Abstract
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.
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Freiling, G., Jank, G. & Lee, S.R. Existence and Uniqueness of Open-Loop Stackelberg Equilibria in Linear-Quadratic Differential Games. Journal of Optimization Theory and Applications 110, 515–544 (2001). https://doi.org/10.1023/A:1017532210579
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DOI: https://doi.org/10.1023/A:1017532210579