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We study the problem of computing correlated strategies to commit to in games with multiple leaders and followers. To the best of our knowledge, this problem is widely unexplored so far, as the majority of the works in the literature focus on games with a single leader and one or more followers. The fundamental ingredient of our model is that a leader can decide whether to participate in the commitment or to defect from it by taking on the role of follower. This introduces a preliminary stage where, before the underlying game is played, the leaders make their decisions to reach an agreement on the correlated strategy to commit to. We distinguish three solution concepts on the basis of the constraints that they enforce on the agreement reached by the leaders. Then, we provide a comprehensive study of the properties of our solution concepts, in terms of existence, relation with other solution concepts, and computational complexity.

The cosmic X-ray background (CXB), which peaks at an energy of ~30 keV, is produced primarily by emission from accreting supermassive black holes (SMBHs). The CXB therefore serves as a constraint on the integrated SMBH growth in the Universe and the accretion physics and obscuration in active galactic nuclei (AGNs). This paper gives an overview of recent progress in understanding the high-energy (>~10 keV) X-ray emission from AGNs and the synthesis of the CXB, with an emphasis on results from NASA's NuSTAR hard X-ray mission. We then discuss remaining challenges and open questions regarding the nature of AGN obscuration and AGN physics. Finally, we highlight the exciting opportunities for a next-generation, high-resolution hard X-ray mission to achieve the long-standing goal of resolving and characterizing the vast majority of the accreting SMBHs that produce the CXB., Comment: Science White paper submitted to Astro2020 Decadal Survey; 5 pages, 3 figures, plus references and cover page

Tissue self-organization into defined and well-controlled three-dimensional structures is essential during development for the generation of organs. A similar, but highly deranged process might also occur during the aberrant growth of cancers, which frequently display a loss of the orderly structures of the tissue of origin, but retain a multicellular organization in the form of spheroids, strands, and buds. The latter structures are often seen when tumors masses switch to an invasive behavior into surrounding tissues. However, the general physical principles governing the self-organized architectures of tumor cell populations remain by and large unclear. In this work, we perform in-vitro experiments to characterize the growth properties of glioblastoma budding emerging from monolayers. Using a theoretical model and numerical tools here we find that such a topological transition is a self-organised, non-equilibrium phenomenon driven by the trade--off of mechanical forces and physical interactions exerted at cell-cell and cell-substrate adhesions. Notably, the unstable disorder states of uncontrolled cellular proliferation macroscopically emerge as complex spatio--temporal patterns that evolve statistically correlated by a universal law.

We present the X-ray properties of 108 Dust-Obscured Galaxies (DOGs; F$_{24 \mu m}$/F$_{R} >$ 1000) in the COSMOS field, all of which detected in at least three far-infrared bands with the Herschel Observatory. Out of the entire sample, 22 are individually detected in the hard 2-8 keV X-ray band by the Chandra COSMOS Legacy survey, allowing us to classify them as AGN. Of them, 6 (27%) are Compton Thick AGN candidates with column densities N$_{H}$$>$10$^{24}$ cm$^{-2}$ while 15 are moderately obscured AGNs with 10$^{22}$ $<$ N$_{H}$ $<$ 10$^{24}$ cm$^{-2}$. Additionally, we estimate AGN contributions to the IR luminosity (8-1000$\mu$m rest-frame) greater than 20% for 19 DOGs based on SED decomposition using Spitzer/MIPS 24$\mu$m and the five Herschel bands (100-500 $\mu$m). Only 7 of these are detected in X-rays individually. We performed a X-ray stacking analysis for the 86 undetected DOGs. We find that the AGN fraction in DOGs increases with 24$\mu$m flux and that it is higher than that of the general 24$\mu$m population. However, no significant difference is found when considering only X-ray detections. This strongly motivates the combined use of X-ray and far-IR surveys to successfully probe a wider population of AGNs, particularly for the most obscured ones., Comment: To be published in ApJ

Initial Coin Offerings (ICO) are public offers of new cryptocurrencies in exchange of existing ones, aimed to finance projects in the blockchain development arena. In the last 8 months of 2017, the total amount gathered by ICOs exceeded 4 billion US$, and overcame the venture capital funnelled toward high tech initiatives in the same period. A high percentage of ICOS is managed through Smart Contracts running on Ethereum blockchain, and in particular to ERC-20 Token Standard Contract. In this work we examine 1388 ICOs, published on December 31, 2017 on icobench.com Web site, gathering information relevant to the assessment of their quality and software development management, including data on their development teams. We also study, at the same date, the financial data of 450 ICO tokens available on coinmarketcap.com Web site, among which 355 tokens are managed on Ethereum blochain. We define success criteria for the ICOs, based on the funds actually gathered, and on the behavior of the price of the related tokens, finding the factors that most likely influence the ICO success likeliness.

