ORCID Claire Carvallo

Nathaly Godoy

Bertha Aguilar Reyes

Claudia Egas

Raul Fuentealba

Bertha Aguilar

Raúl Fuentealba

ORCID Margarita Préndez

Claudia Egas Egas

Orcid Id

Raúl Fuentealba

Long-term monitoring of atmospheric pollution in the Maritime Antarctic with the lichen Usnea aurantiaco-atra (Jacq.) Bory: a magnetic and elemental study

Keywords: environmental magnetism, biomonitoring, anthropogenic particulate matter, elemental composition, Fildes Peninsula, King George Island

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Antarctica continent covers an area of 14062500 km 2 of which only 0.32% of its total area is free of ice. Vegetation grows on a constrained basis in coastal sectors free from ice and snow during the summer and in some rock outcrops located inland.

Despite its remote situation and extreme conditions, human presence has existed in Antarctica for more than 200 years associated to whaling activity [START_REF] Tin | Review Impacts of local human activities on the Antarctic environment[END_REF]) and for more than 70 years associated to science and tourism. Nowadays, the human presence is represented by 112 research bases throughout the continent, which accommodate over 5000 people during the summer. In winter, 40 stations are in operation, which could accommodate 3916 people, who need water, fossil fuel consumption and management of wastes. Moreover, tourism has been continuously increasing; this caused the visit of over 50000 people during the season 2017-2018, with obvious environmental implications.

King George Island, Antarctic Peninsula, Maritime Antarctica stands out as "hot spot" receiving increasing pressure from human activities. Fildes Peninsula with about 29 km 2 of surface has a high biodiversity and is a major logistic centre for the northern Antarctic Peninsula; it supports the largest number of scientific bases (5 permanent from Chile, China, Russia, Uruguay) and shelters and an airfield for aircrafts arriving from the city of Punta Arenas, Chile.

There is no monitoring of air quality and certainly no inventory of sources in the Fildes Peninsula [START_REF] Tin | Review Impacts of local human activities on the Antarctic environment[END_REF]. Studies by [START_REF] Mishra | Aerosol composition and its sources at the King Sejong Station, Antarctic peninsula[END_REF] on total atmospheric aerosols (TSP, suspended particles smaller than 45-50 µm) from King Sejong base (Barton Peninsula) indicate that presence of the elements Bi, Cd, Co, Cr, Cu, Ni, V and Zn is caused by anthropogenic activities associated with local sources. For Fildes peninsula, information is available on fractionated aerosols smaller than 3 µm. The elements Na, Mg, Sr, Mn, Fe, Cr, Co and Ca have a natural origin and Pb, Ni, Zn, Cu and Cd, on the contrary, have an anthropogenic origin. The elements Pb, Cr, Sb, Cu and Mn are partially of extra continental origin. Ca, Ba, Ni and partially Zn and Cu are generated by the human activities in the island [START_REF] Préndez | PM 2.5 Aerosols Collected in the Antarctic Peninsula with a Solar Powered Sampler during Austral Summer Periods[END_REF][START_REF] Carrasco | Element distribution of some soils of continental Chile and the Antarctic Peninsula. Projection to atmospheric pollution[END_REF].

Mosses and lichens constitute the main biotic components of terrestrial ecosystems in Antarctica. Exposed biological surfaces, e.g., lichens, mosses and leaves, accumulate atmospheric particles, and are able to provide a record of location-specific and timeintegrated information on local air quality [START_REF] Bajpai | Suitability of lichens to monitor climate change[END_REF]. The relatively high humidity and mild temperatures found in maritime Antarctica, together with the absence of competition from higher plants, create the perfect place for stress-tolerant organisms such as lichens [START_REF] Sancho | Ecología vegetal en la Antártida[END_REF]. Lichens are good bio-accumulators of chemical elements transported by air [START_REF] Préndez | Cuantificación de elementos pesados en un potencial biomonitor de la calidad del aire en la península antártica[END_REF][START_REF] Sancho | Slowest to fastest: Extreme range in lichen growth rates supports their use as an indicator of climate change in Antarctica[END_REF]) and can be used as bio-monitors of the atmospheric deposition of inorganic pollutant also in Antarctica (Poblet et al.1997). Nutrients and/or material derived from the humid and dry atmospheric precipitation deposit can accumulate on or around the thallus. The incorporation of elements through the thallus surface depends on many factors such as the nature of the element, morphologic characteristics of the surface, and environmental parameters [START_REF] Wolterbeek | Biomonitoring of trace element air pollution: principles, possibilities and perspectives[END_REF]. When using lichens as bioindicators, it is necessary also to evaluate the contribution of the substrate in which they grow in order to determine only the atmospheric contribution in nutrition and accumulation of contaminants in the thallus. [START_REF] Poblet | The use of epilithic Antarctic lichens (Usnea aurantiaco-atra and U. antarctica) to determine deposition patterns of heavy metals in the Shetland Islands, Antarctica[END_REF] have shown that Usnea aurantiaco-atra (Jacq.) Bory (currently Neuropogon aurantiaco-ater) is a convenient organism for developing a biomonitoring of metals in atmospheric aerosols in the area of the South Shetland Islands.

The United States Forest Service (2010) provides an extensive database of air quality in relation to biomonitoring through lichens with tables including threshold levels of chemical elements present in Oregon, Washington and in Alaska's national forests. In our knowledge, there is no similar database for Antarctica.

Atmospheric pollution, in particular urban particulate matter (PM), often contains levels of magnetic minerals, e.g., iron oxides like magnetite, hematite and maghemite (e.g., [START_REF] Matzka | Magnetic biomonitoring of roadside tree leaves: identification of spatial and temporal variations in vehicle-derived particulates[END_REF] that are easily measurable magnetically. This PM can accumulate on exposed biological surfaces and deliver accumulated information on local air quality. A group of trace metals, such as Zn, Cd, Pb, Ni, Cu and Cr, are often directly associated with magnetic PM, due to their incorporation in the mineral structure during combustion processes [START_REF] Cao | Magnetic response to air pollution recorded by soil and dust-loaded leaves in a changing industrial environment[END_REF]. Therefore, the magnetic signal may act not only as a PM proxy but be of direct, often health-related, interest in itself.

Only a few magnetic measurements have been carried out on lichens, in situ or as transplants (see [START_REF] Hofman | Biomagnetic Monitoring of Atmospheric Pollution: A Review of Magnetic Signatures from Biological Sensors[END_REF] for a review). These studies show that magnetic properties are good proxies for anthropogenic atmospheric pollution, though they are species-dependent.

The aim of this study was to evaluate the use of U. aurantiaco-atra lichen as bioindicator of air quality in the Fildes Peninsula over a period of 13 years (1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010) using magnetic and elemental techniques.

Sampling

Five-g samples of U. aurantiaco-atra were carefully collected in some summer/fall seasons between 1997 and 2010. Samples were taken from 1*1 m 2 demarcated and protected sites (Table I), then air-dried in Antarctica, stored in clean paper bags, transported to the laboratory for analyses, and kept isolated from external influences.

Lichen thalli, when dried, do not decompose. In the laboratory, duplicates of all lichen samples were sorted, using only the thinnest thalli, and dead and senescent tissue, superficial dust, soil particles, and small stones were manually removed using a Teflon clamp. Samples were dried at 90ºC for 24 h, mashed in an agate mortar, and stored in sterile Petri dishes to provide samples to the different analysis.

