Social Networks 101

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

2. IMPORTANT ANNOUNCEMENT Lectures are MOVING to Pancoe Auditorium. (so we can accommodate more students, tell your friends to join the class!)

3. The investment game

4. The investment game Experiment: You may invest one of your points in the community. 1. In your envelope is a piece of paper. Write your name and whether you wish to invest or save on the paper. DO NOT SHOW ANYONE. 2. Put the paper in your envelope, pass it to the TAs. We will match the contributions at 50% (hence every invested point becomes 1.5 points) and then redistribute the points evenly among everyone.

5. Lecture Two: How do we play games?

6. What is a game? A set of players their possible strategies, and a function relating strategy choices to payoffs.

7. Normal-form games Hi, my name is Mr. Row. I’m Mrs. Column. Let’s play a game! two players

8. 2-player Investment Game Mr. Row and Mrs. Column each have 4 quarters to invest. Mrs. Column Mr. Row

9. 2-player Investment Game Column strategies Mrs. Column Invest Save Mr. Row

10. 2-player Investment Game Row strategies Mrs. Column Invest Save Invest Save Mr. Row

11. 2-player Investment Game Investments Mrs. Column Invest Save ( ? , ? ) Invest Save Mr. Row

12. 2-player Investment Game Returns Mrs. Column Invest Save ( ? , ? ) ( 3 , 7 ) Invest Save Mr. Row

13. 2-player Investment Game Payoff matrix. Mrs. Column Invest Save ( 6 , 6 ) ( 3 , 7 ) Invest ( 7 , 3 ) ( 4 , 4 ) Save Mr. Row

14. Game Theory Given a game, can we predict which strategies the players will play?

15. What should row do? If Column invests, I am better off not investing. If Column doesn’t invest, I am still better off not investing. I SHOULD NOT INVEST! Same here! Prediction: Players will end up not investing. Mrs. Column Invest Save ( 6 , 6 ) ( 3 , 7 ) Invest > > ( 7 , 3 ) ( 4 , 4 ) Save Mr. Row

16. Conclusion In Investment Game: best strategy is to save, ... no matter what other player does. This is a dominant strategy equilibrium.

17. Conclusion In Investment Game: best strategy is to save, ... no matter what other player does, ... even though it is highly sub-optimal!

18. Social Optimum Social optimum: Each player gets 2 more quarters than in equilibrium! Mrs. Column Invest Save ( 6 , 6 ) ( 3 , 7 ) Invest ( 7 , 3 ) ( 4 , 4 ) Save Mr. Row

19. Price of anarchy How societal value much is lost due to lack of coordination? Mrs. Column Total val. in equil.: 8q Total val. in soc. opt.: 12q PoA: 2/3 Invest Save ( 6 , 6 ) ( 3 , 7 ) Invest ( 7 , 3 ) ( 4 , 4 ) Save Mr. Row

20. What did we do? Results of our investment game.

21. Dominant strategies Dominant Strategy Equilibrium: Each player’s strategy is her best choice no matter what her opponent does. What do you think of this prediction?

22. John Nash

23. Movie Time

24. The dating game Mrs. Column A blonde and two brunettes are sitting in the computer lab … Blonde Brunettes ( 0 , 0 ) ( 2 , 1 ) Blonde ( 1 , 2 ) ( 1 , 1 ) Brunettes Mr. Row

25. The dating game If Column goes for the blonde, Row is better off going for the brunette. Mrs. Column Blonde Brunettes ( 0 , 0 ) ( 2 , 1 ) Blonde ( 1 , 2 ) ( 1 , 1 ) Brunettes Mr. Row

26. The dating game But if Column goes for the brunette, Row definitely wants to go for the blonde. Mrs. Column Blonde Brunettes ( 0 , 0 ) ( 2 , 1 ) Blonde ( 1 , 2 ) ( 1 , 1 ) Brunettes Mr. Row

27. The dating game There is no dominant strategy equilibrium! Mrs. Column Blonde Brunettes ( 0 , 0 ) ( 2 , 1 ) Blonde ( 1 , 2 ) ( 1 , 1 ) Brunettes Mr. Row

28. The dating game Mrs. Column Blonde Brunettes ( 0 , 0 ) ( 2 , 1 ) Blonde ( 1 , 2 ) ( 1 , 1 ) Brunettes Mr. Row

29. How to play the dating game? What did you do?

30. How to play the dating game? If you think the competition is going to go for the blonde, then go for the brunettes. …but if you think the competition will go for the brunettes, hit on the blonde!

31. Nash equilibria Each person is playing a mutual best-response. This is a Nash equilibrium.

32. The dating game Equilibria of the dating game Mrs. Column Are there any other equilibria? Blonde Brunette ( 0 , 0 ) ( 2 , 1 ) Blonde ( 1 , 2 ) ( 1 , 1 ) Brunette Mr. Row

33. Time for Math Corner

34. The dating game Mixed Nash equilibria: Players choose strategies probabilistically. Mrs. Column q (1-q) Blonde Brunette ( 0 , 0 ) ( 2 , 1 ) Blonde (1-p) ( 1 , 2 ) ( 1 , 1 ) p Brunette Mr. Row

35. The dating game Observation: For Row to play both strategies, payoff must be equal. Mrs. Column q 1/2 (1-q) 1/2 Blonde Brunette ( 0 , 0 ) ( 2 , 1 ) Blonde (1-p) ( 1 , 2 ) ( 1 , 1 ) p Brunette Mr. Row

36. The dating game Observation: For Column to play both strategies, payoff must be equal. Mrs. Column 1/2 1/2 Blonde Brunettes ( 0 , 0 ) ( 2 , 1 ) 1/2 Blonde ( 1 , 2 ) ( 1 , 1 ) 1/2 Brunettes Mr. Row

37. The dating game Mixed equilibrium: Each player flips a fair coin to decide whether to chat up the blonde or the brunettes. Mrs. Column 1/2 1/2 Blonde Brunettes ( 0 , 0 ) ( 2 , 1 ) 1/2 Blonde ( 1 , 2 ) ( 1 , 1 ) 1/2 Brunettes Mr. Row

38. Nash Equilibria Nash Equilibrium: Each player’s strategy is a best-response to the strategies of his opponents. (“mixed” if playing probabilistically, else “pure”)

39. Nash Equilibria What do you think of this prediction?

40. Objection to Nash equilibria There may be many Nash equilibria.

41. IMPORTANT ANNOUNCEMENT Lectures are MOVING to Pancoe Auditorium. (so we can accommodate more students, tell your friends to join the class!)

42. Next time markets * * in Pancoe Auditorium.