BUS 525: Managerial Economics Lecture 10 Game Theory: Inside Oligopoly

1. BUS 525: Managerial Economics Lecture 10 Game Theory: Inside Oligopoly

2. Overview 10-2 I. Introduction to Game Theory II. Simultaneous-Move, One-Shot Games III. Infinitely Repeated Games IV. Finitely Repeated Games V. Multistage Games

3. Game Environments 10-3 • Players are individuals who make decisions • Players’ planned decisions are called strategies. • Payoffs to players are the profits or losses resulting from strategies. • Order of play is important: • Simultaneous-move game: each player makes decisions without knowledge of other players’ decisions. • Sequential-move game: one player observes its rival’s move prior to selecting a strategy. • Frequency of rival interaction • One-shot game: game is played once. • Repeated game: game is played more than once; either a finite or infinite number of interactions.

4. Simultaneous-Move, One-Shot Games: Normal Form Game 10-4 • A Normal Form Game consists of: • Set of players i= {1, 2, …n} where n is a finite number. • Each players strategy set or feasible actions consist of a finite number of strategies. • Player 1’s strategies are S1={a, b, c, …}. • Player 2’s strategies are S2 ={A, B, C, …}. • Payoffs, to the players are the profits or losses that results from the strategies. • Due to the interdependence, payoff to a player depends not only the player’s strategy but also on the strategies employed by other players • Player 1’s payoff: π1(a,B) = ? • Player 2’s payoff: π2(b,C) = ?

5. 11,10 10,11 12,12 10,15 10,13 13,14 Normal Form Game 10-5 Normal form game: A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies In game theory, Strategy, is a decision rule that describes the actions a player will take at each decision point Player 2 12,11 11,12 14,13 Player 1

6. A Normal Form Game 10-6 Player B Player A

7. Dominant Strategy 10-7 A strategy that results in highest payoff to a player regardless of opponents action Player B Player A Principle: Check to see if you have a dominant strategy. If you have one, play it.

8. Dominant Strategy in a Simultaneous-Move, One-Shot Game 10-8 • A dominant strategy is a strategy resulting in the highest payoff regardless of the opponent’s action. • If “Up” is a dominant strategy for Player A in the previous game, then: • πA(Up,Left) ≥ πA(Down,Left); • πA(Up,Right) ≥ πA(Down,Right).

9. Secure Strategy 10-9 Does B have a dominant strategy? Play secure strategy, i.e. A strategy that guarantees the highest payoff under the worst possible scenario. i.e choose Right; 8>7 Could you do better? Yes, choose left. Because A will choose Up, A’s dominant strategy; 20>8 Player B Player A Principle: Put yourself in rivals shoes. If your rival has a dominant strategy, anticipate that he or she will play it.

10. Nash Equilibrium 10-10 The Nash equilibrium is a condition in which no player can improve her payoff by unilaterally changing her own strategy, given the other players’ strategy Suppose A chooses Up, B chooses Left. Either player will have no incentive to change his or her strategy. Given that A’s strategy is Up, the best player B can do is to choose Left. Given that B’s strategy is Left, the best player A can do is to choose Up. The Nash equilibrium strategy for player A is Up, and for player B it is Left. Nash Equilibrium Player B Player A

11. Key Insights 10-11 • Look for dominant strategies. • Put yourself in your rival’s shoes.

12. A Pricing Game 10-12 • Two managers want to maximize market share: i= {1,2}. • Strategies are pricing decisions • SA = {Low Price, High Price}. • SB = {Low Price, High Price}. • Simultaneous moves. • One-shot game.

13. A Pricing Game 10-13 Firm B Firm A • Is there Nash equilibrium? • Is Nash equilibrium the best? • Cheating

14. An Advertising game 10-14 Firm B Firm A Dominant strategy for each firm is to advertise. Nash equilibrium for the game is to advertise

15. Key Insight: 10-15 • Game theory can also be used to analyze situations where “payoffs” are non monetary! • We will, without loss of generality, focus on environments where businesses want to maximize profits. • Hence, payoffs are measured in monetary units.

16. Coordination Games 10-16 • In many games, players have competing objectives: One firm gains at the expense of its rivals. • However, some games result in higher profits by each firm when they “coordinate” decisions.

17. A Coordination Game 10-17 Firm B Firm A The game has two Nash equilibria. Better outcome by coordination. If firms produce different outlets, lower demand as consumers needs to spend more money in wiring their houses

18. Key Insights: 10-18 • Not all games are games of conflict. • Communication can help solve coordination problems. • Sequential moves can help solve coordination problems.

19. Games With No Pure Strategy Nash Equilibrium 10-19 Worker Manager Suppose Managers strategy is to monitor the worker. Then the best choice for the worker is to work. But if the worker works, the best strategy for the manager not to monitor. Therefore, monitoring is not a Nash equilibrium strategy. Not monitoring is also not a Nash equilibrium strategy. Because, if manager’s strategy is “don’t monitor;” The worker would choose shirking. Manager should monitor.

