Stationary Equilibrium Research Articles

Stationary equilibria and their stability in a Kuramoto MFG with strong interaction

Recently, R. Carmona, Q. Cormier, and M. Soner proposed a Mean Field Game (MFG) version of the classical Kuramoto model, which describes synchronization phenomena in a large population of “rational” interacting oscillators. The MFG model exhibits several stationary equilibria, but the characterization of these equilibria and their ability to capture dynamic equilibria in long time remains largely open. In this paper, we demonstrate that, up to a phase translation, there are only two possible stationary equilibria: the incoherent equilibrium and the self-organizing equilibrium, given that the interaction parameter is sufficiently large. Furthermore, we present some local stability properties of the self-organizing equilibrium.

Stationary equilibrium torus supported by Weyssenhoff ideal spin fluid in Schwarzschild spacetime: Case of constant specific angular momentum distribution

Stationary equilibrium torus supported by Weyssenhoff ideal spin fluid in Schwarzschild spacetime: Case of constant specific angular momentum distribution

Potts game on graphs: static equilibria

Potts game on graphs: static equilibria

Natural resources, carbon neutrality, and fiscal federalism: Implications for G7 countries amid rising Covid-19 concerns

Growing apprehensions regarding the environment have become a significant focus of recent literature. In this vein, revenue decentralization, expenditure decentralization, natural resource rent, and economic growth gained less prominence in explaining the consumption-based carbon dioxide emissions of the G7 countries amid rising Covid-19 concerns. Thus, the role of these factors is crucial in investigating their impact on environmental sustainability using G7 nations from 1990 to 2021. The preliminary findings assert the existence of cross-sectional dependence, slope heterogeneity, first-order stationary, and long-run equilibrium relationship between the stated variables. The results show that revenue decentralization positively impacts carbon neutrality, while expenditure decentralization, natural resource rent, and economic growth enhance environmental degradation in the region. Regarding the indirect impact, expenditure decentralization positively moderates the influence of natural resource rent; however, the magnitude is less than its direct impact. Concerning the moderating effect of revenue decentralization and natural resource rent, the outcomes revealed that revenue decentralization mitigates the destructive consequences of natural resource dependency and subsidizes the emissions mitigating effects. The study suggests a transparent fiscal decentralized setup could help to achieve environmental sustainability.

The study for optimization strategies on the performance of DCGAN

Since Deep Convolutional Generative Adversarial Networks (DCGAN) was proposed, it has been perceived as a model with difficulty in training due to several factors. To solve this problem, dozens of optimization strategies were presented, but none of them was compared with the others. In this paper, the author chose three representative methods, namely one-label smoothing, the two Time-Scale Update Rule (TTUR), and the Earth-Mover Distance (EMD) or Wasserstein-1 to make a comparison of the optimization effect on the DCGAN model. To be specific, these three approaches were adopted respectively while using MNIST and Fashion-MNIST as datasets. One-side label smoothing was designed to prevent overconfidence in the model by adding a penalty term in the discriminator. TTUR was a simpler update strategy that could help the model find the stationary local Nash equilibrium under mild assumptions. EMD was an alternative loss function that enabled the model to distinguish the difference while the real distribution and generated distribution were not overlapped. Contrast experiments were conducted both vertically and horizontally. The author applied these three methods with the same dataset and the same method with different datasets in order to compare the time of the model collapse, the trend of loss in line graphs, and the impact of different datasets on results. Experimental results indicated that both one-label smoothing and TTUR postponed the model collapse while EMD completely get rid of it. Furthermore, generated images may lose texture information when using more complicated datasets.

Learning to Mitigate Epidemic Risks: A Dynamic Population Game Approach

We present a dynamic population game model to capture the behavior of a large population of individuals in presence of an infectious disease or epidemic. Individuals can be in one of five possible infection states at any given time: susceptible, asymptomatic, symptomatic, recovered and unknowingly recovered, and choose whether to opt for vaccination, testing or social activity with a certain degree. We define the evolution of the proportion of agents in each epidemic state, and the notion of best response for agents that maximize long-run discounted expected reward as a function of the current state and policy. We further show the existence of a stationary Nash equilibrium and explore the transient evolution of the disease states and individual behavior under a class of evolutionary learning dynamics. Our results provide compelling insights into how individuals evaluate the trade-off among vaccination, testing and social activity under different parameter regimes, and the impact of different intervention strategies (such as restrictions on social activity) on vaccination and infection prevalence.

