Nash Equilibrium for 2nd-order Two-Player Non-Zero Sum LQ Games with Executable Decentralized Control Strategies

Author(s):  
Xu Wang ◽  
Jose B. Cruz
2006 ◽  
Vol 2006 ◽  
pp. 1-24 ◽  
Author(s):  
Manuel Jimenez-Lizarraga ◽  
Alex Poznyak

ε-Nash equilibrium or “near equilibrium” for a linear quadratic cost game is considered. Due to inaccurate state information, the standard solution for feedback Nash equilibrium cannot be applied. Instead, an estimation of the players' states is substituted into the optimal control strategies equation obtained for perfect state information. The magnitude of theεin theε-Nash equilibrium will depend on the quality of the estimation process. To illustrate this approach, a Luenberger-type observer is used in the numerical example to generate the players' state estimates in a two-player non-zero-sum LQ differential game.


Author(s):  
Yuntao Han ◽  
Qibin Zhou ◽  
Fuqing Duan

AbstractThe digital curling game is a two-player zero-sum extensive game in a continuous action space. There are some challenging problems that are still not solved well, such as the uncertainty of strategy, the large game tree searching, and the use of large amounts of supervised data, etc. In this work, we combine NFSP and KR-UCT for digital curling games, where NFSP uses two adversary learning networks and can automatically produce supervised data, and KR-UCT can be used for large game tree searching in continuous action space. We propose two reward mechanisms to make reinforcement learning converge quickly. Experimental results validate the proposed method, and show the strategy model can reach the Nash equilibrium.


Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 132
Author(s):  
Valery Y. Glizer

A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented.


Energies ◽  
2019 ◽  
Vol 12 (18) ◽  
pp. 3529 ◽  
Author(s):  
Michael D. Cook ◽  
Eddy H. Trinklein ◽  
Gordon G. Parker ◽  
Rush D. Robinett ◽  
Wayne W. Weaver

This paper presents two control strategies: (i) An optimal exergy destruction (OXD) controller and (ii) a decentralized power apportionment (DPA) controller. The OXD controller is an analytical, closed-loop optimal feedforward controller developed utilizing exergy analysis to minimize exergy destruction in an AC inverter microgrid. The OXD controller requires a star or fully connected topology, whereas the DPA operates with no communication among the inverters. The DPA presents a viable alternative to conventional P − ω / Q − V droop control, and does not suffer from fluctuations in bus frequency or steady-state voltage while taking advantage of distributed storage assets necessary for the high penetration of renewable sources. The performances of OXD-, DPA-, and P − ω / Q − V droop-controlled microgrids are compared by simulation.


2020 ◽  
Vol 40 (1) ◽  
pp. 71-85
Author(s):  
HK Das ◽  
T Saha

This paper proposes a heuristic algorithm for the computation of Nash equilibrium of a bi-matrix game, which extends the idea of a single payoff matrix of two-person zero-sum game problems. As for auxiliary but making the comparison, we also introduce here the well-known definition of Nash equilibrium and a mathematical construction via a set-valued map for finding the Nash equilibrium and illustrates them. An important feature of our algorithm is that it finds a perfect equilibrium when at the start of all actions are played. Furthermore, we can find all Nash equilibria of repeated use of this algorithm. It is found from our illustrative examples and extensive experiment on the current phenomenon that some games have a single Nash equilibrium, some possess no Nash equilibrium, and others had many Nash equilibria. These suggest that our proposed algorithm is capable of solving all types of problems. Finally, we explore the economic behaviour of game theory and its social implications to draw a conclusion stating the privilege of our algorithm. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 71-85


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