ECON241 (Fall 2010) 17. 11. 2010 (Tutorial 9) Chapter 10 Game Theory: Inside Oligopoly Sequential Move Games and Subgame Perfect Equilibrium A sequential move game could be represented by the extensive form (game tree) and the subgame perfect equilibrium could be solved by backward induction Example: Player 1 L R Player 2 U 6 20 Player 2 D U 0 0 D 5 5 10 15 Feasible strategies: Player 1: L, R Player 2: UU, UD, DU, DD (UU means Player 2 chooses U if Player 1 chooses L, and chooses U when Player 1 chooses R) Generally, number of feasible strategies: Sn (where S is the number of strategies in each information set, and n is the number of information set) NE: (L, UU) , (R, DD)and (R, UD) SPE: Player 1: {R}, Player 2: {UD} and SPE outcome is Player one chooses R and Player 2 chooses D with payoffs 10 and 15 respectively (R, UD) is a NE as well as a SPE, while (L, UU) and (R, DD) are NE only as they involves incredible threat. (What is the threat? Why is it incredible?) (How to solve for NE and SPE?) A set of strategies constitutes a subgame perfect equilibrium if (1) It is a Nash equilibrium (How to check?) (2) At each stage of the game (decision node) neither player can improve her payoff by changing her own strategy SPE is a NE that involves only credible threats 1 Example: Rotten Kid Game [Similar examples: Entry Game (P. 379), Innovation (P.380)] Child Go Not go Parents 1 1 Punish Not punish –1 –1 2 0 Feasible Strategies: P if G, NP if G, P if NG, NP if NG Two NEs: (Go, Punish) and (Not go, Not Punish) SPE: (Not go, Not Punish) The NE (Go, Punish) is not a satisfactory one as it involives on incredible threat. The threat by parents to the kid is that “if you choose go, you will get a payoff of 1, but if you choose not go, you will be punished and your payoff will be –1, so it is better for you to go”. (Go, Punish) is a NE because given the parents’ threat, the best response of the child is go. Let’s consider the extensive form and we can see the threat is not credible. Suppose the child chooses not go Parents choose not punish instead of punish (0 –1) if they are rational. Parents would never carry out the threat when the child chooses not go. (Go, Punish) is not a SPE though it’s a NE. The set of equilibria to those rely on credible threat SPE SPE: (Not Go, Not Punish) with payoffs (2, 0) The threat should be incredible if we are dealing with a one-shock game. If the game repeats, then the parents may act irrationally in order to build up their reputation. In the rotten kid example, parents may act on the threat and punish the child if the child refuses to go. (Although the payoff will be –1 for both parties). Example: Entry Game (Baye, P379) Potential Entrant Out In Incumbent 0 10 Hard Soft –1 –1 2 0 Feasible Strategies: H if Out, S if Out, H if In, S if In Two NEs: (Out, Hard) and (In, Soft) SPE: (In, Soft) (Out, Hard): Given the incumbent threat of initiating a price war if the potential entrant enters, potential entrant’s best response is to stay out. Given that the potential entrant stays out, the incumbent may still threaten to play hard if the potential entrant enters. (Out, Hard) is only a NE, but not SPE as it involves an incredible threat. If the incumbent is rational, he will choose to accommodate instead of initiating a price war (0> -1). Again, the incumbent may choose to act irrationally to build up its reputation if the game is not a one-shot game. 2 Example: Sequential Bargaining (Baye, P.381) The manager and the labor union engaged in a negotiation over how to split a surplus of $100. The manager moves first by making an offer ($1, 50 and $99) to the union. The union decides whether to “Accept” or “Reject” the offer. If the offer is rejected, neither party receives anything. Manager $1 $50 Union A 99 1 $99 Union R 0 0 A 50 50 Union R 0 0 A 1 99 R 0 0 Feasible Strategies: Firm: {offer 1, offer 50, offer 99} Union: {AAA, ARR, ARA, RRR, RAA, RAR, AAR, RRA} (Sn = 23 = 8 strategies) 7 NEs: (offer 1, AAA), (offer 1, ARR), (offer 1, ARA), (offer 1, AAR), (offer 50, RAR), (offer 50, RAA), (offer 99, RRA) The labor union threaten the manager to reject the offer made by the manager, however all threats are incredible. (Why?) SPE: (offer 1, AAA). This is the only NE which does not involves incredible threat SPE outcome: The manager make an offer of $1 and the union Accept the offer, payoffs to the two parties are ($99, $1) Is there any first mover advantage in this game? Labor Union Manager AAA ARR ARA AAR RRA RAA RAR RRR $1 (99, 1) (99, 1) (99, 1) (99, 1) (0, 0) (0, 0) (0, 0) (0, 0) $50 (50, 50) (0, 0) (0, 0) (50, 50) (0, 0) (50, 50) (50, 50) (0, 0) $99 (1, 99) (0, 0) (1, 