Masaru Unno

Nash Strategies of Markov Jump Stochastic Systems Applied to Weakly-Coupled Large-Scale Systems

Abstract This paper investigates Nash games for a class of linear stochastic systems governed by Itos differential equation with Markov jump parameters. First, in order to obtain Nash equilibrium strategies, cross-coupled stochastic algebraic Riccati equations (CSAREs) are formulated. Moreover, necessary condition for the existence of solution for CSAREs is also developed. It is noteworthy that this is the first time that conditions for the existence of stochastic equilibria have been derived based on the solutions of sets of CSAREs. As another important application, large-scale weakly-coupled systems are investigated. After establishing an asymptotic structure with positive definiteness for CSAREs solutions, a feasible algorithm that is based on the linear matrix inequality (LMI) for solving CSAREs is considered. Finally, we provide a numerical example to verify the efficiency of the proposed algorithms.

Nash Strategies for Large-Scale Stochastic Delay Systems

Abstract This paper discusses Nash games for a class of large-scale delay systems governed by Itos stochastic differential equation. The sufficient conditions for the existence of Nash strategies are given by means of cross-coupled matrix inequalities. In order to obtain the strategy set, new cross-coupled algebraic equations (CSAEs) are established using Karush-Kuhn-Tucker (KKT) conditions, which constitute the necessary conditions for the existence of Nash strategies. It is noteworthy that the asymptotic structure of cross-coupled matrix inequalities is studied for the first time by using the CSAEs. It is shown that the approximate local state feedback Nash strategies can be obtained by solving the parameter-independent linear matrix inequality (LMI) recursively. Finally, a numerical example shows that the proposed methods can help in attaining an adequate cost bound.

Nash strategy for Markov jump stochastic delay systems

Nash games for a class of linear time-delay system with Markovian jumping parameters are investigated. By using a classical Lyapunov-Krasovskii method and a non-convex optimization approach as a sufficient condition, a strategy set in terms of matrix inequality is established. In order to obtain a strategy set numerically, new cross-coupled stochastic algebraic equations (CSAEs) are given based on Karush-Kuhn-Tucker (KKT) conditions. Furthermore, it is shown that the state feedback strategies can be obtained by solving linear matrix inequalities (LMIs) iteratively. Finally, a numerical example is detailed that shows the effectiveness of the proposed methods.

Pareto-optimal solutions for Markov jump stochastic systems with delay

Pareto-optimal solutions for a class of general class of stochastic systems with both Markovian jumping parameters and time-delay are studied by introducing a linear matrix inequality (LMI) approach. In order to obtain a strategy set, new cross-coupled stochastic algebraic equations (CSAEs) are derived based on Karush-Kuhn-Tucker (KKT) conditions as necessary conditions. Furthermore, it is shown that the state feedback strategies can be obtained by solving LMIs. Finally, a numerical example is detailed that shows the effectiveness of the proposed methods.

Soft-constrained robust equilibria in stochastic differential games

In this paper, infinite-horizon soft-constrained robust equilibria in stochastic differential games are investigated for the Markov jump systems governed by Itô-type stochastic differential equations with state, control and external disturbance-dependent noise. It is shown that the saddle point equilibrium under consideration is associated with the sets of cross-coupled stochastic algebraic Riccati equations (CSAREs). After studying the saddle point solution for a single decision maker case, a multiple decision making problem is considered. A necessary condition for the existence of the robust saddle point equilibrium is found. In particular, it is noteworthy that our new results completely involve some existing results on soft-constrained stochastic Nash game. Finally, a numerical example to verify the efficiency of the proposed algorithms is given.

Dynamic optimal revenue-sharing strategy in e-commerce

In this paper, we consider a dynamic revenue-sharing problem between a platform and a seller in e-commerce. We formulate the revenue-sharing problem as a dynamic principal-agent problem, which is then transformed to a stochastic optimal control problem where the objectives of the platform are to find an optimal revenue-sharing strategy and to advise an incentive-compatible effort to the seller. The sufficient conditions for the existences of the optimal revenue-sharing strategy and the incentive-compatible effort are obtained. A numerical example is solved to show the existences of the strategy and the effort.

Optimal Revenue-Sharing and Network Investment Strategies in Internet Market

Abstract In this paper, we consider a revenue-sharing and network investment problem between an Internet service provider (ISP) and a content provider (CP) by applying dynamic agency theory. We formulate the problem as a principal-agent problem where the ISP is the principal and the CP is the agent. The principal-agent problem is then transformed to a stochastic optimal control problem in which the objective of the ISP is to find an optimal revenue-sharing strategy and a network investment strategy, and to advise an incentive-compatible effort level to the CP. The sufficient conditions for the existence of the optimal revenue-sharing strategy, the optimal network investment strategy and the incentive-compatible effort level are obtained. A numerical example is solved to show the existence of such strategies.