In parallel, 2 or 3 samples of the substrate (first 5 cm of the substrate) were taken together with the lichens. Composite substrate samples were assembled and were airdried in Antarctica and packed in paper bags further stored in plastic containers at -18 C°.

We lack samples of substrate or lichens for some years and/or sites, not because the samples were not taken, but because they were used in other analyzes over the years and there was no more left.

Sampling sites

King George Island is located between 61°50' and 62°15´S and 57°30 and 59°01´W; it belongs to the South Shetlands Islands and is located near the northern part of the Antarctic Peninsula (Fig. 1). The Collins Glacier covers practically the entire island. The rest of the surface, mostly along the coast, is composed of volcanic rocks (basalts and/or andesites) which form cliffs about 100 m high. The substrate is composed of material with coarse granulometry.

The sampling points are localized in sites close to potential human emission sources, such as the aerodrome and scientific bases, and more isolated sites where there should be less human influence, such as the Ardley peninsula and the Collins Glacier (Fig. 1).

The sites that are close to emission sources are: Site Aerodromo, located near the Aerodrome Teniente Rodolfo Marsh, which is used by many the scientific bases on the island; Site LARC, located next to the Antarctic Laboratory of Cosmic Radiation, University of Chile that is installed on top of a hill of about 40 m above sea level (a.s.l), close to two Chilean bases; Site Base China which is located about 100 m southeast of the Great Wall Chinese base station, under construction/extension all along the sampling period; Site Aerodromo MET, located in the vicinity at east side of the Aerodrome T.R.

Marsh.

The sites chosen far away from possible anthropogenic emissions are: Site Ardley, located on a small peninsula located southeast of Fildes and connected by a sand dam, which is a nesting site for a wide variety of birds and one of the most important habitats of penguin, with a rich and extensive vegetation cover; Site Collins, located at the foot of the glacier Collins in an ice-free area about 10-15m from the sea.

Two more sites were chosen as intermediate: Sites Cerro Fosil and Cerro GPS, located on top of a small hill of about 25 m a.s.l., in the vicinity west of the Aerodrome T.R.

Marsh, separated from it by no more than 100 m.

Analytical techniques and methods

Magnetic techniques

Magnetic measurements were carried out with two objectives: to quantify the magnetic minerals through measurements of magnetic susceptibility and remanent magnetization (M r ), and to obtain magnetic parameters that might give information about the nature of magnetic minerals and their magnetic grain size.

In order to perform magnetic measurements, each sample of lichen and substrate was crushed into a powder and tightly packed into a gelcap. A lichen gelcap contains 150-200 mg of material, while a substrate gelcap contains between 300 and 400 mg of material. Then, the gelcaps were used for measurements with two different equipment.

Firstly, the magnetic susceptibility, which is the ability of a sample to be magnetized in a weak field, was measured using an AGICO Kappabridge KLY-3 at the Institut de Physique du Globe de Paris (IPGP). The same samples were then used for the measurement of hysteresis parameters and First-Order Reversal Curve (FORC) diagrams with a Vibrating Sample Magnetometer (VSM) from Lakeshore, which is part of the Mineral Magnetism Analysis Platform situated of IPGP-IMPMC. More detail on the measurement of these properties can be found in the Supplementary Information.

Scanning Electron Microscopy with X-ray Energy Dispersive Spectrometry

This technique allows visualization of the external morphology of the sample. To obtain the image of the analysed surface, the sample is swept with an electron beam in a raster pattern. In the case of MEB-EDS, the backscattered electrons are used to generate the image. An X-ray detector is coupled to this system to perform an elemental qualitative and quantitative analysis of the sample.

The analysis was carried out in the microscopy laboratory of the ENES Morelia unit of the Universidad Nacional Autónoma de México (UNAM). Substrate and stem fragments from lichen samples were analysed using a JSM IT300 scanning electron microscope, JEOL brand with EDS X-Max detector with 20 mm 2 window. The substrate samples were metalized with a gold coating. The analysis was done under low vacuum conditions (50 Pa) for the lichen samples and high vacuum (10 -4 Pa) for the substrate samples. Elemental analysis using X-ray fluorescence (EDS) was done on some individual particles.

Instrumental Neutron Activation Analysis

Elemental quantification by Inductive Coupled Plasma-Optical Emission Spectrometry (ICP-OES)

We selected the elements to be quantified by Inductive Coupled Plama-Optical Emission Spectrometry (ICP-OES) according to these 4 criteria: a) Potentially toxic chemical elements; b) Probability of finding them in the study area in concentrations quantifiable by the technique used, based on previous research; c) Bio accumulative chemical elements; d) Elements of known anthropic origin. Therefore, the quantified elements, both for lichens and for the substrate, were Ag, Al, As, B, Ba, Be, Cd, Co, Cr, Cu, Fe, Mn, Mo, Ni, Pb, Se, V and Zn. However, many of them were found under detection limits (UDL) of the technique. The experimental procedure is detailed in the SI section.

Results

Magnetic measurements

According to the Day-diagrams, hysteresis parameters for the lichen samples (1997 to 2010) all plot in the pseudo-single-domain (PSD) region, between the single-domain (SD)-multidomain (MD) (MD percentage >60%) and the SD-superparamagnetic (SP) mixing lines (SP percentage >30%) (Fig. 2a).

More than 3/4 of the M r /M s ratios are between 0.1 and 0.2, and 85% for the H cr /H c ratios are between 3 and 5. Most of the points that are not in this region are from Ardley site, and they are characterized by a smaller magnetic grain size, with points closer to the SD-SP mixing line, with a higher SP percentage. This tends to show that magnetic particles present on all the sites, except Ardley, have a similar origin. On the contrary, the hysteresis parameters from the substrate samples plot slightly closer to the SD-SP mixing line than the lichen samples, but they are still in the PSD region (Fig. 2b).

Samples from Ardley have a behaviour different from the rest of the samples, identically to what we observed in the lichens, and they plot closer to the SP region.

These results are further confirmed with the good correlation between magnetic susceptibility and M r in the case of the lichen samples (Fig. 3a), where the determination coefficient R 2 =0.889 (95% significance level), but for the substrate samples, there is only little correlation between magnetic susceptibility and remanent magnetization (R 2 =0.378), showing that the magnetic sources in these samples are diverse (Fig. 3b).

The magnetic signal in the substrate is likely caused not only by the anthropogenic signal, but it also takes into account geological sources.

The variation of M r with time shows an increase in M r values in 1997 (when there is available data) and another one in 2006-2007(Fig. 4a). Susceptibility values show a large variation, from almost zero to 2.5×10 -6 SI m 3 /kg. The variation of magnetic susceptibility is similar, again with a peak in 1997 and 2006-2007 (Fig. 4b).

In order to study the spatial variation of magnetic parameters, we plotted the remanent magnetization values and the susceptibility values for lichen samples from 2010 on a cartographic representation of the sites (Fig. 5). That year corresponds to a sampling of all the sites and the results do not seem to be affected by the external causes that increased the magnetization and susceptibilities everywhere. The lowest values correspond to samples from the Collins ice sheet and Ardley peninsula, where there is little human activity. On the contrary, the sites with the highest magnetic susceptibility are the sites close to the airport and the site of Cerro Fósil in particular.

