1,191 research outputs found
Stackelberg strategies in linear-quadratic stochastic differential games
This paper obtains the Stackelberg solution to a class of two-player stochastic differential games described by linear state dynamics and quadratic objective functionals. The information structure of the problem is such that the players make independent noisy measurements of the initial state and are permitted to utilize only this information in constructing their controls. Furthermore, by the very nature of the Stackelberg solution concept, one of the players is assumed to know, in advance, the strategy of the other player (the leader). For this class of problems, we first establish existence and uniqueness of the Stackelberg solution and then relate the derivation of the leader's Stackelberg solution to the optimal solution of a nonstandard stochastic control problem. This stochastic control problem is solved in a more general context, and its solution is utilized in constructing the Stackelberg strategy of the leader. For the special case Gaussian statistics, it is shown that this optimal strategy is affine in observation of the leader. The paper also discusses numerical aspects of the Stackelberg solution under general statistics and develops algorithms which converge to the unique Stackelberg solution
Linear-Quadratic -person and Mean-Field Games with Ergodic Cost
We consider stochastic differential games with players, linear-Gaussian
dynamics in arbitrary state-space dimension, and long-time-average cost with
quadratic running cost. Admissible controls are feedbacks for which the system
is ergodic. We first study the existence of affine Nash equilibria by means of
an associated system of Hamilton-Jacobi-Bellman and
Kolmogorov-Fokker-Planck partial differential equations. We give necessary and
sufficient conditions for the existence and uniqueness of quadratic-Gaussian
solutions in terms of the solvability of suitable algebraic Riccati and
Sylvester equations. Under a symmetry condition on the running costs and for
nearly identical players we study the large population limit, tending to
infinity, and find a unique quadratic-Gaussian solution of the pair of Mean
Field Game HJB-KFP equations. Examples of explicit solutions are given, in
particular for consensus problems.Comment: 31 page
Uniqueness Conditions for the Infinite-Planning Horizon Open-Loop Linear Quadratic Differential Game
In this note we consider the open-loop Nash linear quadratic differential game with an infinite planning horizon.The performance function is assumed to be indefinite and the underlying system affine.We derive both necessary and sufficient conditions under which this game has a unique Nash equilibrium.
Backward Stackelberg Differential Game with Constraints: a Mixed Terminal-Perturbation and Linear-Quadratic Approach
We discuss an open-loop backward Stackelberg differential game involving
single leader and single follower. Unlike most Stackelberg game literature, the
state to be controlled is characterized by a backward stochastic differential
equation (BSDE) for which the terminal- instead initial-condition is specified
as a priori; the decisions of leader consist of a static terminal-perturbation
and a dynamic linear-quadratic control. In addition, the terminal control is
subject to (convex-closed) pointwise and (affine) expectation constraints. Both
constraints are arising from real applications such as mathematical finance.
For information pattern: the leader announces both terminal and open-loop
dynamic decisions at the initial time while takes account the best response of
follower. Then, two interrelated optimization problems are sequentially solved
by the follower (a backward linear-quadratic (BLQ) problem) and the leader (a
mixed terminal-perturbation and backward-forward LQ (BFLQ) problem). Our
open-loop Stackelberg equilibrium is represented by some coupled
backward-forward stochastic differential equations (BFSDEs) with mixed
initial-terminal conditions. Our BFSDEs also involve nonlinear projection
operator (due to pointwise constraint) combining with a Karush-Kuhn-Tucker
(KKT) system (due to expectation constraint) via Lagrange multiplier. The
global solvability of such BFSDEs is also discussed in some nontrivial cases.
Our results are applied to one financial example.Comment: 38 page
- …