NN-player games and mean field games of moderate interactions

We study the asymptotic organization among many optimizing individuals interacting in a suitable "moderate" way. We justify this limiting game by proving that its solution provides approximate Nash equilibria for large but finite player games. This proof depends upon the derivation of a law of large numbers for the empirical processes in the limit as the number of players tends to infinity. Because it is of independent interest, we prove this result in full detail. We characterize the solutions of the limiting game via a verification argument

N-player games and mean-field games with smooth dependence on past absorptions.

Mean-field games with absorption is a class of games that has been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, we allow the state dynamics and the costs to have a very general, possibly innite-dimensional, dependence on the (non-normalized) empirical sub- probability measure of the survivors' states. This includes the particularly relevant case where the mean-eld interaction among the players is done through the empirical measure of the survivors together with the fraction of absorbed players over time. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth in the state variables, hence allowing for more realistic dynamics for players' private states. We prove the existence of solutions of the MFG in strict as well as relaxed feedback form, and we establish uniqueness of the MFG solutions under monotonicity conditions of Lasry-Lions type. Finally, we show in a setting with finite-dimensional interaction that such solutions induce approximate Nash equilibria for the N-player game with vanishing error as N tends to infinity.Mean-field games with absorption is a class of games that has been introduced in (Ann. Appl. Probab. 28 (2018) 2188–2242) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, we allow the state dynamics and the costs to have a very general, possibly infinite-dimensional, dependence on the (non-normalized) empirical sub-probability measure of the survivors’ states. This includes the particularly relevant case where the mean-field interaction among the players is done through the empirical measure of the survivors together with the fraction of absorbed players over time. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth in the state variables, hence allowing for more realistic dynamics for players’ private states. We prove the existence of solutions of the MFG in strict as well as relaxed feedback form, and we establish uniqueness of the MFG solutions under monotonicity conditions of Lasry–Lions type. Finally, we show in a setting with finite-dimensional interaction that such solutions induce approximate Nash equilibria for the N-player game with vanishing error as N → ∞

N-player games and mean field games of moderate interactions

We study the asymptotic organization among many optimizing individuals interacting in a suitable “moderate" way. We justify this limiting game by proving that its solution provides approximate Nash equilibria for large but finite player games. This proof depends upon the derivation of a law of large numbers for the empirical processes in the limit as the number of players tends to infinity. Because it is of independent interest, we prove this result in full detail. We characterize the solutions of the limiting game via a verification argument

Mean-Field Games of Finite-Fuel Capacity Expansion with Singular Controls

We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space into action and inaction regions. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $\varepsilon$ going to zero as $N$ tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework

Kelly betting with quantum payoff: A continuous variable approach

The main purpose of this study is to introduce a semi-classical model describing betting scenarios in which, at variance with conventional approaches, the payoff of the gambler is encoded into the internal degrees of freedom of a quantum memory element. In our scheme, we assume that the invested capital is explicitly associated with the quantum analog of the free-energy (i.e. ergotropy functional by Allahverdyan, Balian, and Nieuwenhuizen) of a single mode of the electromagnetic radiation which, depending on the outcome of the betting, experiences attenuation or amplification processes which model losses and winning events. The resulting stochastic evolution of the quantum memory resembles the dynamics of random lasing which we characterize within the theoretical setting of Bosonic Gaussian channels. As in the classical Kelly Criterion for optimal betting, we define the asymptotic doubling rate of the model and identify the optimal gambling strategy for fixed odds and probabilities of winning. The performance of the model are hence studied as a function of the input capital state under the assumption that the latter belongs to the set of Gaussian density matrices (i.e. displaced, squeezed thermal Gibbs states) revealing that the best option for the gambler is to devote all their initial resources into coherent state amplitude

