The subject of this paper is a linear quadratic case of a differential game model with continuous updating. This class of differential games is essentially new, there it is assumed that at each time instant, players have or use information about the game structure defined on a closed time interval with a fixed duration. As time goes on, information about the game structure updates. Under the information about the game structure we understand information about motion equations and payoff functions of players. A linear quadratic case for this class of games is particularly important for practical problems arising in the engineering of human-machine interaction. The notion of Nash equilibrium as an optimality principle is defined and the explicit form of Nash equilibrium for the linear quadratic case is presented. Also, the case of dynamic updating for the linear quadratic differential game is studied and uniform convergence of Nash equilibrium strategies and corresponding trajectory for a case of continuous updating and dynamic updating is demonstrated.

On Class of Linear Quadratic Non-cooperative Differential Games with Continuous Updating

This paper is devoted to a new class of differential games with continuous updating. It is assumed that at each time instant, players have or use information about the game defined on a closed time interval. However, as the time evolves, information about the game updates, namely, there is a continuous shift of time interval, which determines the information available to players. Information about the game is the information about motion equations and payoff functions of players. For this class of games, the direct application of classical approaches to the determination of optimality principles such as Nash equilibrium is not possible. The subject of the current paper is the construction of solution concept similar to Nash equilibrium for this class of differential games and corresponding optimality conditions, in particular, modernized Hamilton-Jacobi-Bellman equations.

Hamilton-Jacobi-Bellman Equations for Non-cooperative Differential Games with Continuous Updating

In the paper, a new class of dynamic game models with a moving information horizon or dynamic updating is studied. In this class of games, players do not have full information about the game structure (motion equations, payoff functions) on the interval on which the game is defined. It is supposed that the players at each stage of the dynamic game have only truncated information about the game structure defined by the information horizon. Cooperative and noncooperative settings are considered in the paper. Results are illustrated using the oligopoly advertising game model, and comparison between the solution in the initial game model and in the game model with moving information horizon is presented. Simulation results are presented.

https://www.mdpi.com/2227-7390/7/12/1239/pdf

Moving Information Horizon Approach for Dynamic Game Models

We consider a class of cooperative differential games with continuous updating making use of the Pontryagin maximum principle. It is assumed that at each moment, players have or use information about the game structure defined in a closed time interval of a fixed duration. Over time, information about the game structure will be updated. The subject of the current paper is to construct players’ cooperative strategies, their cooperative trajectory, the characteristic function, and the cooperative solution for this class of differential games with continuous updating, particularly by using Pontryagin’s maximum principle as the optimality conditions. In order to demonstrate this method’s novelty, we propose to compare cooperative strategies, trajectories, characteristic functions, and corresponding Shapley values for a classic (initial) differential game and a differential game with continuous updating. Our approach provides a means of more profound modeling of conflict controlled processes. In a particular example, we demonstrate that players’ behavior is braver at the beginning of the game with continuous updating because they lack the information for the whole game, and they are “intrinsically time-inconsistent”. In contrast, in the initial model, the players are more cautious, which implies they dare not emit too much pollution at first.

Transferable Utility Cooperative Differential Games with Continuous Updating Using Pontryagin Maximum Principle

This paper presents a complete description and the results of the Looking Forward Approach for cooperative differential games with transferable utility. The approach is used for constructing game theoretical models and defining solutions for conflict-controlled processes where information about the process updates dynamically or for differential games with dynamic updating. It is supposed that players lack certain information about the dynamical system and payoff function over the whole time interval on which the game is played. At each instant, information about the game structure updates, players receive new updated information about the dynamical system and payoff functions. A resource extraction game serves as an illustration in order to compare a cooperative trajectory, imputations, and the imputation distribution procedure in a game with the Looking Forward Approach and in the original game with a prescribed duration.

About the Looking Forward Approach in Cooperative Differential Games with Transferable Utility

The paper applies Looking Forward Approach to analyze the world oil market with the framework of a differential game model of quantity competition oligopoly Namely Looking Forward Approach is used to take into account dynamically updating information Under the information we understand the forecast of the oil demand dynamics We focus on the period from December 2015 to November 2016 and suppose that during this time interval countries did not cooperate officially on the amounts of oil to be produced Therefore, their behavior can be modeled using the non-cooperative game model As a solution concept for this conflict-controlled process we use feedback Nash equilibrium In order to define the parameters of model open source data is used, results of numerical simulations and comparison with the historical data are presented

Non-cooperative Differential Game Model of Oil Market with Looking Forward Approach

Non-transferable utility game of oil market is considered. Special approach for defining solution is used. This approach enables to construct a real time models of conflicting processes. Connection between the solution in the game with moving information horizon and solutions on the truncated time intervals is shown.

Dmitrii Volf

Papers

Non-cooperative Differential Game Model of Oil Market with Looking Forward Approach

Differential game of oil market with moving informational horizon and non-transferable utility