Stationary Equilibrium Research Articles

Variants of Harsanyi’s tracing procedures for selecting a perfect stationary equilibrium in stochastic games

Purpose. To analyze the relationship between induced innovations, economic growth and technological unemployment in the theory of D. Ricardo, as well as to adapt the mathematical model of L. Pasinetti to test the economic effects of replacing human labor with machines. Methodology. The research methodology is based on a combination of system analysis and economic and mathematical modeling. Specific economic research methods are used: frontier analysis (to model the dynamics of the evolution of capital, employment and other indicators of economic conditions under the influence of changes in other factors); equilibrium analysis (to find the states of stationary equilibrium of the economic system); graphical analysis (for data processing and visual demonstration of the research results). Findings. The main components of D. Ricardo’s economic growth model are analyzed. Using the adapted mathematical model of L. Pasinetti, the pessimistic and optimistic scenarios of economic growth with regard to technological progress are tested. It is proved that D. Ricardo’s system is holistic, logically complete and subject to mathematical formalization. It is demonstrated that under certain conditions, the Ricardian model allows for the simultaneous introduction of innovations and the absence of a reduction in gross product in the short term. Originality. The methodological approaches to studying the relationship between innovation and technological progress, on the one hand, and employment and technological unemployment, on the other hand, have been further developed. An integral part of this relationship is economic growth, the dynamics of which determines the nature of the impact of innovations on the welfare of the population. The analysis of the Ricardian model has shown that with the gradual introduction of machines and their sufficiently high efficiency, the new steady state equilibrium is characterized by higher values of accumulated capital, employment and gross product than the previous one. Practical value. The article proposes a modified mathematical model by L. Pasinetti, which forms a Ricardian methodological framework for analyzing the socio-economic effects of induced innovations. The obtained results can be adapted and applied to analyze modern aspects of technological unemployment, in particular, for further study of the phenomenon of artificial intelligence and digitalization of the economy.

The article develops and analyzes a behaviorally sensitive modification of the SIR model with temporal support via convolutions that reflect delayed negative feedback between epidemic load and adherence to preventive measures. The approach is based on the observation that the classical SIR model with constant intensive behavior does not reproduce the adaptive behavior of the population, while the behavioral-informational circuit significantly changes the process trajectory and causes wave dynamics even with limited seasonal forcing. The modified SIR model demonstrates a transition from spiral convergence to a stationary equilibrium (without memory) and to a limit cycle (with convolutions), while the amplitude and period of oscillation monotonically rise with increasing “inertia” of the behavioral circuit (time constants of memory, intensity of fatigue). The paper shows that fluctuations in people’s behavior are translated into fluctuations of economic activity, budget revenues, social expenditures, etc. It emphasizes the need to combine medical-biological, behavioral, and economic parameters in anti-epidemic policy. The practical model value lies in the possibility of scenario forecasting peaks/intervals between waves and testing guided interventions to reduce the peak load on healthcare and optimize costs. The results confirm that the integration of biological, behavioral, and informational subsystems forms a dynamic with feedback loops capable of generating complex regimes, thereby expanding the scope of applicability of SIR models in the socio-economic analysis of epidemics. Numerical analysis is conducted via the 4th-order Runge-Kutta method, which allowed to construct phase portraits and trace trajectories in the vicinity of stationary regimes. The results demonstrate two qualitatively different regimes of dynamics. In the case of an instantaneous behavioral response, the system exhibits a monotonic relaxation to equilibrium with a single peak of infected individuals. With present convolutions, short oscillations of altered and closed memory orbits are observed in the phase portraits. At the same time, the oscillations are caused not by biological parameters of diseases but by social behavior, inertia, and support in the formation of compliance, which emphasizes information policy and behavior management in accordance with medical measures. Given the demonstrated sensitivity of the dynamics to time lags of behavior, a modification of the SIR model with memory is proposed that serves as a convenient framework for rapid scenario assessment and supports decisions in public health.

Abstract This article concerns the optimal choice of flat taxes on labor and capital income, and on consumption, in a tractable economic model in which agents are subject to idiosyncratic investment risk. We identify the tax rates which maximize welfare in stationary equilibrium while preserving tax revenue, finding that an increase in welfare equivalent to a permanent increase in consumption of nearly 7% can be achieved by only taxing capital income and consumption. The Domar‐Musgrave effect explains cases where it is optimal to tax capital income. We characterize the dynamic response to the substitution of consumption taxation for labor income taxation.

