Equilibrium points for three games based on the Poisson process

An arbitrary number of competitors are presented with independent Poisson streams of offers consisting of independent and identically distributed random variables having the uniform distribution on [0, 1]. The players each wish to accept a single offer before a known time limit is reached and each aim to maximize the expected value of their offer. Rejected offers may not be recalled, but they are passed on to the other players according to a known transition matrix. This paper finds equilibrium points for two such games, and demonstrates a two-player game with an equilibrium point under which the player with the faster stream of offers has a lower expected reward than his opponent.

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WHEN ASPIRING AND RATIONAL AGENTS STRIVE TO COORDINATE

International Game Theory Review ◽

10.1142/s0219198907001539 ◽

2007 ◽

Vol 09 (03) ◽

pp. 461-475 ◽

Cited By ~ 1

Author(s):

JAIDEEP ROY

Keyword(s):

Equilibrium Point ◽

Equilibrium Points ◽

The Other ◽

Static Equilibrium ◽

Common Interest ◽

Rational Agents ◽

Long Run ◽

Speed Of Evolution

The paper studies a game of common interest played infinitely many times between two players, one being aspiration driven while the other being a myopic optimizer. It is shown that the only two long run stationary outcomes are the two static equilibrium points. Robustness of long run behavior is studied to show that whenever the optimizer is allowed to make small mistakes, players are able to coordinate on the Pareto dominant equilibrium point most of the time in the long run if the speed of evolution of aspirations is sufficiently fast. However, when only the aspiring player is allowed to make small mistakes, achieving coordination is inevitable and independent of the speed at which aspirations evolve.

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Evolution of COVID-19 Disease Using a One Prey-Two Predator Mode

Advances in Technology ◽

10.31357/ait.v1i1.4888 ◽

2021 ◽

Vol 1 (1) ◽

Author(s):

Amila Sudu Ambegedara ◽

Asini A. Konpola ◽

Chathurika S. Gunasekara ◽

Indika G. Udagedara

Keyword(s):

Equilibrium Point ◽

Disease Transmission ◽

Equilibrium Points ◽

Optimization Method ◽

The Other ◽

Population Declines ◽

Data Set ◽

Uniqueness Of The Solution ◽

Predator Model ◽

The One

Mathematical modeling is used to understand the dynamics of transmission of infectious diseases such as COVID-19, SARS, Ebola, and Dengue among populations. In this work, a one prey-two predator model has been developed to understand the underlying dynamics of COVID-19 disease transmission. We considered the infected, recovered, and death populations with the fact that an infected person can be transformed into the recovered or death group assuming that the infected ones are the prey, and the other two populations are the two predators in the one prey-two predator model. It was found that the proposed model has four equilibrium points; the vanishing equilibrium point ( ), recovered and death-free equilibrium point ( ), recovered population-free equilibrium point ( ), and the death-free equilibrium point ( ). Stability analysis of the equilibrium points shows that except all the other equilibrium points are locally asymptotically stable. Global asymptotic stability of the recovered population-free equilibrium point and death-free equilibrium point are also analyzed. Moreover, the existence and uniqueness of the solution were proved. The parameters for the model are estimated from a data set that consists of the total number of infected, recovered, and dead populations worldwide in the year 2020 using the Nelder-Mead optimization method. When the time approaches infinity, the infected population converges to a constant value, the recovered population declines and reaches zero, and the death population attains a constant value. However, some modifications to the system are needed. In future work, measures such as health precautions, vaccinations are needed to be considered for the formulation of the mathematical model.

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Language Family Engineering with Product Lines of Multi-level Models

Formal Aspects of Computing ◽

10.1007/s00165-021-00554-3 ◽

2021 ◽

Author(s):

Juan de Lara ◽

Esther Guerra

Keyword(s):

Software Engineering ◽

Arbitrary Number ◽

Formal Theory ◽

The Other ◽

Language Family ◽

Top Down ◽

Product Lines ◽

Modelling Languages ◽

Multi Level ◽

Definition Of

AbstractModelling is an essential activity in software engineering. It typically involves two meta-levels: one includes meta-models that describe modelling languages, and the other contains models built by instantiating those meta-models. Multi-level modelling generalizes this approach by allowing models to span an arbitrary number of meta-levels. A scenario that profits from multi-level modelling is the definition of language families that can be specialized (e.g., for different domains) by successive refinements at subsequent meta-levels, hence promoting language reuse. This enables an open set of variability options given by all possible specializations of the language family. However, multi-level modelling lacks the ability to express closed variability regarding the availability of language primitives or the possibility to opt between alternative primitive realizations. This limits the reuse opportunities of a language family. To improve this situation, we propose a novel combination of product lines with multi-level modelling to cover both open and closed variability. Our proposal is backed by a formal theory that guarantees correctness, enables top-down and bottom-up language variability design, and is implemented atop the MetaDepth multi-level modelling tool.