The discoveries made over the past 20 years by Chandra and XMM-Newton surveys in conjunction with multiwavelength imaging and spectroscopic data available in the same fields have significantly changed the view of the supermassive black hole (SMBH) and galaxy connection. These discoveries have opened up several exciting questions that are beyond the capabilities of current X-ray telescopes and will need to be addressed by observatories in the next two decades. As new observatories peer into the early Universe, we will begin to understand the physics and demographics of SMBH infancy (at $z>6$) and investigate the influence of their accretion on the formation of the first galaxies ($\S$ 2.1). We will also be able to understand the accretion and evolution over the cosmic history (at $z\sim$1-6) of the full population of black holes in galaxies, including low accretion rate, heavily obscured AGNs at luminosities beyond the reach of current X-ray surveys ($\S$2.2 and $\S$2.3), enabling us to resolve the connection between SMBH growth and their environment., Comment: 5 pages, 3 figures; Science white paper submitted to the Astro2020 Decadal Survey

In this work we study line arrangements consisting in lines passing through three non-aligned points. We call them triangular arrangements. We prove that any combinatorics of a triangular arrangement is always realized by a Roots-of-Unity-Arrangement, which is a particular class of triangular arrangements. Among these Roots-of Unity-Arrangements, we provide conditions that ensure their freeness. Finally, we give two triangular arrangements having the same weak combinatorics, such that one is free but the other one is not., Comment: 20 pages. Published in \'Epijournal de G\'eom\'etrie Alg\'ebrique. Section 4 has been deeply revised due to an incorrect statement pointed out by the referee

We investigate the multi-wavelength properties of host galaxies of 3701 X-ray-selected active galactic nuclei (AGNs) out to z~5 in the Chandra-COSMOS Legacy Survey. Thanks to the extensive multi-wavelength photometry available in the COSMOS field, we derive AGN luminosities, host stellar masses, and star formation rates (SFRs) via a multi-component SED fitting technique. Type 1 and Type 2 AGNs follow the same intrinsic Lx-L6um relation, suggesting that mid-infrared emission is a reasonably good measure of the AGN accretion power regardless of obscuration. We find that there is a strong increase in Type 1 AGN fraction toward higher AGN luminosity, possibly due to the fact that Type 1 AGNs tend to be hosted by more massive galaxies. The AGN luminosity and SFR are consistent with an increase toward high stellar mass, while both the Mstellar-dependence is weaker towards the high-mass end, which could be interpreted as a consequence of quenching both star formation and AGN activity in massive galaxies. AGN host galaxies tend to have SFRs that are consistent with normal star-forming galaxies, independent of AGN luminosities. We confirm that black hole accretion rate and SFR are correlated up to z~5, when forming stars. The majority (~73%) of our AGN sample are faint in the far-infrared, implying that the moderate-luminosity AGNs seem to be still active after the star formation is suppressed. It is not certain whether AGN activity plays a role in quenching the star formation. We conclude that both AGN activity and star formation might be more fundamentally related to host stellar mass., Comment: 30 pages, 18 figures, accepted for publication in ApJ

Active galactic Nuclei (AGNs) with their relativistic jets pointed towards the observer, form a subclass of luminous gamma-ray sources commonly known as blazars. The study of blazars is essential to improve our understanding on the AGNs emission mechanisms and evolution, as well as to map the extragalactic background light. To do so, however, one needs to correctly classify and measure a redshift for a large sample of these sources. The Third {\it Fermi}--LAT Catalog of High-Energy Sources (3FHL) contains $\approx1160$ blazars reported at energies greater than $10$\,GeV. However $\sim$25\% of these sources are unclassified and $\sim$50\% lack of redshift information. To increase the spectral completeness of the 3FHL catalog, we are working on an optical spectroscopic follow up campaign using 4--m and 8--m telescopes. In this paper, we present the results of the second part of this campaign, where we observed 23 blazars using the 4\,$m$ telescope at CTIO in Chile. We report all the 23 sources to be classified as BL Lacs, a confirmed redshift measurement for 3 sources, a redshift lower limit for 2 sources and a tentative redshift measurement for 3 sources., Comment: Figure 2 is made up of 23 individual plot figures and while Figure 3 is made up of 8 individual plot figures. APJ accepted 01/24/2019

