Zhu Han, Dusit Niyato, Walid Saad, and Tamer Basar Game Theory in Wireless and Communication Networks: Theory, Models, and Applications Lecture 12 Variational Inequalities, Mathematical Programs with Equilibrium Constraints (MPEC) and Equilibrium Programs with Equilibrium Constraints (EPEC) Mathematical Programs with Equilibrium Constraints and its Application in Solving Stackelberg Games Zhu Han, Dusit Niyato, Walid Saad, and Tamer Basar Thanks for Xiao Tang, Xi’an Jiaotong University

Overview of Lecture Notes Introduction to Game Theory: Lecture 1, book 1 Non-cooperative Games: Lecture 1, Chapter 3, book 1 Bayesian Games: Lecture 2, Chapter 4, book 1 Differential Games: Lecture 3, Chapter 5, book 1 Evolutionary Games: Lecture 4, Chapter 6, book 1 Cooperative Games: Lecture 5, Chapter 7, book 1 Auction Theory: Lecture 6, Chapter 8, book 1 Matching Game: Lecture 7, Chapter 2, book 2 Contract Theory, Lecture 8, Chapter 3, book 2 Learning in Game, Lecture 9, Chapter 6, book 2 Stochastic Game, Lecture 10, Chapter 4, book 2 Game with Bounded Rationality, Lecture 11, Chapter 5, book 2 Equilibrium Programming with Equilibrium Constraint, Lecture 12, Chapter 7, book 2 Zero Determinant Strategy, Lecture 13, Chapter 8, book 2 Mean Field Game, Lecture 14, UCLA course, book 2 Network Economy, Lecture 15, Dr. Jianwei Huang, book 2

Outline Introduction Variational Inequalities MPEC EPEC Stackelberg Game Variational Inequalities Definition Examples MPEC EPEC

Stackelberg Game Definition and features Modeling sequential decision-makings; Solution by backward induction. Leader’s Problem Decision making: Follower chooses the action by reacting to leader’s action Backward induction: Leader makes decisions by considering follower’s reaction Follower’s Problem 2018/11/12

Basics on Variational Inequality

Basics on Variational Inequality

Basics on Variational Inequality

Basics on Variational Inequality

Example: IterativeWater-Filling

IterativeWater-Filling

IterativeWater-Filling

IterativeWater-Filling

IterativeWater-Filling

IterativeWater-Filling

Numerical result 2018/11/12

Xi’an Jiaotong University | University of Houston Basics on Quasi-VI 2018/11/12 Xi’an Jiaotong University | University of Houston

Energy-Efficient Multi-Channel Power Allocation 2018/11/12

Energy-Efficient Multi-Channel Power Allocation 2018/11/12

Energy-Efficient Multi-Channel Power Allocation 2018/11/12

Energy-Efficient Multi-Channel Power Allocation 2018/11/12

Energy-Efficient Multi-Channel Power Allocation 2018/11/12

Outline Introduction Variational Inequalities MPEC EPEC Stackelberg Game Variational Inequalities Definition Power Control with Aggregated Interference Constraints MPEC Examples EPEC

MPEC Problem Definition A mathematical optimization with part of the constraints being in the form of complementarity conditions complementarity condition w.r.t. y (parameterized by x) 2018/11/12

MPEC Problem Definition A mathematical optimization with part of the constraints being in the form of another optimization (bi-level optim.) Optimization w.r.t. y as constraint (parametrized by x) 2018/11/12

MPEC Problem Features Bi-level (multi-level) structure; Suit for the Stackelberg game analysis Leader’s problem Follower’s problem 2018/11/12

Two-Tier Power Control System Model Two-tier heterogeneous networks Overlapping frequency reuse Multi-channel transmissions Rate maximization 2018/11/12

Two-Tier Power Control Problem Formulation Stackelberg game Leader: Macro-cell users; Follower: Small-cell users Hierarchical power control Maximize: w.r.t.: 2018/11/12

Two-Tier Power Control Analysis Follower’s power allocation Water-filling: 2018/11/12

Two-Tier Power Control Analysis Leader’s power allocation MPEC reformulation as an optimization: MPEC formulation: Follower’s optimal as constraints Backward induction 2018/11/12

Two-Tier Power Control Analysis Leader’s power allocation MPEC reformulation as an optimization: Water-filling based solution Based on follower’s optimal: p is an implicit function of q 2018/11/12

Two-Tier Power Control Analysis Water-filling based part Implicit function part: Denote the implicit function as and apply KKT: Leader’s power allocation as a fixed-point iteration 2018/11/12

Two-Tier Power Control Analysis Structural results on lower water-filling Linear structure 2018/11/12

Two-Tier Power Control Analysis Solution to the MPEC (leader’s problem) Note The structural results of the follower’s problem is the crucial to solve the MPEC problem. 2018/11/12

Jamming-Aided Eavesdropping System Model Multi-channel communications Full duplex eavesdropper Improve eavesdropping by jamming attacks 2018/11/12

Jamming-Aided Eavesdropping Problem Formulation Stackelberg game Leader’s problem: Follower’s problem: 2018/11/12

Jamming-Aided Eavesdropping Analysis Follower’s problem Convex problem and solved with Lagrange Generalized water-level: Implicit function of PE 2018/11/12

Jamming-Aided Eavesdropping Analysis Leader’s problem backward induction: intractable optimization problem Intractable optimization due to the lack of explicit analytical expression 2018/11/12

Jamming-Aided Eavesdropping Analysis MPEC problem Changed optimization parameters The lower optimal condition as a constraint rather than part of the objective 2018/11/12

