1. Chapter 12 Game Theory Presented by Nahakpam 1213504 PhD Student 1Game Theory

2. Brief Contents Game Theory Strong or weak pig Prisoner’s Dilemma System for studying strategic behavior – Positive – Normative Pareto Optima Sequential Games 2Game Theory

3. What is Game theory? Game theory is a study of strategic decision making. The study of how people behave in strategic situations. Strategic decisions are those in which each person, in deciding what actions to take, must consider how others might respond to that action. 3Game Theory

4. Game Theory and Economics If the market is composed by a small number of firms, each firm must act strategically. Each firm affects the market price changing the quantity produced. Suppose 2 firms are producing 100 units. – If one of the firms decides to increase the production by 10 units. – The market supply will increase from 200 to 210 and the price has to drop to reach an equilibrium. Therefore, it also affects the profits of other firms. Each firm knows that its profit depends not only on how much it produced but also on how much the other firms produce. 4Game Theory

5. What is a Game? A game is a situation where the participants’ payoffs depend not only on their decisions, but also on their rivals’ decisions. This is called Strategic Interactions: – My optimal decisions will depend on what others do in the game. 5Game Theory

6. A Game Four elements to describe a game: – players; – rules: when each player moves, what are the possible moves, what is known to each player before moving; – outcomes of the moves; – payoffs of each possible outcome: how much money each player receive for any specific outcome. 6Game Theory

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8. Boeing-Airbus game Boeing and Airbus have to decide whether to invest in the development of a Super Jumbo for long distance travel; If they both develop successfully the new plane, their profits will drop by 50 millions a year; If only one develop the Super Jumbo, it will make 80 millions a year in additional profits, whereas the profits of the other firm will drop by 30 millions a year; If no firm develops the plane, nothing changes. 8Game Theory

9. Matrix Representation of Boeing- Airbus game Airbus Develop Do not develop Boeing Develop-50,-5080,-30 Do not develop -30,800,0 9Game Theory

10. Solutions of the Games To predict what will be the solution/outcome of the game we need some tools: – dominated and dominant strategies; – Nash equilibrium. 10Game Theory

11. 11Game Theory Prisoners Dilemma

12. Tom and Mary have been arrested for possession of guns. The police suspects that they have committed 10 bank robberies. If nobody confesses the police, they will be jailed for 2 years. If only one confesses, she’ll go free and her partner will be jailed for 40 years. If they both confess, they get 16 years. 12Game Theory

13. Matrix Representation of Prisoners Dilemma Mary Confess Do not Confess Tom Confess16,160,40 Do not Confess 40,02,2 13Game Theory

14. We want to predict the outcome of the game Mary Confess Do not Confess Tom Confess16,160,40 Do not Confess 40,02,2 Suppose that Tom decides to confess. What is the best decision for Mary? 14Game Theory

15. We want to predict the outcome of the game Mary Confess Do not Confess Tom Confess16,160,40 Do not Confess 40,02,2 Suppose that Tom decides to remain silent. What is the best decision for Mary? 15Game Theory

16. Dominated and Dominant Strategy Dominant Strategy: – a strategy that gives higher payoffs no matter what the opponent does; Dominated Strategy: – a strategy is dominated if there exists another strategy that is dominant. So far we have only assumed that each player is rational to determine the outcome of the game. 16Game Theory

17. We want to predict the outcome of the game Mary Confess Do not Confess Tom Confess16,160,40 Do not Confess 40,02,2 Suppose that Mary decides to confess. What is the best decision for Tom? 17Game Theory

18. We want to predict the outcome of the game Mary Confess Do not Confess Tom Confess16,160,40 Do not Confess 40,02,2 Suppose that Mary decides to remain silent. What is the best decision for Tom? 18Game Theory

19. Outcome of the Game Mary Confess Do not Confess Tom Confess16,160,40 Do not Confess 40,02,2 19Game Theory

20. No Dominant Strategies In most games there are no dominant strategies for all players. We cannot use this method to predict the outcome of the game. 20Game Theory

21. Nash Equilibrium The decisions of the players are a Nash Equilibrium if no individual prefers a different choice. In other words, each player is choosing the best strategy, given the strategies chosen by the other players. 21Game Theory

22. Nash equilibrium 22 Each player’s predicted strategy is the best response to the predicted strategies of other players No incentive to deviate unilaterally Strategically stable or self-enforcing Game Theory

23. Nash’s Theorem 23 Existence – Any finite game will have at least one Nash equilibrium possibly involving mixed strategies Finding a Nash equilibrium is not easy – Not efficient from an algorithmic point of view Game Theory

24. “Battle of Sexes” or “Bach or Stravinsky” 24 A couple deciding how to spend the evening Wife would like to go for a opera Husband would like to go for a boxing match Both however want to spend the time together Scope for strategic interaction No means of communication Game Theory

25. Battle of Sexes 25 Normal Form representation – Payoff Matrix Game Theory

26. The Copycat Game 26 Dot is happy as long as she is alone; Ditto is happy as long as he is with Dot. There is no Nash equilibrium in this game. Game Theory

27. Choosing Strategies Player choosing columns or rows Adjusting choice Nash equilibrium – No deviation from choice by either player forms outcome – Take other player’s behavior as given Any outcome that survives this process of elimination is called a Nash equilibrium outcome. An outcome is a Nash equilibrium if neither player would want to deviate from it, taking his opponent’s behavior as given. 27Game Theory

28. Mixed Strategies in Sports In the international tournaments organized by the World Rock Paper Scissors Society The best players are the least predictable players. In Nash equilibrium, everyone plays a mixed strategy—1/3 “Rock,” 1/3 “Paper,” and 1/3 “Scissors.” 28Game Theory

29. Pareto Optima An outcome is Pareto-optimal if nothing sits above it in the tree. The tree shows that outcomes A and D are Pareto-preferred to C and B, and C is Pareto-preferred to B. A and D are Pareto optima, because nothing sits above them in the tree. 29Game Theory

30. Sequential Games Simultaneous games Second player advantage Oligopoly problem – Stackelberg equilibrium Player commits to strategy Importance of commitment 30Game Theory

31. Dynamic games 31 Sequential moves – One player moves – Second player observes and then moves Examples – Industrial Organization – a new entering firm in the market versus an incumbent firm; a leader-follower game in quantity competition – Sequential bargaining game - two players bargain over the division of a pie of size 1 ; the players alternate in making offers – Game Tree Game Theory

32. 32Game Theory The only Nash equilibrium is in the center square, where Kodak and Fuji each earn profits of $15. But if the game is played sequentially and Kodak moves first, then Kodak announces a policy of producing 100 rolls of film. Fuji’s best response is to produce 50, leading to the upper right-hand square. Cournot equilibrium: A Nash equilibrium in a game where each company chooses its quantity.

33. Economic applications of game theory The study of oligopolies (industries containing only a few firms) The study of cartels, e.g., OPEC The study of externalities, e.g., using a common resource such as a fishery The study of military strategies The study of international negotiations Bargaining 33Game Theory

34. Summary Game Theory34 Strategic situations can be represented by game matrices, – showing the outcome that results from each combination of strategies that the players can choose. A Nash equilibrium is an outcome from which neither player would deviate, taking the other’s behavior as given. A game can have one Nash equilibrium, no Nash equilibrium, or many Nash equilibria.

35. Summary Game Theory35 A dominant strategy is a strategy that a player would want to adopt regardless of his beliefs about the other player’s strategy choice. The Prisoner’s Dilemma is an example of a game where both players have dominant strategies.

36. Thank you Game Theory36