3 Presentations

• 
  11:15 AM - 11:40 AM

  First order, stationary mean-field games with congestion

  • David Evangelista, presenter, KAUST
  • Diogo Gomes, King Abdullah University of Science and Technology
  • Levon Nurbekyan, KAUST
  • Rita Ferreira, KAUST
  • Vardan Voskanyan,

  We present stationary mean-field games (MFGs) with congestion with quadratic or power-like Hamiltonians. First, using explicit examples, we illustrate two main difficulties: the lack of classical solutions and the existence of areas with vanishing density. Our main contribution is a new variational formulation for MFGs with congestion. With this formulation, we prove the existence and uniqueness of solutions. Finally, we consider applications to numerical methods.

• 
  11:40 AM - 12:05 PM

  Exploitation of a Productive Asset in the Presence of Strategic Behavior and Pollution externalities

  • Baris Vardar, presenter, HEC Montréal / GERAD
  • Georges Zaccour, Chair in Game Theory and Management, GERAD and HEC Montréal

  This paper studies the strategic behavior of firms competing in the exploitation of a common-access productive asset, in the presence of pollution externalities. We consider a differential game with two state variables (asset stock and pollution stock), and, by using a piecewise-linear approximation of the nonlinear asset growth function, we provide a tractable characterization of the symmetric feedback-Nash equilibrium, which is globally asymptotically stable. The results show that the firm's strategy takes three forms depending on the pair of state variables, and that different options for the model parameters lead to contrasting outcomes in both the short- and long-run equilibria.

• 
  12:05 PM - 12:30 PM

  Nash Equilibrium for Stochastic Differential-Algebraic Games with Applications in Control and Economy

  • Quanyan Zhu, presenter, New York University
  • Tanwani Aneel, LAAS

  Several applications require us to study game-theoretic problems where the state variable not only follow a differential equation, but are also constrained by algebraic equations; and the parameters describing these differential-algebraic constraints may change randomly with time. Thus, motivated by the need to develop a framework for stochastic differential games with algebraic constraints, we study dynamic noncooperative games where the constraints are described by Markov jump differential-algebraic equations (DAEs). The switching in system dynamics is governed by a finite state Markov chain, giving rise to piecewise-deterministic Markov DAEs. The switching can represent the structural changes in the system dynamics and constraints caused by natural disasters or cyber attacks, e.g., the cyber attack on the power plants, and the flooding of lower Manhattan due to Hurricane Sandy.

  We start by developing theoretical tools for analysis of Markov jump DAEs. A rigorous derivation of the infinitesimal generator associated with a cost functional is provided. We then use this result in the context of dynamic programming to derive the Hamilton-Jacobi-Bellman equation associated with Markov jump DAEs under consideration. These results are then used for deriving the optimal strategies of the individual players in the game setting. In the general case, with nonlinear dynamics, these strategies rely on solving the coupled partial differential equations. For the case of linear dynamics and quadratic cost functionals, these strategies can be obtained by solving coupled Riccati differential equations.

  As an application of our results, we design robust controllers for Markov jump DAEs subject to some disturbances in the dynamics. This is done be studying the two-player zero sum game where the performance index is the expected value of a quadratic function of the state, input, and the disturbance. The control input that we design minimizes the disturbance-to-state gain. An overview of robust control problems for DAEs in deterministic setting without switching can be found in \cite{Mehr91}, and this article generalizes those results when the dynamics undergo switching driven by a Markov process.

  Application in the context of economic system are also envisaged where different suppliers aim to maximize their profits.