A Computable Theory of Dynamic Games and Its Applications

Open Access
Author:
Mookherjee, Reetabrata
Graduate Program:
Industrial Engineering
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
October 03, 2006
Committee Members:
  • Terry Lee Friesz, Committee Chair
  • David Arthur Nembhard, Committee Member
  • Tao Yao, Committee Member
  • Venkataraman Shankar, Committee Member
  • Georgia Perakis, Committee Member
Keywords:
  • Differential variational inequalities
  • Optimal control
  • Nash equilibrium
  • State dependent time shifts
Abstract:
Recently dynamic competitive service sector problems such as revenue management, dynamic pricing, supply chain management, combined energy generation and distribution, and transportation network flow prediction and control have been the focus of much research activity. In this thesis we describe a general computable theory of dynamic games that relaxes most of the restrictive assumptions of classical nonzero-sum differential game theory, thought to be mandatory in the study of service and utility network problems. We have shown that differential variational inequalities (DVIs), which are infinite dimensional variational inequalities that include state dynamics, controls as well as control constraints, have the same notions of minimum principle, adjoint equations and transversality conditions familiar from the theory of optimal control when relatively mild regularity conditions are imposed. We also show that DVIs may conveniently be used to compute Cournot-Nash-Bertrand equilibria of broad classes of dynamic games where the game-theoretic agents have a forward-looking or anticipatory perspective. We have also shown that when state-dependent time shifts - such as those encountered in some applied problems namely modeling vehicular traffic and supply chain flows are present, the resulting problems remain surprisingly tractable. A simple fixed point algorithm combined with descent in Hilbert space for which sub-problem solutions are expressed as pure functions of time allowed us to compute solutions efficiently. Our numerical examples suggest the fixed-point-descent-in-Hilbert-space algorithm may be practical for intermediate size problems without special structure. We have also showed that the fixed-point-descent-in-Hilbert-space algorithm is convergent when suitable small step sizes are employed. We have applied this framework to study a wide class of applied problems. In particular we have studied (a) dynamic revenue management competition, (b) city logistics and supply chain competition, (c) electric power generation-distribution game, and (d) the transportation network flow prediction and congestion option games. In each case we present some structural properties of the games as well as numerical examples that further characterize respective Nash equilibria.