Social Networks 101

1. Social Networks 101 Prof. Jason Hartline and Prof. Nicole Immorlica

2. Lecture Thirteen: Normal form games and equilibria notions.

3. Let’s play a game Experiment: The median game. 1. Guess an integer between 1 and 100, inclusive. 2. Write your number and name on your card. P R I Z E : The people whose numbers are closest to 2/3 of the median win 5 points.

4. The Median Game 25 45 0 50 69 Alok Brent Casey Dirk Ela Calculating the winner: 1. Sort the numbers: 0, 25, 45, 50, 69 2. Pick the middle one (the median): 45 3. Compute 2/3 of the median: 30

5. The Median Game Median is 45, and Alok wins because his guess is closest to 2/3 of the median, or 30. 25 45 0 50 69 Alok Brent Casey Dirk Ela

6. How did you play?

7. Reasoning in games Imagine what everyone else will do, decide how to act based on that assumption.

8. Bi-matrix games Example: prisoners’ dilemma Mrs. Column Confess Deny ( -4 , -4 ) ( 0 , -10 ) Confess ( -10 , 0 ) ( -1 , -1 ) Deny Mr. Row

9. Prisoners’ dilemma Q. If Row confesses, what should Column do? Mrs. Column Confess Deny ( -4 , -4 ) ( 0 , -10 ) Confess ( -10 , 0 ) ( -1 , -1 ) Deny Mr. Row

10. Prisoners’ dilemma Q. If Row denies, what should Column do? Mrs. Column Confess Deny ( -4 , -4 ) ( 0 , -10 ) Confess ( -10 , 0 ) ( -1 , -1 ) Deny Mr. Row

11. Dominant strategies Row’s best-response was Confess no matter what Column did. Confess is a dominant strategy for row.

12. Normal form games Definition. A normal form game for a set N of n players is described by 1. A set of strategies Si for each player i. 2. A payoff function ¼i for each player i and each profile of strategies (s1, …, sn) indicating player i’s reward for every strategy profile.

13. Best responses Definition. A strategy si* is a best-response to strategies sj of players i ≠ j if ¼(s1, …, si*, …, sn) ¸¼(s1, …, si, …, sn) for all strategies si in Si.

14. Dominant strategies Definition. A strategy si is a dominant strategy for player i if it is a best-response to all strategy profiles of the other players.

15. Finding dominant strategies To find a dominant strategy for a row player, compare vectors of payoffs in each row. If (and only if) some row vector dominates coordinate-wise, it is a dominant strategy for the row player.

16. Prisoners’ dilemma Q. Is there a dominant strategy? Mrs. Column Confess Deny ( -4 , -4 ) ( 0 , -10 ) Confess ( -10 , 0 ) ( -1 , -1 ) Deny Mr. Row

17. Dominant strategy equilibria Definition. A strategy profile (s1, …, sn) is a dominant strategy equilibrium if, for each player i, si is a dominant strategy.

18. Another game Q. Is there a dominant strategy? Mrs. Column High Low ( 2 , 2 ) ( 0 , 3 ) High ( 3 , 2 ) ( 5 , 1 ) Low Mr. Row

19. Nash equilibrium Definition: A strategy profile (s1, …, sn) is a Nash equilibrium (NE)if for each player i, si is a best-response to strategies sj of players j ≠ i.

20. Chicken

21. Chicken Q. Is there a Nash equilibrium? Mrs. Column Swerve Stay ( 1 , 1 ) ( 0 , 2 ) Swerve ( 2 , 0 ) ( -1 , -1 ) Stay Mr. Row

22. Finding Nash equilibria Method: Best-response (directed) graph 1. For each strategy profile s create a node su. 2. Connect node su to node sv if for some player i, his strategy sviin v is a best response to the other players’ strategies in u and for all other players j, suj= svj. 3. Search for a node with no out-going links.

23. Chicken Swerve Stay ( 1 , 1 ) ( 0 , 2 ) Swerve (swerve, swerve) ( 2 , 0 ) ( -1 , -1 ) Stay (swerve, stay) (stay, swerve) (stay, stay)

24. Chicken Q. Is there a Nash equilibrium? Mrs. Column Swerve Stay ( 1 , 1 ) ( 0 , 2 ) Swerve ( 2 , 0 ) ( -1 , -1 ) Stay Mr. Row

25. Matching pennies Q. Is there a Nash equilibrium? Mrs. Column Heads Tails ( -1 , 1 ) ( 1 , -1 ) Heads ( 1 , -1 ) ( -1 , 1 ) Tails Mr. Row

26. Matching pennies Heads Tails ( -1 , 1 ) ( 1 , -1 ) Heads (heads, heads) ( 1 , -1 ) ( -1 , 1 ) Tails (heads, tails) (tails, heads) (tails, tails)

27. Next time Mixed Nash equilibria and fixed points.