Unfortunately, the only sampling years for which we have substrate samples available for this study are 1997, 1998 and 2006, which are the years corresponding to the possible occurrences of external sources of magnetic input. It is therefore difficult to make any interpretation on the variations of magnetic parameters with time (Fig. SI-2).

FORC diagrams-Comparison magnetic signal lichen/substrate

Because they are calculated with a second derivative of magnetization, FORC diagrams are extremely sensitive to noise, and therefore can only be measured on the magnetically strongest samples. We measured FORC diagrams on the strongest lichen samples (most of them from the 1997 sampling) and on corresponding lichen and substrate samples whenever available. show FORC diagrams measured on the substrate samples. The first three pairs were measured on corresponding lichen and substrate samples. Firstly, patterns characteristic of a SD-PSD magnetic grain size are visible on all FORC diagrams. The SD contribution is identified with the high coercivity tail, and the PSD contribution by the peak of FORC distribution close to the origin of the diagram and the fact that most contours do not close but instead they intersect the vertical axis. The spread along the vertical axis is quite limited, except for sample ALL97 (Lichen-LARC 1997), which shows that magnetic interactions are moderate.

Elemental concentrations in the substrate determined by INAA and ICP-OES

Elemental concentrations determined by SEM-EDS

Figure 8 shows examples of SEM micrographs and semi-quantitative EDS analysis corresponding to individual particles that contain heavy elements, and therefore could carry the magnetic signal as well as heavy metals, both in lichen and the substrate. SEM micrographs of lichens (Fig. 8a-b) show the presence of particles of a wide range of sizes (from sub-micrometres to several tens of micrometres), dispersed in the sample.

EDS analysis in lichens all show the presence of natural elements such as Si, Al, Mg.

Among the possible anthropogenic elements, Cr is present on two sites sampled in 1997 (Collins and Base China). Small amounts of K, P and V are also measured in various sites and sampling years. Calcium is also present in small amounts but widely throughout the different sites.

Angular particles between 1-20 µm can be seen on SEM micrographs of substrate samples (year 1997) (Fig. 8c-d). EDS measurements also show the presence of the same natural elements (Si, Al, Mg) that were identified in the lichen samples. Titanium is also present in the soil; the origin of this is probably natural. Other elements present in small amounts are Ca, V and K. These elements were also present in small amounts in the lichens.

Elemental concentrations in lichens determined by ICP-OES

Discussion

Magnetic measurements

The average magnetic susceptibility in lichen samples for the year 2010 is 7.5×10 -8 m 3 /kg, and the average SIRM for that same year is around 1 to 10 mAm 2 /kg. These values are one to two orders of magnitude lower than the values measured on lichens from Argentinian cities by [START_REF] Chaparro | Biomonitors of urban air pollution: Magnetic studies and SEM observations of corticolous foliose and microfoliose lichens and their suitability for magnetic monitoring[END_REF], and about the same order of magnitude as values measured on lichens sampled close to a cement factory in Slovakia [START_REF] Paoli | Vitality of the cyanolichen Peltigera praetextata exposed around a cement plant (SW Slovakia): a comparison with green algal lichens[END_REF]. For the years 1997 and 2006-2007, when both susceptibilities and SIRM values are the highest, the SIRM values are comparable to the lowest values measured in these three studies, and susceptibilities are comparable to the highest values. However, it should be kept in mind that the lichen species are all different in all these studies so direct comparison with these previous results is difficult.

The cartography of magnetic parameters enables us to identify the sites that receive the most magnetic particles. Regardless of the sampling year, the highest values are measured for the sites that are close to the airstrip: either one of the three sites the closest to the airstrip, or the site Aerodromo MET, or either the site of Cerro Fosil or Cerro GPS. These two sites are slightly further from the airstrip but they are higher in altitude and therefore could be impacted by the emissions from planes during take-off or landing.

At the other side of the spectrum, the lichen samples from Collins ice-sheet and Ardley peninsula have a very low magnetic signal. They are almost not impacted by direct human activities because they are far from all the scientific bases. Furthermore, hysteresis parameters for samples from Ardley peninsula are different from other samples: on the Day-plot, they all plot closer to the SP mixing line and SD region. The Peninsula hosts a penguin colony, and this difference in hysteresis parameters could be linked to an incipient formation process of soil by organic matter accumulation and associated phosphatization [START_REF] Machado | Ornithogenic Gelisols (Cryosols) from Maritime Antarctica[END_REF]. Actually, high levels of SP grains in soils are often due to magnetic enhancement by secondary magnetite/maghemite produced by pedogenic processes [START_REF] Dearing | Secondary ferrimagnetic minerals in Welsh soils: a comparison of mineral magnetic detection methods and implications for mineral formation[END_REF].

The FORC diagrams measured in this study can be compared with those measured by [START_REF] Salo | Biomonitoring of air pollution in SW Finland by magnetic and chemical measurements of moss bags and lichens[END_REF] on lichens from urban and industrial sites. Despite the low resolution of their measurements, it is easy to see that their FORC diagrams are much more characteristic of multidomain (MD) magnetic grains than ours are. The FORC diagrams measured by Sagnotti & Winkler (2012) on tree leaves from Rome are also MD-like and therefore quite different from the diagrams we measured.

The only published paleomagnetic study from Antarctic Peninsula also includes FORC diagrams [START_REF] Gao | New paleomagnetic and 40Ar/39Ar geochronological results for the South Shetland Islands, West Antarctica, and their tectonic implications[END_REF]. Measurements on basalt samples from three different formations show patterns that are MD-like, with no high-coercivity component and a large vertical spread. On the other hand, a FORC diagram measured on volcanoclastic rocks from Fildes Peninsula is SD-like. In any case, our FORC diagrams show patterns that are very different from those measured on rocks from the Antarctic Peninsula, which tends to indicate that the magnetic signal we have measured is not mostly caused by geological input.

Comparison of magnetic results between corresponding lichen and substrate samples show a few differences. The substrate samples from sites close to the airstrip still have higher values than the samples from the other sites, but unlike for lichen samples, substrate samples from LARC and Collins have high M r and susceptibilities. FORC diagrams for lichen and substrate samples also bear some differences. Even though they bear similar SD-PSD patterns, there are two significant differences: the SD tail is more prominent and longer, and the spread along the vertical axis more limited for the substrate than for the lichen samples. This shows that the magnetic particles deposited on the substrate are different from those on the lichen. It is probable that the magnetic signal in the substrate samples contain a geological part as well as anthropogenic part.

Another explanation could also be that coarser-grain particles from anthropogenic origin are more efficiently fixed on the lichens, that in the substrate. It also seems that magnetic particles are more dispersed in substrate than in lichens, perhaps because they become more strongly fixed when deposited on lichen than on substrate. These observations are further confirmed by the fact that there is no correlation between M r or magnetic susceptibility on sister substrate/lichen samples (Fig. 11). This shows that the signal from the lichen U. aurantiaco-atra is likely to be more representative of the anthropogenic contribution than the signal from the substrate.

