Measures for the Importance of Information Exchange in Linear Quadratic Differential Games Over Networks

Abstract

The objective of this paper is to study and characterize the role and the importance of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information</i> in achieving a feedback (Nash) equilibrium strategy in linear quadratic (LQ) differential games whenever the underlying players are distributed over a (physical or logic) network. It is assumed that each player should achieve a desired goal, quantified by an individual cost functional, in a competitive dynamic environment - captured by an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interconnection network</i> - by relying only on the information and data exchanged with other players according to a prescribed <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information network:</i> the objective of the paper is to establish the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">value</i> of such an information exchange pattern towards achieving a more favorable (social) equilibrium. Interestingly, it is not assumed that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">interconnection network</i> and the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">information network</i> coincide. Moreover, since the ability of achieving a certain Nash equilibrium strategy may be lost even by removing a single communication link in the network - thus partially limiting the use of the metrics discussed above - in the second part of the paper we also consider the value of the information in forming approximate Nash equilibrium strategies, namely the so-called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\varepsilon$</tex-math></inline-formula> -Nash equilibria. Finally, the newly defined metrics are corroborated - together with a few suggested constructive results to characterize such values - by means of numerical examples.