and deterministically on a finite or infinite directed graph, and the key notion is that of winning strategy. For a decade, though, computer scientists such as Ummels [START_REF] Ummels | Rational behaviour and strategy construction in infinite multiplayer games[END_REF] have been considering multi-player games to represent more complex verification or synthesis problems, e.g. relating to distributed systems. The notion of winning strategy is specific to two-player win-lose games, but in a multi-player setting it may be replaced with a faithful extension, namely the famous notion of (pure) Nash equilibrium. It does not only accommodate more than two players, it also allows for refined/quantitative objectives.

The deterministic turn-based games on graphs may be unfolded, usually without much loss of information, into deterministic turn-based games on finite or infinite trees, which have been widely studied in game theory. It is one reason why this article focuses on perfect information games in extensive form (i.e. played on trees) and their deterministic strategies, unless otherwise stated.

Kuhn [START_REF] Kuhn | Extensive games and the problem of information[END_REF] proved the existence of Nash equilibrium (NE) in finite games with real-valued payoffs. His proofs uses backward induction and constructs a special kind of NE that was later called subgame perfect equilibrium (SPE) by Selten [START_REF] Selten | Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit[END_REF]. An extension of Kuhn's result [START_REF] Le | Acyclic preferences and existence of sequential Nash equilibria: a formal and constructive equivalence[END_REF] characterizes the preferences that always yield finite games with NE/SPE: the acyclic preferences. Also, Escardó and Oliva [START_REF] Escardó | Selection functions, bar recursion and backward induction[END_REF] studied generalizations of backward induction in possibly infinite yet well-founded game trees, i.e. without infinite plays. The SPE have nice extra properties and are usually preferred over the more general NE: for psychology in a broad sense an SPE amounts to the absence of empty threats, and for system engineering in a broad sense it amounts to stability of a system regardless of the initial state.

The concept of infinite horizon is convenient in economics and also in computer science, e.g., for liveness. Gale and Stewart [START_REF] Gale | Infinite games with perfect information[END_REF] studied infinite two-player win-lose games, but backward induction is no longer applicable since there may not be leaves to start the induction at. Nevertheless, they proved that if the winning set of each player is open or closed (with the usual topology), one player has a wining strategy. This result was extended by Wolfe [START_REF] Wolfe | The strict determinateness of certain infinite games[END_REF] for Σ 0 2 and Π 0 2 sets, then by other people to more complex sets, and eventually by Martin for Borel [START_REF] Martin | Borel determinacy[END_REF] and even quasi-Borel [START_REF] Martin | An extension of Borel determinacy[END_REF] sets. This is called (quasi-)Borel determinacy. Mertens and Neymann [21, p. 1567] found that Borel determinacy can be used to show existence of -NE in infinite games with bounded Borel-measurable realvalued payoffs. By generalizing their technique, an abstract result about pointclasses and determinacy [START_REF] Le | Infinite sequential Nash equilibrium[END_REF] implies a characterization of the preferences that always yield games with NE, in games with (quasi-)Borel-measurable outcome functions, countably many outcomes, and an arbitrary cardinality of players: the preferences whose inverses are well-founded. Then it was shown [START_REF] Le | Infinite sequential games with real-valued payoffs[END_REF] that two-player antagonistic games (i.e. abstract zero-sum games) with finitely many outcomes and (quasi-)Borel-measurable outcome function have SPE.

When the outcome function is a continuous real-valued payoff function, Fudenberg and Levine [START_REF] Fudenberg | Subgame-perfect equilibria of finite-and infinite-horizon games[END_REF] showed that there is always an SPE in multi-player games. Similar results were obtained recently via an abstract and uniform density argument [START_REF] Le | Infinite sequential games with real-valued payoffs[END_REF]. The continuity assumption may be slightly relaxed if one is willing to accept approximate SPE. Indeed existence of -SPE was proved for lower-semicontinuous [START_REF] Flesch | Perfect-information games with lower-semicontinuous payoffs[END_REF] and upper-semicontinuous [START_REF] Purves | Perfect information games with upper-semicontinuous payoffs[END_REF] payoffs.

However, when the real-valued payoff function is discontinuous enough and the preferences are not antagonistic, there may be no ( -)SPE, as in the following example which is similar to [START_REF] Solan | Deterministic multi-player dynkin games[END_REF]Example 3]. Let a and b be two players with preferences z < a y < a x and x < b z < b y. They are alternatively given the possibility to stop and yield outcomes y and z, respectively, but the outcome is x if no one ever stops. In addition, a real-valued two-player game [START_REF] Flesch | Non-existence of subgame-perfect εequilibrium in perfect information games with infinite horizon[END_REF] was recently designed with the following features: it has a similar preference pattern as in the example above, and it has no -SPE for small enough even when the players are allowed to use mixed strategies at every node of the game tree. All this shows that Mertens [START_REF] Mertens | Repeated games[END_REF] was right when writing that "Subgame perfectness is a completely different issue" from NE. This article solves the issue partially, and the contribution is two-fold. First, it characterizes the linear preferences that always yield SPE in games with outcome functions in the Hausdorff difference hierarchy of the open sets: the preferences that are void of infinite ascending chains (of course) and of the SPE killer from the example above. Said otherwise, if

¬(z < a y < a x ∧ x < b z < b y)
holds for all players a and b and all outcomes x, y and z of a multi-player game with outcome function in the difference hierarchy (and preferences without infinite ascending chains), the game has an SPE, and even a global-Pareto one, as in Definition 7. Second contribution, the characterization still holds for twoplayer games with strict weak order preferences, but no longer for three-player games. (Strict weak orders are important since they are an abstraction of the usual preference order over the real-valued payoff functions.) Section 2 consists of definitions; Section 3 proves the characterization for many players and linear preferences; Section 4 proves the characterization for two players and strict weak order preferences.