Comparison magnetic-elemental concentration measurements

The concentrations of Fe determined by INAA in the substrate samples of 1997 from Aerodromo, LARC and Ardley sites are consistent with the saturation magnetization measurements: for both data sets, the highest values are observed for the Aerodromo site, then an intermediate value for LARC and a lower value for Ardley. Similarly, the increase of Fe concentration from 1997 to 2006 also corresponds to an increase of saturation magnetization. However, when looking at the concentrations of the other elements, only the Ni concentrations for the year 1997 show similar variations of concentrations among sites.

In the lichen samples, the concentrations of the various elements measured by ICP-OES can be compared with the magnetic proxies (saturation magnetization and magnetic susceptibility). The element that shows the highest correlation with magnetic parameters is Ni (R 2 =0.832 for M r and 0.6173 for MS). Some elements show a weak correlation: Cr (R 2 =0.334 for M r and 0.349 for MS); Co (R 2 =0.420 for M r and 0.243 for MS); V (R 2 =0.342 for M r and 0.154 for MS) and Ag (R 2 =0.088 for M r and 0.254 for MS). The correlation for Cu is even weaker (R 2 =0.069 for M s and 0.136 for MS). Nickel and Cr are elements for which an association with magnetic parameters has been described (e.g., [START_REF] Cao | Magnetic response to air pollution recorded by soil and dust-loaded leaves in a changing industrial environment[END_REF][START_REF] Salo | Moss bag (Sphagnum papillosum) magnetic and elemental properties for characterising seasonal and spatial variation in urban pollution[END_REF]), however it is somewhat surprising that no stronger correlation is observed for the other elements such as Cu, Pb, Cd, Ba, Mo.

Possible sources of the various elements

Possible explanations for the presence of the various elements in sometimes fairly high concentration, both in the lichen and the substrate, are presented in Table II.

Conclusion

We carried out for the first time magnetic and elemental analysis on Usnea aurantiacoatra (Jacq.) Bory lichen and its substrate from various sites on the Antarctic Peninsula with various levels of influence from human activities. The main results are the following:

-Several anthropogenic elements have been identified on sites close to human activities: Ni, Pb, Mo, Cd, and Zn are the most prevalent. Some of the elements were found in sites that should have very little human influence, which shows that elements coming from other continents can travel long distances and contribute to the pollution in Antarctica.

-Magnetic proxies from lichen show a spatial correlation with human influence (scientific bases or airstrips). The temporal variation shows that on average, magnetic parameters started with high values in 1997, then decreased with time until 2007 but increased again in 2006-2007. The most recent measurements indicated nonetheless a general decreasing trend.

-It seems that the magnetic signal from the lichen is likely to be more representative of the anthropogenic contribution than that from the substrate.

-We observed a correlation between magnetic parameters and Ni, and to a lesser extent with Cr, Co, V, Ag. Increasing use of these elements in catalytic systems of the new motor vehicles used in all other continents and arriving to Antarctica. P (Ardley)

Figure captions

Presence of the penguin colony: the substrate is formed by organic matter accumulation and associated phosphatization [START_REF] Machado | Ornithogenic Gelisols (Cryosols) from Maritime Antarctica[END_REF]). Cr (Collins, 2006) Extra-Antarctic source* *The earth crust and rocks contain about 100 mg/kg of Cr. Soils contain normally on the average about 40 ppm of Cr, sometimes up to some hundreds mg/kg. [START_REF] Merian | Introduction on environmental chemistry and global cycles of chromium, nickel, cobalt beryllium, arsenic, cadmium and selenium, and their derivatives[END_REF]. In this work we find around 20 mg/kg in substrate. Some plants can take up higher Cr amounts, for instance lichens, mosses, ferns, grasses and others. We find lower concentrations of Cr in the lichen. On the other hand, practically all chromium compounds used by man originate from chromite, of which largest resources are placed in South Africa, Zimbabwe and Russia.

Figures

Instrumental

  Neutron Activation Analysis (INAA) is a multi-elemental and nondestructive technique that uses characteristic nuclear reactions. Neutron flux interacts with the sample, producing a nuclear reaction, mainly of the type (n, γ). The characteristic gamma rays emitted by each radionuclide are captured by a highresolution detector. INAA technique have been used widely to study the elemental composition of samples from Antarctica (e.g. Mroz et al. 2018), whether they are sediments, lichens or mosses. In this work, we used the INAA technique to measure the concentrations of the elements Na, Sc, Cr, Fe, Co, As, La, Ce, Sm, Eu, Yb, Hf in the substrate of the sites LARC, Aerodromo and Ardley peninsula sampled at the same time than the lichens in the years 1997 and 2006. Experimental details can be found in the SI.

Fig. 6

 6 Fig.6shows some of the FORC diagrams measured, over three different sampling years.

Figure 7

 7 Figure 7 shows the elemental concentrations determined by INAA and ICP-OES of

Figure 9

 9 Figure9shows the elements Fe, Al, Mn and Zn that are present with an important

Figure 10

 10 Figure10shows the presence of 13 elements in concentrations between 0.01 to 5 mg/kg,

Figure 1 .

 1 Figure 1. Sample sites in Fildes peninsula, King George Island, and lichen Usnea

Figure 2 .

 2 Figure 2. Day-plot showing the hysteresis parameters with the mixing lines of Dunlop

Figure 3 .

 3 Figure 3. Susceptibility vs remanent magnetization for: a: lichen Usnea aurantiaco-atra

Figure 4 .

 4 Figure 4. Variation of a: remanent magnetization; b: magnetic susceptibility of lichen

Figure 5 .

 5 Figure 5. Cartography of: a: remanent magnetization; b: magnetic susceptibility for

Figure 6 .

 6 Figure 6. FORC diagrams. Top row: Usnea aurantiaco-ater(Jacq.) Bory samples;

Figure 7 .

 7 Figure 7. Elemental concentrations determined by INAA (Fe, Na, Zn, Sc, Cr, Ce, Co,

Figure 8 .

 8 Figure 8. Examples of representative micrographs and EDS semi-quantitative analysis

Figure 9 .

 9 Figure 9. Concentrations of major elements determined by ICP-OES of Usnea

Figure 10 .

 10 Figure 10. Concentrations of 13 minor elements determined by ICP-OES of Usnea

Figure. 11

 11 Figure. 11. a: M r of substrate sample vs. M r of the corresponding Usnea aurantiaco-

Figure 1

 1 Figure 1

Figure

  Figure 2

  Figure 2

  Figure 3

Table II -

 II Possible origins of the elements detected in lichen Usnea aurantiaco-atra (Jacq.) Bory and the corresponding substrate samples Element Possible source Fe, Al (2010) Dust from road construction and extension of Chinese scientific base Ni Oil and diesel combustion(Pereira et al. 2010); use of lubricating oil (Song et al. 2010); additives in airplane fuel to increase motor efficiency (Marín-Flores et al. 2010); electric batteries Pb Airplane fuel burning (Pereira et al. 2010), abrasion of brake pads (Hong et al. 2020) and vehicle tires (Lough et al. 2005), oil lubricants (Song et al. 2010, Altıntaş et al. 2019); electric batteries Cd Elimination of wastes from incinerators (Boccaccini and Ondracek, 1995, Sekito et al. 2014); electric batteries Mo Elimination of wastes from incinerators (Boccaccini and Ondracek, 1995, Sekito et al. 2014); fossil fuel combustion (Pereira et al. 2010, Celo et al. 2015) and use of lubricating oil with molybdenite (Song et al. 2010) used in Antarctica in airplanes and machines such as snowploughs and construction machines; additives in airplane fuel (Marín-Flores et al. 2010) Zn Electric batteries, building construction work. V Use of diesel to produce electric energy (Hope 1997) Be Used with Cu or Ni alloys (Goddard et al. 2016) La, Hf, Sm (Ardley)

Acknowledgements: Authors acknowledge projects URedes Consolidación-URC-026/17, VID, Universidad de Chile, Conicyt-Inach 0897, 04/03 and T-02/07 for financial support. They also thank the editor and two anonymous reviewers for their constructive comments.

