1. Introduction to Game Theory Xudong 2013/1/12

2. A: Playing Games B: Concepts & Theory C: Discussions

3. Game 1: Prisoners’ dilemma CooperateDefect Cooperate(2,2)(-1,3) Defect(3,-1)(0,0) Player II Player I The rational choice is defect The payoff of defect is consistently better No matter of the action of other players

4. Dominated strategy Definition – Suppose two strategies A and B – If the payoff of strategy A consistently smaller than the payoff of strategy B, no matter what strategies other players use – Then strategy A is dominated Example – Cooperation is a dominated strategy in prisoners’ dilemma game

5. Discussion Invisible hand v.s. Market failure – The wealth of the nation Adam Smith (1776) – Selfish motivation benefits the whole society Externality – Traffic jam in Beijing – Tragedy of commons – Pollution Negotiation – Transaction expense Coase, Ronald – The fable of the bee Cheung, S. N. (1973).

6. Definition of a game

7. John Nash There is at least an equilibrium in every finite game

8. Nash Equilibrium It is a state in a game, in which nobody himself will gain by making a deviation alone

9. Game 2: Collective Investment A profitable project for investors If over x% of the population invest, then each investor will get $10 Otherwise, the investor will lose his money

10. Discussion Two equilibriums in the investment – All of the players invest – None of the player invest Communication plays an important role Bank runs – Morgan’s role in saving the bank run in 1907 Getting out of Recession – Encouraging public speech – Fiscal policy and Monetary policy

11. Game 3: 捡石子 Background – Nim a more well-known name – It is said originated from china (wiki) Game description – Two players – Actions Two piles of stones with x and y stones in each pile Player can pick the any number of stones from ONLY one pile – The player who take the final stone wins

12. 捡石子 If the number of stones of the two piles are even, the first mover will lose Otherwise, the first mover will win Backward induction

13. Extensive-form Games Using a tree to represent a game 捡石子 as an example (1,1) (0,1) (1,0) (2,2) (0,2) (2,0) (1,2) (2,1) … … (2,0) (1,1) (0,1) (2,0) (0,0) (1,0) … … … … Player blue Player red Player blue wins Player red wins

14. … (2,0) (0,1) Backward induction for solving equilibrium (2,2) (0,2) (2,0) (1,2) (2,1) … … (1,1) … … … Cutting off the branches which lead to worse payoffs comparing with other braches Backwards: from the leafs to the root Example: For the red player at the (2,1) node the branches(dash line) leading to (2,0) or (0,1) are cut off because red player will win by choosing (1,1), otherwise he will lose

15. Subgame (2,3) (0,3) (2,0) (2,2) (0,2) (2,0) (1,2) (2,1) … … (2,0) (1,1) (0,1) … … … … … Subgame is – The subtree of original game tree – The information sets are self-contained

16. Game 4: Pirate Game 5 pirates (A,…,E) Actions – 100 Gold coins – Making proposal sequentially on how to distribute the gold coins – The proposal will pass if more than half of the pirates support it – Otherwise the proposer will be killed The pirates who are still alive will get the coins according to the approved proposal

17. Solving Pirate Game by Backward Induction Two players left – Payoff (E100, D-1) – E will take all gold and kill D, regardless of D’s decision Three players left – Payoff(E0,D0,C100) – D will agree, otherwise he will be killed in the next round after C is killed Four players left – Payoff(E1,D1,C0,B98) – E and D will agree, otherwise, their payoff will be 0 in the next round Five players left – Payoff(E2,D0,C1,B0,A97) – E and C will agree, because their payoff is better than the payoff next round

18. Discussion First Mover – Facebook v.s. Google+ – Google v.s. Bing Different solutions for query correction in Google and Bing Late Mover Advantage – The Chinese Miracle Lin, J. Y. et al (2003) – Tencent

19. Game 5: Rock, Paper and Scissors RockPaperScissors Rock(0,0)(-1,1)(1,-1) Paper(1,-1)(0,0)(-1,1) Scissors(-1,1)(1,1)(0,0) Equilibrium ?? NO Equilibrium ??

20. Mixed strategy Rock, Paper and Scissors – Determinate – Pure strategy Mixed strategy – indeterminate – Assigning probabilities to the pure strategies – e.g. (1/3 Rock ， 1/3 Paper ， 1/3 Scissors) is the Nash Equilibrium

21. Indifference Choices: A trick for solving equilibrium mixed strategy Suppose your opponent uses the equilibrium mixed strategy Your revenues under different pure strategies are identical Otherwise, your best response is pure strategy Abbreviating it as ICTrick

22. An example of using ICTrick Consider the game: Rock, Paper and Scissors Suppose the equilibrium mixed strategy of player 2 is (p1, p2,p3) RockPaperScissors Rock(0,0)(-1,1)(1,-1) Paper(1,-1)(0,0)(-1,1) Scissors(-1,1)(1,1)(0,0) Player 2 (p1, p2,p3) Player 1

23. An example of using ICTrick cont.

24. Game 6: Hawk and Dove HawkDove Hawk(V/2-C, V/2-C)(V,0) Dove(0,V)(V/2, V/2) A population Each individuals have two choices when they distribute the food with others – Hawk(aggressive, fight with others) – Dove(mild, equally sharing with others) The fraction of Hawk and Dove?

25. Equilibrium of Hawk and Dove Game Using the ICTrick, the equilibrium of mixed strategy – Hawk V/(2C) – Dove 1 – V/(2C) The probabilities of the mixed strategy – Not only represents the odds of the certain pure strategy taken by an individual – But also the proportion of certain population

26. Evolutionary Stable Strategy(ESS): Historical reviews Smith, J. M., & Price, G. R. The logic of animal conflict (1973) Richard Dawkins Selfish Gene (1976) Axelrod, R. The evolution of cooperation (1984)

27. ESS: Definition

28. New fitness & Original fitness

29. Evolutionary stable strategy criterions

30. Relationship with Nash Equilibrium

31. Discussions Revisiting ESS criterions – When E(S,S) > E(S,T), strategy S is equilibrium stable, even when E(T,T) > E(S,S) – Progressive reform fail to jump out the local minimum – Shock Therapy for the reform of Russia in 1990s Is the mixed strategy in Hawk and Dove Game a evolutionary stable strategy?

32. Game 7: Migration Two towns: East and West Town Strategies Two Types of people: Black and White Players Choose which town they will live (0.5, 1) (1, 0.5) payoff Proportion of same type people

33. Equilibrium Equilibrium I – All white in one town, all black in the other town – Racial segregation, even it is not the willingness of anybody Schelling, T. C. (1969) – Stable Equilibrium II – Half white and half black in the one town – Unstable, need additional force to achieve better equilibrium – Randomization improve the outcomes Schelling, T. C. (1969). Models of segregation. The American Economic Review, 59(2), 488-493.

34. Discussion The segregation is a stable equilibrium, even both the black and the white are willing to live together with each other Other scenarios? – Boy and girl segregation in high school?

35. Useful links http://gametheory101.com/ http://v.163.com/special/gametheory/ http://en.wikipedia.org/wiki/Game_theory