scholarly journals Numerical Solution of Open-Loop Nash Differential Games Based on the Legendre Tau Method

Games ◽  
2020 ◽  
Vol 11 (3) ◽  
pp. 28
Author(s):  
Mojtaba Dehghan Banadaki ◽  
Hamidreza Navidi
Keyword(s):  
Differential Game ◽  
Time Horizon ◽  
Renewable Resource ◽  
Open Loop ◽  
Tau Method ◽  
Point Boundary ◽  
Algebraic Equations ◽  
Runge Kutta Method ◽  
Finite Time Horizon ◽  
Nash Differential Games

In this paper, an efficient implementation of the Tau method is presented for finding the open-loop Nash equilibrium of noncooperative nonzero-sum two-player differential game problems with a finite-time horizon. Regarding this approach, the two-point boundary value problem derived from Pontryagin’s maximum principle is reduced to a system of algebraic equations that can be solved numerically. Finally, a differential game arising from bioeconomics among firms harvesting a common renewable resource is included to illustrate the accuracy and efficiency of the proposed method and a comparison is made with the result obtained by fourth order Runge–Kutta method.

A Duopoly with Common Renewable Resource and Incentives

2017 ◽  
Vol 19 (04) ◽  
pp. 1750018 ◽  
Author(s):  
Luca Grilli ◽  
Michele Bisceglia
Keyword(s):  
Finite Time ◽  
Time Horizon ◽  
Renewable Resource ◽  
Open Loop ◽  
Linear Feedback ◽  
The Social ◽  
Duopoly Model ◽  
Resource Exhaustion ◽  
Finite Time Horizon ◽  
Homogeneous Good

In this paper, we study a duopoly model in which two symmetric firms exploit the same public renewable resource as an input for the production of a homogeneous good. We consider the case where the firms are provided with public incentives in order to prevent the resource exhaustion in a finite time horizon which coincides with the harvesting-license period. As a consequence, we consider a differential game in finite time horizon and compute the Open Loop and linear Feedback Nash Equilibria of the game. We study the social welfare and the optimal incentives polices derived from the solutions.

scholarly journals Linear Quadratic Nash Differential Games of Stochastic Singular Systems

2014 ◽  
Vol 2 (6) ◽  
pp. 553-560
Author(s):  
Haiying Zhou ◽  
Huainian Zhu ◽  
Chengke Zhang
Keyword(s):  
Differential Games ◽  
Finite Time ◽  
Time Horizon ◽  
Singular Systems ◽  
Existence Condition ◽  
Infinite Time ◽  
Infinite Time Horizon ◽  
Finite Time Horizon ◽  
Stochastic Singular Systems ◽  
Nash Differential Games

AbstractIn this paper, we deal with the Nash differential games of stochastic singular systems governed by Itô-type equation in finite-time horizon and infinite-time horizon, respectively. Firstly, the Nash differential game problem of stochastic singular systems in finite time horizon is formulated. By applying the results of stochastic optimal control problem, the existence condition of the Nash strategy is presented by means of a set of cross-coupled Riccati differential equations. Similarly, under the assumption of the admissibility of the stochastic singular systems, the existence condition of the Nash strategy in infinite-time horizon is presented by means of a set of cross-coupled Riccati algebraic equations. The results show that the strategies of each players interact.

PARAMETER ANALYSIS OF A ZERO-SUM DIFFERENTIAL GAME APPROXIMATING A NONZERO-SUM GAME

2008 ◽  
Vol 10 (04) ◽  
pp. 437-459
Author(s):  
ARIK MELIKYAN ◽  
GEERT JAN OLSDER ◽  
ANDREI AKHMETZHANOV
Keyword(s):  
Exact Solution ◽  
Numerical Methods ◽  
Differential Game ◽  
Time Horizon ◽  
Parameter Analysis ◽  
The Other ◽  
Feedback Controls ◽  
Finite Time Horizon ◽  
Two Parameters ◽  
Zero Sum

In this paper we consider a zero-sum differential game in feedback setting with one state coordinate and finite time horizon. The problem is characterized by two parameters. This game arises as an approximation to a nonzero-sum game. Using analytical and numerical methods we solve the HJBI equation and give the description of the optimal feedback controls for both players. These feedbacks are given in terms of switching curves and singular universal lines. We also determine the change (bifurcations) of the optimal phase portrait of the game depending upon the parameters. In conclusion, we show that for some values of the parameters the solution of the considered zero-sum game leads to the exact solution of the corresponding nonzero-sum game, and for the other values of the parameters it supplies just some approximation.

scholarly journals A Constrained Markovian Diffusion Model for Controlling the Pollution Accumulation

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1466
Author(s):  
Beatris Adriana Escobedo-Trujillo ◽  
José Daniel López-Barrientos ◽  
Javier Garrido-Meléndez
Keyword(s):  
Dynamic Programming ◽  
Dirichlet Problem ◽  
Stochastic Control ◽  
Finite Time ◽  
Time Horizon ◽  
Closed Loop ◽  
Programming Techniques ◽  
Pollution Accumulation ◽  
Finite Time Horizon ◽  
The Cost

This work presents a study of a finite-time horizon stochastic control problem with restrictions on both the reward and the cost functions. To this end, it uses standard dynamic programming techniques, and an extension of the classic Lagrange multipliers approach. The coefficients considered here are supposed to be unbounded, and the obtained strategies are of non-stationary closed-loop type. The driving thread of the paper is a sequence of examples on a pollution accumulation model, which is used for the purpose of showing three algorithms for the purpose of replicating the results. There, the reader can find a result on the interchangeability of limits in a Dirichlet problem.

Unique Open Loop Nash Equilibria on the Infinite Time Horizon

2001 ◽  
Vol 34 (20) ◽  
pp. 29-34
Author(s):  
Gerhard Jank ◽  
Dirk Kremer ◽  
Gábor Kun
Keyword(s):  
Time Horizon ◽  
Nash Equilibria ◽  
Open Loop ◽  
Infinite Time ◽  
Infinite Time Horizon

scholarly journals Note on the Solution of Transport Equation by Tau Method and Walsh Functions

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Abdelouahab Kadem ◽  
Adem Kilicman
Keyword(s):  
Transport Equation ◽  
Legendre Polynomials ◽  
Neutron Transport ◽  
Three Dimensional ◽  
Walsh Function ◽  
Dimensional Case ◽  
Angular Variable ◽  
Tau Method ◽  
Algebraic Equations ◽  
Neutron Transport Equation

We consider the combined Walsh function for the three-dimensional case. A method for the solution of the neutron transport equation in three-dimensional case by using the Walsh function, Chebyshev polynomials, and the Legendre polynomials are considered. We also present Tau method, and it was proved that it is a good approximate to exact solutions. This method is based on expansion of the angular flux in a truncated series of Walsh function in the angular variable. The main characteristic of this technique is that it reduces the problems to those of solving a system of algebraic equations; thus, it is greatly simplifying the problem.

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