Author contribution

MP performed the fieldwork and supervised the entire work. RF, CE, NG and CC performed the different analysis of the samples. CC, MP, BA and NG interpreted the obtained results. CC and MP prepared the manuscript and created the figures with input from BA and NG.

Details of data deposit

The article will be deposited with free access in the libraries of the Universidad de Chile, the Sorbonne Université and the Universidad Nacional Autónoma de México.

Tables

Table I


José Neuman De Souza

Petre Dini

Pascal Lorenz

New Trends in Telecommunications and Networking

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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ICT 2004 was sponsored by the IEEE Communications Society and the International Federation for Information Processing (IFIP) Technical Committee 6. The theme for ICT 2004 was "Telecommunications and Networking." The theme was chosen to cover a wide range of challenging topics in telecommunications and networking to motivate discussions and evaluate the newest solutions.

ICT 2004 attracted a total of 430 submissions from 36 countries in Asia, the Middle East, Europe, South America, and North America. Of these submissions, only 178 papers were selected. The selective acceptance rate (41.4 percent) is an indication of the very high quality of the technical sessions.

The organization of the Technical Program of ICT 2004 had 44 technical paper sessions, two keynote sessions, two panel sessions, two invited speakers, seven tutorials, and 39 posters. The program included a new workshop named Service Assurance with Partial and Intermittent Resources (SAPIR), which had six technical paper sessions. Full accepted papers were published by Springer-Verlag as Lecture Notes in Computer Science (LNCS-3124 for ICT and LNCS-3126 for SAPIR). A CD-ROM with papers, posters, and tutorials was distributed to the attendees.

The Technical Paper Sessions covered the latest research results in multimedia services, antennas, transmission technologies and wirelessnetworks, communication theory, telecommunications pricing and billing, network performance and telecommunications services, active networks and mobile agents, optical photonic technologies, optical networks, ad hoc networks, signal processing, network performance and MPLS, traffic engineering and Internet, SIP, QoS and switches, network operation and management, network management theory and telecommunications networks, mobility and broadband wireless, cellular system evolution, personal communication, terrestrial radio systems and satellites, mobility management, image processing, ATM and Web services, communication security, switching and routing, and next-generation systems.

The SAPIR Workshop was the first event in a series introducing the concept of pi-resources and bringing it with the emerging and important field of distributed and heavily shared resources.

Two Panels were organized. The first focused on the new trends in telecommunications and networking; it was moderated by Prof. José Neuman de Souza (UFC, Brazil) with the following guests: Nikolai Nefedov (Nokia Research Center, Finland), Rui Aguiar (Aveiro University, Portugal), F. R. P. Cavalcanti (UFC, Brazil), W. Kabaciñski (Poznan University of Technology, Poland), and Rossana Andrade (UFC, Brazil). The second had as its main subject the pi-resources in grid networks; it was moderated by Prof. Petre Dini (Cisco Systems, Inc., United States), with the following guests: Ram Khare (Telcordia Technologies, Inc., United States), Claude Jard (IRISA/ENS, France), Antonella Molinaro (Calabria University, Italy), E. den Boef (Philips Research Laboratories, Netherlands) and Pascal Lorenz (Haute Alsace University, France).

Seven Tutorial Sessions were held on the following themes: next-generation optical networks from network layer and physical layer perspectives (speaker: Djafar Mynbaef, City University of New York, United States); IP-oriented QoS in the next-generation networks: application to wireless networks (speaker: Pascal Lorentz, Haute Alsace University, France); UMTS/IMT2000 (speaker: Hamid Aghvami, King's College London, United Kingdom); proactive management through key performance indicators for 3G systems (speaker: Said Soulhi, Ericsson , Canada); standards for wireless data communications: 802.11, 802.15, 802.16

  (speaker: T. Cooklev, San Francisco State University, United States); signal-processing-based strategies for packet collision resolution (speaker: Charles Cavalcante and Rodrigo Cavalcanti, Federal University of Ceara , Brazil); and telegraphic performance evaluation of wireless cellular networks (speaker: Francisco Barceló, University Polytechnic of Catalunya, Spain).Prof. Farokh Marvasti (King's College, London, United Kingdom) and Dr. José Leite Pereira Filho (ANATEL, Brazil) were the keynote speakers, talking about telecommunications in developing countries and universal services deregulating in Telecommunications, respectively.Finally, the organizing committee would like very much to thank the technical program committee members, who had hard work reviewing papers, and all the other participants who contributed to a successful conference.José Neuman de Souza is an associate professor at the Federal University of Ceará, Brazil. He received a Ph.D. degree from Paris VI (Pierre et Marie Curie) University, France, in 1994. He is the Brazilian representative to IFIP TC6 and a member of the administrative council of the National Research Network. He was General Chair of IEEE/IFIP ICT 2004 and IEEE/IFIP MMNS 2000. Petre Dini is a senior technical leader with Cisco Systems, Inc., United States, at the NMTG CTO Office. He is Chair of the Policy-Driven Control Work Group of the IEEE Emerging Technologies Group, Vice-Chair of the IEEE CommSoft TC, and Rapporteur for ITU-T/SG4. He received his M.Eng. from Polytechnic Institute of Timisoara, Romania, in computer engineering, and a Ph.D. in computer science from the University of Montreal, Canada. He is currently a Senior IEEE member, and an ACM member. He founded the International Academy, Research and Industry Association (IARIA) to promote young researchers [http://www.iaria.org].

	New Trends in Telecommunications and Networking:
	A Report on ICT 2004
	By José Neuman de Souza, Petre Dini, and Pascal Lorenz

José Neuman de Souza is General

Chair of ICT 2004 (e-mail: neuman@ufc.br)

Petre Dini is Co-Chair of

ICT 2004 TPC and General Chair of SAPIR 2004 (e-mail: pdini@cisco.com)

Pascal Lorenz is Co-Chair of ICT 2004 TPC (email: lorenz@ieee.org)


Ehud Lehrer

Eilon Solan

Yannick Viossat
email: vios-sat@ceremade.dauphine.fr.