Related works

Characterizing the preferences that guarantee existence of NE in interesting classes of games is not a new idea: earlier than the two examples above ( [START_REF] Le | Acyclic preferences and existence of sequential Nash equilibria: a formal and constructive equivalence[END_REF], [START_REF] Le | Infinite sequential Nash equilibrium[END_REF]), Gimbert and Zielonka [START_REF] Gimbert | Games where you can play optimally without any memory[END_REF] "characterise the family of payoff mappings for which there always exist optimal positional strategies for both players" in some win-lose games played on finite graphs. Also, [START_REF] Le | From winning strategy to Nash equilibrium[END_REF] characterizes the preferences that guarantee existence of NE in determined, countable twoplayer game forms.

The notion of SPE has also been studied in connection with system verification and synthesis, at low levels of the Borel hierarchy: in [START_REF] Ummels | Rational behaviour and strategy construction in infinite multiplayer games[END_REF] with qualitative objectives, in [START_REF] Brihaye | On subgame perfection in quantitative reachability games[END_REF] with quantitative objective for reachability, and in [START_REF] Brihaye | Weak subgame perfect equilibria and their application to quantitative reachability[END_REF] with quantitative objectives and a weak variant of SPE.

Finally, some specific infinite games in extensive form (such as the dollar auction) and especially their SPE have been studied using co-algebraic methods in [START_REF] Lescanne | Backward coinduction, Nash equilibrium and the rationality of escalation[END_REF] and [START_REF] Abramsky | Coalgebraic analysis of subgame-perfect equilibria in infinite games without discounting[END_REF].

Technical background

The games in this article are built on infinite trees, which may be defined as prefix-closed sets of finite sequences. The elements of a tree are called nodes. Intuitively, a node represents both a position in the tree and the only path from the root to this position. Definition 1 (Tree). Let Σ be a set.

-Σ * (Σ ω ) is the set of finite (infinite) sequences over Σ, and a tree over Σ is a subset T of Σ * such that γ δ (i.e. γ is a prefix of δ) and δ ∈ T implies γ ∈ T . -For a node γ in a tree T , let succ(T, γ)

:= {δ ∈ T | γ δ ∧ |δ| = |γ| + 1},
where |γ| is the length of γ. -A tree T is pruned if succ(T, γ) = ∅ for all γ ∈ T .

-Let T be a tree over Σ. The set [T ] is made of the infinite paths of T , namely the elements of Σ ω whose every finite prefix is in T . -Let T be a tree over Σ. For γ ∈ T let

T γ := {δ ∈ Σ * | γδ ∈ T }.
In this article the outcomes of a game correspond to some partition of the infinite paths of the game tree, and the subsets of the partition are restricted to the Hausdorff difference hierarchy of the open sets. This hierarchy is defined in, e.g., [START_REF] Kechris | Classical Descriptive Set Theory[END_REF][START_REF] Purves | Perfect information games with upper-semicontinuous payoffs[END_REF].E], but a probably-folklore result [START_REF] Le | Weihrauch degrees of finding equilibria in sequential games[END_REF]Section 2.4] gives an equivalent presentation, which is in turn rephrased in Definition 4 below. These new definitions facilitate the proofs by induction in this article. Then, the Hausdorff-Kuratowski theorem (see, e.g., [START_REF] Kechris | Classical Descriptive Set Theory[END_REF]Theorem 22.27]) implies that, in the Baire space (i.e. N N ), the difference hierarchy is equal to ∆ 0 2 , the sets that are both countable unions of closed sets and countable intersections of open sets. This equality tells how low the difference hierarchy lies in the Borel hierarchy.

For every pruned tree T , let {γ[T γ ] | γ ∈ T } be a basis for the open subsets of [T ]. Definition 2 below is a special case of a more general definition that can be found, e.g., in [11, 22.E].

Definition 2 (Difference hierarchy). Let T be a pruned tree, θ > 0 be a countable ordinal, and (A η ) η<θ be an increasing sequence of open subsets of [T ]. D θ ((A η ) η<θ ) is defined as below.

x ∈ D θ ((A η ) η<θ ) :⇔ x ∈ ∪ η<θ A η and the least η < θ with x ∈ A η has parity opposite to that of θ. -Let (γ i ) i∈I be pairwise non-comparable nodes in T , and for all i ∈ I let

And D θ ([T ]) := {D θ ((A η ) η<θ ) | ∀η < θ, A η is an open subset of [T ]}. Observation 3 D θ+1 ((A η ) η<θ+1 ) = A θ \ D θ ((A η ) η<θ ) Definition 4,
D i ∈ D([T ]) be such that D i ⊆ γ i [T γi ]. Then ∪ i∈I D i ∈ D([T ]). -The complement in [T ] of a set in D([T ]) is also in D([T ]).
One can prove Observation 5 below by induction along Definition 4.

Observation 5 Given a node γ of a pruned tree T , and

A ∈ D([T ]), γ[T γ ]∩A ∈ D([T ]). Lemma 1. D θ ⊆ D([T ]
) for all non-zero countable ordinal θ and all pruned tree T .

Proof. By induction on θ, which holds for θ = 1 since D 1 ([T ]) is made of the open subsets of [T ]. Let θ > 1 be an ordinal and let A ∈ D θ , so A = D θ ((A η ) η<θ ) for some family (A η ) η<θ of open subsets of [T ] that is increasing for the inclusion. Every A η is open so it can be written ∪ i∈I γ η,i [T γη,i ] where γ η,i and γ η,j are not proper prefixes of one another (but are possibly equal) for all η < θ and i, j ∈ I. We can also require some minimality for all γ η,i , more specifically γ γ η,i ⇒ ¬(γ[T γ ] ⊆ A η ). Let F consists of the minimal prefixes among {γ η,i | i ∈ I ∧ η < θ}, and for all γ ∈ F let f (γ) be the least η such that γ = γ η,i for some i ∈ I. So f (γ) < θ for all γ ∈ F , and

A ⊆ ∪ η<θ A η = ∪ γ∈F γ[T γ ], so A = ∪ γ∈F A ∩ γ[T γ ]. Let γ ∈ F and let us make a case disjunction to show that A ∩ γ[T γ ] ∈ D([T ]). First case, f (γ) and θ have the same parity, so A ∩ γ[T γ ] = D f (γ) ((A η ∩ γ[T γ ]) η<f (γ) ) ∈ D f (γ) ([T ]), so A ∩ γ[T γ ] ∈ D([T ]
) by induction hypothesis. Second case, f (γ) and θ have opposite parity, so

A ∩ γ[T γ ] = D f (γ)+1 ((A η ∩ γ[T γ ]) η<f (γ)+1 ) by Definition 2, = (A f (γ) ∩ γ[T γ ]) \ D f (γ) ((A η ∩ γ[T γ ]) η<f (γ) )) by Observation 3, = γ[T γ ] \ D f (γ) ((A η ∩ γ[T γ ]) η<f (γ) )) by definition of F γ and f (γ),

∈ D([T ]) by induction hypothesis and Observation 5.