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On the Complexity of Deadlock Recovery

Fundamenta Informaticae ◽

10.3233/fi-1986-9304 ◽

1986 ◽

Vol 9 (3) ◽

pp. 323-342

Author(s):

Joseph Y.-T. Leung ◽

Burkhard Monien

Keyword(s):

Computational Complexity ◽

Polynomial Time ◽

Arbitrary Number ◽

The Other ◽

Np Hard ◽

Total Cost ◽

Other Hand ◽

Input Parameters ◽

Resource Type

We consider the computational complexity of finding an optimal deadlock recovery. It is known that for an arbitrary number of resource types the problem is NP-hard even when the total cost of deadlocked jobs and the total number of resource units are “small” relative to the number of deadlocked jobs. It is also known that for one resource type the problem is NP-hard when the total cost of deadlocked jobs and the total number of resource units are “large” relative to the number of deadlocked jobs. In this paper we show that for one resource type the problem is solvable in polynomial time when the total cost of deadlocked jobs or the total number of resource units is “small” relative to the number of deadlocked jobs. For fixed m ⩾ 2 resource types, we show that the problem is solvable in polynomial time when the total number of resource units is “small” relative to the number of deadlocked jobs. On the other hand, when the total number of resource units is “large”, the problem becomes NP-hard even when the total cost of deadlocked jobs is “small” relative to the number of deadlocked jobs. The results in the paper, together with previous known ones, give a complete delineation of the complexity of this problem under various assumptions of the input parameters.

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Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽

10.3390/sym13050785 ◽

2021 ◽

Vol 13 (5) ◽

pp. 785

Author(s):

Hasan S. Panigoro ◽

Agus Suryanto ◽

Wuryansari Muharini Kusumawinahyu ◽

Isnani Darti

Keyword(s):

Hopf Bifurcation ◽

Fractional Derivative ◽

Equilibrium Point ◽

Power Law ◽

Equilibrium Points ◽

Essential Difference ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

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Research of Trajectory Optimization Approaches in Synthesized Optimal Control

Symmetry ◽

10.3390/sym13020336 ◽

2021 ◽

Vol 13 (2) ◽

pp. 336

Author(s):

Askhat Diveev ◽

Elizaveta Shmalko

Keyword(s):

Optimal Control ◽

Equilibrium Point ◽

Trajectory Optimization ◽

Stable Equilibrium ◽

Equilibrium Points ◽

Control Object ◽

Symbolic Regression ◽

Optimal Location ◽

Stable Equilibrium Point ◽

Stabilization System

This article presents a study devoted to the emerging method of synthesized optimal control. This is a new type of control based on changing the position of a stable equilibrium point. The object stabilization system forces the object to move towards the equilibrium point, and by changing its position over time, it is possible to bring the object to the desired terminal state with the optimal value of the quality criterion. The implementation of such control requires the construction of two control contours. The first contour ensures the stability of the control object relative to some point in the state space. Methods of symbolic regression are applied for numerical synthesis of a stabilization system. The second contour provides optimal control of the stable equilibrium point position. The present paper provides a study of various approaches to find the optimal location of equilibrium points. A new problem statement with the search of function for optimal location of the equilibrium points in the second stage of the synthesized optimal control approach is formulated. Symbolic regression methods of solving the stated problem are discussed. In the presented numerical example, a piece-wise linear function is applied to approximate the location of equilibrium points.

