Logical Dynamic Games: Models, Equilibria, and Potentials

Logical dynamic games (LDGs) are a class of dynamic games that incorporate logical dynamics to describe the evolution of external states. Such games can be found in a wide range of natural and engineered systems, such as the Boolean network of lactose operon in Escherichia coli . However, little att...

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Published in:	IEEE transactions on automatic control Vol. 69; no. 11; pp. 7584 - 7599
Main Authors:	Li, Changxi, Li, Aming, Wu, Yuhu, Wang, Long
Format:	Journal Article
Language:	English
Published:	New York IEEE 01.11.2024 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:	Algorithms Boolean functions Cost function Decomposition dynamic games Dynamic Nash equilibrium (DNE) Dynamic programming Dynamical systems E coli finite-horizon optimization Game theory Games Heuristic algorithms Lactose logical dynamic systems (LDSs) Mathematical models Nash equilibrium Optimal control potential games Power system dynamics Recursive methods Stochastic processes Verification
ISSN:	0018-9286, 1558-2523
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Abstract	Logical dynamic games (LDGs) are a class of dynamic games that incorporate logical dynamics to describe the evolution of external states. Such games can be found in a wide range of natural and engineered systems, such as the Boolean network of lactose operon in Escherichia coli . However, little attention has been paid to LDGs in the control community. This article aims at developing a framework for the analysis and synthesis of LDGs under finite-horizon criteria. First, a general mathematical model of LDGs is constructed. Using dynamic programming theory, we prove that an LDG can be decomposed as a series of time-sliced static games, and the existence of pure dynamic Nash equilibrium (DNE) is proved to be equivalent to the existence of pure Nash equilibria of the decomposed time-sliced static games. To disentangle the circular dependence in the DNE calculation, a backward recursive method is proposed. Second, the concept of logical dynamic potential games (LDPGs) is proposed, and the connection between an LDG and its corresponding optimal control problem is established. Three verification conditions for a given LDG to be LDPG are presented, including time-sliced condition, closed-path condition, and potential equation condition. And a recursive algorithm is further designed for the verification of LDPGs via potential equation conditions. Third, to seek time-independent verification conditions, LDGs with action-independent transition properties are investigated. We prove that, if the auxiliary game constructed by the stage cost function is a state-based potential game, then the LDG is an LDPG. Finally, the effectiveness of the theoretical results is demonstrated by some numerical examples.
AbstractList	Logical dynamic games (LDGs) are a class of dynamic games that incorporate logical dynamics to describe the evolution of external states. Such games can be found in a wide range of natural and engineered systems, such as the Boolean network of lactose operon in Escherichia coli . However, little attention has been paid to LDGs in the control community. This article aims at developing a framework for the analysis and synthesis of LDGs under finite-horizon criteria. First, a general mathematical model of LDGs is constructed. Using dynamic programming theory, we prove that an LDG can be decomposed as a series of time-sliced static games, and the existence of pure dynamic Nash equilibrium (DNE) is proved to be equivalent to the existence of pure Nash equilibria of the decomposed time-sliced static games. To disentangle the circular dependence in the DNE calculation, a backward recursive method is proposed. Second, the concept of logical dynamic potential games (LDPGs) is proposed, and the connection between an LDG and its corresponding optimal control problem is established. Three verification conditions for a given LDG to be LDPG are presented, including time-sliced condition, closed-path condition, and potential equation condition. And a recursive algorithm is further designed for the verification of LDPGs via potential equation conditions. Third, to seek time-independent verification conditions, LDGs with action-independent transition properties are investigated. We prove that, if the auxiliary game constructed by the stage cost function is a state-based potential game, then the LDG is an LDPG. Finally, the effectiveness of the theoretical results is demonstrated by some numerical examples.
Author	Wang, Long Wu, Yuhu Li, Changxi Li, Aming
Author_xml	– sequence: 1 givenname: Changxi orcidid: 0000-0002-8071-8439 surname: Li fullname: Li, Changxi email: lichangxi@pku.edu.cn organization: Center for Systems and Control, College of Engineering, Peking University, Beijing, China – sequence: 2 givenname: Aming orcidid: 0000-0003-4045-8721 surname: Li fullname: Li, Aming email: amingli@pku.edu.cn organization: Center for Systems and Control, College of Engineering, Peking University, Beijing, China – sequence: 3 givenname: Yuhu orcidid: 0000-0001-9317-1404 surname: Wu fullname: Wu, Yuhu email: wuyuhu@dlut.edu.cn organization: School of Control Science and Engineering, Dalian University of Technology, Dalian, China – sequence: 4 givenname: Long orcidid: 0000-0001-5600-8157 surname: Wang fullname: Wang, Long email: longwang@pku.edu.cn organization: Center for Systems and Control, College of Engineering, Peking University, Beijing, China
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SubjectTerms	Algorithms Boolean functions Cost function Decomposition dynamic games Dynamic Nash equilibrium (DNE) Dynamic programming Dynamical systems E coli finite-horizon optimization Game theory Games Heuristic algorithms Lactose logical dynamic systems (LDSs) Mathematical models Nash equilibrium Optimal control potential games Power system dynamics Recursive methods Stochastic processes Verification
Title	Logical Dynamic Games: Models, Equilibria, and Potentials
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