Time-Consistency of an Imputation in a Cooperative Hybrid Differential Game

Ekaterina Gromova,Anastasiia Zaremba,Shimai Su

doi:10.3390/math9151830

Abstract

This work is aimed at studying the problem of maintaining the sustainability of a cooperative solution in an n-person hybrid differential game. Specifically, we consider a differential game whose payoff function is discounted with a discounting function that changes its structure with time. We solve the problem of time-inconsistency of the cooperative solution using a so-called imputation distribution procedure, which was adjusted for this general class of differential games. The obtained results are illustrated with a specific example of a differential game with random duration and a hybrid cumulative distribution function (CDF). We completely solved the presented example to demonstrate the application of the developed scheme in detail. All results were obtained in analytical form and illustrated by numerical simulations.

Highlights

This contribution aims at bringing together two different concepts: the notion of sustainable cooperation from game theory and the notion of hybrid optimal control from the theory of hybrid systems
In [2], the notion of the imputation distribution procedure (IDP) was extended to the class of differential games with infinite duration and a discounting function of a rather general form
In this work we extend and generalize the results of [17] and put them in the context of hybrid optimal control

Summary

Introduction

This contribution aims at bringing together two different concepts: the notion of sustainable cooperation from game theory and the notion of hybrid optimal control from the theory of hybrid systems. In [2], the notion of the IDP was extended to the class of differential games with infinite duration and a discounting function of a rather general form. There have been a number of papers devoted to the analysis of optimal problems with different types of discounting functions and their extension to the class of differential games (see, e.g., [3,4,5], where this problem was considered in both deterministic and stochastic settings, and [6] for the very recent results.). We consider the problem of the sustainability of a cooperative solution in an n-person hybrid differential game. We will consider a differential game such that the payoff function changes its structure at specific time instants.

Subgame

Cooperative Differential Game

Optimal Solution

Optimal Solutions for Subgames

Conclusions