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A Discontinuity of the Energy of Quantum Walk in Impurities

Symmetry ◽

10.3390/sym13071134 ◽

2021 ◽

Vol 13 (7) ◽

pp. 1134

Author(s):

Kenta Higuchi ◽

Takashi Komatsu ◽

Norio Konno ◽

Hisashi Morioka ◽

Etsuo Segawa

Keyword(s):

Unit Circle ◽

Stationary State ◽

Discrete Time ◽

Quantum Walk ◽

Time Limit ◽

The Other ◽

Unitary Matrix ◽

Initial State ◽

Time Quantum ◽

Long Time

We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

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A Ternary, Two-Phase, Mathematical Model of Oil Recovery With Surfactant Systems

Society of Petroleum Engineers Journal ◽

10.2118/12934-pa ◽

1984 ◽

Vol 24 (06) ◽

pp. 606-616 ◽

Cited By ~ 19

Author(s):

Charles P. Thomas ◽

Paul D. Fleming ◽

William K. Winter

Keyword(s):

Mathematical Model ◽

Oil Recovery ◽

Arbitrary Number ◽

The Other ◽

Phase System ◽

Chemical Flooding ◽

Two Phase ◽

Oil Displacement ◽

Surfactant Systems ◽

And Diffusion

Abstract A mathematical model describing one-dimensional (1D), isothermal flow of a ternary, two-phase surfactant system in isotropic porous media is presented along with numerical solutions of special cases. These solutions exhibit oil recovery profiles similar to those observed in laboratory tests of oil displacement by surfactant systems in cores. The model includes the effects of surfactant transfer between aqueous and hydrocarbon phases and both reversible and irreversible surfactant adsorption by the porous medium. The effects of capillary pressure and diffusion are ignored, however. The model is based on relative permeability concepts and employs a family of relative permeability curves that incorporate the effects of surfactant concentration on interfacial tension (IFT), the viscosity of the phases, and the volumetric flow rate. A numerical procedure was developed that results in two finite difference equations that are accurate to second order in the timestep size and first order in the spacestep size and allows explicit calculation of phase saturations and surfactant concentrations as a function of space and time variables. Numerical dispersion (truncation error) present in the two equations tends to mimic the neglected present in the two equations tends to mimic the neglected effects of capillary pressure and diffusion. The effective diffusion constants associated with this effect are proportional to the spacestep size. proportional to the spacestep size. Introduction In a previous paper we presented a system of differential equations that can be used to model oil recovery by chemical flooding. The general system allows for an arbitrary number of components as well as an arbitrary number of phases in an isothermal system. For a binary, two-phase system, the equations reduced to those of the Buckley-Leverett theory under the usual assumptions of incompressibility and each phase containing only a single component, as well as in the more general case where both phases have significant concentrations of both components, but the phases are incompressible and the concentration in one phase is a very weak function of the pressure of the other phase at a given temperature. pressure of the other phase at a given temperature. For a ternary, two-phase system a set of three differential equations was obtained. These equations are applicable to chemical flooding with surfactant, polymer, etc. In this paper, we present a numerical solution to these equations paper, we present a numerical solution to these equations for I D flow in the absence of gravity. Our purpose is to develop a model that includes the physical phenomena influencing oil displacement by surfactant systems and bridges the gap between laboratory displacement tests and reservoir simulation. It also should be of value in defining experiments to elucidate the mechanisms involved in oil displacement by surfactant systems and ultimately reduce the number of experiments necessary to optimize a given surfactant system.

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Analysis of Smooth Implementation of Industry Poverty Alleviation considering Government Supervision

Mathematical Problems in Engineering ◽

10.1155/2021/5554595 ◽

2021 ◽

Vol 2021 ◽

pp. 1-13

Author(s):

Haifeng Yao ◽

Jiangyue Fu

Keyword(s):

Equilibrium Point ◽

Evolutionary Game Theory ◽

Poverty Alleviation ◽

Equilibrium Points ◽

Evolutionary Game ◽

Parameter Design ◽

Game Model ◽

Government Supervision ◽

Proposed Model ◽

The Government

Vigorous implementation of industrial poverty alleviation is the fundamental path and core power of poverty alleviation in impoverished areas. Enterprises and poor farmers are the main participants in industry poverty alleviation. Government supervision measures regulate their behaviors. This study investigates how to smoothly implement industry poverty alleviation projects considering government supervision. A game model is proposed based on the evolutionary game theory. It analyses the game processes between enterprises and poor farmers with and without government supervision based on the proposed model. It is shown that poverty alleviation projects will fail without government supervision given that the equilibrium point (0, 0) is the ultimate convergent point of the system but will possibly succeed with government supervision since the equilibrium points (0, 0) and (1, 1) are the ultimate convergent point of the system, where equilibrium point (1, 1) is our desired results. Different supervision modes have different effects on the game process. This study considers three supervision modes, namely, only reward mode, only penalty mode, and reward and penalty mode, and investigates the parameter design for the reward and penalty mode. The obtained results are helpful for the government to develop appropriate policies for the smooth implementation of industry poverty alleviation projects.

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