Asymptotic equivalence between Cournot–Nash and Walras equilibria in exchange economies with atoms and an atomless part

In this paper, we consider an exchange economy à la Shitovitz (Econometrica 41:467–501, 1973), with atoms and an atomless set. We associate with it a strategic market game of the kind first proposed by Lloyd S. Shapley, known as the Shapley window model. We analyze the relationship between the set of the Cournot–Nash allocations of the strategic market game and the Walras allocations of the exchange economy with which it is associated. We show, with an example, that even when atoms are countably infinite, any Cournot–Nash allocation of the game is not a Walras allocation of the underlying exchange economy. Accordingly, in the original spirit of Cournot (Recherches sur les principes mathématiques de la théorie des richesses. Hachette, Paris, 1838), we partially replicate the mixed exchange economy by increasing the number of atoms, without affecting the atomless part, and ensuring that the measure space of agents remains finite. Our main theorem shows that any sequence of Cournot–Nash allocations of the strategic market games associated with the partial replications of the exchange economy has a limit point for each trader and that the assignment determined by these limit points is a Walrasian allocation of the original economy

Cournot-Walras Equilibrium as a Subgame Perfect Equilibrium

In this paper, we investigate the problem of the strategic foundation of the Cournot-Walras equilibrium approach. To this end, we respecify a'la Cournot-Walras the mixed version of a model of simultaneous, noncooperative exchange, originally proposed by Lloyd S.Shapley. We show, through an example, that the set of the Cournot-Walras equilibrium allocations of this respecification does not coincide with the set of the Cournot-Nash equilibrium allocations of the mixed version of the original Shapley's model. As the nonequivalence, in a one-stage setting, can be explained by the intrinsic two-stage nature of the Cournot-Walras equilibrium concept, we are led to consider a further reformulation of the Shapley's model as a two-stage game, where the atoms move in the first stage and the atomless sector moves in the second stage. Our main result shows that the set of the Cournot-Walras equilibrium allocations coincides with a specific set of subgame perfect equilibrium allocations of this two-stage game, which we call the set of the Pseudo-Markov perfect equilibrium allocations.

Noncooperative oligopoly in markets with a continuum of traders : a limit theorem

In this paper, in an exchange economy with atoms and an atomless part, we analyze the relationship between the set of the Cournot-Nash equilibrium allocations of a strategic market game and the set of the Walras equilibrium allocations of the exchange economy with which it is associated. In an example, we show that, even when atoms are countably infinite, Cournot-Nash equilibria yield different allocations from the Walras equilibrium allocations of the underlying exchange economy. We partially replicate the exchange economy by increasing the number of atoms without affecting the atomless part while ensuring that the measure space of agents remains finite. We show that any sequence of Cournot-Nash equilibrium allocations of the strategic mar- ket game associated with the partially replicated exchange economies approximates a Walras equilibrium allocation of the original exchange econom

Noncooperative Oligopoly in Markets with a Continuum of Traders

In this paper, we study three prototypical models of noncooperative oligopoly in markets with a continuum of traders : the model of Cournot-Walras equilibrium of Codognato and Gabszewicz (1991), the model of Cournot-Nash equilibrium of Lloyd S. Shapley, and the model of Cournot-Walras equilibrium of Busetto et al. (2008). We argue that these models are all distinct and only the Shapley's model with a continuum of traders and atoms gives an endogenous explanation of the perfectly and imperfectly competitive behavior of agents in a one-stage setting. For this model, we prove a theorem of existence of a Cournot-Nash equilibrium.

Noncooperative oligopoly in markets with a continuum of traders

In this paper, we study three prototypical models of noncooperative oligopoly in markets with a continuum of traders : the model of Cournot-Walras equilibrium of Codognato and Gabszewicz (1991), the model of Cournot-Nash equilibrium of Lloyd S. Shapley, and the model of Cournot-Walras equilibrium of Busetto et al. (2008). We argue that these models are all distinct and only the Shapley's model with a continuum of traders and atoms gives an endogenous explanation of the perfectly and imperfectly competitive behavior of agents in a one-stage setting. For this model, we prove a theorem of existence of a Cournot-Nash equilibrium

