1. Social Networks 101 P ROF. J ASON H ARTLINE AND P ROF. N ICOLE I MMORLICA

2. Lecture Fifteen: Mixed Nash equilibria and price of anarchy.

3. Time for

4. How good is this? ( 8, 2 ) LeftRight Down Up ( 6, 3 )( 10, 1 ) ( 0, 6 ) 5/61/6 1/3 2/3 welfare of NE = 9.33 best social welfare = 11

5. How good is this? welfare of NE = 9.33 best social welfare = 11 = 0.85 Society gets only 85% of the max it could achieve.

6. The invisible hand In a free market, an individual pursuing his own self-interest tends to also promote the good of his community as a whole. Adam Smith (plus others like Ayn Rand)

7. Price of anarchy Capitalism can suck. Sometimes we need (governmental(?)) regulation to design rules with good social value. (happy May Day).

8. Next Multiple equilibria with multiple values. Which do we play?

9. Min income game (for two) ( 1, 1 ) ( 0, 1 )( 2, 2 ) ( 1, 0 ) ( 0, 1 ) ( 0, 2 ) ( 1, 0 ) ( 2, 0 ) ( 1, 0 ) ( 3, 3 ) ( 0, 3 )( 4, 4 ) ( 3, 0 ) one two three four onetwothreefour

10. Question What are the Nash equilibria of the min income game?

11. Min income game (for two) ( 1, 1 ) ( 0, 1 )( 2, 2 ) ( 1, 0 ) ( 0, 1 ) ( 0, 2 ) ( 1, 0 ) ( 2, 0 ) ( 1, 0 ) ( 3, 3 ) ( 0, 3 )( 4, 4 ) ( 3, 0 ) one two three four onetwothreefour

12. Question Which outcome do you like best?

13. Min income game ( 1, 1 ) ( 0, 1 )( 2, 2 ) ( 1, 0 ) ( 0, 1 ) ( 0, 2 ) ( 1, 0 ) ( 2, 0 ) ( 1, 0 ) ( 3, 3 ) ( 0, 3 )( 4, 4 ) ( 3, 0 ) one two three four onetwothreefour

14. Question Which outcome do you like best? Everyone likes the all-four outcome, it is the best possible outcome for society, and it’s an equilibrium in this game!

15. Min income game Experiment: In your envelope is a blank card. 1. On your card, write your name and an integer between one and four. 2. The people whose integer equals the minimum will receive a number of points equal to their guess.

16. The value of society Definition. The social welfare of an outcome is the value of that outcome for society (i.e., the total value to society of the outcome).

17. Min income game ( 1, 1 ) ( 0, 1 )( 2, 2 ) ( 1, 0 ) ( 0, 1 ) ( 0, 2 ) ( 1, 0 ) ( 2, 0 ) ( 1, 0 ) ( 3, 3 ) ( 0, 3 )( 4, 4 ) ( 3, 0 ) one two three four onetwothreefour Social welfare: 4 + 4 = 8 Social welfare: 2 + 0 = 2 Social welfare: 3 + 3 = 6 Social welfare: 2 + 2 = 4 Social welfare: 1 + 1 = 2

18. Min-income game has multiple Nash equilibria with varying social welfare. ( 1, 1 ) ( 2, 2 ) ( 3, 3 ) ( 4, 4 ) 8 6 4 2 (Also some mixed Nash)

19. Questions How much welfare will society get? What’s the worst welfare society may get?

20. Price of anarchy Definition. The price of anarchy is the ratio of value( worst Nash equilibrium ) value( best social welfare )

21. Min income game ( 1, 1 ) ( 0, 1 )( 2, 2 ) ( 1, 0 ) ( 0, 1 ) ( 0, 2 ) ( 1, 0 ) ( 2, 0 ) ( 1, 0 ) ( 3, 3 ) ( 0, 3 )( 4, 4 ) ( 3, 0 ) one two three four onetwothreefour worst Nash equilibrium best social outcome

22. Example: Min income game Price of anarchy value( worst Nash equilibrium ) value ( best social welfare ) value( all-one ) value( all-four ) 1 4

23. Min income game ( 1, 1 ) ( 0, 1 )( 2, 2 ) ( 1, 0 ) ( 0, 1 ) ( 0, 2 ) ( 1, 0 ) ( 2, 0 ) ( 1, 0 ) ( 3, 3 ) ( 0, 3 )( 4, 4 ) ( 3, 0 ) one two three four onetwothreefour best Nash equilibrium best social outcome

24. Equilibrium selection Which outcome will we see? In some games, certain equilibria are more likely than others. Sometimes this is a sub-optimal equilibrium (all-one in min-income).

25. Chicken Nash welfare ( 1, 1 ) StaySwerve Stay ( 2, 0 )( -1, -1 ) ( 0, 2 ) mixed NE: p (1) + (1-p) (0) = p (2) + (1-p) (-1)  p = 1/2 p (1-q) (1-p) q Social welfare: 2

26. Social welfare of mixed NE: ¼ (2) + ¼ (2) + ¼ (2) + ¼ (-2) = 1 Chicken Nash welfare StaySwerve Stay 1/2 ( 1, 1 ) ( 2, 0 )( -1, -1 ) ( 0, 2 ) Social welfare: 2

27. Chicken social welfare StaySwerve Stay 1/2 ( 1, 1 ) ( 2, 0 )( -1, -1 ) ( 0, 2 ) Best social welfare = 2

28. Chicken anarchy Price of anarchy value( worst Nash equilibrium ) value ( best social welfare ) value( (½, ½) ) value( (stay, swerve) ) 1 2

29. The price of anarchy of chicken is ½.

30. Can a central designer induce a good outcome? e.g., min-income with suggestions. e.g., chicken with stop-lights.

31. Next time Selfish routing.

32. Example: Penalty kicks ( 0.58, 0.42 ) LeftRight Left ( 0.93, 0.07 )( 0.70, 0.30 ) ( 0.95, 0.05 ) 39% 61% 42%58% No matter what outcome occurs, value is the same (one). Price of anarchy is one.