Therefore A ∈ D([T ]).

Proposition 1 below shows that the quasi-difference sets coincide with the sets in the difference hierarchy for countable trees, just like quasi-Borel sets [START_REF] Martin | An extension of Borel determinacy[END_REF] and Borel sets coincide on Polish spaces.

Proposition 1. D([T ]) = ∪ θ<ω1 D θ ([T ]) for all countable pruned tree T . Proof. ∪ θ<ω1 D θ ⊆ D([T ]) was already proved in Lemma 1, so let A ∈ D([T ]). Let us prove that A ∈ D θ ([T ]) for some θ, by induction on the definition of D([T ]). Base case, if A is open, A ∈ D 1 ([T ]).
Second case, let (γ i ) i∈I be pairwise non-comparable nodes in T , and let

A i ∈ D([T ])∩γ i [T γi ] for all i ∈ I, such that A = ∪ i∈I A i . By induction hypothesis A i ∈ D θi for some θ i < ω 1 . Let θ := sup i∈I θ i , so θ < ω 1 by countability of T , and A i ∈ D θ for all i ∈ I. For all i ∈ I let (A i,η ) η<θ be an increasing sequence of open sets such that A i = D θ ((A i,η ) η<θ ). So A = ∪ i∈I D θ ((A i,η ) η<θ ) = D θ ((∪ i∈I A i,η ) η<θ ), which shows that A ∈ D θ ([T ]). Third case, A is the complement of B ∈ D θ ([T ]) for some θ < ω 1 . So B = D θ ((B η ) η<θ ) for some increasing sequence of open sets of [T ]. Let B θ = [T ], so A = D θ+1 ((B η ) η<θ+1 ).
Informally, a play of a game starts at the root of an infinite game tree and at each stage of the game the unique owner of the current node chooses a child of the node. The articles [START_REF] Le | Infinite sequential Nash equilibrium[END_REF] and [START_REF] Le | Infinite sequential games with real-valued payoffs[END_REF] used game trees of the form C * because it was more convenient and done without much loss of generality, but this article works with general pruned trees because they will be cut in a non-uniform way. Moreover pruned trees are general enough, since leaves in infinite games can be simulated by the pseudo-leaves defined below. Definition 6 (Game, subgame, pseudo-leaf ). An infinite game g is a tuple A, T, d, O, v, (≺ a ) a∈A complying with the following.

-A is a non-empty set (of players).

-T is a non-empty pruned tree (of possible finite plays).

d : T → A (assigns a decision maker to each stage of the game).

-O is a non-empty set (of possible outcomes of the game).

v : [T ] → O (uses outcomes to value the infinite plays in the tree).

-Each ≺ a is a binary relation over O (modelling the preference of player a).

For γ ∈ T , the subgame g γ is defined by dγ :

T γ → A such that d γ (δ) := d(γδ) and by v γ : [T γ ] → O such that v γ (p) := v(γp). A x-pseudo-leaf of g is a shortest node γ ∈ T such that only the outcome x occurs in g γ .
Definition 7 (strategy profile, induced play, global-Pareto equilibrium). Let g = A, T, d, O, v, (≺ a ) a∈A be a game.

-A strategy profile is a function s : T → T such that s(γ) ∈ succ(T, γ) for all γ ∈ T . Let S g be the set of the strategy profiles for g. For γ ∈ T and s ∈ S g the subprofile s γ : T γ → T γ is defined by the equality γs γ (δ) = s(γδ). -For γ ∈ T and s ∈ S g , the play p = p γ (s) induced by s at γ is defined inductively by p 0 . . . p |γ|-1 := γ and p n := s(p 0 . . . p n-1 ) for all n > |γ|. -A Nash equilibrium is a profile s ∈ S g such that

N E g (s) := ∀s ∈ S g , ∀a ∈ A, ¬(v•p (s) ≺ a v•p (s ) ∧ (∀γ ∈ T, s(γ) = s (γ) ⇒ d(γ) = a)) A subgame perfect equilibrium is a profile s ∈ S g such that N E gγ (s γ ) for all γ ∈ T . -Let O ⊆ O. One says that x ∈ O is Pareto-optimal in O if for all y ∈
O and a ∈ A such that x ≺ a y there exists b ∈ B such that y ≺ b x. A global-Pareto Nash equilibrium (GP-NE) is an NE whose induced outcome is Pareto-optimal in the outcomes occurring in the underlying game. A GP-SPE is a profile that induces a GP-NE in every subgame.

The proofs in this article do not build SPE by mere backward induction, but more generally by recursively refining rational behavioral promises. At each stage the refinement is optimal given the existing promises and regardless of the future ones. Since a promise not to choose a specific successor of a given node cannot be represent by a strategy profile, the more general notion of quasi-profile is defined below.

Definition 8 (quasi profile). Let g = A, T, d, O, v, (≺ a ) a∈A be a game.

-A quasi profile is a multivalued function q : T T such that ∅ = q(γ) ⊆ succ(γ) for all γ ∈ T . Let Q g be the set of the quasi profiles for g. For γ ∈ T and q ∈ Q g the sub-quasi-profile q γ : T γ T γ is defined by the equality γq γ (δ) = q(γδ).