A stochastic optimal stopping model for storable commodity prices

In this paper, we propose a continuous time version of the well-known speculative storage model for commodity prices. But from the mathematical point of view this is not a trivial extension and needs careful consideration of the theory of stochastic stopping time combined with fixed point theory. We formulate the problem in a manner that the main objective of the storage model, known as the stationary rational expectations equilibrium (SREE), becomes a fixed-point of an operator which solves a free boundary problem and show that this operator under some conditions is a contraction. We also demonstrate the benefits of our continuous time model through a numerical algorithm.

On the Nash Equilibria of a Duel with Terminal Payoffs

We formulate and study a two-player duel game as a terminal payoffs stochastic game. Players P1,P2 are standing in place and, in every turn, each may shoot at the other (in other words, abstention is allowed). If Pn shoots Pm (m≠n), either they hit and kill them (with probability pn) or they miss and Pm is unaffected (with probability 1−pn). The process continues until at least one player dies; if no player ever dies, the game lasts an infinite number of turns. Each player receives a positive payoff upon killing their opponent and a negative payoff upon being killed. We show that the unique stationary equilibrium is for both players to always shoot at each other. In addition, we show that the game also possesses “cooperative” (i.e., non-shooting) non-stationary equilibria. We also discuss a certain similarity that the duel has to the iterated Prisoner’s Dilemma.

A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations

We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori, each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field.

Stationary Strong Stackelberg Equilibrium in Discounted Stochastic Games

Stationary Strong Stackelberg Equilibrium in Discounted Stochastic Games

Detection Possibility of Continuous Gravitational Waves from Rotating Magnetized Neutron Stars

In recent decades, several neutron stars (NSs), particularly pulsars, with masses of M > 2 M ⊙ have been observed. On the other hand, the existence of massive white dwarfs, even violating the Chandrasekhar mass limit, was inferred from the peak luminosities of Type Ia supernovae. Hence, there is a generic question of the origin of massive compact objects. Here we explore the existence of massive, magnetized, rotating NSs with the soft and steep equations of state by solving axisymmetric stationary stellar equilibria in general relativity. For our purposes, we consider the Einstein equation solver for stellar structure XNS code. Such rotating NSs with magnetic fields and rotation axes misaligned, and hence a nonzero obliquity angle, can emit continuous gravitational waves (GWs), which can be detected by upcoming detectors, e.g., the Einstein Telescope, etc. We discuss the decay of the magnetic field, angular velocity, and obliquity angle with time due to angular momentum extraction by GWs and dipole radiation, which determine the timescales related to the GW emission. Further, in the Alfvén timescale, a differentially rotating, massive proto-NS rapidly settles into a uniformly rotating, less massive NS due to magnetic braking and viscosity. These explorations suggest that detecting massive NSs is challenging and sets a timescale for detection. We calculate the signal-to-noise ratio of GW emission, which confirms that any detector cannot detect them immediately, but that they are detectable by the Einstein Telescope and Cosmic Explorer over months of integration time, leading to direct detection of NSs.

The Logarithmic Stochastic Tracing Procedure: A Homotopy Method to Compute Stationary Equilibria of Stochastic Games

We introduce the logarithmic stochastic tracing procedure, a homotopy method to compute stationary equilibria for finite and discounted stochastic games. We build on the linear stochastic tracing procedure but introduce logarithmic penalty terms as a regularization device, which brings two major improvements. First, the scope of the method is extended: it now has a convergence guarantee for all games of this class rather than just generic ones. Second, by ensuring a smooth and interior solution path, computational performance is increased significantly. A ready-to-use implementation is publicly available. As demonstrated here, its speed compares quite favorably to other available algorithms, and it allows us to solve games of considerable size in reasonable times. Because the method involves the gradual transformation of a prior into equilibrium strategies, it is possible to search the prior space and uncover potentially multiple equilibria and their respective basins of attraction. This also connects the method to established theory of equilibrium selection. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms – Continuous. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.0360 .

Qualitative analysis of a prey–predator model with prey refuge and intraspecific competition among predators

In this study, we consider a prey–predator model with prey refuge and intraspecific competition between predators using the Crowley–Martin functional response and investigate the dynamic characteristics of spatial and nonspatial prey–predator systems via both analytical and numerical methods. The local stability of nontrivial interior equilibrium, the existence of a Hopf bifurcation, and the stability of bifurcating periodic solutions are obtained in the absence of diffusion. For the spatial system, the Turing and non-Turing patterns are evaluated for some set of parametric belief functions, and we obtain some interesting results in terms of prey and predator inhabitants. We present the results of numerical simulations that demonstrate that both prey and predator populations do not converge to a stationary equilibrium state at any foreseeable future time when the parametric values are processed in the Turing domain.

