Conditions For Existence Of Equilibrium Research Articles

The existence of equilibrium states in dynamic systems with attractive interaction

Nowadays, science is characterized by needs of the study of various complex processes and phenomena’s. Today’s research of complex and dynamical systems is one of the most advanced ways of research and evolution of the modern world. Models of biology and ecology, physical models, various economic and social models are typical examples of dynamic systems. The concept of an interactive complex system in modern science is a main tool for construction of mathematical models for solving modern civilization problems and development. The dynamical systems approach to conflict is relatively new, but it has beginning in different research fields. Theory of dynamic systems helps us to understand the experiments, build the mathematical model of iterations and examine behavior and relations between opponents, like distribution of resources and territory, population growing etc. This is a challenging problem of finding and achieving a compromise state for opponents on a common territory has different options to define the task and to choose conflict interaction. In 2016, the monograph by V. Koshmanenko where was introduced new approach for dynamic system of conflict that based on interactions of the opponents in the form probability distribution in the disputed area was published. In particular, presented the concept of a complex dynamic system with attractive interaction. The relevance of this research is improving new dynamical system and researching for a new application of abstract models in everyday life. In this paper briefly fundamentals of the theory of dynamical systems described and the theorem on the existence of a equilibrium state in a the new, perspective for research, dynamical system with attractive interaction in terms of probability distributions (measures) and their densities, formulated and proved.

Quantum majorization and a complete set of entropic conditions for quantum thermodynamics

What does it mean for one quantum process to be more disordered than another? Interestingly, this apparently abstract question arises naturally in a wide range of areas such as information theory, thermodynamics, quantum reference frames, and the resource theory of asymmetry. Here we use a quantum-mechanical generalization of majorization to develop a framework for answering this question, in terms of single-shot entropies, or equivalently, in terms of semi-definite programs. We also investigate some of the applications of this framework, and remarkably find that, in the context of quantum thermodynamics it provides the first complete set of necessary and sufficient conditions for arbitrary quantum state transformations under thermodynamic processes, which rigorously accounts for quantum-mechanical properties, such as coherence. Our framework of generalized thermal processes extends thermal operations, and is based on natural physical principles, namely, energy conservation, the existence of equilibrium states, and the requirement that quantum coherence be accounted for thermodynamically.

Existence of Perturbed Equilibrium States in Some Two-Layer Systems with Density Inversion

In the system formed by a heavy elastic layer (lithosphere) and a half-space filled by an ideal incompressible fluid (asthenosphere), the possibility of the existence of equilibrium states with curved boundaries near an equilibrium of a system with rectilinear boundaries is investigated. Using an analysis of the characteristic equation, we obtain a relationship between the wave number of the desired static perturbation and the dimensionless parameters of the problem, namely, the dimensionless shear modulus, Poisson’s ratio, and the decompression. The assumption that the deformations are small imposes conditions on the ranges of modification of the quantities. For example, for a moderately compressible elastic material, the equilibrium which is called tectonic waves in geophysical applications is possible only in the long-wavelength range in the presence of the inversion of density and very strong decompression. The stability problem with respect to small dynamical perturbations of an (obtained) equilibrium with curved boundaries is stated. A wave dispersion relation connecting the complex frequency of oscillations with the wave number of perturbations and with the above dimensionless parameters of the system is derived.

Technical Note—The Competitive Facility Location Problem in a Duopoly: Advances Beyond Trees

We consider a competitive facility location problem on a network where consumers located on vertices wish to connect to the nearest facility. Knowing this, each competitor locates a facility on a vertex, trying to maximize market share. We focus on the two-player case and study conditions that guarantee the existence of a pure-strategy Nash equilibrium for progressively more complicated classes of networks. For general graphs, we show that attention can be restricted to a subset of vertices referred to as the central block. By constructing trees of maximal bi-connected components, we obtain sufficient conditions for equilibrium existence. Moreover, when the central block is a vertex or a cycle (for example, in cactus graphs), this provides a complete and efficient characterization of equilibria. In that case, we show that both competitors locate their facilities in a solution to the 1-median problem, generalizing a well-known insight arising from Hotelling’s model. We further show that an equilibrium must solve the 1-median problem in other classes of graphs, including grids, which essentially capture the topology of urban networks. In addition, when both players select a 1-median, the solution must be at equilibrium for strongly-chordal graphs, generalizing a previously known result for trees. The electronic companion is available at https://doi.org/10.1287/opre.2017.1694 .