-For γ ∈ T and q ∈ Q g , the tree induced by q starting at γ is defined inductively by ∈ T γ (q), where is the empty sequence, and δ ∈ T γ (q) ⇒ q γ (δ) ⊆ T γ (q). -Let q ∈ Q g . Let (γ i ) i∈I be the nodes of T such that γ i / ∈ q(γ) for all γ ∈ T , and let G(g, q) := {g γi | Tγ i (q) } i∈I .

Making a promise in a game g by defining a quasi-profile q splits the game into "smaller" games, formally via G(g, q). If the promise is rational, these "smaller" games can be processed independently since gluing any of their respective SPE will yield an SPE for g. Towards this, Observation 9 below suggests that the recursive refinement will lead to a fully defined strategy profile of g, if performed a sufficiently great (ordinal) number of times.

Observation 9 Let g be a game on a tree T , let q be a quasi profile for g, and let G(g, q) = {g γi | Tγ i (q) } i∈I . Then {γ i T γi (q)} i∈I is a partition of T .

Many players with linearly ordered preferences

This section characterizes the linear preferences that always yield SPE in games with D ω1 -measurable outcome functions: the families of preferences without infinite ascending chains and without the SPE killer, i.e. the pattern z < a y < a x ∧ x < b z < b y for some players a and b and outcomes x, y, z. The main difficulty is tackled by Lemma 2 and corollary 1 below. It consists in slightly generalizing an existing result [START_REF] Le | Infinite sequential games with real-valued payoffs[END_REF] stating that two-player Borel games with antagonist preferences have SPE by considering preferences that are almost antagonist, but in addition there is an outcome y that is the worst one for both players, and the set of plays with outcome y is a closed set (union an open set). This is then generalized for a set in D ω1 by induction, and eventually to multi-player games without the SPE killer thanks to a combinatorial result.

Lemma 2. Let a game involve two players a and b, preferences

y < a x n < a • • • < a x 1 and y < b x 1 < b • • • < b x n
for some n, such that all plays without pseudo-leaves have outcome y. The game has a global-Pareto subgame perfect equilibrium.

Proof. Let us consider only infinite games involving two players a and b, preferences

y < a x n < a • • • < a x 1 and y < b x 1 < b • • • < b x n for some n.
Let us call a game weak-stop if every play without pseudo-leaves has outcome y, and strongstop if in addition every node that does not lie on a play with outcome y has a prefix that is a pseudo-leaf. Note the following: in every weak-stop game the plays with outcome y form a closed set; for every quasi profile q for a weak-stop game g, the set G(g, q) contains only weak-stop games; and modifying the outcome function of a weak-stop game such that it is constant on given subgames yields a weak-stop game. (But the same does not hold for strong-stop games.) Let us call a node of a strong-stop game an a k (b k ) node if it is owned by player a (b), and if it is the parent of a x k -pseudo-leaf. Let us call every a k (b k ) node a a-stop (b-stop) node, and furthermore let us call every a-stop or b-stop node a stop node.

Let us prove the claim by induction on n, which holds for n = 0 and n = 1, so let us assume that 1 < n. Five transformations on games are defined below, and they are meant to be applied recursively to a weak-stop game.

1. "Weak-stop towards strong-stop": Let g be a weak-stop game on tree T .

Let γ be a node such that g γ involves more than one outcome but not y. By construction g γ is an antagonist game, and it amounts, when seeing a pseudo-leaf as a leaf, to a game without infinite plays, but possibly without uniform bound on the length of the plays. By [START_REF] Escardó | Selection functions, bar recursion and backward induction[END_REF] it has a GP-SPE s γ nonetheless, which induces some x k . Let us derive a weak-stop game g from g by modification of the outcome function: for all p ∈ [T ] let v (p) := x k if γ p and v (p) := v(p) otherwise. Pasting s γ at node γ on a GP-SPE s for g yields a GP-SPE s for g. Formally s(γδ) := s γ (δ) for all δ ∈ T γ and s(δ) := s (δ) for all δ ∈ T such that γ δ. 2. "Emptying the interior of y": Let g be a strong-stop game on tree T and let γ be a y-pseudo-leaf. If γ = all profiles for g are GP-SPE. If γ is not the root of g let us define a quasi profile q for g by letting the owner of the parent of γ ignore γ. Formally, q(δ) := succ(T, δ)\{γ} for all δ ∈ T . Since y is the worst outcome for both players, none will have an incentive to deviate from this promise, regardless of the future choices at the other nodes. G(g, q) contains g γ and a weak-stop game g . Combining any profile s γ for g γ and a GP-SPE for g yields a GP-SPE for g. 3. "b chooses x n ": Let g be a strong-stop game on tree T , let γ be a b n node and let δ ∈ succ(T, γ) be an x n -pseudo-leaf. Let us define a quasi profile q for g by letting b choose δ at γ. Formally, q(γ) := {δ} and q(α) := succ(T, α) for all α ∈ T \{γ}. Since x n is b's preferred outcome, she will have no incentive to deviate from this choice, regardless of the choices at the other nodes. So, finding a GP-SPE for every weak-stop game in G(g, q) will complete the definition of a GP-SPE for g. 4. "a ignores x n ": Let g be a strong-stop game. Let γ be a node in g such that g γ involves outcome y but no b-stop nodes, and such that every subgame of g γ involving outcome y has an a k node for some k < n. Let us define a quasi profile q for g by letting a ignore all x n -pseudo-leaves at all a n nodes below γ. The set G(g, q) is made of games involving only outcome x n and of one g such that g γ does not involve x n . By induction hypothesis g γ has an GP-SPE s γ , which induces some x k . Let us define g by modification of the outcome function of g:

for all p ∈ γ[T γ ] let v (p) := x k and for all p ∈ [T ]\γ[T γ ] let v (p) := v(p).
Pasting s γ on a GP-SPE s for g yields a GP-SPE for g. 5. "a chooses x n ": Let g be a strong-stop game. Let γ be such that g γ involves outcome y and every subgame of g γ involving y has some a n nodes but no a k node for all k < n. To build for g γ a GP-SPE s γ inducing x n on all of its subprofiles, it suffices, first, to choose arbitrary profiles for the subgames rooted at the x n -pseudo-leaves of g γ , and second, to fix consistently paths from each node to an x n -pseudo-leaf. This second step can be done by letting player a choose an x n -pseudo-leaf at some node, which defines a quasi profile q, and by repeating it recursively for the games in G(g γ , q). This s γ is a GP-SPE because every subprofile induces outcome x n , which is b's preferred outcome, and the only alternative for a in every subprofile is outcome y since all a-stop nodes are a n nodes. Let us define g by modification of the outcome function of g: for all p ∈ γ[T γ ] let v (p) := x n and for all p ∈ [T ]\γ[T γ ] let v (p) := v(p). Pasting s γ on a GP-SPE s for g yields a GP-SPE for g.