Introduction: Currently, the theory of conflict management is in its infancy and needs further development. This is due to the emergence, presence, and development of various types of conflicts in different subject areas of the functioning of socio-economic organizations, forcing them to search for adaptive development strategies about the destructive effects of the competitive environment. The profit of organizations is characterized by the criterion of conflict stability - a dynamic trajectory of equilibrium application (a set of states) with a given efficiency. To achieve it, forecasting strategies of the behavior of competing organizations is carried out based on a variety of their considered strategies and solving the problem of rational allocation of available resources to choose the optimal one for the implementation of a given level of profit based on reducing the destructive actions of competing systems. This determines the special relevance of the task of ensuring the conflict stability of the interaction of organizations in modern conditions. Materials and Methods: A method is proposed to substantiate the points of conflict stability of the functioning of the organizational and technical system by allocating individual and group resources for defensive and offensive actions about a structured hierarchical system of typical cores of conflict interaction between the parties. Results: The conducted research confirms the conclusions of well-known scientists that the stable functioning of the organizational and technical system is achievable not at the point of the minimax value of the objective function (the upper price of the game) with a priori awareness of the interacting systems, but in the form of a saddle point with a completely open game when the actions of the parties are a priori known. Discussion: A stationary equilibrium state can be asserted if, when the parameters of an organizational and technical system change as a result of the influence of external or internal disturbances, it returns to a stable equilibrium state. For a long time, only its technical elements can be in equilibrium, which does not have feedback from the external environment, whereas for the system itself, equilibrium can only be a moment in the process of continuous changes. Conclusion: The process of functioning of organizational and technical systems in conditions of conflict interaction can be characterized by a mobile dynamic change in the points of stable equilibrium determined by the ability of these systems to adapt and transition to a qualitatively higher level of development. The proposed methodological approach makes it possible to find these points for various conditions of such interaction.

Abstract We show here that a stationary Markov equilibrium can be computed for an n-person stochastic game by using an iterative procedure often used in machine learning when the state space is finite and the action sets of the players are finite. The iterative procedure involves computing the equilibrium of the constituent game (possibly mixed) after each iteration. The constituent game in each iteration is the game in which the payoff of a player consists of two terms. One is the payoff of the player from the underlying single-period game and the second is the discounted equilibrium payoff of the player from the constituent game of the previous iteration. We show that the equilibrium payoffs of the constituent games converge, and lead to a constituent game in the limit, whose equilibrium payoffs are the same as those from the previous iteration. The equilibrium payoffs from the iterative procedure thus converge to stationary equilibrium payoffs. These equilibrium payoffs are then the payoffs of a stationary Markov perfect equilibrium of the infinite horizon stochastic game. We provide an algorithm based on this result to compute the stationary Markov perfect equilibrium of the infinite horizon stochastic game. To evaluate our proposed algorithm, we conduct several experiments to derive their stationary Markov perfect equilibrium, including a version of the cybersecurity game analyzed in Lye and Wing and a version of the price-setting dynamic game of market competition analyzed in Besanko, Daralzeski, Kryukov, and Satterthwaite.

We propose a new mean-field game model with two states to study synchronization phenomena, and we provide a comprehensive characterization of stationary and dynamic equilibria along with their stability properties. The game undergoes a phase transition with increasing interaction strength. In the subcritical regime, the uniform distribution, representing incoherence, is the unique and stable stationary equilibrium. Above the critical interaction threshold, the uniform equilibrium becomes unstable and there is a multiplicity of stationary equilibria that are self-organizing. Under a discounted cost, dynamic equilibria spiral around the uniform distribution before converging to the self-organizing equilibria. With an ergodic cost, however, unexpected periodic equilibria around the uniform distribution emerge. Funding: This work was supported by the National Science Foundation [Grant DMS 2406762].

This work examines the behaviors of the online projected gradient ascent (OPGA) algorithm and its variant in a repeated oligopoly price competition under reference effects. In particular, we consider that multiple firms engage in a multiperiod price competition, where consecutive periods are linked by the reference price update and each firm has access only to its own first-order feedback. Consumers assess their willingness to pay by comparing the current price against the memory-based reference price, and their choices follow the multinomial logit (MNL) model. We use the notion of stationary Nash equilibrium (SNE), defined as the fixed point of the equilibrium pricing policy, to simultaneously capture the long-run equilibrium and stability. We first study the loss-neutral reference effects and show that if the firms employ the OPGA algorithm—adjusting the price using the first-order derivatives of their log-revenues—the price and reference price paths attain last-iterate convergence to the unique SNE, thereby guaranteeing the no-regret learning and market stability. Moreover, with appropriate step-sizes, we prove that this algorithm exhibits a convergence rate of [Formula: see text] in terms of the squared distance and achieves a constant dynamic regret. Despite the simplicity of the algorithm, its convergence analysis is challenging due to the model lacking typical properties such as strong monotonicity and variational stability that are ordinarily used for the convergence analysis of online games. The inherent asymmetry nature of reference effects motivates the exploration beyond loss-neutrality. When loss-averse reference effects are introduced, we propose a variant of the original algorithm named the conservative-OPGA (C-OPGA) to handle the nonsmooth revenue functions and show that the price and reference price achieve last-iterate convergence to the set of SNEs with the rate of [Formula: see text]. Finally, we demonstrate the practicality and robustness of OPGA and C-OPGA by theoretically showing that these algorithms can also adapt to firm-differentiated step-sizes and inexact gradients. This paper was accepted by Chung Piaw Teo, optimization and decision analytics. Funding: J. Lavaei acknowledges the support from the U.S. Army Research Laboratory and the U.S. Army Research Office under Grant W911NF2010219, Office of Naval Research under Grant N000142412673, AFOSR, NSF, and the UC Noyce Initiative. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.03464 .

