DCS/CSCI 2350: Social & Economic Networks Games and game theory: - PDF document

4/10/16 ¡ DCS/CSCI 2350: Social & Economic Networks Games and game theory: A brief introduction Reading: Ch. 6 of EK Mohammad T . Irfan Game Theory u “Game” u Ernst Zermelo (1913): In any chess game that does not end in a draw, a player has a winning strategy u Mathematical theory of strategic decision making u John von Neumann (1944) 1 ¡

4/10/16 ¡ Example: Split or Steal u https://www.youtube.com/watch? v=yM38mRHY150 u Rules of the game u Outcome One possible model Lucy Payoff Split Steal matrix $0+fr., Tony $33K, $33K Split $66K $66K, $0, $0 Steal $0+fr. u What will happen? 2 ¡

4/10/16 ¡ Why did they end up with 0? Lucy Payoff Split Steal matrix $0+fr., $33K, $33K Tony Split $66K $66K, $0, $0 Steal $0+fr. Nash Equilibrium John F . Nash Nobel Prize, 1994 Everyone plays his/her best response simultaneously Nash equilibrium Practical scenarios = Stable outcome = Nash equilibrium 3 ¡

4/10/16 ¡ Applications u Application: market equilibria u Predict where the market is heading to u Mechanism design for the Internet u Google and Yahoo apply game-theoretic techniques u Auctions u Example– spectrum allocation 7 Other Applications u Understanding the Internet u Selfish routing is a constant-factor off from optimal u Load balancing and resource allocation u p2p and file sharing systems u Cryptography and security u Social and economic networks u Many other … 8 4 ¡

4/10/16 ¡ Next: Formal discussion Game u One-shot games (simultaneous move) u 3 components u Players Call these pure strategies u Strategies/actions u Payoffs Lucy Payoff Split Steal matrix $0+fr., Tony $33K, $33K Split $66K Pure-strategy NE $66K, $0, $0 Steal $0+fr. 5 ¡

4/10/16 ¡ Famous example: prisoner's dilemma u What will they do? Suspect 2 Payoff Not Confess Confess matrix Not -1, -1 -10, 0 Suspect 1 Confess 0, -10 -5, -5 Confess Assumptions u Payoffs reflect player’s preference u Payoffs are known to all u Actions are known to all (different players could have different actions– but everyone knows everyone’s actions) u Each player wants to maximize own payoff 6 ¡

4/10/16 ¡ Solution concepts Best response u Best strategy of a player, given the other players’ strategies u Always exists! 7 ¡

4/10/16 ¡ (Strictly) dominant strategy u A strategy of a player that is (strictly) better than any of his other strategies, no matter what the other players do u Does not always exist Nash equilibrium (NE) u A joint strategy (one strategy/player) s.t. each player plays his best response to others simultaneously u (Equiv.) A joint strategy s.t. no player gains by deviating unilaterally 8 ¡

4/10/16 ¡ Pure-strategy Nash equilibrium u Players do not use any probability in choosing strategies as they do in "mixed-strategy" (to be covered later) Quiz u What is the difference between a dominant strategy and a best response? 9 ¡

4/10/16 ¡ Quiz u Here’s a clip from the movie A Beautiful Mind that “tries to” portray John Nash’s discovery of Nash equilibrium. u Is this actually a Nash equilibrium? Misconceptions u Equilibrium signifies a tie/draw/balance u Equilibrium outcome is the best possible outcome for all players ( A Beautiful Mind ) u Self-interested players want to hurt each other 10 ¡

4/10/16 ¡ Questions u Does NE always exist? (Answer later ...) u If it exists, is it unique? (Next) Games with multiple NE 1. Coordination game/battle of the sexes 2. Stag hunt game (coordination) 3. Hawk-dove game (anti-coordination) 11 ¡

4/10/16 ¡ Does NE always exist? Mixed-strategy NE 1. Normandy Landing 2. Matching pennies game Can there be both pure- strategy and mixed- strategy NE? Hawk-dove game 12 ¡

4/10/16 ¡ More on mixed-strategy NE Penalty kick game What does playing mixed strategy mean? Penalty kick game 29 13 ¡

4/10/16 ¡ Penalty kick game (continued) 30 Penalty kick game (continued) 31 14 ¡

4/10/16 ¡ Penalty Kick Game- Equilibrium u E[GK plays Left] Zero-sum Game = p(1) + (1-p)(-1) Goalkeeper = 2p – 1 Right (1- Left (q) q) u E[GK plays Right] Shooter Left (p) -1, +1 +1, -1 = p(-1) + (1-p)(1) = 1 – 2p Right (1- +1, -1 -1, +1 p) u 2p – 1 = 1 – 2p è p = ½ u Similarly, q = ½ u “Professionals Play Minimax”- Ignacio Palacios-Huerta 32 Penalty kick game (real-world) Equilibrium probabilities match real-world probabilities from data! Goalkeeper Left Right (0.42) (0.58) Shooter 0.58, 0.95, Left (0.38) 0.42 0.05 0.93, 0.70, Right 0.07 0.30 (0.62) 15 ¡

4/10/16 ¡ What does mixed strategy mean? u Active randomization – tennis, soccer u Proportion interpretation – evolutionary biology u Probabilities of player 1 are the belief of player 2 about what player 1 is doing (Bob Aumann) u Misconception u Players just choose probabilities u Correct u players play pure strategies selected according to these probabilities Von Neumann’s Theorem (1928) u Every finite 2-person zero-sum game has a mixed equilibrium John von Neumann (1903 – 1957) 35 16 ¡

4/10/16 ¡ Theorem of Nash (1950) u Every finite game has an equilibrium in mixed strategies John F . Nash (1928 – 2015) Nobel Prize, 1994 36 Other solution concepts u Socially optimal solution u Joint strategy that maximizes the sum of payoffs u The sum of payoffs is called social welfare u Pareto-optimal solution u A joint-strategy such that there is no other strategy where (1) everyone gets payoff at least as high (2) at least one player gets strictly higher payoff u None of the above might be NE (Extremely rarely they are NE) 17 ¡