Given a weak-stop game g, let us apply to it the five transformations above, sequentially, non-deterministically whenever they are applicable, and until none of them is applicable, i.e. possibly an ordinal number of times. This yields a set G of strong-stop games (otherwise Transformation 1 could be applied) whose subgames that involve outcome y all have stop nodes (otherwise Transformation 2 could be applied), without b n nodes (otherwise Transformation 3 could be applied), such that every subgame that involves y but no b-stop nodes has a subgame without a k nodes for all k < n (otherwise Transformation 4 could be applied), and such that every subgame h of every game in G has the following property (otherwise Transformation 5 could be applied): if every subgame of h has a a-stop node, h has an a k node for some k < n.

Let h be a subgame of h ∈ G, and that involves y. If h has no b-stop nodes, combining the properties above shows that all of its subgames have astop nodes, so one of them has only a n nodes, contradiction, so every subgame of h that involves y has a b-stop node.

For every h ∈ G let us define the quasi profile q by letting a ignore all the x n -pseudo-leaves. G(h, q) is made of games involving the outcome x n only and of one h void of x n . Since every subgame of h that involves y has a b-stop node, it also holds for h . By induction hypothesis, h has a GP-SPE, which is easily extended to a GP-SPE for h, thus completing the definition of a GP-SPE for the original g. i.e. a disjoint union of clopen balls with defined by the prefixes (γ i ) i∈I . Every game g γi is antagonist and quasi-Borel, so it has an SPE s i by [START_REF] Le | Infinite sequential games with real-valued payoffs[END_REF]. Let us define g by modification of the outcome function of g : every play going through γ i yields the outcome induced by s i . This g has a GP-SPE s by Lemma 2, and together with the s i it can be used to build a GP-SPE for g.

Let us extend Corollary 1 from open union closed to the difference hierarchy.

Lemma 3. Let be a game on a tree T , with two players a and b and preferences For the inductive case, let us make a case disjunction depending on the last step of the construction of Y . For the union case, Y = ∪ i∈N γ i Y i for some Y i that have lower levels than Y in the difference hierarchy, and where the γ i are not prefixes of one another. By induction hypothesis each g γi has a GP-SPE s i inducing either y or an outcome x k(i) . Let us start the construction of a profile s for g by fixing the s i as the respective subprofiles for the g | γiTγ i . Let us define g by modification of the outcome function of g: let each play going through γ i yield the outcome induced by s i . This is a quasi-Borel game and the plays with outcome y form an open set, so it has a GP-SPE s by Corollary 1, which we use to complete the definition of s. It is easy to check that s is a GP-SPE for g.

y < a x n < a • • • < a x 1 and y < b x 1 < b • • • < b x n ,

For the complementation case, Y = [T ]\([T ]\Y )

, where [T ]\Y is equal to ∪ i∈N γ i X i for some X i that have lower levels than [T ]\Y (and Y ) in the difference hierarchy, and where the γ i are not prefixes of one another. Since all Y ∩[γ i T γi ] = [γ i T γi ]\γ i X i have lower levels than Y in the difference hierarchy, by induction hypothesis each g γi has a GP-SPE s i inducing either y or an outcome x k(i) . Let us start the construction of a profile s for g by fixing the s i as the respective subprofiles for the g | γiTγ i . Let us define g by modification of the outcome function of g: let each play going through γ i yield the outcome induced by s i . This is a quasi-Borel game and the plays with outcome y form the union of an open set and the closed set [T ]\∪ i∈I [γ i T γi ], so it has a GP-SPE s by Corollary 1, which we use to complete the definition of s. It is easy to check that s is a GP-SPE for g.

The combinatorial Lemma 4 below shows that the "local" absence of the SPEkiller amounts to a very simple "global" structure. Lemma 4. Let A be a non-empty set and for all a ∈ A let < a be a strict linear order over some non-empty set O. The following assertions are equivalent.

1. ∀a, b ∈ A, ∀x, y, z ∈ O, ¬(z < a y < a x ∧ x < b z < b y).
2. There exists a partition {O i } i∈I of O and a linear order < over I such that:

(a) i < j implies x < a y for all a ∈ A and x ∈ O i and y

∈ O j . (b) < b | Oi =< a | Oi or < b | Oi =< a | -1
Oi for all a, b ∈ A.

If 1. and 2. hold, we may also assume that

< b | Oi =< a | -1
Oi is witnessed for all i ∈ I. Also, O i is always a < a -interval for all (i, a) ∈ I × A, as implied by 2a.

Proof. 2 ⇒ 1 is straightforward, so let us assume 1. Let x ∼ y stand for ∃a, b ∈ A, x ≤ a y ≤ b x, which defines a reflexive and symmetric relation, and note that due to the SPE killer z < a y < a x ∧ x < b z < b y the following holds: if x < a y and y < b x, then x < a z < a y iff y < b z < b x. To show that ∼ is transitive too, let us assume that x ∼ y ∼ z. If x, y, z are not pairwise distinct, x ∼ z follows directly, so let us assume that they are pairwise distinct, so by assumption there exist a, b, c, d ∈ A such that y < a x < b y < c z < d y. To show that x ∼ z there are three cases depending on where z lies with respect to y < a x, all cases invoking the (above-mentioned) forbidden-pattern argument: if y < a z < a x then x < b z < b y, and x ∼ z follows; if y < a x < a z then z < d x < d y and subsequently y < c x < c z, by invoking twice the forbidden-pattern argument, and x ∼ z follows; third case, let us assume that z < a y < a x. If x < b z then x ∼ z follows, and if z < b x then z < b x < b y, so y < c x < c z, and x ∼ z follows. Therefore ∼ is an equivalence relation; let {O i } i∈I be the corresponding partition of O. Now let us show that the ∼-classes are < a -intervals for all a, so let x ∼ y and x < a z < a y. By definition of ∼, there exists b such that y < b x, in which case y < b z < b x by the forbidden-pattern argument, so x ∼ z by definition.