In this data paper we present and characterise the multi-component radio sources identified in the VLA-COSMOS Large Project at 3 GHz (0.75 arcsec resolution, 2.3 {\mu}Jy/beam rms), i.e. the radio sources which are composed of two or more radio blobs.The classification of objects into multi-components was done by visual inspection of 351 of the brightest and most extended blobs from a sample of 10,899 blobs identified by the automatic code blobcat. For that purpose we used multi-wavelength information of the field, such as the 1.4-GHz VLA-COSMOS data and the UltraVISTA stacked mosaic available for COSMOS. We have identified 67 multi-component radio sources at 3 GHz: 58 sources with AGN powered radio emission and 9 star-forming galaxies. We report 8 new detections that were not observed by the VLA-COSMOS Large Project at 1.4 GHz, due to the slightly larger area coverage at 3 GHz. The increased spatial resolution of 0.75 arcsec has allowed us to resolve (and isolate) multiple emission peaks of 28 extended radio sources not identified in the 1.4-GHz VLA-COSMOS map. We report the multi-frequency flux densities (324 MHz, 325 MHz, 1.4 GHz & 3 GHz), star-formation-rates, and stellar masses of these objects. Multi-component objects at 3-GHz VLA-COSMOS inhabit mainly massive galaxies (>10^10.5 Msun). The majority of the multi-component AGN lie below the main-sequence of star-forming galaxies (SFGs), in the green valley and the quiescent region. We provide detailed description of the objects: amongst the AGN there are 2 head-tail, 10 core-lobe, 9 wide-angle-tail (WAT), 8 double-double or Z-/X-shaped, 3 bent-tail radio sources, and 26 symmetric sources, while amongst the SFGs we find the only star-forming ring seen in radio emission in COSMOS. We report a large number (32/58) of disturbed/bent multi-component AGN, 18 of which do not lie within X-ray groups in COSMOS (0.08 < z < 1.53). [abridged], Comment: A&A accepted; 32 pages, 14 figures, 5 tables, 3 Appendices

We present the results of an identification campaign of unassociated sources from the Fermi Large Area Telescope 3FHL catalog. Out of 200 unidentified sources, we selected 110 sources for which archival Swift-XRT observations were available, 52 of which were found to have exactly one X-ray counterpart within the 3FHL 95\% positional uncertainty. In this work, we report the X-ray, optical, IR, and radio properties of these 52 sources using positional associations with objects in various catalogs. The Wide-field Infrared Survey Explorer color-color plot for sources suggests that most of these belong to the blazar class family. The redshift measurements for these objects range from z=0.277 to z=2.1. Additionally, under the assumption that the majority of these sources are blazars, three machine-learning algorithms are employed to classify the sample into flat spectrum radio quasars or BL Lacertae objects. These suggest that the majority of the previously unassociated sources are BL Lac objects, in agreement with the fact the BL Lac objects represent by far the most numerous population detected above 10 GeV in 3FHL., Comment: accepted in ApJ

We present the analysis of a sample of 35 candidate Compton thick (CT-) active galactic nuclei (AGNs) selected in the nearby Universe (average redshift ~0.03) with the Swift-BAT 100-month survey. All sources have available NuSTAR data, thus allowing us to constrain with unprecedented quality important spectral parameters such as the obscuring torus line-of-sight column density (N_{H, z}), the average torus column density (N_{H, tor}) and the torus covering factor (f_c). We compare the best-fit results obtained with the widely used MyTorus (Murphy et al. 2009) model with those of the recently published borus02 model (Balokovic et al. 2018) used in the same geometrical configuration of MyTorus (i.e., with f_c=0.5). We find a remarkable agreement between the two, although with increasing dispersion in N_{H, z} moving towards higher column densities. We then use borus02 to measure f_c. High-f_c sources have, on average, smaller offset between N_{H, z} and N_{H, tor} than low-f_c ones. Therefore, low f_c values can be linked to a "patchy torus" scenario, where the AGN is seen through an over-dense region in the torus, while high-f_c objects are more likely to be obscured by a more uniform gas distribution. Finally, we find potential evidence of an inverse trend between f_c and the AGN 2-10 keV luminosity, i.e., sources with higher f_c values have on average lower luminosities., Comment: 35 Pages, 23 Figures. Accepted for publication in ApJ