IL VERDE PER MIGLIORARE LA QUALITÀ DELL’ARIA OUTDOOR NEI CONTESTI URBANI

Introduzione: I contesti urbani presentano elevati livelli di inquinamento atmosferico, responsabili di patologie acute e di malattie cronico-degenerative. La vegetazione è in grado di assorbire e trattenere alcune sostanze, tuttavia nelle aree densamente costruite è spesso difficile creare nuovi spazi verdi, se non ipotizzando di utilizzare le coperture. Scopo della ricerca è stato l’elaborazione di un modello di calcolo per la quantificazione delle possibili superfici trasformabili a verde e la relativa capacità di assorbimento degli inquinanti di uno specifico contesto urbano. Metodologia: Il confronto con la letteratura scientifica ha consentito da definizione di correlazioni tra patologie, inquinanti atmosferici e specie arboree in grado di abbatterli. In seguito, è stato sviluppato un modello di calcolo in grado di quantificare gli inquinanti abbattuti in relazione ad uno specifico aumento di verde di un’area urbana. Nella terza parte, il modello di calcolo, è stato sperimentato e validato in un quartiere di Milano. Risultati: L’applicazione dello strumento è avvenuta nel quartiere città studi, a Milano. Le ipotetiche trasformazioni di coperture piane in tetti verdi, porterebbero ad una riduzione degli inquinanti fino a 3,9 μg/m³ di PM10, 2,92 μg/m³ di PM2. 5 e 1,57 μg/m³ di O3 ogni giorno in tutto il quartiere. Se tale ipotesi fosse estesa a tutta la città di Milano si arriverebbe ad un abbattimento di circa 18,09 μg/m³ di PM10, 13,56 μg/m³ di PM2. 5 e 7,24 μg/m³ di O3. Lo strumento ipotizza anche una stima dei costi di realizzazione e i benefici sanitari in termini di riduzione del n. di ricoveri. Conclusioni: Considerando un progetto di riqualificazione urbana la possibilità di recuperare spazi verdi dalle coperture diviene una strategia fondamentale non solo per i benefici psicologici-percettivi, energetici e climatico-ambientali ma anche per la riduzione dell’inquinamento atmosferico urbano, con ricadute positive sulla salute della popolazione

The detail is more pleasant than the whole: global and local prime affect esthetic appreciation of artworks showing whole-part ambiguity

Esthetic experience is the result of the coordination of different cognitive processes. It has been widely reported that top-down processes of orienting of attention interact with bottom-up perceptual facilitation occurring during esthetic experience of artworks. Here we use whole-part ambiguity as a tool to test the effect of global and local prime on esthetic appreciation of complex visual artworks. To this aim 139 healthy young individuals completed an esthetic judgment of Arcimboldo’s ambiguous artworks, which were preceded by a local or global prime. Their perceptual style was also assessed using a Navon task. We found that local prime significantly enhanced esthetic appreciation of ambiguous portraits. Also, we found that prime level interacted with individual’s perceptual style: participants showing local perceptual style liked less ambiguous portraits when they were preceded by global prime. Overall, the present findings shed some light on the processes involved in esthetic experience, pointing towards a pivotal role of re-direction of attention towards perceptual features of the artworks and its interaction with individual factors, such as perceptual style

Bone Marrow Foot Oedema in Adolescents: The Role of Vitamin D

BACKGROUND: Bone marrow oedema (BMO) in children/adolescents is a rare clinical condition without an etiologic cause. It is associated with typical increased signal intensity on T2-weighted magnetic resonance images (MRI) and an increase in bone turnover in which vitamin D plays a pivotal role. No treatment guidelines for these young patients are to date available. METHODS: We performed a retrospective study in a pediatric setting of 13 patients with diagnosis of primary BMO of the foot on the basis of clinical and radiological findings. Data collection included sex, age, patient history, symptoms at presentation, clinical examination, laboratory bone turnover markers, vitamin D levels, MRI, treatment, and outcome. RESULTS: Vitamin D deficiency or insufficiency was found in 76.9% of cases. All patients were treated with adequate vitamin D daily intake, a short course of analgesic therapy, physical therapy, avoiding detrimental feet and ankle immobilization. All fully recovered in 3-month lag period. CONCLUSIONS: Our data highlight that environmental factors, such as underestimated articular or bone microtraumatisms, as well as joint hyper mobility, in a bone turnover milieu of vitamin D deficiency could be the cause of this clinical conditions. Adequate vitamin D supplementation, associated with physical and analgesic therapy, is crucial in the management of BMO