Let x ∈ O i and y ∈ O j be such that x < a y. Since O i and O j are intervals, x < a y for all x ∈ O i and y ∈ O j . Since ¬(x ∼ y ) by assumption, x < b y for all b ∈ A, by definition of ∼. In this case defining i < j meets the requirements.

Before proving 2b let us prove that if x < a y and y < b x and z ∼ y, then z < a y iff y < b z: this is trivial if z equals x or y, so let us assume that x = z = y, and also that z < a y. If z < b y, then y < c z for some c since z ∼ y, and wherever x may lie with respect to y < c z, it always yields a SPE killer using < a or < b , so y < b z. The converse is similar, it follows actually from the application of this partial result using < -1 b and < -1 a instead of < a and < b . Now assume that < b | Oi =< a | Oi for some O i , so x < a y and y < b x for some x, y ∈ O i . Let z, t ∈ O i . By the claim just above z < a y iff y < b z, so by the same claim again z < a t iff t < b z, which shows that

< b | Oi =< a | -1
Oi . This proves the equivalence.

Finally, let us assume that the assertions hold. By definition of ∼, if O i is not a singleton, x < a y and y < b x for some a, b ∈ A and x, y

∈ O i , so < b | Oi =< a | -1
Oi is witnessed.

Theorem 1 extends Lemma 3 to many players and more complex preferences. Theorem 1. Let g be a quasi-Borel game with players in A, outcomes in O, and linear preferences < a for all a ∈ A. Let us assume that the inverses of the < a are well-ordered, and that there exists a partition {O i } i∈I of O and a linear order < of I such that:

-i < j implies x < a y for all a ∈ A and x ∈ O i and y ∈ O j . -< b | Oi =< a | Oi or < b | Oi =< a | -1
Oi for all a, b ∈ A and i ∈ I.

Let us further assume that for all i ∈ I the plays with outcome in ∪ j<i O j form a D ω1 set. Then g has a global-Pareto subgame perfect equilibrium.

Proof. By Lemma 4 let us further assume wlog that

< b | Oj =< a | -1
Oj is witnessed for all j ∈ I, so the O j are finite by well-ordering. Let us build a GP-SPE for g as the limit of a recursive procedure: Let O i be such that some outcome of O i occurs in g and such that for all j > i no outcome from O j occurs in g. Let (γ k ) k∈K be the shortest nodes of g such that the outcomes occuring in g γ k are in ∪ j<i O j only. Let us define a quasi profile q for g by having the γ k ignored by their parents. G(g, q) consists of the g γ k and of a game g . By Lemma 3 there is a GP-SPE s for g . (To see this, replace ∪ j<i O j with one single fresh outcome y, i.e. y / ∈ O and set y < a x for all a ∈ A and x ∈ O i .) Combining s with GP-SPE for the g γ k (obtained recursively) yields a GP-SPE for g, since the choices made in g hold regardless of the choices made in the g γ k .

Corollary 2. Let A and O be non-empty finite sets (of players and of outcomes) and for all a ∈ A let < a be a linear preference. The following are equivalent.

1. ∀a, b ∈ A, ∀x, y, z ∈ O, ¬(z < a y < a x ∧ x < b z < b y).
2. Every D ω1 -Gale-Stewart game using A, O and the < a has a GP-SPE.

Proof. For 1. ⇒ 2. invoke Lemma 4 and Theorem 1, and prove 2. ⇒ 1. by contraposition with the following folklore example which is detailed, e.g., in [START_REF] Le | Infinite sequential games with real-valued payoffs[END_REF]. Proof. 1. follows Theorem 2. For 2. let ≺ a be a linear extension of ≺ a . If x ≺ a y then z ≺ a x ≺ a y and y ≺ b z ≺ b x. If y ≺ a x then t ≺ a y ≺ a x and y ≺ b x ≺ b t.

Two players with strict weak order preferences

The preferences considered in Proposition 2 are strict weak orders. Informally, strict weak orders are strict partial orders that can be seen as strict linear orders from afar, i.e. up to an equivalence relation. Traditionally in game theory the outcomes are real-valued payoff functions f, g : A → R and the preferences are defined by f ≺ a g iff f (a) < g(a). These preferences are not strict linear orders but they are strict weak orders, so the results from Section 3 are worth generalizing. Strict weak orders are defined below.

Definition 10 (Strict weak order). A strict weak order is a strict partial order whose complement is transitive, i.e. is satisfies ¬(x ≺ x) and x ≺ y ∧ y ≺ z ⇒ x ≺ z and ¬(x ≺ y) ∧ ¬(y ≺ z) ⇒ ¬(x ≺ z).

Lemma 4 above describes the structure of strict linear orders void of the SPE killer. A similar result for strict weak orders will be useful. Lemma 5 below is part to it, and the other part appears directly in the proof of Lemma 6.