We study firms size distribution in a mean-field model of Cournot competition in a commodity market, where price follows an inverse power demand function. Firms face irreversible investment decisions and constant depreciation of production capacity. Output is affected by Gaussian productivity shocks, whose volatility and the price function can shift due to rare macroeconomic events modeled by a two-state Markov chain. Firms aim to maximize expected discounted profits, net of investment and operating costs, based on the long-run stationary price. We establish existence and uniqueness of a stationary mean-field equilibrium and characterize it through a barrier-type investment strategy with endogenous thresholds for each economic regime. A quasi-closed form for the stationary distribution of firms’ states is provided. The model generates Pareto-distributed firm sizes, consistent with empirical industry data. It also shows that downturns raise market concentration and that firm performance depends on depreciation rates and the persistence of economic fluctuations.

In this paper, the problem of transporting an inverted or a suspended pendulum on a quadrotor, subject to external crosswind disturbances is addressed. The proposed solution is based on the flatness property of the tangent linearisation model, obtained around a stationary equilibrium point of the system; such an approach allows to define auxiliary outputs called flat outputs associated with the horizontal translational dynamics of the systems, thus simplifying the system model. The simplified dynamics of the two incremental flat outputs are used for the design of a Reduced Order Extended State Observer-based Active Disturbance Rejection Control (ROESO-ADRC) scheme in a trajectory tracking task. The ROESO-ADRC scheme estimates, and properly compensates, the unknown total disturbance; constituted by the crosswind forces, regarded as exogenous disturbances, and neglected nonlinear terms, regarded as endogenous disturbances. An equivalence between the ROESO-ADRC with flat filters is shown. The stability of the trajectory and estimation errors is formally studied and numerical simulations are carried out to verify the performance of the controlled system.

ABSTRACTOnly one trip planner is needed for a group of friends to enjoy a pleasant trip and only one country is needed to coordinate on international talks that can be beneficial for all participating countries. We study a dynamic volunteering dilemma game in which two players choose to volunteer or wait given there have not been any volunteering actions in the past. The players can be procrastinators and the benefits of volunteering arrive later than the costs. We fully characterize the stationary Markov Strotz‐Pollak equilibria. When the cost of volunteering is sufficiently small or agents' present‐bias parameters are sufficiently close, there always exists an equilibrium in which both players randomize. This equilibrium features stochastic delay, and the delay is exacerbated if one or both agents become more present‐biased. However, if the agents differ significantly in their present‐bias parameters, this difference may act as a 'natural coordination device' and the unique stationary equilibrium predicts that only the less severe procrastinator volunteers, and this may result in an even quicker provision compared with the case of two exponential discounters.

In this paper, we investigate the robustness of stationary mean-field equilibria in the presence of model uncertainties, specifically focusing on infinite-horizon discounted cost functions. To achieve this, we initially establish convergence conditions for value iteration-based algorithms in mean-field games. Subsequently, utilizing these results, we demonstrate that the mean-field equilibrium obtained through this value iteration algorithm remains robust even in the face of system dynamics misspecifications. We then apply these robustness findings to the finite model approximation problem in mean-field games, showing that if the state space quantization is fine enough, the mean-field equilibrium for the finite model closely approximates the nominal one.

We address the counting of level crossings for the generalized Brownian motion in which the motion of a Brownian particle (unbound as well as linearly bound) is governed by an integrodifferential stochastic equationwith a memory dissipation kernel: the generalized Langevin equation. We suppose that the driving noise is internal and the fluctuation-dissipation relation holds. The most common assumption is that the driving noise is Gaussian, which allows us to make a complete analytical treatment of the whole problem, including crossing statistics. We obtain the general asymptotic behavior for regular driving noises having finite intensity and fast decaying correlations, as well as for fractional noises with long-time tail correlations and slow decay, which are related to anomalous diffusion. For the generalized Brownian oscillator we study the stationary (equilibrium) state and the approach to equilibrium.

