CSCI 3210: Computational Game Theory www.mtirfan.com/CSCI-3210 - PDF document

9/28/20 CSCI 3210: Computational Game Theory www.mtirfan.com/CSCI-3210 Mohammad T . Irfan Email: mirfan@bowdoin.edu Website: www.mtirfan.com 1 Syllabus and required background u Course website u www.mtirfan.com/CSCI-3210 10 1

9/28/20 You said you are here because: u Be intellectually challenged in a class that is outside of my comfort zone! u The game theory concept! u new algorithms! u I’ve been fascinated by game theory in the past, so I’m excited to take a look at it through a cs lens. u Learning something new 😂 u The difference between human decisions and rational thinking u Don’t know a lot about game theory, but most curious about application to human choice 11 Games and game theory: A brief introduction Reading: 1. Ch 6 of Easley-Kleinberg (pdf on class website) 2. [EGT] Ch 1-3 12 2

9/28/20 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) 13 Example: Split or Steal u https://www.youtube.com/watch?v=yM38mR HY150 u Rules of the game u Outcome 14 3

9/28/20 One possible model Payoff Split Steal matrix $0+fr., $33K, $33K Split $66K $66K, $0, $0 Steal $0+fr. u What will happen? 15 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. John F . Nash Nash Equilibrium Nobel Prize, 1994 Everyone plays their best response to others simultaneously 16 4

9/28/20 Nash equilibrium Practical scenarios = Stable outcome = Nash equilibrium 17 Applications u Application: market equilibria u Predict where the market is heading to u Mechanism design and auctions u Google and Yahoo apply game-theoretic techniques u Keyword search auction u Spectrum allocation among wireless companies 18 18 5

9/28/20 Applications u Understanding the Internet u Selfish routing is a constant-factor off from optimal 19 19 Other applications u Load balancing and resource allocation u p2p and file sharing systems u Cryptography and security u Social and economic networks, etc. 20 6

9/28/20 Next: Formal discussion (Without math) 21 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. 22 7

9/28/20 Nash equilibrium (NE) u A joint strategy (one strategy/player) s.t. every player plays their best response to others simultaneously u (Equiv.) A joint strategy s.t. no player gains by deviating unilaterally u Useful for checking whether a cell is NE 23 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 24 8

9/28/20 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 subject to others' actions 25 Commonly Used Terms 26 9

9/28/20 Best response u Best strategy of a player, given the other players’ strategies u Always exists! 27 (Strictly/weakly) dominant strategy u A strategy of a player that is (strictly/weakly) better than any of their other strategies, no matter what the other players do u Does not always exist 28 10

9/28/20 Pure-strategy Nash equilibrium (PSNE) u Players do not use any probability in choosing strategies as they do in "mixed- strategy" (to be covered later) u Every player plays their best response to others simultaneously 29 Checkpoint u What is the difference between a dominant strategy and a best response? u What is the difference between best response and PSNE? u What is the difference between weakly and strictly dominant strategies? Will a player always have one? 30 11

9/28/20 Checkpoint: “Generalization” u Watch the following 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? 32 33 12

9/28/20 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 34 Questions u Does NE always exist? (Answer later ...) u If it exists, is it unique? 35 13

9/28/20 Games with multiple NE 1. Battle of the sexes (Coordination) 2. Hawk-dove game (anti-coordination) 36 Does NE always exist? Mixed-strategy NE 44 14

9/28/20 Penalty kick game 45 45 Penalty kick game (continued) 46 46 15

9/28/20 Penalty kick game (continued) 47 47 Penalty kick game- first model 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] Left (p) -1, +1 +1, -1 Shooter = 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 48 48 16

9/28/20 Penalty kick game (real-world) u “Professionals Play Minimax”- Ignacio Palacios-Huerta Equilibrium probabilities (computed by solving equations) 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 From real- world data 0.93, 0.70, Right (0.62) 0.07 0.30 49 What does mixed strategy mean? u Active randomization – tennis, soccer u Proportion interpretation – evolutionary biology u Probabilities of player 1 are the beliefs 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 chosen according to these probabilities 50 17

9/28/20 Von Neumann’s Theorem (1928) u Every finite 2-person zero-sum game has a mixed equilibrium John von Neumann (1903 – 1957) 51 51 Theorem of Nash (1950) u Every finite game has an equilibrium in mixed strategies u Reading John F . Nash (1928 – 2015) Nobel Prize, 1994 52 52 18

9/28/20 Pure-strategy vs. mixed-strategy NE Hawk-dove game 53 Key take-away messages u Players act simultaneously, but NE outcome is stable in the sense that there is no incentive for unilateral deviation. u There is always at least one mixed-strategy NE. A pure-strategy NE is not guaranteed. u The concept of NE doesn't say how NE happens. u NE is not a balance or tie. It is often times a socially-inefficient outcome. 59 19

9/28/20 Formal (Mathematical) Definitions 61 Background: discrete math u Set theory u Sets u Representation: list, describe properties u Belongs to u Subset u Empty set u Power set u Operations: union, intersection, difference, product u Sum, product 62 20

9/28/20 Normal form games 63 Example u Idea: Connor Marrs u Assume: Cars don’t want to wait; all turns/S same Car1 Car2 L/R One way L/R/S St Photo modified from topdriver.com 64 21

9/28/20 1. Who are the players? u N = {1, 2} 2. What can each player do? u A1 = {L, R, S}; A2 = {L, R} u Set of action profiles A= A 1 x A 2 = {(L, L), (L, R), (R, L), (R, R), (S, L), (S, R)} u Each element (e.g., (L,R)) is called an action profile 3. What are the payoffs? 65 L/R One way L/R/S St Car 2 L R L 1, 1 0, 0 Car 1 R 0, 0 1, 1 S 0, 0 1, 1 66 22

9/28/20 Zero-sum/constant-sum game action profile u Is constant sum the same as zero sum? 67 Example (zero-sum game) u You and opponent flip two coins u Same parity (both heads up or tails up)=> you win (keep both coins) u Otherwise, opponent wins Opponent You 68 23

9/28/20 Mixed-strategy NE u Mixed strategy of a player and mixed- strategy profile of all players u Expected utility u Best response u Nash equilibrium 69 Mixed strategy • A mixed strategy is an element of S i 70 24

9/28/20 Expected utility 71 Best response 72 25

9/28/20 Nash equilibrium 73 26