∈ O u × O l , then ≺ a ∩ ≺ b = ∅. 2. If ≺ a ∩ ≺ b = ∅, there exists a linear extension < of ≺ a such that ≺ b ⊆< -1 .
Proof. 1. Let x be ≺ b -minimal among the ≺ a -maximal elements, and let y be ≺ a -minimal among the ≺ b -maximal elements, so x = y by the partition assumption. Towards a contradiction let us assume that, e.g., x is not ≺ bminimal, and let z be ≺ b -minimal. For all t ≺ b x, it follows that t is not ≺ a -maximal by definition of x, and ¬(y ≺ a t) by absence of the SPE killer. So y is not ≺ a -minimal, otherwise z is also ≺ a -minimal, thus contradicting the partition assumption. So likewise, for all t ≺ a y, it follows that t is not Proof. By induction on the number of outcomes, which holds up to two outcomes. Let us make a case disjunction for the inductive case. First main case, there is no partition

{O u , O l } of O such that ∀(x, y) ∈ O u × O l , ¬(x ≺ a y) ∧ ¬(x ≺ b y).
Let us make a nested case distinction. First nested case, there exists a ≺ a -nonextremal element and a ≺ b -non-extremal element, so by Lemma 5 let < be a linear extension of ≺ a such that ≺ b ⊆< -1 . By [START_REF] Le | Infinite sequential games with real-valued payoffs[END_REF] the antagonist game with preference < has an SPE, which is also an SPE for ≺ a and ≺ b .

Second nested case, one preference, e.g., ≺ a has only extremal elements. By the partition assumption let y be ≺ b -maximal and ≺ a -minimal. Let Y be the set of plays with outcome y, and let us define a quasi-profile q as follows. Let γ be the parent of a y-pseudo-leaf. If γ is owned by a, let a ignore the y-pseudo-leaves at γ; otherwise let b choose a y-pseudo-leaf at γ. Let us apply this construction recursively (an ordinal number of times) to the games in G(g, q) that do not involve y, until the Y of each remaining game has empty interior. It is easy to check that Y is also closed since Y is the union of an open set and a closed set, by assumption. So let (γ i ) i∈I be the shortest nodes that are not on any play with outcome y. The g γi do not involve y, so by induction hypothesis they have suitable SPE s i inducing some x i ∈ O, which allow us to start the definition of a suitable SPE for g. Let us define g by modification of the outcome function of g: let v (γ i p) := x i for all i ∈ I and let v (p) := v(p) when γ i p for all i ∈ I. Let M a be the ≺ a -maximal outcomes, and let us define a quasi-profile q for g as follows. Let γ be the parent of a M a -pseudo-leaf, i.e. a shortest node involving outcome in M a only. If γ is owned by a, let a choose a M a -pseudo-leaf at γ; otherwise let b ignore the M a -pseudo-leave at γ. Let us apply this recursively to the games in G(g , q) that involve y. In the remaining games a can deviate from a play with outcome y only to reach a (O\)M a -pseudo-leaf, so every profile that follows plays with outcome y whenever possible is a suitable SPE.

Second main case, there exist partitions {O u , O l } of O such that ∀(x, y) ∈ O u × O l , ¬(x ≺ a y) ∧ ¬(x ≺ b y). Among these partitions let us consider the one with the smallest possible O u . Let us make a further case disjunction. First case, |O l | > 1. As is now customary, let us start defining a suitable SPE for g by using the induction hypothesis on the maximal subgames involving only outcomes in O l , and on the game derived from g by replacing outcomes in O l with a fresh outcome y that is the new ≺ a and ≺ b -minimum.

Second case,

|O l | = {y}. By minimality of |O u |, there is no partition {O uu , O ul } of O u such that ∀(x, y) ∈ O uu × O ul , ¬(x ≺ a y) ∧ ¬(x ≺ b y).
Therefore the situation is reminiscent of the first main case above, but for ≺ a | Ou and ≺ b | Ou instead of ≺ a and ≺ b . In both nested cases from the first main case, there exists some x n that is, e.g., ≺ a | Ou -minimal and ≺ b | Ou -maximal. Applying the proof of Lemma 2 almost verbatim yields a suitable SPE for g Theorem 2. Let g be a game with two players a and b, finitely many outcomes, a D ω1 -measurable outcome function, strict weak order preferences such that ¬(z ≺ a y ≺ a x ∧ x ≺ b z ≺ b y) for all outcomes x, y and z. Then the game has an SPE. Furthermore, for every node γ of g let {O γ 1 , . . . , O γ nγ } be a partition of the outcomes of g γ such that ¬(x ≺ a y) and ¬(x ≺ b y) for all 1 ≤ k < n γ and (x, y) ∈ O γ k+1 × O γ k . There exists an SPE for g such that the outcome induced at every node γ belongs to O γ nγ .

Proof. By induction on the levels in the difference hierarchy of the sets of plays corresponding to the outcomes. The base case holds by Lemma 6.

For the inductive case, let y be an outcome whose corresponding set Y has level more than one in the difference hierarchy, and let us make a case disjunction depending on the last step of the construction of Y . The remainder of the proof can be taken almost verbatim from the proof of Lemma 3, but by replacing "GP-SPE" with "suitable SPE", and by invoking Lemma 6 or the induction hypothesis instead of Corollary 1.

Theorem 2 considers two-player games only. Observation 11 shows that absence of the SPE killer is no longer a sufficient condition for a three-player game with strict weak order preferences to have an SPE. Proof. For 2. Towards a contradiction let us assume that there exists an SPE for the game. Let us consider a node where player a chooses y. Then at the node right above it c chooses to continue to benefit from y, and at the node above b chooses to continue, too. The induced outcome at the node further above is y regardless of the choice of a, and so on up to the root. Let us make a case disjunction: first case, there exists infinitely many nodes where a chooses y, so b and c always continue by the remark above, so a has an incentive to continue too, to induce outcome x, contradiction. Second case, there exists a node below which a always continues. From then on, one player must stop at some point, otherwise the outcome is x and c has an incentive to stop. The first player to stop cannot be b, otherwise a would stop before b, and it cannot be c, otherwise b would stop before c, contradiction.

  Lemma 1, and Proposition 1 relate to [17, Section 2.4]. Definition 4 (Quasi-difference sets). Let T be a pruned tree. The set D([T ]) is defined by transfinite induction below. -Every open set of [T ] is in D([T ]).

Corollary 1 .

