University of Pisa – Feb. 1-4, 2016 Game theory and Optimization in Communications and Networking University of Pisa – PhD Program in Ingegneria dell’Informazione Game Theory and Optimization in Communications and Networking Luca Sanguinetti, Marco Luise {luca.sanguinetti, marco.luise}@unipi.it 2 Special classes of static games February 2016 dell’Informazione PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 3 Special classes of static games Among noncooperative static games, there are special classes of games which are particularly relevant to address problems in wireless and communication networks: February 2016 • Supermodular games • Potential games dell’Informazione • Generalized Nash games PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 4 Supermodular games (1/2) A supermodular game is a static noncooperative game, which has: (1) a set of players K = [1, …, K] (2) Ak is the set of strategies (actions) available to player k February 2016 (3) uk(a) is the utility (payoff) for player k where, for any and for any (component-wise), dell’Informazione The utility function has increasing differences: if a player takes a higher action, then the other players are better off when taking higher actions as well PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 5 Supermodular games (2/2) Checking that a game is supermodular is attractive because: The set of pure-strategy Nash equilibria is not empty February 2016 The best-response dynamics converges to the smallest element-wise vector in the set of Nash equilibria dell’Informazione PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 6 Supermodular games: An example [24] (1/6) Let’s consider again the near-far power control game with continuous power sets h1 h2 player 1 Ak=[0, p]: player 2 AP February 2016 The inefficiency of the Nash equilibrium occurs because each ★ terminal ignores the interference dell’Informazione u2(a ) it generates to the other ★ PhD Program in Ing. PhD Program u (a ) 1 Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 7 Supermodular games: An example (2/6) Pricing (a.k.a. taxation) is an effective tool used in microeconomics to deal with this problem. We can in fact take advantage of a modified utility: February 2016 pricing factor dell’Informazione where the pricing factor a is used to introduce some form of externality, that charges players proportionally to the powers they radiate (that causes multiple-access interference) PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 8 Supermodular games: An example (3/6) Unlike the game without pricing (a = 0), the utility function is not quasi- concave in ak However, has increasing differences, and hence the game is supermodular. To check it, it is sufficient to prove that is twice February 2016 differentiable and such that dell’Informazione Exercise 1: Compute in the case solution: PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 9 Supermodular games: An example (4/6) In the case , February 2016 and thus dell’Informazione PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 10 Supermodular games: An example (5/6) To include pricing in our power-control h1 game, we can modify the definitions of h2 player 1 player 2 the game ingredients as follows: AP February 2016 (1) K = [1, 2] (2) dell’Informazione (3) PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 11 Supermodular games: An example (6/6) This modified formulation improves the efficiency of the Nash equilibrium (NE): a★ : NE of the original game a~★ : NE of the pricing game February 2016 ~★ u2(a ) ★ u2(a ) dell’Informazione Intuitively, this form of externality encourages players to use the ★ ~★ u1(a ) u1(a ) resources more efficiently PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 12 Potential games (1/2) A potential game is a static noncooperative game, which has: (1) a set of players K = [1, …, K] (2) Ak is the set of strategies a (actions) available to player k February 2016 (3) uk(a) is the utility (payoff) for player k where, for all players and all other players’ profile a\k, dell’Informazione potential no dependence (exact potential game) function on user index! PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 13 Potential games (2/2) A potential game is a static noncooperative game, which has: (1) a set of players K = [1, …, K] (2) Ak is the set of strategies (actions) available to player k February 2016 (3) uk(a) is the utility (payoff) for player k where, for all players and all other players’ profile a\k, dell’Informazione (ordinal potential game) PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 14 Potential games: An example [26] (1/5) Uplink power allocation for an OFDMA cellular network, with N subcarriers and K terminals user 1’s channel user 2’s channel … February 2016 user K’s channel N subcarriers dell’Informazione Each terminal k’s objective: allocate the power so as to maximize its own achievable rate given a constraint on the maximum total radiated power pk to exploit the channel frequency diversity PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 15 Potential games: An example (2/5) The strategic-form representation of the game is the following: (1) players: K = [1, …, K] terminals in the network February 2016 (2) strategies: (3) utilities: Shannon capacity dell’Informazione where and hk(n) are terminal k’s SINR and channel power gain over subcarrier n, resp. PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 16 Potential games: An example (3/5) This is an exact potential game, with potential function February 2016 Exercise 2: Check that dell’Informazione PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special classes of static games 17 Potential games: An example (4/5) In an infinite potential game (as occurs in this case), a pure-strategy Nash equilibrium exists if: • the strategy sets Ak are compact • the potential function F is upper semi-continuous on A=A1×…×AK February 2016 The interest in potential games stems from the guarantee of existence of pure-strategy Nash equilibria, and from the study of a single-variable function (the potential one) dell’Informazione If A is a compact and convex set, and F(a) is a continuously diffentiable function and stricly concave, then the Nash equilibrium of the potential game is unique PhD Program in Ing. PhD Program Dip. Ingegneria dell’Informazione Luca Sanguinetti and Marco Luise University of Pisa, Pisa, Italy Game Theory and Optimization in Communications and Networking Basics of noncooperative game theory: special