Decomposition of Acoustic and Entropy Modes in a Non-Isothermal Gas Affected by a Mass Force

Diagnostics and decomposition of atmospheric disturbances in a planar flow are considered in this work. The study examines a situation in which the stationary equilibrium temperature of a gas may depend on the vertical coordinate due to external forces. The relations connecting perturbations are analytically established. These perturbations specify acoustic and entropy modes in an arbitrary stratified gas affected by a constant mass force. The diagnostic relations link acoustic and entropy modes, and are independent of time. Hence, they provide an ability to decompose the total vector of perturbations into acoustic and non-acoustic (entropy) parts, and to establish the distribution of energy between the sound and entropy modes, uniquely at any instant. The total energy of a flow is hence determined in its parts which are connected with acoustic and entropy modes. The examples presented in this work consider the equilibrium temperature of a gas, which linearly depends on the vertical coordinate. Individual profiles of acoustic and entropy parts for some impulses are illustrated with plots.

Rescue policies for small businesses in the COVID-19 recession

While the COVID-19 pandemic had a large and asymmetric impact on firms, many countries quickly enacted massive business rescue programs which are specifically targeted to smaller firms. Little is known about the effects of such policies on business entry and exit, investment, factor reallocation, and macroeconomic outcomes. This paper builds a general equilibrium model with heterogeneous and financially constrained firms in order to evaluate the short- and long-term consequences of small firm rescue programs in a pandemic recession. We calibrate the stationary equilibrium and the pandemic shock to the U.S. economy, taking into account the factual Paycheck Protection Program (PPP) as a specific policy. We find that the policy has only a modest impact on aggregate output and employment because (i) jobs are saved predominately in the smallest firms that account for a minor share of employment and (ii) the grant reduces the reallocation of resources towards larger and less impacted firms. Much of the reallocation effects occur in the aftermath of the pandemic episode. By preventing inefficient liquidations, the policy dampens the long-term declines of aggregate consumption and of the real wage, thus delivering small welfare gains.

Stationary equilibrium of test particles near charged black branes with the hyperscaling violating factor

Stationary equilibrium of test particles near charged black branes with the hyperscaling violating factor

A Self-Contained Karma Economy for the Dynamic Allocation of Common Resources

This paper presents karma mechanisms, a novel approach to the repeated allocation of a scarce resource among competing agents over an infinite time. Examples include deciding which ride hailing trip requests to serve during peak demand, granting the right of way in intersections or lane mergers, or admitting internet content to a regulated fast channel. We study a simplified yet insightful formulation of these problems where at every instant two agents from a large population get randomly matched to compete over the resource. The intuitive interpretation of a karma mechanism is “If I give in now, I will be rewarded in the future.” Agents compete in an auction-like setting where they bid units of karma, which circulates directly among them and is self-contained in the system. We demonstrate that this allows a society of self-interested agents to achieve high levels of efficiency without resorting to a (possibly problematic) monetary pricing of the resource. We model karma mechanisms as dynamic population games and guarantee the existence of a stationary Nash equilibrium. We then analyze the performance at the stationary Nash equilibrium numerically. For the case of homogeneous agents, we compare different mechanism design choices, showing that it is possible to achieve an efficient and ex-post fair allocation when the agents are future aware. Finally, we test the robustness against agent heterogeneity and propose remedies to some of the observed phenomena via karma redistribution.

Stationary Equilibria in Discounted Stochastic Games

Stationary Equilibria in Discounted Stochastic Games

Preface

Preface

Constrained-Efficient Capital Reallocation

We characterize efficiency in an equilibrium model of investment and capital reallocation with heterogeneous firms facing collateral constraints. The model features two types of pecuniary externalities: collateral externalities, because the resale price of capital affects collateral constraints, and distributive externalities, because buyers of old capital are more financially constrained than sellers, consistent with empirical evidence. We prove that the stationary equilibrium price of old capital is inefficiently high because the distributive externality exceeds the collateral externality, by a factor of two when we calibrate the model. New investment reduces the future price of old capital, providing a rationale for new-investment subsidies. (JEL D21, D24, D25, D62, E22, G31, G32)