Ekaterina Gromova

On a Class of Hybrid Differential Games

This paper is intended to present a systematic application of the hybrid systems framework to differential games. A special class of bimodal linear-quadratic differential games is presented and illustrated with examples; two particular classes of switching rules, time-dependent and state-dependent switches are discussed. The main contribution of the paper consists in formulating necessary optimality conditions for determining optimal strategies in both cooperative and non-cooperative cases. A practically relevant hybrid differential game of pollution reduction is considered to illustrate the developed framework.

On an approach to constructing a characteristic function in cooperative differential games

We propose a novel approach to constructing characteristic functions in cooperative differential games. A characteristic function of a coalition S is computed in two stages: first, optimal control strategies maximizing the total payoff of the players are found, and next, these strategies are used by the players from the coalition S, while the other players, those from NS, use strategies minimizing the total payoff of the players from S. The characteristic function obtained in this way is superadditive. In addition, it possesses a number of other useful properties. As an example, we compute values of a characteristic function for a specific differential game of pollution control.

The Shapley Value as a Sustainable Cooperative Solution in Differential Games of Three Players

The contribution of the paper is twofold: first, it has been shown that the Yeung’s conditions can be used to construct a strongly time-consistent core. In this core there is a supporting imputation which has the property that a single deviation from this imputation in favor of any other imputation from the core still leads to the payment from the core. The obtained results were formulated for the Shapley value taken as the supporting imputation. Second, a particular class of differential games was considered. For this class of games the δ-characteristic function turns out to be superadditive and the Yeung’s conditions are satisfied without any additional restrictions on the parameters of the model. All results are presented in the analytic form.

A Differential Game Model for The Extraction of Nonrenewable Resources with Random Initial Times — The Cooperative and Competitive Cases

In this work, we propose a model for the extraction of a nonrenewable resource in an economy where, initially, only one agent is enabled to perform extraction tasks. However, at certain nonpredictable (random) times, more companies receive the government’s approval for extracting the country’s resources. We provide a setup suitable for the use of standard dynamic programming results for both, the competitive and cooperative schemes; we develop the corresponding HJB equations, prove a verification theorem, and give an example. Our framework is inspired by the trends that oil industries are experiencing in countries like Mexico and Russia.

Control of information horizon for cooperative differential game of pollution control

In this paper, we present a novel approach to defining the solution of a differential game. This approach allows to define optimal strategies in the differential games when players do not have the full information about the structure of the game along the time interval [t0, T] on which the game is defined. It assumed that at each time instant the players have information about the structure of the game (payoff functions, motion equations) only for the next fixed time interval T̅̅, which is called the information horizon. Based on this information the players make their decision. Also in this work we consider the case when the players have the opportunity to control the informational horizon in order to receive the better joint payoff for the cooperating players. The problem is to define T̅̅ such that the imputation of each player computed on the basis of restricted information deviates from the imputation in classical cooperative differential game (with full information about the structure of the game known at the beginning of the game) by a value that does not exceed given M. The developed concept is applied to a cooperative differential game of pollution control.

A Dynamic Game of Mobile Agent Placement in a MANET

In this paper, we describe a novel game-theoretic formulation of the optimal mobile agents placement problem which arises in the context of Mobile Ad-hoc Networks (MANETs). In particular, we consider two classes of multistage games: sequential and simultaneous. For such games, the definitions of the Nash equilibria and the cooperative solution are given. The described games exhibit a number of interesting features. For instance, the Nash equilibrium may turn out to be unattainable in both a simultaneous and a sequential game. In this case, the game dynamics may exhibit the behaviour similar to that of a limit cycle albeit in a discrete space. A modelling environment for the analysis of different strategies of the players was developed in MATLAB. The programme generates various game situations and determines each players move by solving respective optimisation problems. Using the developed environment, two specific game scenarios were considered in detail.

S-strongly Time-Consistency in Differential Games

In the paper the definition of S-strongly time-consistency in differential games is introduced. The approach of the construction of S-strong time-consistent subcore of the classical core on the base of characteristic function obtained by normalization of classical characteristic function is formulated. Its relation to another characteristic function obtained by an integral extension of the original characteristic function is studied.

On the superadditivity of a characteristic function in cooperative differential games with negative externalities

In this paper, a 3-player cooperative differential game with negative externalities is considered. We assume constant constraints on controls. For this game, a delta-characteristic function is constructed. Numerical example of such delta-characteristic function is presented.

STABLE MARKETING COOPERATION IN A DIFFERENTIAL GAME FOR AN OLIGOPOLY

In this paper, we develop a dynamic model of an oligopoly playing an advertising game of goodwill accumulation with random terminal time. The goal is to find a cooperative solution that is time-con...

Risk and deviation measures for a class of optimal control problems with random time horizon

This paper reports on an ongoing research project devoted to the investigation of different risk and deviation measures for a class of optimal control problems with random time horizon. Three optimal control problems related to the optimization of the expected value, the variance, and the second moment of the cost functional are described in detail. The results are illustrated by a numerical example.