 1 Given a game g with two players a and b, a quasi-Borel measurable outcome function, and preferencesy < a x n < a • • • < a x 1 and y < b x 1 < b • • • < b x n for some n.If the plays with outcome y form the union of an open set and a closed set, the game has a global-Pareto subgame perfect equilibrium. Proof. Let the plays with outcome y be the union Y = Y o ∪Y c of an open set and a closed set. Wlog Y o and Y c are disjoint. Let us derive g from g by removing the plays in Y o . So the plays of g that do not yield outcome y form an open set,

  and let us assume that each set of plays with outcome x i is quasi-Borel and that Y the set of plays with outcome y is in D ω1 ([T ]). Then the game has a global-Pareto subgame perfect equilibrium. Proof. By transfinite induction on the level of Y in the difference hierarchy. The base case where Y is open or closed is solved by Corollary 1.

Proposition 2 .

 2 Corollary 2 and the results that lead to it considers linear preference only. Proposition 2 below show that this restriction incurs a loss of generality, which is partly solved in Section 4. Let us define two binary relations by z, t ≺ a x, y and y ≺ b z ≺ b x ≺ b t. 1. D ω1 infinite games with players a and b and preferences ≺ a and ≺ b have SPE. 2. The SPE killer occurs in any strict linear extensions of ≺ a and ≺ b .

Lemma 5 .

 5 Let ≺ a and ≺ b be two strict weak orders over some finite O.1. If ≺ a and ≺ b are void of the SPE killer, if there exists a ≺ a -non-extremal element and a ≺ b -non-extremal element, and if there is no partition {O u , O l } of O such that ¬(x ≺ a y) and ¬(x ≺ b y) for all (x, y)

Lemma 6 .

 6 ≺ b -maximal by definition of y, and ¬(x ≺ b t) by absence of the SPE killer. So the two-element partition induced by {t ∈ O | t ≺ a y ∨ t ≺ b x} contradicts the partition assumption. This shows that x is ≺ b -minimal and y is ≺ aminimal. Towards a contradiction let us assume that t(≺ a ∩ ≺ b )z for some t, z ∈ O. So {x, y} ∩ {z, t} = ∅. By the partition assumption z is not both ≺ a and ≺ b -maximal, so, e.g., z ≺ a x, and t is not both ≺ a and ≺ b -minimal. By absence of the SPE killer t is ≺ b -minimal, so y ≺ a t by the partition assumption., and subsequently z is ≺ b -maximal. By assumption there exists γ that is neither ≺ b -maximal nor ≺ b -minimal. Wherever γ lies wrt ≺ a , the SPE killer occurs. 2. By induction on the cardinality of O, which holds for |O| = 0. Let x be ≺ b -minimal among the ≺ a -maximal elements, so x is also ≺ b -minimal since ≺ a ∩ ≺ b = ∅. By induction hypothesis let < x witness the claim for ≺ a | O\{x} and ≺ b | O\{x} . The linear order < := < x ∪{(y, x) | y ∈ O\{x}} witnesses the claim.Lemma 6 below is a generalization of Corollary 1 from an order-theoretic point of view and a special case thereof from a topological point of view. Due lack of space and strong similarities with the proof of Lemma 2, the proof of Lemma 6 is in appendix. Let g be a game with two players a and b, finitely many outcomes O, and strict weak order preferences void of the SPE killer. If each outcome corresponds to the union of an open set and a closed set, the game has an SPE. Furthermore, for every node γ of g let {O γ 1 , . . . , O γ nγ } be a partition of the outcomes of g γ such that ¬(x ≺ a y) and ¬(x ≺ b y) for all 1 ≤ k < n γ and (x, y) ∈ O γ k+1 × O γ k . There exists an SPE for g such that the outcome induced at every node γ belongs to O γ nγ .

Observation 11

 11 Let three players a,b and c have preferences z ≺ a y ≺ a x and t ≺ b z ≺ b y and x ≺ c t ≺ c y. (and, e.g., y ∼ a t, z ∼ b x, and y ∼ c z or x ∼ c z) 1. The SPE killer does not occur in the strict weak orders ≺ a and ≺ b and ≺ c . 2. The following game with ≺ a and ≺ b and ≺ c has no SPE.

Proposition 3 Proposition 3 .

 33 below shows that considering only strict weak orders incurs a loss of generality. Let us define two binary relations by γ ≺ a y ≺ a x and z ≺ a β ≺ a α and x ≺ b z ≺ y and α ≺ b γ ≺ b β. 1. The SPE killer occurs in every strict weak order extensions of ≺ a and ≺ b . 2. D ω1 -games with players a and b and preferences ≺ a and ≺ b have SPE. Proof. 1. Let ≺ a be a strict weak order extension of ≺ a . If z ≺ a y, the SPE killer occurs, with x. If ¬(z ≺ a y), then γ ≺ a β and the SPE killer occurs, with α. 2. It suffices to prove the claim for games where each outcome set is the union of an open set and a closed set. (Then using a transfinite induction as in the proof of Lemma 3 will do.) The techniques from Lemmas 2 and 6 are suitable here, and used without details. If the outcome x does not occur in the game, note that ≺ a | O\{x} and ≺ b | O\{x} can be extended into the strict weak orders z ≺ a γ ∼ a β ≺ a y ∼ a α and z ∼ b α ≺ b γ ≺ b y ∼ b β, respectively, and that the SPE killer is absent from these. So by Theorem 2 the game has an SPE wrt ≺ a and ≺ b and therefore also wrt ≺ a and ≺ b . (And likewise if the outcome α does not occur in the game.) Now one can reduce the set for x to a closed set by letting a choose the clopen balls with constant outcome x and by letting b ignore them. The plays with outcomes different from x can be seen as belonging to a union of subgames without x, so by the remark above they have SPE. It allows us to replace these subgames with (pseudo)-leaves with outcomes the ones induced by the SPE. Now one can let a choose the pseudo-leaves with outcome α and b ignore them, which yields a game without outcome α. So by the remark above there is an SPE for the game.

Acknowledgements

I thank Vassilios Gregoriades and Arno Pauly for useful discussions. The author is supported by the ERC inVEST (279499) project.