SUPG-stabilized stabilization-free VEM: a numerical investigation

We numerically investigate the possibility of defining stabilization-free Virtual Element (VEM) discretizations of advection-diffusion problems in the advection-dominated regime. To this end, we consider a SUPG stabilized formulation of the scheme. Numerical tests comparing the proposed method with standard VEM show that the lack of an additional arbitrary stabilization term, typical of VEM schemes, that adds artificial diffusion to the discrete solution, allows to better approximate boundary layers, in particular in the case of a low order scheme.Comment: 15 page

Cournot-Walras equilibrium as a subgame perfect equilibrium

In this paper, we investigate the problem of the strategic foundation of the Cournot-Walras equilibrium approach. To this end, we respecify à la Cournot-Walras the mixed version of a model of simultaneous, noncooperative exchange, originally proposed by Lloyd S. Shapley. We show, through an example, that the set of the Cournot- Walras equilibrium allocations of this respecifcation does not coincide with the set of the Cournot-Nash equilibrium allocations of the mixed version of the original Shapley's model. As the nonequivalence, in a one-stage setting, can be explained by the intrinsic two-stage nature of the Cournot-Walras equilibrium concept, we are led to consider a further reformulation of the Shapley's model as a two-stage game, where the atoms move in the first stage and the atomless sector moves in the second stage. Our main result shows that the set of the Cournot-Walras equilibrium allocations coincides with a specific set of subgame perfect equilibrium allocations of this two-stage game, which we call the set of the Pseudo-Markov perfect equilibrium allocations

Mitofusin-2 Down-Regulation Predicts Progression of Non-Muscle Invasive Bladder Cancer

Identification of markers predicting disease outcome is a major clinical issue for non-muscle invasive bladder cancer (NMIBC). The present study aimed to determine the role of the mitochondrial proteins Mitofusin-2 (Mfn2) and caseinolytic protease P (ClpP) in predicting the outcome of NMIBC. The study population consisted of patients scheduled for transurethral resection of bladder tumor upon the clinical diagnosis of bladder cancer (BC). Samples of the main bladder tumor and healthy-looking bladder wall from patients classified as NMIBC were tested for Mfn2 and ClpP. The expression levels of these proteins were correlated to disease recurrence, progression. Mfn2 and ClpP expression levels were significantly higher in lesional than in non-lesional tissue. Low-risk NMIBC had significantly higher Mfn2 expression levels and significantly lower ClpP expression levels than high-risk NMIBC; there were no differences in non-lesional levels of the two proteins. Lesional Mfn2 expression levels were significantly lower in patients who progressed whereas ClpP levels had no impact on any survival outcome. Multivariable analysis adjusting for the EORTC scores showed that Mfn2 downregulation was significantly associated with disease progression. In conclusion, Mfn2 and ClpP proteins were found to be overexpressed in BC as compared to non-lesional bladder tissue and Mfn2 expression predicted disease progression

Effects of an intensive inpatient rehabilitation program in elderly patients with obesity

Objective: The aim of this study was to assess the short-term effectiveness of an intensive inpatient multidimensional rehabilitation program (MRP), including diet, exercise, and behavioral therapy, in elderly patients with severe obesity. Methods: Forty-four elderly patients (old; age 69.3 \ub1 3.5 years, BMI 41.9 \ub1 14.9) were analyzed against 215 younger patients (young; age 48.2 \ub1 18.5 years, BMI 43.9 \ub1 9.4), who were used as controls. All patients underwent MRP, based on group therapy guided by a multidisciplinary team (physicians, dietitians, exercise trainers, psychologists). We evaluated changes in anthropometry, cardiovascular risk factors, physical fitness, quality of life, and eating behavior. Results: After 3 weeks of MRP, we observed a reduction in body weight (old -3.8%, young -4.3%), BMI (old -3.9%, young -4.4%), waist circumference (old -3.4%, young -4.1%), total cholesterol (old -14.0%, young -15.0%), and fasting glucose (old -8.3%, young -8.1%), as well as improved performance in the Six-Minute-Walk Test (old +28.7%, young +15.3%), chair-stand test (old +24.8%, young +26.9%), and arm-curl test (old +15.2%, young +27.3%). Significant improvement was registered in all other analyzed domains. Conclusion: Our 3-week MRP provided significant clinical and functional improvement, which was similar between elderly and younger patients with severe obesity. In the long-term, this may be translated into better quality of life, through better management of obesity-associated morbidities and reduced frailty