Iterative computation of noncooperative equilibria in nonzero-sum differential games with weakly coupled players

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Abstract

The authors study the Nash equilibria of a class of two-person nonlinear, deterministic differential games where the players are weakly coupled through the state equation and their objective functionals. The weak coupling is characterized in terms of a small perturbation parameter ε. With ε = 0, the problem decomposes into two dependent standard optimal control problems, while for ε ≠ 0, even though it is possible to derive the necessary and sufficient conditions to be satisfied by a Nash equilibrium solution, it is not always possible to construct such a solution. An iterative scheme is developed to obtain an approximate Nash solution when ε lies in a small interval around zero. Further, after requiring strong time consistency of the Nash equilibrium solution when at least one of the players uses dynamic information, the issue of existence and uniqueness of these solutions is studied for the cases when both players use the same information, either closed-loop or open-loop. It is also shown that even though the original problem is nonlinear, the higher (than zero'th) order terms in the Nash equilibria can be obtained as solutions to LQ (linear quadratic) optimal control problems, or static quadratic optimization problems.

Original languageEnglish (US)
Pages (from-to)252-257
Number of pages6
JournalProceedings of the IEEE Conference on Decision and Control
Volume1
StatePublished - Dec 1 1990
EventProceedings of the 29th IEEE Conference on Decision and Control Part 6 (of 6) - Honolulu, HI, USA
Duration: Dec 5 1990Dec 7 1990

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Nonzero-sum Games
Differential Games
Nash Equilibrium
Equilibrium Solution
Optimal Control Problem
Time Consistency
Quadratic Optimization
Strong Consistency
Weak Coupling
Zero
State Equation
Existence and Uniqueness of Solutions
Iterative Scheme
Small Perturbations
Closed-loop
Person
Optimization Problem
Necessary Conditions
Decompose
Interval

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

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title = "Iterative computation of noncooperative equilibria in nonzero-sum differential games with weakly coupled players",
abstract = "The authors study the Nash equilibria of a class of two-person nonlinear, deterministic differential games where the players are weakly coupled through the state equation and their objective functionals. The weak coupling is characterized in terms of a small perturbation parameter ε. With ε = 0, the problem decomposes into two dependent standard optimal control problems, while for ε ≠ 0, even though it is possible to derive the necessary and sufficient conditions to be satisfied by a Nash equilibrium solution, it is not always possible to construct such a solution. An iterative scheme is developed to obtain an approximate Nash solution when ε lies in a small interval around zero. Further, after requiring strong time consistency of the Nash equilibrium solution when at least one of the players uses dynamic information, the issue of existence and uniqueness of these solutions is studied for the cases when both players use the same information, either closed-loop or open-loop. It is also shown that even though the original problem is nonlinear, the higher (than zero'th) order terms in the Nash equilibria can be obtained as solutions to LQ (linear quadratic) optimal control problems, or static quadratic optimization problems.",
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T1 - Iterative computation of noncooperative equilibria in nonzero-sum differential games with weakly coupled players

AU - Srikant, Rayadurgam

AU - Basar, M Tamer

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N2 - The authors study the Nash equilibria of a class of two-person nonlinear, deterministic differential games where the players are weakly coupled through the state equation and their objective functionals. The weak coupling is characterized in terms of a small perturbation parameter ε. With ε = 0, the problem decomposes into two dependent standard optimal control problems, while for ε ≠ 0, even though it is possible to derive the necessary and sufficient conditions to be satisfied by a Nash equilibrium solution, it is not always possible to construct such a solution. An iterative scheme is developed to obtain an approximate Nash solution when ε lies in a small interval around zero. Further, after requiring strong time consistency of the Nash equilibrium solution when at least one of the players uses dynamic information, the issue of existence and uniqueness of these solutions is studied for the cases when both players use the same information, either closed-loop or open-loop. It is also shown that even though the original problem is nonlinear, the higher (than zero'th) order terms in the Nash equilibria can be obtained as solutions to LQ (linear quadratic) optimal control problems, or static quadratic optimization problems.

AB - The authors study the Nash equilibria of a class of two-person nonlinear, deterministic differential games where the players are weakly coupled through the state equation and their objective functionals. The weak coupling is characterized in terms of a small perturbation parameter ε. With ε = 0, the problem decomposes into two dependent standard optimal control problems, while for ε ≠ 0, even though it is possible to derive the necessary and sufficient conditions to be satisfied by a Nash equilibrium solution, it is not always possible to construct such a solution. An iterative scheme is developed to obtain an approximate Nash solution when ε lies in a small interval around zero. Further, after requiring strong time consistency of the Nash equilibrium solution when at least one of the players uses dynamic information, the issue of existence and uniqueness of these solutions is studied for the cases when both players use the same information, either closed-loop or open-loop. It is also shown that even though the original problem is nonlinear, the higher (than zero'th) order terms in the Nash equilibria can be obtained as solutions to LQ (linear quadratic) optimal control problems, or static quadratic optimization problems.

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