Jamming-Aided Eavesdropping Analysis Equivalent optimization problem Changed optimization parameters A well-defined optimization: Solved with regular method violating the constraint qualification (CQ) KKT condition in replace of the optimality condition 2018/11/12

Jamming-Aided Eavesdropping Note Replacing the optimality constraints with its KKT equivalent is very useful in solving MPEC; The violation of CQ is a common problem which should be tackled carefully. 2018/11/12

Smoothing Method for General MPEC General Problem Convex problem 2018/11/12

Smoothing Method for General MPEC Equivalent Reformulation violating the constraint qualification (CQ) Well-defined optimization with slack variable z 2018/11/12

Smoothing Method for General MPEC Smoothing Function Definition: Properties: 2018/11/12

Smoothing Method for General MPEC Smoothing Approximation -Parameterized optimization Well-defined, smooth optimization 2018/11/12

Outline Introduction Variational Inequalities MPEC EPEC Stackelberg Game Variational Inequalities Definition Power Control with Aggregated Interference Constraints MPEC Examples EPEC

Reduce latency and transmission costs Introduction Internet of Things Big Data Compu ting and Storage Cloud Computi ng Fog Computin g Reduce latency and transmission costs Data Service Operators (DSOs) Data Centers Fog Nodes (FNs) Authorized Data Service Subscribers (ADSSs) 47

Motivation Large number of FNs deployed at various locations How to manage computing resources? How to perform large-scale optimization? Equilibrium Problem with Equilibrium Constraints (EPEC) Competition among multiple DSOs and ADSSs How to deal with multiple conflicting entities? Alternating Direction Method of Multipliers (ADMM) Proposed method!!! 48

Real Time Data Service Scenario K DSOs, N ADSSs Computing resources from data centers or FNs – Computing Resource Blocks (CRBs) DSOs manage CRBs for serving the ADSSs ADSSs request for CRBs from the DSOs 49

Utility Functions of DSOs and ADSSs Price set for one CRB by DSO i for ADSS j – Ɵi,j Number of CRBs purchased from DSO i by ADSS j – xi,j Utility function for DSO i Utility function for ADSS j Revenue from the CRBs provided to ADSSs Operational and measurement costs Revenue from workload data Cost of service delay 50

Maximization of Profits for DSOs and ADSSs Optimization problem for DSO i Optimization problem for ADSS j Constraint on budget to purchase CRBs Constraint on available CRBs Constraint on service delay Constraint on service delay DSOs and ADSSs aim to maximize their own profits (optimal Ɵi,j and xi,j) DSOs provide incentives to ADSSs, to choose xi,j to obtain optimal Ɵi,j 51

Equilibrium Problem with Equilibrium Constraints (EPEC) Optimization for the DSOs Optimization for the ADSSs Optimization of the utilities of DSOs, while considering the utilities of ADSSs (conflicting objectives) Two level hierarchical optimization problem Equilibria and constraints exist at both upper and lower levels 52

Alternating Direction Method of Multipliers (ADMM) Objective function of this form Constraints of this form Method for large scale optimization Fast convergence when objective function is convex EPEC + ADMM Conflicting objectives + Large network!!! 53

ADMM based EPEC in Fog Computing Convergence check for outer loop Optimization for the ADSSs Optimization for the DSOs The outer loop terminates when the total profit of the DSOs converges to an optimal value, i.e., Value at current iteration – Value at previous iteration < Error threshold 54

Total Profit of ADSSs vs. Number of ADSSs Fog Computing Cloud Computing Data Center Service Optimization using ADMM increases the total profit of the ADSSs in Fog Computing compared to Cloud Computing 55

Summary VI MPEC and EPEC Use GNEP to tackle the common constraints in games; VI assisted convergence analysis in distributed algorithms; Use Quasi-NE to tackle the non-convex games. The VI theory is a very powerful mathematical tool which has applications in many areas. By building up the equivalence between VI and game, the VI theory provides us alternative manner in the investigation on the properties of Nash equilibrium, particularly on the uniqueness and convergence. MPEC and EPEC Provide a tool to facilitate the discussion on more complicated hierarchical game model; 2018/11/12

Xi’an Jiaotong University | University of Houston References [1]. S. Guruacharya, D. Niyato, D. I. Kim, and E. Hossain, “Hierarchical competition for downlink power allocation in OFDMA femtocell networks,” IEEE Trans. Wireless Commun., vol. 12, no. 4, pp. 1543–1553, Apr. 2013. [2]. K. Zhu, E. Hossain, and A. Anpalagan, “Downlink power control in two-tier cellular ofdma networks under uncertainties: a robust Stackelberg game,” IEEE Trans. Commun., vol. 63, no. 2, pp. 520–543, Feb. 2015. [3]. J. Wang, M. Peng, S. Jin, and C. Zhao, “A generalized nash equilibrium approach for robust cognitive radio networks via generalized variational inequalities,” IEEE Trans. Wireless Commun., vol. 13, no. 7, pp. 3701–3714, Jul. 2014. [4]. Q. Han, B. Yang, X. Wang, K. Ma, C. Chen, and X. Guan, “Hierarchical-game-based uplink power control in femtocell networks,” IEEE Trans. Veh. Technol., vol. 63, no. 6, pp. 2819–2843, Jul. 2014. [5]. Z. Luo, J. Pang, and D. Ralph, Mathematical Programs with Equilibrium Constraints. Cambridge, UK: Cambridge University Press, 1996. 2018/11/12 Xi’an Jiaotong University | University of Houston