On the existence of a positive definite solution of the matrix equation X+AXA=I

Matrices;mathematics

Multicriteria Dynamic Optimization Problems and Cooperative Dynamic Games

We survey some recent research results in the field of dynamic cooperative differential games with non-transferable utilities. Problems which fit into this framework occur for instance if a person has more than one objective he likes to optimize or if several persons decide to combine efforts in trying to realize their individual goals. We assume that all persons act in a dynamic environment and that no side-payments take place. For these kind of problems the notion of Pareto efficiency plays a fundamental role. In economic terms, an allocation in which no one can be made better-off without someone else becoming worseoff is called Pareto efficient. In this paper we present as well necessary as sufficient conditions for existence of a Pareto optimum for general non-convex games. These results are elaborated for the special case that the environment can be modeled by a set of linear differential equations and the objectives can be modeled as functions containing just affine quadratic terms. Furthermore we will consider for these games the convex case. In general there exists a continuum of Pareto solutions and the question arises which of these solutions will be chosen by the participating persons. We will flash some ideas from the axiomatic theory of bargaining, which was initiated by Nash [16, 17], to predict the compromise the persons will reach.Dynamic Optimization;Pareto Efficiency;Cooperative Differential Games;LQ The- ory;Riccati Equations;Bargaining

A Numerical Algorithm to find Soft-Constrained Nash Equilibria in Scalar LQ-Games

In this paper we provide a numerical algorithm to calculate all soft-constrained Nash equilibria in a regular scalar indefinite linear-quadratic game.The algorithm is based on the calculation of the eigenstructure of a certain matrix.The analysis follows the lines of the approach taken by Engwerda in [7] to calculate the solutions of a set of scalar coupled feedback Nash algebraic Riccati equations.C63;C72;C73

On the Sensitivity Matrix of the Nash Bargaining Solution

In this note we provide a characterization of a subclass of bargaining problems for which the Nash solution has the property of disagreement point monotonicity.While the original d-monotonicity axiom and its stronger notion, strong d-monotonicity, were introduced and discussed by Thomson [15], this paper introduces local strong d-monotonicity and derives a necessary and sufficient condition for the Nash solution to be locally strong d-monotonic.This characterization is given by using the sensitivity matrix of the Nash bargaining solution w.r.t. the disagreement point d.Moverover, we present a sufficient condition for the Nash solution to be strong d-monotonic.Nash bargaining solution;d-monotonicity;diagonally dominant Stieltjes matrix

Linear Quadratic Games: An Overview

In this paper we review some basic results on linear quadratic differential games.We consider both the cooperative and non-cooperative case.For the non-cooperative game we consider the open-loop and (linear) feedback information structure.Furthermore the effect of adding uncertainty is considered.The overview is based on [9].Readers interested in detailed proofs and additional results are referred to this book.linear-quadratic games;Nash equilibrium;affine systems;solvability conditions;Riccati equations

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.linear-quadratic games;open-loop Nash equilibrium;affine systems;solvability conditions;Riccati equations