20. Strategies for Games With No Pure Strategy Nash Equilibrium 10-20 • In games where no pure strategy Nash equilibrium exists, players find it in there interest to engage in mixed (randomized) strategies. • This means players will “randomly” select strategies from all available strategies.

21. A Nash Bargaining Game 10-21 Union Management How much of a $100 surplus is to be given to the union?. The parties write the amount (either 0,50 or100). If the sum exceeds 100, the bargaining ends in a stalemate. Cost of delay -$1 both for the union and the management. Three Nash equilibrium to this bargaining game(100,0), (50,50), and (0,100). Six of the nine potential payoffs are inefficient as they result in total payoff lesser than amount to be divided. Three negative payoffs. Settle for 50-50 split which is fair, look at history

22. Lessons in Simultaneous Bargaining 10-22 • Simultaneous-move bargaining results in a coordination problem. • Experiments suggests that, in the absence of any “history,” real players typically coordinate on the “fair outcome.” • When there is a “bargaining history,” other outcomes may prevail.

23. Infinitely Repeated Games 10-23 • Infinitely Repeated Games • A game that is played over and over again forever and in which players receive payoffs during each play of the game • Recall key aspects of present value analysis • Value of the firm for T period with different payoffs at different periods • Value of the firm for ∞ period with same payoffs at different periods • Trigger strategy • A strategy that is contingent on the past play of a game and in which some particular past action “triggers” a different action by a player

24. A Pricing Game that is Repeated 10-24 Firm B Firm A Nash equilibrium in one shot play of this game is for each firm to charge low prices and earn zero profits Impact of repeated play on the equilibrium outcome of the game Trigger strategy: ”Cooperate provided no player has ever cheated in the past. If any players cheats, punish the player by choosing one shot Nash equilibrium strategy forever after.”

25. A Pricing Game that is Repeated 10-25 • The benefit of cheating today on the collusive agreement is earning $50 instead of $10 today. The cost of cheating today is earning $0 instead of $10 in each future period. If the present value of cost of cheating exceeds one time benefit of cheating, it does not pay for a firm to cheat, and high prices can be sustained. • In the example, if the interest rate is less than 25 percent both firms A and B will loose more (in present value) from cheating. That’s why we see collusion in oligopoly. • Condition for sustaining cooperative outcome with trigger strategies (ΠCheat- πCoop)/(πCoop- πN)≤ 1/i or ΠCheat- πCoop ≤ 1/i(πCoop- πN) • The LHS represents the one time gain from cheating. The RHS represents the present value of what is given up in the future by cheating today. In one-shot games there is no tomorrow, and threats have no bite. When interest rate is low firms may find it in their interest to collude and charge high prices

26. Factor Affecting Collusion in Pricing Games 10-26 To sustain collusive arrangements via punishment strategies firms must know: (i) who the rivals are; (ii) who the rivals customers are; (iii) When the rivals deviate from collusive arrangement; and (iv) must be able to successfully punish rivals for cheating. These factors are related to: (1) Number of firms; (2) Firm size; (3) History of the market; (4) Punishment mechanism

27. Finitely Repeated Games 10-27 • Finitely Repeated Games • Games in which players do notknow when the game will end • Games in which players know when the game will end • Games with an uncertain final period • When the game is repeated a finite but uncertain times, the benefits of cooperating look exactly the same as benefits from cooperating in an infinitely repeated game except that 1-θ plays the role of 1/(1+i); PLAYERS DISCOUNT THEIR FUTURE NOT BECAUSE OF THE INTEREST RATE BUT BECAUSE THEY ARE NOT CERTAIN FUTURE PLAYS WILL OCCUR • In our example Firm A has no incentive to cheat if ΠFirm ACheat=50 ≤10/θ = ΠFirm ACoperate, which is true if θ ≤1/5, probability the game will end is less than 20 percent, If θ=1, one shot game, dominant strategy for both firm is to charge the low price

28. Repeated Games with a Known Final period 10-28 • Provided the players know when the game will end, the game has only one Nash equilibrium. • Same outcome as one shot game • Trigger strategies would not work • Rival cannot be punished in the last period • Trigger strategies wont work • Backward unraveling. • End of period problem • Workers work hard due to threat of firing • Announces plans to quit, the cost of shirking is reduced • Manager’s response, fire immediately after announcements • Worker responds by telling you at the very end • What to do? Incentives, gratuity Same outcome as one shot game

29. Multistage Games 10-29 • Multistage framework permits players to make sequential rather than simultaneous decisions • Differs from one-shot and infinitely repeated games • Extensive-form game A representation of a game that summarizes the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payoff resulting from alternative strategies

30. Pricing to Prevent Entry: An Application of Game Theory 10-30 • Two firms: an incumbent and potential entrant. • Potential entrant’s strategies: • Enter. • Stay Out. • Incumbent’s strategies: • {if enter, play hard}. • {if enter, play soft}. • {if stay out, play hard}. • {if stay out, play soft}. • Move Sequence: • Entrant moves first. Incumbent observes entrant’s action and selects an action.