Equilibrium states and zero temperature limit on topologically transitive countable Markov shifts

Consider a topologically transitive countable Markov shift and, let $f$ be a summable potential with bounded variation and finite Gurevic pressure. We prove that there exists an equilibrium state $\mu_{tf}$ for each $t > 1$ and that there exists accumulation points for the family $(\mu_{tf})_{t>1}$ as $t \to \infty$. We also prove that the Kolmogorov-Sinai entropy is continuous at $\infty$ with respect to the parameter $t$, that is $\lim_{t \to \infty} h(\mu_{tf})=h(\mu_{\infty})$, where $\mu_{\infty}$ is an accumulation point of the family $(\mu_{tf})_{t>1}$. These results do not depend on the existence of Gibbs measures and, therefore, they extend results of \cite{MaUr01} and \cite{Sar99} for the existence of equilibrium states without the BIP property, \cite{JMU05} for the existence of accumulation points in this case and, finally, we extend completely the result of \cite{Mor07} for the entropy zero temperature limit beyond the finitely primitive case.

A variational principle for free semigroup actions

In this paper we introduce a notion of measure theoretical entropy for a finitely generated free semigroup action and establish a variational principle when the semigroup is generated by continuous self maps on a compact metric space and has finite topological entropy. In the case of semigroups generated by Ruelle-expanding maps we prove the existence of equilibrium states and describe some of their properties. Of independent interest are the different ways we will present to compute the metric entropy and a characterization of the stationary measures.

Equilibrium states for random lattice models of hyperbolic type

We study the structural stability of random coupled map lattice models of hyperbolic type under certain metrics. Furthermore, by describing the thermodynamic formalism of the underlying random spin lattice system, we prove the existence of equilibrium states for equi-Hölder continuous random functions on lattice models under the conditions of random weak interaction and translation invariance and present some partial results on the uniqueness of equilibrium states.

Explicitly Solvable Models of Redistribution of the Conflict Space

We analyze a class of explicitly solvable models for the problems of redistribution of the conflict space between two alternative opponents. The existence of equilibrium state is proved for a complex nonlinear system whose time evolution is generated by the conflict interaction between its components. Explicit formulas are obtained for the limit compromise distributions in terms of the densities of probability measures. We consider a number of specific model examples of the dynamics of redistribution of the conflict territory and formation of an equilibrium (compromise) distribution of the space. We propose an interpretation of the results for the case of social and territorial conflicts.

Distributive Politics with Other-Regarding Preferences

This paper analyzes a non-smooth model of probabilistic voting with two parties and a broad family of other-regarding behavior, including fairness and quasi-maximin preferences, income-dependent altruism, and inequity aversion. The paper provides conditions for equilibrium existence and uniqueness. It also characterizes the Nash equilibrium in pure strategies when parties hold either symmetric payoffs, or minor forms of asymmetries. The characterization shows that the two parties converge to an equilibrium policy that maximizes a mixture of a ``self-regarding utilitarian'' social welfare function and an aggregate of society's other-regarding preferences. These results are shown to be applicable to other non-smooth frameworks, such as probabilistic voting with loss averse voters. The characterization also shows that the direction and the size of the inefficiencies emerging from electoral competition depend in a subtle way on the nature of the other-regarding preferences (and resp., loss aversion)

Kinetic Theory beyond the Stosszahlansatz

In a recent paper (Chliamovitch, et al., 2015), we suggested using the principle of maximum entropy to generalize Boltzmann’s Stosszahlansatz to higher-order distribution functions. This conceptual shift of focus allowed us to derive an analog of the Boltzmann equation for the two-particle distribution function. While we only briefly mentioned there the possibility of a hydrodynamical treatment, we complete here a crucial step towards this program. We discuss bilocal collisional invariants, from which we deduce the two-particle stationary distribution. This allows for the existence of equilibrium states in which the momenta of particles are correlated, as well as for the existence of a fourth conserved quantity besides mass, momentum and kinetic energy.

Subgame-perfect equilibria in stochastic timing games

We develop a notion of subgames and the related notion of subgame-perfect equilibrium – possibly in mixed strategies – for stochastic timing games. To capture all situations that can arise in continuous-time models, it is necessary to consider stopping times as the starting dates of subgames. We generalize Fudenberg and Tirole’s (Rev. Econom. Stud. 52, 383–401, 1985) mixed-strategy extensions to make them applicable to stochastic timing games and thereby provide a sound basis for subgame-perfect equilibria of preemption games. Sufficient conditions for equilibrium existence are presented, and examples illustrate their application as well as the fact that intuitive arguments can break down in the presence of stochastic processes with jumps.