We present the 2--78 keV spectral analysis of the deep NuSTAR and XMM-Newton observation of a nearby Seyfert 2 galaxy, ESO 116-G018, which is selected as a candidate Compton-thick (CT-) active galactic nucleus (AGN) based on a previous Chandra-Swift/BAT study. Through our analysis, the source is for the first time confirmed to be a CT-AGN at a $>$3 $\sigma$ confidence level, with the 'line-of-sight' column density N$_{\rm H,Z}$ = [2.46-2.76] $\times$ $10^{24}$ cm$^{-2}$. The 'global average' column density of the obscuring torus is N$_{\rm H,S}$ = [0.46-0.62] $\times$ $10^{24}$ cm$^{-2}$, which suggests a clumpy, rather than uniform, distribution of the obscuring material surrounding the accreting supermassive black hole. The excellent-quality data given by the combined NuSTAR and XMM-Newton observations enable us to produce a strong constraint on the covering factor of the torus of ESO 116-G018, which is found to be $f_c$ = [0.13-0.15]. We also estimate the bolometric luminosity from the broad-band X-ray spectrum, being L$\rm _{bol}$ = [2.57-3.41] $\times$ 10$^{44}$ erg s$^{-1}$., Comment: 11 pages, 5 figures

We present the combined NuSTATR and XMM-Newton 0.6-79 keV spectral analysis of a Seyfert 2 galaxy, NGC 1358, which we selected as a candidate Compton thick (CT-) active galactic nucleus (AGN) on the basis of previous Swift/BAT and Chandra studies. According to our analysis, NGC 1358 is confirmed to be a CT-AGN using physical motivated models, at >3 $\sigma$ confidence level. Our best-fit shows that the column density along the 'line-of-sight' of the obscuring material surrounding the accreting super-massive black hole is N$\rm _H$ = [1.96--2.80] $\times$ 10$^{24}$ cm$^{-2}$. The high-quality data from NuSTAR gives the best constraints on the spectral shape above $\sim$10 keV to date on NGC 1358. Moreover, by combining NuSTAR and XMM-Newton data, we find that the obscuring torus has a low covering factor ($f_c$ <0.17), and the obscuring material is distributed in clumps, rather than uniformly. We also derive an estimate of NGC 1358's Eddington ratio, finding it to be $\lambda_{\rm Edd}$ $\sim$$4.7_{-0.3}^{+0.3}$ $\times$ 10$^{-2}$, which is in acceptable agreement with previous measurements. Finally, we find no evidence of short-term variability, over a $\sim$100 ks time-span, in terms of both 'line-of-sight' column density and flux., Comment: 12 pages, 6 figures

We report on the investigation of a very high energy (VHE), Galactic gamma-ray source recently discovered at >50GeV using the Large Area Telescope (LAT) on board the Fermi-LAT. This object, 2FHL J0826.1-4500, displays one of the hardest >50GeV spectra (gamma-ray photon index ~1.6) in the 2FHL catalog, and a follow-up observation with XMM-Newton has uncovered diffuse, soft thermal emission at the position of the gamma-ray source. A detailed analysis of the available multi-wavelength data shows that this source is located on the Western edge of the Vela supernova remnant (SNR): the observations and the spectral energy distribution modelling support a scenario where this gamma-ray source is the byproduct of the interaction between the SNR shock and a neutral Hydrogen cloud. If confirmed, this shock-cloud interaction would make 2FHL J0826.1-4500 a promising candidate for efficient particle acceleration., Comment: 9 pages, 7 figures, Accepted for publication in ApJ on November 12, 2018. Published Jan. 3, 2019