Equilibrium payoffs in finite games

Keywords:

come

Introduction

It is well known that the set of correlated equilibrium payoffs of a n-player finite game is a polytope in R n [START_REF] Aumann | Subjectivity and Correlation in Randomized Strategies[END_REF]. We show that, conversely, any polytope in R n is the set of correlated equilibrium payoffs of a n-player game. Moreover, for any n-player game G and any polytope P in R n containing the correlated equilibrium payoffs of G, there exists a n-player game with the same set of Nash equilibrium payoffs as G and P as set of correlated equilibrium payoffs. If G is a two-player game, it suffices to assume that P contains the Nash equilibrium payoffs of G (but not necessarily its correlated equilibrium payoffs). We also show that a subset of R 2 is the set of Nash equilibrium payoffs of a finite game if and only if it is a finite union of rectangles. Together with the previous result, this implies that: for any nonempty finite union of rectangles U and any polytope P containing U, there exists a bimatrix game with U as set of Nash equilibrium payoffs and P as set of correlated equilibrium payoffs.

Finally, we study the robustness of our results to perturbation of the payoff matrices. Since almost all games have a finite set of equilibria, all we can hope to show is that, for any finite set U in R n and any polytope P in R n containing U, there exists an open set of games whose set of Nash equilibrium payoffs is arbitrarily close to U and whose set of correlated equilibrium payoffs is arbitrarily close to P . We show that this is indeed the case.

Notations and definitions: throughout, n denotes an integer weakly greater than 2. Let G be a n-player game with pure strategy set S i and payoff function u i : S := × 1≤i≤n S i → R for player i. Let S -i := × j =i S j . A correlated equilibrium of G [START_REF] Aumann | Subjectivity and Correlation in Randomized Strategies[END_REF] is a probability distribution µ on the set of strategy profiles S of G such that, for every player i in {1, .., n} and every pure strategy s i in S i :

∀t i ∈ S i , s -i ∈S -i µ(s i , s -i )[u i (s i , s -i ) -u i (t i , s -i )] ≥ 0 (1)
The set of correlated equilibria of G is a polytope, which contains the Nash equilibria. An extreme correlated equilibrium is an extreme point of this polytope.

Let u i (µ) := s∈S µ(s)u i (s) denote the average payoff of player i in the correlated equilibrium µ. The set of correlated equilibrium payoffs of G is the set of n-tuples (u 1 (µ), ..., u n (µ)) where µ is a correlated equilibrium of G. This is a polytope in R n . It is denoted by CEP (G). Similarly, NEP (G) denotes the set of Nash equilibrium payoffs of G, and we may write NE and CE for "Nash equilibrium" and "correlated equilibrium", respectively. Finally, if A is a subset of R n then Conv(A) denotes its convex hull.

Main results

We begin with a result on correlated equilibria: Proposition 1 For any polytope P in R n , there exists a n-player game whose set of correlated equilibrium payoffs (resp. convex hull of the set of Nash equilibrium payoffs) is P .

Proof. We prove the result for two-player games; see the appendix for the n-player case. Let P be a polytope in R 2 . Let (x 1 , y 1 ),...,(x m , y m ) be its extreme points. Assume that for every i in {1, 2, .., m}, x i and y i are strictly positive. This is without loss of generality, since adding a constant to all payoffs of a game does not change the set of Nash equilibria and correlated equilibria. Consider the m × m game with payoff matrix (a ij , b ij ) 1≤i,j≤m such that, for every i in {1, 2, ..., m}, a ii = a mi = x i , b ii = b im = y i , and all other payoffs are zero. For m = 4 this gives :

     x 1 , y 1 0, 0 0, 0 0, y 1 0, 0 x 2 , y 2 0, 0 0, y 2 0, 0 0, 0 x 3 , y 3 0, y 3 x 1 , 0 x 2 , 0 x 3 , 0 x 4 , y 4      (2)
Clearly, any diagonal square corresponds to a pure Nash equilibrium. Furthermore, choosing the last row (column) is a weakly dominant strategy for the row (column) player, and it gives a strictly higher payoff than choosing row (column) i = m whenever the column (row) player does not choose column (row) i. It follows that in every correlated equilibrium, the probability of every off-diagonal square is zero, hence there are no other Nash equilibria or extreme correlated equilibria than the diagonal squares. Therefore, the convex hull of the set of Nash equilibrium payoffs coincides with the set of correlated equilibrium payoffs and is equal to P .

To state our next result, we first need a definition: let us say that a subset of

R 2 is a rectangle if it is of the form [a, b] × [c, d],
for some real numbers a, b, c, d, with a ≤ b, c ≤ d.

Proposition 2 In any bimatrix game, the set of Nash equilibrium payoffs is a finite union of rectangles. Conversely, for any nonempty finite union of rectangles U, there exists a bimatrix game whose set of Nash equilibrium payoffs is U. Moreover, for any bimatrix game G, there exists a bimatrix game

G ′ with NEP (G ′ ) = NEP (G) and CEP (G ′ ) = Conv(NEP (G)).
Proof. Consider a bimatrix game with pure strategy set I for player 1 and J for player 2. Let NEP ⊂ R 2 denote its set of Nash equilibrium payoffs. For any subsets I ′ of I and J ′ of J, let NEP (I ′ , J ′ ) denote the set of payoffs of Nash equilibria with support I ′ × J ′ . For any subset A of R 2 , let Ā denote the closure of A. We have

NEP = I ′ ⊂I,J ′ ⊂J NEP (I ′ , J ′ ) = I ′ ⊂I,J ′ ⊂J NEP (I ′ , J ′ )
where the second equality holds because NEP is closed. Thus, to show that NEP is a finite union of rectangles, it suffices to show that NEP (I ′ , J ′ ) is a rectangle. This follows easily from the fact that, in bimatrix games, Nash equilibria with the same support are exchangeable.

Conversely, let m ∈ N * and for 1

≤ i ≤ m, let a i , b i , c i , d i be real numbers. Let U = 1≤i≤m [a i , b i ] × [c i , d i ]. Assuming w.l.o.g. that the numbers a i , b i , c i , d i
are all positive, we build below a bimatrix game with U as set of Nash equilibrium payoffs. Consider first the game with payoff matrices:

(A i , B i ) = a i , c i b i , c i a i , d i b i , d i
In this game, a player does not influence its own payoffs and the set of Nash equilibrium payoffs is

[a i , b i ] × [c i , d i ]. Let (A i , 0) = a i , 0 b i , 0 a i , 0 b i , 0 , (0, B i ) = 0, c i 0, c i 0, d i 0, d i
and consider the game built by blocks:

       (A 1 , B 1 ) 0 .. 0 (0, B 1 ) 0 (A 2 , B 2 ) .. 0 (0, B 2 ) .. .. .. .. .. 0 0 .. (A m-1 , B m-1 ) (0, B m-1 ) (A 1 , 0) (A 2 , 0) .. (A m-1 , 0) (A m , B m )        (3)
where an isolated 0 represents a 2 × 2 block of payoffs (0, 0). This game has the same structure as (2), but the payoffs x i and y i have been replaced by the blocks A i and B i , respectively. Any mixed strategy profile with support in one of the blocks (A i , B i ) is a Nash equilibrium. Furthermore, it is easy to prove along the lines of the proof of proposition 1 that there are no other Nash equilibria and that the set of correlated equilibria is the convex hull of the set of Nash equilibria. It follows that: first, the set of NE payoffs of (3) is equal to U, which shows that any finite union of rectangles is the set of NE payoffs of a bimatrix game; and second, the set of CE payoffs of (3) is equal to Conv(U). Since the set of NE payoffs of a bimatrix game is always a finite union of rectangles, this implies that for any bimatrix game G, there exists a bimatrix game G ′ (of type (3)) such as NEP (G ′ ) = NEP (G) and

CEP (G ′ ) = Conv(NEP (G)). Lemma 3 Let (x 1 , x 2 , ..., x n ) ∈ R n .
For any n-player game G, there exists a n-player game with the same set of Nash equilibrium payoffs as G and whose set of correlated equilibrium payoffs is the convex hull of (x 1 , x 2 , ..., x n ) and of the set of correlated equilibrium payoffs of G.

Proof. We prove the result in the two-player case. For the n-player case, see the appendix. Let G be a two-player m 1 × m 2 game and (x, y) ∈ R 2 . Assume w.l.o.g. that x, y, and all the payoffs in G are positive. Consider the

(3 + m 1 ) × (3 + m 2 ) game Γ =      0, 0 x + 1, y -1 x -1, y + 1 x -1, y + 1 0, 0 x + 1, y -1 x + 1, y -1 x -1, y + 1 0, 0 [0, y] [x, 0] G      (4)
where [x, 0] denotes a block of payoffs (x, 0) of appropriate size (same notations for player 2). Let ν denote the correlated strategy putting probability 1/6 on every off-diagonal square of the top-left block, and probability 0 on every other square of the whole payoff matrix. Clearly, ν is a correlated equilibrium of Γ, with payoff (x, y), and every correlated equilibrium of G induces a correlated equilibrium of Γ. We claim that any correlated equilibrium of Γ is a convex combination of ν and of a correlated equilibrium of G.

To see this, let µ be a correlated equilibrium of Γ. Clearly, µ 12 ≥ µ 13 , otherwise player 1 would have an incentive to deviate from his first to his last strategy (recall that all payoffs in G are positive). Repeating this reasoning with other strategies and with player 2 leads to the chain of inequalities

µ 12 ≥ µ 13 ≥ µ 23 ≥ µ 21 ≥ µ 31 ≥ µ 32 ≥ µ 12
Since the first and last terms are equal, this is a chain of equalities, hence µ puts equal weight on all off-diagonal squares of the top-left block. It is then easy to see that µ puts probability zero on the diagonal of the top-left block as well as on the top-right and bottom-left blocks. This implies that µ is a convex combination of ν and of a correlated equilibrium of G, proving the claim.

It follows that : (i) a fortiori, any CE payoff of Γ is a convex combination of the payoff of ν and of a correlated equilibrium payoff of G; and (ii) Γ and G have the same Nash equilibria, hence a fortiori the same NE payoffs. This concludes the proof.

Proposition 4 For any n-player game G and for any polytope P in R n containing CEP (G), there exists a game with the same set of Nash equilibrium payoffs as G and P as set of correlated equilibrium payoffs.

Proof. Let P be a polytope containing CEP (G), with q extreme points. Applying iteratively lemma 3 (q times), we obtain a game with the same NE payoffs as G and whose set of CE payoffs is the convex hull of P and CEP (G), hence is equal to P .

In the two-player case, we get a stronger result:

Corollary 5 For any bimatrix game G and for any polytope P in R n containing NEP (G), there exists a game with the same set of Nash equilibrium payoffs as G and P as set of correlated equilibrium payoffs. 1Proof. By proposition 2, there exists a game G ′ with NEP (G ′ ) = NEP (G) and CEP (G ′ ) = Conv(NEP (G)). Applying proposition 4 to G ′ gives the result.

In light of proposition 2, corollary 5 may also be stated as follows : for any nonempty finite union of rectangles U in R 2 and for any polytope P in R 2 containing U, there exists a two-player game with U as set of Nash equilibrium payoffs and P as set of correlated equilibrium payoffs. Proposition 2 and corollary 5 fully characterize the possible shapes of the set of Nash equilibrium payoffs and correlated equilibrium payoffs for two-player games. For n-player games, the general shape of the set of Nash equilibrium payoffs remains to be understood. Furthermore, we do not know whether, for any n-player game G, there exists a n-player game with the same set of Nash equilibrium payoffs as G and whose set of correlated equilibrium payoffs is the convex hull of its Nash equilibrium payoffs.

Genericity

The games used above are highly non generic. For instance, a small perturbation of the payoffs of (2) is enough to eliminate all its Nash and correlated equilibria but one. This raises the issue of the robustness of our results. Ideally, to show that, for instance, corollary 5 is robust, one would like to show that for any nonempty finite union of rectangles U and for any polytope P in R 2 containing U, there exists an open set of games whose set of Nash equilibrium payoffs is "close" to U and whose set of correlated equilibrium payoffs is "close" to P . This is hopeless however, since almost all games have a finite set of Nash equilibria. Thus, all we can hope to prove is the same result but with U being a finite set. This is the object of this section.

We first need some definitions. Let ǫ > 0. For any

x in R n , let ||x|| = max 1≤i≤n |x i |. Let A and A ′ be subsets of R n . Recall that A and A ′ are ǫ-close in the Hausdorff distance sense if ∀x ∈ A, ∃x ′ ∈ A ′ , ||x -x ′ || < ǫ (5) and ∀x ′ ∈ A ′ , ∃x ∈ A, ||x -x ′ || < ǫ (6)
Let G be a finite game with payoff function u i for player i and let α > 0.

The open ball of center G and radius α, denoted by B(G, α), is the set of all games G ′ with the same sets of players and strategies as in G and such that for every player i and every pure strategy profile s, Proposition 6 Let U be a finite set in R n . Let P ⊂ R n be a polytope containing U. There exists a (nonempty) open set of n-player games whose set of Nash equilibrium payoffs is ǫ-close to U and whose set of correlated equilibrium payoffs is ǫ-close to P .

|u ′ i (s) -u i (s)| < α,
Proof. We prove the result for two-player games. The proof for n-player games is similar (see the appendix). Let U = {(x 1 , y 1 ), ..., (x m , y m )}, let P ⊂ R 2 be a polytope containing U, with vertices (x ′ 1 , y ′ 1 ),...,(x ′ q , y ′ q ). Assume w.l.o.g. that, for all i in {1, .., q}, x i , y i , x ′ i and y ′ i are all positive. Let G α denote the m × m game with payoff matrix (a ij , b ij ) 1≤i,j≤m such that: for every i in {1, 2, ..., m}, a ii = x i and b ii = y i ; for every i in {1, 2, ..., m -1}, a mi = x i -α and b im = y i -α; and all other payoffs are zero. For m = 4 this gives:

G α =      x 1 , y 1 0, 0 0, 0 0, y 1 -α 0, 0 x 2 , y 2 0, 0 0, y 2 -α 0, 0 0, 0 x 3 , y 3 0, y 3 -α x 1 -α, 0 x 2 -α, 0 x 3 -α, 0 x 4 , y 4     
Thus, G 0 is the game used in the proof of proposition 1 and NEP (G 0 ) = U.

For every (x, y) in R 2 , let C(x, y) denote the game corresponding to the top-left block of (4).