Existence of stationary bargaining equilibria

This paper addresses the question of existence of stationary Markov perfect equilibria in a class of dynamic games that includes many known bargaining models and models of coalition formation. General sufficient conditions for existence of equilibria are currently lacking in a number of interesting environments, e.g., models with non-convexities, consumption lower bounds, or an evolving state variable. The main result establishes existence of equilibrium under compactness and continuity conditions, without the structure of convexity or strict comprehensiveness used in the extant literature. The proof requires a precise selection of voting equilibria following different proposals using a generalization of Fatou's lemma.

On Existence and Stability of Equilibria of Linear Time-Invariant Systems With Constant Power Loads

The problem of existence and stability of equilibria of linear systems with constant power loads is addressed in this paper. First, we correct an unfortunate mistake in our recent paper pertaining to the sufficiency of the condition for existence of equilibria in multiport systems given there. Second, we give two necessary conditions for existence of equilibria. The first one is a simple linear matrix inequality hence it can be easily verified with existing software. Third, we prove that the latter condition is also sufficient if a set defined by the problem data is convex, which is the case for single and two-port systems. Finally, sufficient conditions for stability and instability for a given equilibrium point are given. The results are illustrated with two benchmark examples.

On the Influence of the Price Dependence of a Goods Consumption Structure on the Equilibrium of an Economy

Equilibrium conditions for an open economy with monopolies are investigated. The presence of insatiable consumers in the economy is taken into account. The influence of the price dependence of a goods consumption structure on equilibrium states is investigated. The existence of equilibrium states in such an economy is proved. A relation is established between the selection of a taxation strategy and realization of a concrete equilibrium state of the economy.

Efficiency in decentralized oligopolistic markets

The paper analyzes quantity competition in economies in which a network describes the set of feasible trades. A model is presented in which the identity of buyers, of sellers, and of intermediaries is endogenously determined by the trade flows in the economy. The analysis first considers small economies, and provides sufficient conditions for equilibrium existence, a characterization of prices and flows, and some negative results relating welfare to network structure. The second and central part of the analysis considers behavior in large markets, and presents necessary and sufficient conditions on the network structure for equilibria to be approximately efficient when the number of players is large.

Conditions for Existence of Equilibria of Systems With Constant Power Loads

In this paper we investigate the sine qua non condition of existence of equilibria for electrical systems with external (AC or DC) sources furnishing constant power to the loads, which is a scenario encountered in modern applications. Two general cases are considered, when the system is i) linear time-invariant or ii) nonlinear, with dynamic behavior described by a port-Hamiltonian model with constant dissipation and switching interconnection matrix. The latter class includes the practically important case of power converters. For both cases necessary and sufficient conditions for existence of equilibria are given, which give an upper bound on the power dissipated in steady-state that should exceed the extracted constant power. The existence of the equilibrium is ensured if and only if the inequality is satisfied.

Complex bifurcations in a nonlinear system of moving belt

A kind of one-degree-of-freedom nonlinear moving belt system is considered. The analytical research of sliding region and existence conditions of equilibrium are first derived by the theory of piecewise-smooth dynamical system. Then, using numerical method, one- or two-parameter continuation of several types of periodic orbits of the system is calculated. We obtain codimension-1 sliding bifurcation curves, codimension-2 sliding bifurcation points, and global bifurcation diagram in parameter space for the system. The investigation of bifurcation behavior shows that the speed of moving belt and amplitude of friction have a great influence on dynamic behavior, and reveals the complex nonlinear dynamic phenomenon of the moving belt system.

Existence of equilibrium states of hollow hollomon cylinders submerged in a fluid

Existence of equilibrium states of hollow hollomon cylinders submerged in a fluid

Nice Inducing Schemes and the Thermodynamics of Rational Maps

We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential \({-t {\rm ln} \left|f^{\prime} \right|}\) , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in \({\mathbb{R}}\) , with at most two exceptions.

Bidding in a possibly common-value auction

We analyze a second-price auction with two bidders in which only one of the bidders is informed as to whether the object is valued commonly. We show that any equilibrium strategy of the bidder who is uninformed must be part of an equilibrium when both bidders instead know that the auction is not common value, regardless of the way in which the values are different. We derive sufficient conditions for equilibrium existence.