Equilibrium refinements are important in extensive-form (i.e., tree-form) games, where they amend weaknesses of the Nash equilibrium concept by requiring sequential rationality and other beneficial properties. One of the most attractive refinement concepts is quasi-perfect equilibrium. While quasi-perfection has been studied in extensive-form games, it is poorly understood in Stackelberg settings---that is, settings where a leader can commit to a strategy---which are important for modeling, for example, security games. In this paper, we introduce the axiomatic definition of quasi-perfect Stackelberg equilibrium. We develop a broad class of game perturbation schemes that lead to them in the limit. Our class of perturbation schemes strictly generalizes prior perturbation schemes introduced for the computation of (non-Stackelberg) quasi-perfect equilibria. Based on our perturbation schemes, we develop a branch-and-bound algorithm for computing a quasi-perfect Stackelberg equilibrium. It leverages a perturbed variant of the linear program for computing a Stackelberg extensive-form correlated equilibrium. Experiments show that our algorithm can be used to find an approximate quasi-perfect Stackelberg equilibrium in games with thousands of nodes.

This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds threedimensional, with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on smooth weighted projective complete intersection Fano manifolds threefold. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable., Comment: 18 pages; Improved exposition; To Israel Vainsencher on the occasion of his 70th birthday

As matter accretes onto the central supermassive black holes in active galactic nuclei (AGN), X-rays are emitted. We present a population synthesis model that accounts for the summed X-ray emission from growing black holes; modulo the efficiency of converting mass to X-rays, this is effectively a record of the accreted mass. We need this population synthesis model to reproduce observed constraints from X-ray surveys: the X-ray number counts, the observed fraction of Compton-thick AGN [$\log$ (N$_{\rm H}$/cm$^{\rm -2}$) $>$ 24] and the spectrum of the cosmic X-ray background (CXB), after accounting for selection biases. Over the past decade, X-ray surveys by {\it XMM-Newton}, {\it Chandra}, \textit{NuSTAR} and \textit{Swift}-BAT have provided greatly improved observational constraints. We find that no existing X-ray luminosity function (XLF) consistently reproduces all these observations. We take the uncertainty in AGN spectra into account, and use a neural network to compute an XLF that fits all observed constraints, including observed Compton-thick number counts and fractions. This new population synthesis model suggests that, intrinsically, 50$\pm$9\% (56$\pm$7\%) of all AGN within z $\simeq$ 0.1 (1.0) are Compton-thick., Comment: Accepted by ApJ

Exploring how radio-emitting active galactic nuclei (AGN) behave and evolve with time is critical for understanding how AGN feedback impacts galaxy evolution. In this work, we investigate the relationship between 1.4 GHz radio continuum AGN luminosity ($L^{\rm AGN}_{\rm 1.4}$), specific black hole accretion rate (s-BHAR, defined as the accretion luminosity relative to the galaxy stellar mass) and redshift, for a luminosity-complete sample of radio-selected AGN in the VLA COSMOS 3 GHz Large Project. The sample was originally selected from radio-continuum observations at 3 GHz, and includes about 1800 radio AGN identified via ($>2\sigma$) radio-excess relative to the infrared-radio correlation of star-forming galaxies. We further select a subsample of over 1200 radio AGN that is complete in $L^{\rm AGN}_{\rm 1.4}$ over different redshift ranges, out to z~4, and use X-ray stacking to calculate the average s-BHAR in each $L^{\rm AGN}_{\rm 1.4}$-$z$ bin. We find that the average s-BHAR is independent of $L^{\rm AGN}_{\rm 1.4}$, at all redshifts. However, we see a strong increase of s-BHAR with redshift, at fixed $L^{\rm AGN}_{\rm 1.4}$. This trend resembles the strong increase in the fraction of star-forming host galaxies (based on the $(NUV-r)$ / $(r-J)$ colours) with redshift, at fixed $L^{\rm AGN}_{\rm 1.4}$. A possible explanation for this similarity might imply a link between average AGN radiative power and availability of cold gas supply within the host galaxy. This study corroborates the idea that radio-selected AGN become more radiatively efficient towards earlier epochs, independently of their radio power., Comment: 13 pages, 6 figures, 2 tables. Accepted for publication in MNRAS

