Computational Game Theory Amos Fiat Modified Slides prepared for Yishay Mansour’s class

1. Computational Game TheoryAmos FiatModified Slides prepared for YishayMansour’s class Lecture 1 - Introduction

2. Agenda • Introduction to Game Theory • Examples • Matrix form Games • Utility • Solution concepts • Dominant Strategies • Nash Equilibria • Complexity • Mechanism Design: reverse game theory

3. Computational Game Theory • The study of Game Theory in the context of Computer Science, in order to reason about problems from the perspective of computability and algorithm design.

4. CGT in Computer Science • Computing involves many different selfish entities. Thus involves game theory. • The Internet, Intranet, etc. • Many players (end-users, ISVs, Infrastructure Providers) • Players wish to maximize their own benefit and act accordingly • The trick is to design a system where it’s beneficial for the player to follow the rules

5. CGT in Computer Science • Theory • Algorithm design • Complexity • Quality of game states (Equilibrium states in particular) • Study of dynamics • Industry • Sponsored search • Other auctions

6. Game Theory • Rational Player • Prioritizes possible actions according to utility or cost • Strives to maximize utility or to minimize cost • Competitive Environment • More than one player at the same time Game Theory analyzes how rational players behave in competitive environments

7. The Prisoner’s Dilema • Matrix representation of the game 2 < 3 5 < 6 Row Player Column Player

8. The Prisoner’s Dilema • It is a dominant strategy to confess • A dominant strategy is a “solution concept” 6,10

9. ISP Routing • Internet Service Providers (ISP) often share their physical networks for free • In some cases an ISP can either choose to route traffic in its own network or via a partner network

10. ISP Routing • ISP 1 needs to route traffic from s1 to t1 • ISP 2 needs to route traffic from s2 to t2 • The cost of routing along each edge is one A B

11. ISP Routing • ISP1 routes via B: • Cost for ISP1: 1 • Cost for ISP2: 4

12. ISP Routing B,A: s1 to t1 B,A: s2 to t2 • Cost matrix for the game: ISP 2 ISP 1 Prisoners Dilemma Again

13. Strategic Games • The game consists of only one ‘turn’ • All the players play simultaneously and are unaware of what the other players do • Players are selfish, seek to maximize their own benefit

14. Strategic Games – Formal Model • N = {1,…,n} players • Player i has actions We will say “action” or “strategy” • The space of all possible action vectors is • A joint action is the vector a∈A • Player i has a utility function If utility is negative we may call it cost

15. Strategic Games – Formal Model • A strategic game: Utility of each player Players Actions of each player

16. Dominant Strategies • Action ai of player i is a weakly dominant strategy if: • Action ai of player i is a strongly dominant strategy if:

17. Pareto Optimality • An outcome a of a game is Pareto optimal if for every other outcome b, some player will lose by changing to b Vilfredo Pareto

18. Bernulli Utility St. Petersburg Paradox: • Toss a coin until tails, I pay you • What will you pay me to play? “Utility of Money”, “Bernulli Utility”

19. Von Neumann–Morgenstern Rationality Axioms (1944) Preferences over lotteries Completeness: Transitivity: Continuity: Independence:

20. Rationality Axioms Utility function over lotteries, real valued, expected utility maximization

21. Allias Paradox (1953) Gamble A or B? Gamble A: 100% € 1,000,000 Gamble B: 10% € 5,000,000 89% € 1,000,000 1% Nothing Gamble C: 11% € 1,000,000 89% Nothing Gamble D: 10% € 5,000,000 90% Nothing Experimental ”Fact”: Gamble C or D? Experimental “Fact”:

22. Allias Paradox Gamble A: 100% € 1,000,000 Gamble B: 10% € 5,000,000 89% € 1,000,000 1% Nothing Gamble C: 11% € 1,000,000 89% Nothing Gamble D: 10% € 5,000,000 90% Nothing “Fact”: “Fact”:

23. Expected Utility Theory VNM Axioms Expected Utility Maximization Mixed Nash Equilibrium exists

24. Tragedy of the commons • Assume there’s a shared resource (network bandwidth) and N players. • Each player “uses” the common resource, by choosing Xifrom [0,1]. If Otherwise,

25. Tragedy of the commons Given that the other players are fixed, what Is the best response?

26. Tragedy of the commons This is an equilibrium No player can improve

27. Tragedy of the commons The case for Privatization or central control of commons

28. Nash Equilibrium • A Nash Equilibrium is an outcome of the game in which no player can improve its utility alone: • Alternative definition: every player’s action is a best response:

29. Battle of the Sexes • The payoff matrix:

30. Battle of the Sexes • The payoff matrix: Row player has no incentive to move up

31. Battle of the Sexes • The payoff matrix: Column player has no incentive to move left

32. Battle of the Sexes • The payoff matrix: So this is an Equilibrium state

33. Battle of the Sexes • The payoff matrix: Same thing here

34. Routing Game • 2 players need to send a packet from point O to the network. • They can send it via A (costs 1) or B (costs 2)

35. Routing Game • The cost matrix:

36. Routing Game • The cost matrix: Equilibrium states

37. Matching Pennies • 2 players, each chooses Head or Tail • Row player wins if they match the column player wins if they don’t • Utility matrix:

38. Matching Pennies • 2 players, each chooses Head or Tail • Row player wins if they match the column player wins if they don’t • Utility matrix: Row player is fine, but Column player wants to move left

39. Matching Pennies • 2 players, each chooses Head or Tail • Row player wins if they match the column player wins if they don’t • Utility matrix: Column player is fine, but Row player wants to move up

40. Matching Pennies • 2 players, each chooses Head or Tail • Row player wins if they match the column player wins if they don’t • Utility matrix: Row player is fine, but Column player wants to move right

41. Matching Pennies • 2 players, each chooses Head or Tail • Row player wins if they match the column player wins if they don’t • Utility matrix: Column player is fine, but Row player wants to move down

42. Matching Pennies • 2 players, each chooses Head or Tail • Row player wins if they match the column player wins if they don’t • Utility matrix: No equilibrium state!

43. Mixed Strategies • Players do not choose a pure strategy (one specific strategy) • Players choose a distribution over their possible pure strategies • For example: with probability p choose Heads, and with probability 1-p choose Tails

44. Matching Pennies • Row player chooses Heads with probability p and Tails with probability 1-p • Column player chooses Heads with probability q and Tails with probability 1-q • Row plays Heads: • Row plays Tails:

45. Mixed Strategy • Each player selects where is the set of all possible distributions over Ai • An outcome of the game is the Joint Mixed Strategy • An outcome of the game is a Mixed Nash Equilibrium if for every player

46. Mixed Strategy • 2nd definition of Mixed Nash Equilibrium: • Definition: • Definition: • Property of Mixed Nash Equilibrium:

47. Rock Paper Scissors • No pure strategy Nash Equilibrium, only Mixed Nash Equilibrium, for mixed strategy (1/3, 1/3, 1/3) .

48. Location Game • N ice cream vendors are spread on the beach • Assume that the beach is the line [0,1] • Each vendor chooses a location Xi, which affects its utility (sales volume). • The utility for player i : X0 = 0, Xn+1 = 1

49. Location Game • For N=2 we have a pure Nash Equilibrium: No player wants to move since it will lose space • For N=3 no pure Nash Equilibrium: The player in the middle always wants to move to improve its utility 1 0 1/2 1 0 1/2

50. Location Game • If instead of a line we will assume a circle, we will always have a pure Nash Equilibrium where every player is evenly distanced from each other: