Journal of Optimization Theory and Applications

, Volume 110, Issue 3, pp 515–544 | Cite as

Existence and Uniqueness of Open-Loop Stackelberg Equilibria in Linear-Quadratic Differential Games

  • G. Freiling
  • G. Jank
  • S. R. Lee


We present existence and uniqueness results for a hierarchical or Stackelberg equilibrium in a two-player differential game with open-loop information structure. There is a known convexity condition ensuring the existence of a Stackelberg equilibrium, which was derived by Simaan and Cruz (Ref. 1). This condition applies to games with a rather nonconflicting structure of their cost criteria. By another approach, we obtain here new sufficient existence conditions for an open-loop equilibrium in terms of the solvability of a terminal-value problem of two symmetric Riccati differential equations and a coupled system of Riccati matrix differential equations. The latter coupled system appears also in the necessary conditions, but contrary to the above as a boundary-value problem. In case that the convexity condition holds, both symmetric equations are of standard type and admit globally a positive-semidefinite solution. But the conditions apply also to more conflicting situations. Then, the corresponding Riccati differential equations may be of H-type. We obtain also different uniqueness conditions using a Lyapunov-type approach. The case of time-invariant parameters is discussed in more detail and we present a numerical example.

Noncooperative games differential games Stackelberg games hierarchical games linear-quadratic games open-loop Stackelberg equilibrium matrix Riccati differential equations nonsymmetric Riccati differential equations 


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© Plenum Publishing Corporation 2001

Authors and Affiliations

  • G. Freiling
  • G. Jank
  • S. R. Lee

There are no affiliations available

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