We present the results of the optical spectroscopy follow-up of a sample of 28 unclassified blazars from the Third Fermi-LAT Catalog of High-Energy Sources (3FHL). All the spectra were taken with the 4m Mayall telescope at Kitt Peak. With this follow-up program we are able to classify 27 out of 28 objects as BL Lacs, while the remaining one is a flat spectrum radio quasar. We determine a redshift (z) for three of these objects and a lower limit on z for other four sources: the farthest object for which we obtain a redshift has z>0.836. These results are part of a more extended campaign of optical spectroscopy follow-up of 3FHL blazars, aimed to obtain a complete sample of blazars at >10 GeV which will then be used to extend our knowledge on blazar emission mechanisms and on the extragalactic background light., Comment: 12 pages, 6 figures. Accepted for publication on the Astronomical Journal Supplement Series. The spectra analyzed in this work are available at the following link: https://clemson.app.box.com/s/uu1hk6g4qy0ow9j4nst4nifm3mmrs19f

We study the problem of computing Stackelberg equilibria Stackelberg games whose underlying structure is in congestion games, focusing on the case where each player can choose a single resource (a.k.a. singleton congestion games) and one of them acts as leader. In particular, we address the cases where the players either have the same action spaces (i.e., the set of resources they can choose is the same for all of them) or different ones, and where their costs are either monotonic functions of the resource congestion or not. We show that, in the case where the players have different action spaces, the cost the leader incurs in a Stackelberg equilibrium cannot be approximated in polynomial time up to within any polynomial factor in the size of the game unless P = NP, independently of the cost functions being monotonic or not. We show that a similar result also holds when the players have nonmonotonic cost functions, even if their action spaces are the same. Differently, we prove that the case with identical action spaces and monotonic cost functions is easy, and propose polynomial-time algorithm for it. We also improve an algorithm for the computation of a socially optimal equilibrium in singleton congestion games with the same action spaces without leadership, and extend it to the computation of a Stackelberg equilibrium for the case where the leader is restricted to pure strategies. For the cases in which the problem of finding an equilibrium is hard, we show how, in the optimistic setting where the followers break ties in favor of the leader, the problem can be formulated via mixed-integer linear programming techniques, which computational experiments show to scale quite well.

The search problem of computing a \textit{leader-follower equilibrium} has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the \textit{optimistic} and \ textit{pessimistic} versions of the problem are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers' game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader's commitment, play a Nash equilibrium. The corresponding search problem, both in the optimistic and pessimistic versions, is known to be not in Poly-$\textsf{APX}$ unless $\textsf{P}=\textsf{NP}$ and exact algorithms are known only for the optimistic case. We focus on the case where the followers play in pure strategies under the assumption of pessimism. After casting this search problem as a \italic{pessimistic bilevel programming problem}, we show that, with two followers, the problem is $\textsf{NP}$-hard and, with three or more followers, it is not in Poly-$\textsf{APX}$ unless $\textsf{P}=\textsf{NP}$. We propose a single-level mathematical programming reformulation which calls for the maximisation of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exact \textit{ad hoc} algorithm, which we also embed within a branch-and-bound scheme, capable of computing the supremum of the problem.

We present an X-ray stacking analysis of $\sim$75,000 star-forming galaxies between $0.11.3$ exhibit a significant X-ray excess indicating nuclear emission. The AGN emission is strongly correlated with stellar mass but does not exhibit an additional correlation with SFR. The hardness ratios of the high-redshift stacks indicate that the AGN are substantially obscured ($N_{\mathrm{H}}\sim10^{23}$ cm$^{-2}$). These obscured AGN are not identified by IRAC color selection and have $L_X\sim10^{41}-10^{43}$ erg s$^{-1}$, consistent with accretion at an Eddington rate of $\sim10^{-3}$ onto $10^7-10^8 M_{\odot}$ black holes. Combining our results with other X-ray studies suggests that AGN obscuration depends on stellar mass and an additional variable, possibly the Eddington rate., Comment: 24 pages, 12 figures, 3 tables (2 tables available in full online or by request)

Leadership games provide a powerful paradigm to model many real-world settings. Most literature focuses on games with a single follower who acts optimistically, breaking ties in favour of the leader. Unfortunately, for real-world applications, this is unlikely. In this paper, we look for efficiently solvable games with multiple followers who play either optimistically or pessimistically, i.e., breaking ties in favour or against the leader. We study the computational complexity of finding or approximating an optimistic or pessimistic leader-follower equilibrium in specific classes of succinct games---polymatrix like---which are equivalent to 2-player Bayesian games with uncertainty over the follower, with interdependent or independent types. Furthermore, we provide an exact algorithm to find a pessimistic equilibrium for those game classes. Finally, we show that in general polymatrix games the computation is harder even when players are forced to play pure strategies.