The dynamics of a Bertrand duopoly with differentiated products: Synchronization,intermittency and global dynamics

We study the dynamics of a duopoly game à la Bertrand with horizontal product differentiation as proposed by Zhang et al. (2009) [35] by introducing opportune microeconomic foundations. The final model is described by a two-dimensional non-invertible discrete time dynamic system T. We show that synchronized dynamics occurs along the invariant diagonal being T symmetric; furthermore, we show that when considering the transverse stability, intermittency phenomena are exhibited. In addition, we discuss the transition from simple dynamics to complex dynamics and describe the structure of the attractor by using the critical lines technique. We also explain the global bifurcations causing a fractalization in the basin of attraction. Our results aim at demonstrating that an increase in either the degree of substitutability or complementarity between products of different varieties is a source of complexity in a duopoly with price competition.     

A note on the long time behavior for the drift-diffusion-Poisson system

In this note we analyze the long time behavior of a drift-diffusion-Poisson system with a symmetric definite positive diffusion matrix, subject to Dirichlet boundary conditions. This system models the transport of electrons in semiconductor or plasma devices. By using a quadratic relative entropy obtained by keeping the lowest order term of the logarithmic relative entropy, we prove the exponential convergence to the equilibrium. To cite this article: N. Ben Abdallah et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Split equilibrium problems for related games and applications to economic theory

ABSTRACT

In this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan-KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As an application of this existence to economics, an example is provided that studies the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual (playing twice) extended Bertrand duopoly model of price competition.     

Population equilibrium with support in evolutionary matrix games

In this paper we consider symmetric bimatrix games [A, A^T]. We use a matrix operator s(A), defined as the sum of the cofactors of the given matrix A, for finding the population equilibrium and its fitness in evolutionarily matrix games with all supported strategies, and to complete Bishop-Cannings Theorem.     

Dirac submanifolds and Poisson involutions

Dirac submanifolds are a natural generalization in the Poisson category of the symplectic submanifolds of a symplectic manifold. They correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable loci of Poisson involutions. In this paper, we make a general study of these submanifolds including both local and global aspects.In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable loci. In particular, we discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups, and symmetric Poisson groupoids. Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoid structures arising from dynamical r-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we prove that the Dubrovin Poisson structure on the space of Stokes matrices U₊ (appearing in Dubrovin's theory of Frobenius manifolds) is a Poisson symmetric space.     

Decentralization versus coordination in competing supply chains under retailers’ extended warranties

This paper studies a two-tier duopoly competing supply chain system consisting of two manufacturers and two exclusive retailers. Both manufacturers produce differentiated products and both retailers provide extended warranties for the products they sell. Two types of channel-structure strategy options are considered: a decentralized structure with a wholesale price contract and a coordinated structure with a sophisticated contract. We first derive the equilibrium outcomes under three possible chain-to-chain competition scenarios. Subsequently, we reveal how manufacturers control their retail channels to gain more supply chain system profit under an interactive environment with supply chain competition and retailers’ extended warranties. We find that pure coordinated channel competition and pure decentralized channel competition may both reach equilibrium. Furthermore, the interaction forces of supply chain competition and extended warranty service significantly impact the characteristics of the equilibria. Finally, we analyze the competing supply chain’s coordination contract design by using the example of a two-part tariff contract, and determine the feasible contract parameter range that results in a win-win solution for supply chain members.

     

On the stability of a class of noncooperative games

The Nash equilibrium of a class of games generated from a market is examined. Demands are assumed linear, and production constraints are imposed. The equilibrium is shown to be solvable as a complementarity problem. If the demand matrix is a positive definite symmetric z-matrix, then the Nash equilibrium is stable. If the demand matrix is not symmetric, an additional condition yielding stability is developed.     

A model structure for coloured operads in symmetric spectra

We describe a model structure for coloured operads with values in the category of symmetric spectra (with the positive model structure), in which fibrations and weak equivalences are defined at the level of the underlying collections. This allows us to treat R-module spectra (where R is a cofibrant ring spectrum) as algebras over a cofibrant spectrum-valued operad with R as its first term. Using this model structure, we give sufficient conditions for homotopical localizations in the category of symmetric spectra to preserve module structures.     

Oligopoly games under asymmetric costs and an application to energy production

Oligopolies in which firms have different costs of production have been relatively under-studied. In contrast to models with symmetric costs, some firms may be inactive in equilibrium. (With symmetric costs, the results trivialize to all firms active or all firms inactive.) We concentrate on the linear demand structure with constant marginal but asymmetric costs. In static one-period models, we compare the number of active firms, i.e. the number of firms producing a positive quantity in equilibrium, across four different models of oligopoly: Cournot and Bertrand with homogeneous or differentiated goods. When firms have different costs, we show that, for fixed good type, Cournot always results in more active firms than Bertrand. Moreover, with a fixed market type, differentiated goods result in more active firms than homogeneous goods. In dynamic models, asymmetric costs induce different entry times into the market. We illustrate with a model of energy production in which multiple producers from costly but inexhaustible alternative sources such as solar or wind compete in a Cournot market against an oil producer with exhaustible supply.     

Strict <Emphasis Type="Italic">K</Emphasis>-monotonicity and <Emphasis Type="Italic">K</Emphasis>-order continuity in symmetric spaces

This paper is devoted to strict K-monotonicity and K-order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K-monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K-order continuity in a symmetric space E on \([0,\infty )\) implies that the embedding \(E\hookrightarrow {L^1}[0,\infty )\) does not hold. We present a complete characterization of an equivalent condition to K-order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K-monotonicity and K-order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec–Klee property for global convergence in measure in a symmetric space E.     