C(x, y) =    0, 0 x + 1, y -1 x -1, y + 1 x -1, y + 1 0, 0 x + 1, y -1 x + 1, y -1 x -1, y + 1 0, 0    (7)
Finally, let Γ α denote the following game:

Γ α =        C(x ′ 1 , y ′ 1 ) 0 ... 0 [0, y ′ 1 -α] 0 C(x ′ 2 , y ′ 2 ) ... 0 [0, y ′ 2 -α] ... ... ... ... ... 0 0 ... C(x ′ q , y ′ q ) [0, y ′ q -α] [x ′ 1 -α, 0] [x ′ 2 -α, 0] ... [x ′ q -α, 0] G α       
where [x ′ i -α, 0] means a block of payoffs (x ′ i -α, 0) of appropriate size. Γ 0 is a slight modification of the game obtained from G 0 by iterative applications (q times) of lemma 3. Along the lines of the proof of lemma 3, it is easy to show that its Nash equilibria correspond to the Nash equilibria of G 0 and that its extreme correlated equilibria are: (i) its Nash equilibria; (ii) the probability distributions with support in one of the blocks C(x ′ i , y ′ i ) and putting probability 1/6 on every off-diagonal square of this block. It follows that NEP (Γ 0 ) = U and CEP (Γ 0 ) = P .

Moreover, for any α small enough and any game Γ in B(Γ α , α/2), every Nash equilibrium of Γ 0 is a Nash equilibrium of Γ; therefore,

∀(a 0 , b 0 ) ∈ NEP (Γ 0 ), ∃(a, b) ∈ NEP (Γ), ||(a, b) -(a 0 , b 0 )|| < 3α/2 (8)
(a closer look shows that we may replace 3α/2 by α/2 in the RHS, but this is not needed). Finally, let ǫ > 0. By upper semi-continuity of the Nash equilibrium correspondence, for α small enough, for any game Γ in B(Γ 0 , 3α/2),

∀(a, b) ∈ NEP (Γ), ∃(a 0 , b 0 ) ∈ NEP (Γ 0 ), ||(a 0 , b 0 ) -(a, b)|| < ǫ (9)
It follows from ( 8) and ( 9) that, for any α small enough and any game Γ in B(Γ α , α/2), NEP (Γ) and NEP (Γ 0 ) are ǫ-close. The same argument (up to replacement of Nash equilibrium by correlated equilibrium everywhere) shows that for every α small enough and for every game Γ in B(Γ α , α/2), CEP (Γ) and CEP (Γ 0 ) are ǫ-close. Recalling that NEP (Γ 0 ) = U and CEP (Γ 0 ) = P , this completes the proof.

Note that, in the above proof, for every game Γ in B(Γ α , α/2), any Nash equilibrium (resp. extreme correlated equilibrium) of Γ 0 is actually a strict Nash equilibrium (resp. strict correlated equilibrium2 ) of Γ. Since furthermore a convex combination of strict correlated equilibria is a strict correlated equilibrium, it follows that for every finite set U in R n , every polytope P containing U and every ǫ > 0, there exists an open set of games Γ such that: first, both the set of Nash equilibrium payoffs and the set of strict Nash equilibrium payoffs of Γ are ǫ-close to U; second, both the set of correlated equilibrium payoffs and the set of strict correlated equilibrium payoffs of Γ are ǫ-close to P .

A Proofs of lemma 3 and propositions 1 and 6 in the n-player case.

Proof of lemma 3. Let G be a three-player game and let (x, y, z) ∈ R 3 . Assume w.l.o.g. that x, y, z, and all the payoffs in G are positive. Let C(x, y, z) denote the 3 × 3 × 1 game (player 3 is a dummy) where the payoffs of players 1 and 2 are as in (7) and the payoff of player 3 is always z.

C(x, y, z) =    0, 0, z x + 1, y -1, z x -1, y + 1, z x -1, y + 1, z 0, 0, z x + 1, y -1, z x + 1, y -1, z x -1, y + 1, z 0, 0, z    (10) Let Γ = C(x, y, z) [0, y, 0] [x, 0, 0] [x, y, 0] [0, 0, z] [0, y, z] [x, 0, z] G
where [x, 0, 0] denotes a block of payoffs (x, 0, 0) of appropriate size. The same proof as in the two-player case shows that Γ has the same set of NE payoffs as G and that its set of CE payoffs is the convex hull of (x, y, z) and of the set of CE payoffs of G.

In the n-player case, the generalization of C(x, y, z) simply consists in adding more dummy players with constant payoff. The generalization of Γ should be clear from the following description of the three-player case: the pure strategy set of player i is S ′ i ⊔ S i (disjoint union), where S i and S ′ i are respectively the pure strategy set of player i in G and in C(x, y, z). If for every i, player i chooses a strategy in S i (resp. S ′ i ), then the payoffs are as in G (resp. as in C(x, y, z)). If there exist players i and j such that i chooses a strategy in S i and j a strategy in S ′ j , then the payoff of player 1 (resp. 2, 3) is x (resp. y, z) if he chooses a strategy in S 1 (resp. S 2 , S 3 ) and 0 otherwise.

Proof of propositions 1 and 6. Let U = {(x i , y i , z i ), 1 ≤ i ≤ m} and let P be the convex hull of {(x ′ i , y ′ i , z ′ i ), 1 ≤ i ≤ q}. Assume w.l.o.g. that the numbers x i , x ′ i , etc. are all positive. Let

G α =       
x 1 , y 1 , z 1 0, 0, 0 .. 0, 0, 0 0, y 1 -α, 0 0, 0, 0

x 2 , y 2 , z 2 .. 0, 0, 0 0, y 2 -α, 0 .. .. .. .. .. 0, 0, 0 0, 0, 0 .. x m-1 , y m-1 , z m-1 0, y m-1 -α, 0 x 1 -α, 0, 0 x 2 -α, 0, 0 .. x m-1 -α, 0, 0

x m , y m , z m       
The same argument as in the two-player case shows that the Nash equilibria of G 0 are equal to its extreme correlated equilibria and correspond to the

  where u ′ i is the payoff function of player i in G ′ . A set of games Σ is open if for every game G in Σ, Σ contains an open ball of center G and positive radius.

The difference with the n-player case in that we only assume P ⊃ N EP (G) instead of P ⊃ CEP (G).

A correlated equilibrium µ is strict if for every pure strategy s i with positive marginal probability under µ, the inequalities in (1) are strict.

Instead of C(x, y, z), Γ, G α , Γ α , and their n-player version, it is possible to use games in which the roles of the players are symmetric, but this is less parcimonious.

diagonal squares. It follows that NEP (G 0 ) = U and that CEP (G 0 ) is the convex hull of U. This proves proposition 1 in the three-player case (for the n-player case, just add more dummy players). Now recall (10) and let G ′ denote the (3q + m) × (3q + m) × 1 game with block diagonal payoff matrix

Let x, y, z be positive real numbers. Let Γ α denote the (3q+m)×(3q+m)×2 game:

This should be read as follows: if player 1 chooses row i > 3q and player 2 chooses column j ≤ 3q with 3p + 1 ≤ j ≤ 3p + 3, then the payoffs are (x ′ p -α, 0, 0) if player 3 chooses the left matrix and (x, 0, z) if player 3 chooses the matrix on the right.

Fix ǫ > 0 and assume that P ⊃ U. The same arguments as in the two-player case show that, for every α small enough, and for every game Γ in B(G α , α/2), NEP (Γ) is ǫ-close to U and CEP (Γ) is ǫ-close to P . The n-player case is similar. 3