Dynamic matching games: Stationary equilibria under varying commitments

Abstract We study the behavior of dynamic equilibria in mean field games with large time horizons in a dynamic consumer choice model. We show that if the stationary equilibrium in the associated infinite horizon game is unique, the dynamic equilibria of the finite horizon games converge to the stationary equilibrium of the infinite horizon game as the time horizon tends to infinity. If the stationary equilibrium is not unique, however, the situation becomes more involved. In this case, we show that in addition to convergence to the stationary equilibria, in the long run, the dynamic equilibria circle around randomized stationary equilibria for certain choices of boundary data.

We propose a class of overlapping generations models that can serve as a workhorse for policy analysis. Recent literature identifies several features of key observable economic variables in Europe and the U.S.: the life-cycle path of earnings is hump-shaped, while the aggregate variables — per-capita consumption and labor hours — exhibit a clear time trend. Our class generates non-monotonic life-cycle behavior of labor supply and the desired aggregate trends in all its balanced-growth equilibria (BGE). There is a finite number of these equilibria and at least one of them exists provided a single-generation consumer problem has a solution. The model has a constant-returns-to-scale production, non-trivial depreciation of capital, exogenous labor-saving growth and an arbitrary individual life-cycle productivity. The necessary restrictions imposed on preferences are consistent with those generating the aggregate trends in a representative-agent economy, while ruling out popular specifications such as Cobb–Douglas or CES. We characterize BGE with MaCurdy preferences and solve a parametrized model that yields at least two stationary equilibria with reasonable interest rates. The hump-shaped life-cycle consumption profile can be generated with an additional parameter interpreted as reflecting family structure or a desire for immediate gratification that peaks at mid-life.

We study Asynchronous Dynamic games and show that in games with a finite state space and finite action sets, one can obtain the pure strategy Markov perfect equilibrium by using a simple backward induction method when the time period for the game is finite. The equilibrium strategies for games with an infinite horizon are then obtained as the point-wise limit of the equilibrium strategies of a sequence of finite horizon games, where the finite horizon games are truncated versions of the original game with successively longer time periods. We also show that if the game has a fixed K-period cycle, then there is a stationary Markov equilibrium. Using these results, we derive an algorithm to compute the equilibrium strategies. We test the algorithm in three experiments. The first is a two-player asynchronous game with three states and three actions. In the second experiment, we compute the equilibrium of a cybersecurity game in which there are two players, an attacker and a defender. In the third experiment, we compute the stationary equilibrium of a duopoly game with two firms that choose an output in alternate periods.

We develop a new class of high-order accurate well-balanced finite difference (FD) weighted essentially nonoscillatory (WENO) methods for numerical general relativity (GR), which can be applied to any first-order reduction of the Einstein field equations, even if nonconservative terms are present. We choose the first-order nonconservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. By introducing auxiliary variables, the vacuum Einstein equations in first-order form constitute a partial differential equation system of 54 equations that is naturally nonconservative. We show how to design FD-WENO schemes that can handle nonconservative products. Different variants of FD WENO are discussed, with an eye to their suitability for higher-order accurate formulations for numerical GR. We successfully solve a set of fundamental tests of numerical GR with up to ninth-order spatial accuracy. Due to their intrinsic robustness, flexibility, and ease of implementation, FD-WENO schemes can effectively replace traditional central finite differencing in any first-order formulation of the Einstein field equations, without any artificial viscosity. When used in combination with well-balancing, the new numerical schemes preserve stationary equilibrium solutions of the Einstein equations exactly. This is particularly relevant in view of the numerical study of the quasi-normal modes of oscillations of relevant astrophysical sources. In conclusion, general relativistic high-energy astrophysics could benefit from this new class of numerical schemes and the ecosystem of desirable capabilities built around them.

Using the total chemical potential to generalize the capillary pressure concept and therefrom derive a governing equation for two-phase flow in porous media

ABSTRACT In modelling a relativistic disc around a compact object, the self-gravity of the disc is often neglected while it needs to be incorporated for more accurate descriptions in several circumstances. Extending the Komatsu–Eriguchi–Hachisu self-consistent field method, we present numerical models of a rapidly rotating neutron star with a self-gravitating disc in stationary equilibrium. In particular, our approach allows us to obtain numerical solutions involving a massive disc with the rest mass $\mathcal {O}(10^{-1})-\mathcal {O}(10^0)\, \mathrm{ M}_\odot$ closely attached to a rotating neutron star, given that the disc is mainly supported by the relativistic electron degeneracy pressure. We also assess the impact of self-gravity on the internal structure of the disc and the neutron star. These axisymmetric, stationary solutions can be employed for simulations involving the neutron star–disc system in the context of high-energy transients and gravitational-wave emissions.