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CMU 15-896 Noncooperative games 1: Basic concepts Teacher: Ariel Procaccia Normal-Form Game A game in normal form consists of: Set of players o Strategy set o For each , utility function : if o each plays the strategy

  1. CMU 15-896 Noncooperative games 1: Basic concepts Teacher: Ariel Procaccia

  2. Normal-Form Game • A game in normal form consists of: Set of players o Strategy set o � For each , utility function � : if o each plays the strategy � , the utility of player is � � � • Next example created by taking screenshots of http://youtu.be/jILgxeNBK_8 15896 Spring 2016: Lecture 17 2

  3. Selling ice cream at the beach. One day your cousin Ted shows up. His ice cream is identical! You split the beach in half; you set up at 1/4. 50% of the customers buy from you. 50% buy from Teddy. One day Teddy sets up at the 1/2 point! Now you serve only 37.5%! 15896 Spring 2016: Lecture 17 3

  4. The Ice Cream Wars • � � �� � • � � � � � �� � • � � � � � � � � � � • To be continued… 15896 Spring 2016: Lecture 17 4

  5. The prisoner’s dilemma • Two men are charged with a crime • They are told that: If one rats out and the other does not, the o rat will be freed, other jailed for nine years If both rat out, both will be jailed for six o years • They also know that if neither rats out, both will be jailed for one year 15896 Spring 2016: Lecture 17 5

  6. The prisoner’s dilemma Cooperate Defect -1,-1 -9,0 Cooperate Defect 0,-9 -6,-6 What would you do? 15896 Spring 2016: Lecture 17 6

  7. Prisoner’s dilemma on TV http://youtu.be/S0qjK3TWZE8 15896 Spring 2016: Lecture 17 7

  8. The professor’s dilemma Class Listen Sleep 10 6 ,10 6 -10,0 Professor Make effort Slack off 0,-10 0,0 Dominant strategies? 15896 Spring 2016: Lecture 17 8

  9. Nash equilibrium • Each player’s strategy is a best response to strategies of others • Formally, a Nash equilibrium is a vector of � strategies � � such that � � � � �� � � 15896 Spring 2016: Lecture 17 9

  10. Nash equilibrium http://youtu.be/CemLiSI5ox8 15896 Spring 2016: Lecture 17 10

  11. Russel Crowe was wrong 15896 Spring 2016: Lecture 17 11

  12. End of the Ice Cream Wars Day 3 of the ice cream wars… Teddy sets up south of you! You go south of Teddy. Eventually… 15896 Spring 2016: Lecture 17 12

  13. Rock-paper-scissors R P S R 0,0 -1,1 1,-1 P 1,-1 0,0 -1,1 S -1,1 1,-1 0,0 Nash equilibrium? 15896 Spring 2016: Lecture 17 13

  14. Mixed strategies • A mixed strategy is a probability distribution over (pure) strategies • The mixed strategy of player is � , where � � � • The utility of player is � � � � � , … , � � � � � � � � , … , � � ⋅ � � � �� � � �� � ,…,� � �∈� � ��� 15896 Spring 2016: Lecture 17 14

  15. Nash’s Theorem • Theorem [Nash, 1950]: if everything is finite then there exists at least one (possibly mixed) Nash equilibrium • We’ll talk about computation some other time 15896 Spring 2016: Lecture 17 15

  16. Does NE make sense? • Two players, strategies are • If both choose the same number, that is what they get • If one chooses , the other , and , the former player gets , and the latter gets • Poll 1: what would you choose? 95 96 97 98 99 100 15896 Spring 2016: Lecture 17 16

  17. Correlated equilibrium • Let for simplicity • A mediator chooses a pair of strategies � � according to a distribution over � • Reveals � to player and � to player • When player gets � , he knows that the distribution over strategies of is � � � � � � � � � � ∈� � � � 15896 Spring 2016: Lecture 17 17

  18. Correlated equilibrium � • Player is best responding if for all � � � � � � � � � � � � � � ∈� � � ∈� • Equivalently, � � � � � � � � � � � � � ∈� � � ∈� is a correlated equilibrium (CE) if both • players are best responding 15896 Spring 2016: Lecture 17 18

  19. Game of chicken http://youtu.be/u7hZ9jKrwvo 15896 Spring 2016: Lecture 17 19

  20. Game of chicken • Social welfare is the sum of utilities Dare Chicken • Pure NE: (C,D) and (D,C), 0,0 4,1 Dare social welfare = 5 • Mixed NE: both social welfare = 4 1,4 3,3 Chicken • Optimal social welfare = 6 15896 Spring 2016: Lecture 17 20

  21. Game of chicken • Correlated equilibrium: Dare Chicken (D,D): o � (D,C): 0,0 4,1 Dare o � � (C,D): o � � 1,4 3,3 Chicken (C,C): o � �� • Social welfare of CE = � 15896 Spring 2016: Lecture 17 21

  22. Implementation of CE • Instead of a mediator, use a hat! • Balls in hat are labeled with “chicken” or “dare”, each D D C C C blindfolded player takes a ball D C C Which balls implement the distribution of the previous slide? 15896 Spring 2016: Lecture 17 22

  23. CE vs. NE • Poll 2: What is the relation between CE and NE? CE NE 1. NE CE 2. NE CE 3. NE CE 4. 15896 Spring 2016: Lecture 17 23

  24. CE As LP • Can compute CE via linear programming in polynomial time! find � � � � � s.t. � , � � � � � � � , � � � � � � , � � � � � � � , � � � � �� � � � � � � ∈� � � ∈� � � � � � � � , � � � � � � , � � � � � � � , � � � � �� � , � � � � � � � ∈� � � ∈� � � � � , � � � 1 � � ,� � ∈� � � � � 15896 Spring 2016: Lecture 17 24

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