31. The Pricing to Prevent Entry Game in Extensive Form 10-31 -1, 1 Price war B (Incumbent) Enter Share 5, 5 A(Entrant) Out 0, 10

32. Identify Nash and Subgame Perfect Equilibria 10-32 -1, 1 Price war B (Incumbent) Enter Share 5, 5 A (Entrant) Out 0, 10

33. Two Nash Equilibria 10-33 -1, 1 Price war B (Incumbent) Enter Share 5, 5 A (Entrant) Out 0, 10

34. One Subgame Perfect Equilibrium 10-34 -1, 1 Price war B (Incumbent) Enter Share 5, 5 A (Entrant) Out 0, 10 Subgame Perfect Equilibrium Strategy: {enter; If enter, Share market}

35. Insights 10-35 • Establishing a reputation for being unkind to entrants can enhance long-term profits. • It is costly to do so in the short-term, so much so that it isn’t optimal to do so in a one-shot game.

36. An Innovation Game: An Application of Game Theory 10-36 • Class Exercise • Two firms: Innovator and Rival • Innovator’s strategies: • Introduce. • Don’t introduce. • Rival’s strategies: • {if introduced, clone}. • {if introduced, don’t clone}. • Move Sequence: • Innovator moves first. Rival observes innovator’s action and selects an action.

37. Class Exercise • If you do not introduce the new product, you and your rival will earn $1 million each • If you introduce the new product • Rival clones : you will loose $ 5 million and your rival will earn $20million • Rival does not clone: you will make $100 million and your rival will make $0 million • Set up the extensive form of the game • Should you introduce the new product? • How would your answer change if your rival has promised not to clone the product? • What would you do if patent law prevented your rival from cloning the product?

38. An Innovation Game: An Application of Game Theory 10-38 -5,20 Clone B (Rival) Introduce Don’t clone 100, 0 A (Innovator) Don’t Introduce 1, 1

39. Simultaneous-Move Bargaining 10-39 • Management and a union are negotiating a wage increase. • Strategies are wage offers & wage demands. • Successful negotiations lead to $600 million in surplus, which must be split among the parties. • Failure to reach an agreement results in a loss to the firm of $100 million and a union loss of $3 million. • Simultaneous moves, and time permits only one-shot at making a deal.

40. Fairness: The “Natural” Focal Point 10-40 Union Management

41. Single Offer Bargaining 10-41 • Now suppose the game is sequential in nature, and management gets to make the union a “take-it-or-leave-it” offer. • Analysis Tool: Write the game in extensive form • Summarize the players. • Their potential actions. • Their information at each decision point. • Sequence of moves. • Each player’s payoff.

42. Step 1: Management’s Move 10-42 10 5 Firm 1

43. Step 2: Add the Union’s Move 10-43 Accept Union Reject 10 Accept 5 Firm Union Reject 1 Accept Union Reject

44. Step 3: Add the Payoffs 10-44 Accept 100, 500 Union Reject -100, -3 10 Accept 300, 300 5 Firm Union Reject -100, -3 1 Accept 500, 100 Union Reject -100, -3

45. The Game in Extensive Form 10-45 Accept 100, 500 Union Reject -100, -3 10 Accept 300, 300 5 Firm Union Reject -100, -3 1 Accept 500, 100 Union Reject -100, -3

46. Step 4: Identify the Firm’s Feasible Strategies 10-46 • Management has one information set and thus three feasible strategies: • Offer $10. • Offer $5. • Offer $1.

47. Step 5: Identify the Union’s Feasible Strategies 10-47 • The Union has three information set and thus eight feasible strategies: • Accept $10, Accept $5, Accept $1 • Accept $10, Accept $5, Reject $1 • Accept $10, Reject $5, Accept $1 • Accept $10, Reject $5, Reject $1 • Reject $10, Accept $5, Accept $1 • Reject $10, Accept $5, Reject $1 • Reject $10, Reject $5, Accept $1 • Reject $10, Reject $5, Reject $1

48. Step 6: Identify Nash Equilibrium Outcomes 10-48 • Outcomes such that neither the firm nor the union has an incentive to change its strategy, given the strategy of the other.

49. Finding Nash Equilibrium Outcomes 10-49 $1 Yes $5 Yes $1 Yes $1 Yes $10 Yes $5 Yes $1 Yes $10, $5, $1 No

50. Step 7: Find the Subgame Perfect Nash Equilibrium Outcomes 10-50 • Outcomes where no player has an incentive to change its strategy, given the strategy of the rival, and • The outcomes are based on “credible actions;” that is, they are not the result of “empty threats” by the rival.