Calabi's diastasis function for Hermitian symmetric spaces

In this paper we study the Calabi diastasis function of Hermitian symmetric spaces. This allows us to prove that if a complete Hermitian locally symmetric space (M,g) admits a Kähler immersion into a globally symmetric space (S,G) then it is globally symmetric and the immersion is injective. Moreover, if (S,G) is symmetric of a specified type (Euclidean, noncompact, compact), then (M,g) is of the same type. We also give a characterization of Hermitian globally symmetric spaces in terms of their diastasis function. Finally, we apply our analysis to study the balanced metrics, introduced by Donaldson, in the case of locally Hermitian symmetric spaces.     

Continuous Time Finite State Mean Field Games

In this paper we consider symmetric games where a large number of players can be in any one of d states. We derive a limiting mean field model and characterize its main properties. This mean field limit is a system of coupled ordinary differential equations with initial-terminal data. For this mean field problem we prove a trend to equilibrium theorem, that is convergence, in an appropriate limit, to stationary solutions. Then we study an N+1-player problem, which the mean field model attempts to approximate. Our main result is the convergence as N→∞ of the mean field model and an estimate of the rate of convergence. We end the paper with some further examples for potential mean field games.     

Bifurcation and forced symmetry breaking in Hamiltonian systems

We consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian system on a symplectic manifold. In particular we study the persistence of an initial relative equilibrium subjected to this forced symmetry breaking. We see that, under certain nondegeneracy conditions, an estimate can be made on the number of bifurcating relative equilibria. To cite this article: F. Grabsi et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

New Turaev Braided Group Categories and Group Schur–Weyl Duality

For a group G, we describe a new construction of a Turaev braided G-category with a particular braided monoidal subcategory and then we study the structure of a Hopf algebra in this subcategory. As an application, we establish a generalized G-Schur–Weyl duality between certain Turaev G-algebra and the symmetric group algebra.     

The Leitmann-Schmitendorf advertising game with n players and time discounting

The extension of the Leitmann-Schmitendorf advertising game to n players and positive time discounting is investigated. We show that the strong time consistency of the open-loop Nash equilibrium is preserved. As to optimal controls, while the boundary solution is unaffected by the number of firms as well as discounting, the inner solution depends on industry structure. The fully symmetric version of the game allows us to identify the parameter regions wherein both solutions are sustainable.     

Simple centrifugal incentives in spatial competition

This paper studies the effects of introducing centrifugal incentives in an otherwise standard Downsian model of electoral competition. First, we demonstrate that a symmetric equilibrium is guaranteed to exist when centrifugal incentives are induced by any kind of partial voter participation (such as abstention due to indifference, abstention due to alienation, etc.) and, then, we argue that: (a) this symmetric equilibrium is in pure strategies, and it is hence convergent, only when centrifugal incentives are sufficiently weak on both sides; (b) when centrifugal incentives are strong on both sides (when, for example, a lot of voters abstain when they are sufficiently indifferent between the two candidates) players use mixed strategies—the stronger the centrifugal incentives, the larger the probability weight that players assign to locations near the extremes; and (c) when centrifugal incentives are strong on one side only—say for example only on the right—the support of players’ mixed strategies contain all policies except from those that are sufficiently close to the left extreme.     

Symmetric orthogonal filters and wavelets with linear-phase moments

In this paper we study symmetric orthogonal filters with linear-phase moments, which are of interest in wavelet analysis and its applications. We investigate relations and connections among the linear-phase moments, sum rules, and symmetry of an orthogonal filter. As one of the results, we show that if a real-valued orthogonal filter a is symmetric about a point, then a has sum rules of order m if and only if it has linear-phase moments of order 2m. These connections among the linear-phase moments, sum rules, and symmetry help us to reduce the computational complexity of constructing symmetric real-valued orthogonal filters, and to understand better symmetric complex-valued orthogonal filters with linear-phase moments. To illustrate the results in the paper, we provide many examples of univariate symmetric orthogonal filters with linear-phase moments. In particular, we obtain an example of symmetric real-valued 4-orthogonal filters whose associated orthogonal 4-refinable function lies in C²(R).     

On zero-pattern invariant properties of structured matrices

It is interesting that inverse M-matrices are zero-pattern (power) invariant. The main contribution of the present work is that we characterize some structured matrices that are zero-pattern (power) invariant. Consequently, we provide necessary and sufficient conditions for these structured matrices to be inverse M-matrices. In particular, to check if a given circulant or symmetric Toeplitz matrix is an inverse M-matrix, we only need to consider its pattern structure and verify that one of its principal submatrices is an inverse M-matrix.     

The higher-order derivatives of spectral functions

We are interested in higher-order derivatives of functions of the eigenvalues of real symmetric matrices with respect to the matrix argument. We describe a formula for the k-th derivative of such functions in two general cases.The first case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times differentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues.The second case describes the derivatives of the composition of a k-times differentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix. We show that the formula significantly simplifies when the separable symmetric function is k-times continuously differentiable.As an application of the developed techniques, we re-derive the formula for the Hessian of a general spectral function at an arbitrary symmetric matrix. The new tools lead to a shorter, cleaner derivation than the original one.To make the exposition as self contained as possible, we have included the necessary background results and definitions. Proofs of the intermediate technical results are collected in the appendices.     

On local spectral properties of complex symmetric operators

In this paper we study properties of complex symmetric operators. In particular, we prove that every complex symmetric operator having property (β) or (δ) is decomposable. Moreover, we show that complex symmetric operator T has Dunford?s property (C) and it satisfies Weyl?s theorem if and only if its adjoint does.