On the Open-loop Nash Equilibrium in LQ-games

Abstract

In this paper we consider open-loop Nash equilibria of the linear-quadratic differential game. We present both necessary and sufficient conditions for existence of a unique solution for the finite-planning horizon case, and show that there exist situations where the set of associated Riccati differential equations has no solution, whereas the problem does have an equilibrium. The pursued analyses allows a simple study of convergence of the equilibrium strategy if the planning horizon expands. Conditions are given under which this strategy converges. A detailed study of the infinite planning horizon case is given. In particular we show that, in general, this problem has not a unique equilibrium. Furthermore, we show that the limit of the above mentioned converged strategy may be not an equilibrium for the infinite planning horizon problem. Particular attention is paid to computational aspects and the scalar case.

Linear–Quadratic Mean-Field-Type Games: A Direct Method

“…In the LQ-mean-field-type game problems, the state process can be modeled by a set of linear stochastic differential equations of McKean-Vlasov, and the preferences are formalized by quadratic cost functions with mean-field terms. These game problems are of practical interest, and a detailed exposition of this theory can be found in [7,12,[22][23][24][25]. The popularity of these game problems is due to practical considerations in signal processing, pattern recognition, filtering, prediction, economics, and management science [26][27][28][29].…”

Debt Stabilization Games in the Presence of Risk Premia

“…Weeren et al [23] and Engwerda [8] analyze the existence and asymptotic behaviour of feedback Nash and openloop Nash equilibria of LQ dynamic games and the lastmentioned paper applies this analysis to the Tabellini debt stabilization game. Van den Broek [2] situates the debt stabilization game in a moving horizon dynamic game that could e.g.…”

Uniqueness conditions for the infinite-planning horizon Open-Loop Linear Quadratic Differential Game.

“…Furthermore, since Q i are assumed to be indefinite, the obtained results can be directly used to (re)derive properties for the zero-sum game, which plays, e.g., an important role in robustness analysis. If players discount their future loss, similar to [6], it follows from Theorem 3.6 that if the discount factor is "large enough" the game has generically a unique open-loop Nash equilibrium. Finally we conclude from (23) that the conclusion in [13], that if the game has an open-loop Nash equilibrium for every initial state either there is a unique equilibrium or an infinite number of equilibria, applies in general.…”

“…This problem has been considered by many authors and dates back to the seminal work of Starr and Ho in [16] (see, e.g., [14], [15], [5], [12], [11], [1], [17], [6], [7], [3] and [13]). More specifically, we study in this paper the (regular indefinite) infinite-planning horizon case.…”

“…More specifically, we study in this paper the (regular indefinite) infinite-planning horizon case. The corresponding regular definite (that is the case that the state weighting matrices Q i (see below) are semi-positive definite) problem has been studied, e.g., extensively in [6] and [7]. Whereas [13] studied the regular indefinite case using a functional analysis approach, under the assumption that the uncontrolled system is stable.…”