99) (0, 0) (1, 99) (1, 99) (0, 0) (0, 0) 3 Problems and Examples: Question 4 (Baye, P.387) Player 2 Player 1 A B C 10, 10 -5, 60 D 60, -5 50, 50 (a) Identify the one-shot NE (b) Suppose the players know this game will be repeated exactly 3 times. Can they achieve payoffs that are better than the one-shot NE? Explain. (c) Suppose this game is infinitely repeated and the interest rate is 5%. Can the players achieve payoffs that are better than the one-shot NE? Explain. (d) Suppose the players do not know exactly how many times this game will be repeated, but they do know that the probability the game will end after a given play is . If is sufficiently low, can players earn more than they could in the one-shot NE? (a) NE is (A, C) and each player gets a payoff of 10. (b) No. Collusion of any would not be stable for any game with certain finite time horizon. As eventually it will come to a point that both players are certain there is no tomorrow and there is no way to punish/ can be punished for having broken the promise, both players will cheat in every period up to the known last stage. (c) If firms adopt the trigger strategies, higher payoffs can be achieved if Cheat Coop 1 . Coop N i Given an interest rate of 5%, πCheat = 60, πCoop = 50, πN = 10, Cheat Coop 60 50 1 1 1 0.25 < 20 Coop N 50 10 4 i .05 Thus, each firm can indeed earn a payoff of 50 in each period via the trigger strategies. (d) Yes. If the probability the game will end after a given play is sufficiently low ( 0.8), then players can earn more than they could in the one-shot NE 4 Question 8 (Baye, P.388) Player 1 A B Player 2 X Y 5, 5 0, -200 -200, 0 20, 20 (a) Determine the NE outcomes that arise if the players make decisions independently, simultaneously, and without any communication. Which of these outcomes would you consider most likely? Explain. (b) Suppose Player 1 is permitted to “communicate” by uttering one syllable before the players simultaneously and independently make their decisions. What should Player 1 utter, and what outcome do you think would occur as a result? (c) Suppose player 2 can choose its strategy before Player 1, that Player 1 observes Player 2’s choice before making her decision, and that this move structure is known by both players. What outcome would you expect? Explain. (a) There are two Nash equilibria: (A, X) and (B, Y). The (A, X) equilibrium would seem most likely since the other equilibrium entails considerable risk if the players don’t coordinate on the same equilibrium. (b) “B”. This would signal to player 2 that player 1 is going to use strategy B, and therefore permit the players to coordinate on the (20, 20) equilibrium. (c) Player 2 would choose Y and player 1 would follow by choosing B. This is the subgame perfect equilibrium. Question 17 (Baye, P.391) Ata time when demand for ready-to-eat cereal was stagnant, a spokesperson for the cereal maker Kellogg’s was quoted as saying, “…for the past several years, our individual company growth has come out of the other fellow’s hide.” Kellogg’s has been producing cereal since 1906 and continues to implement strategies that make it a leader in the cereal industry. Suppose that when Kellogg’s and its largest rival advertise, each company earns $0 billion in profits. When neither company advertises, each company earns profits of $8billion. If one company advertises and the other does not, the company that advertises earns $48 billion and the company that does not advertise loses $1 billion. Under what conditions could these firms use trigger strategies to support the collusive level of advertising? Kellogg’s Advertise Not Advertise Advertise 8, 8 48, -1 Rival Not Advertise -1, 48 0, 0 The NE is for both firms not to advertise if it is a one-shot game. If it is not a one shot game, collusion is profitable under the usual trigger strategies if $48 $8 1 Cheat Coop 1 $5 . Thus, one requirement is for the interest rate to be , or Coop N $8 0 i i less than 20 percent. Another requirement includes the ability of firms to monitor (observe) potential deviations by rivals. 5 Question 13 (Baye, P.389) Coca-cola and PepsiCo are the leading competitors in the market for cola products. In 1960 Coca-cola introduced Sprite, which today is the worldwide leader in lemon-lime soft drink and ranks fourth among all soft drinks worldwide. Prior to 1999, PepsiCo did not have a product that competed against Sprite and had to decide whether to introduce such a soft drink. By not introducing a lemon-lime soft drink, PepsiCo would continue to earn$200 million profit and Coca-cola could continue to earn a $300 million profit. By introducing a new lemon-lime soft drink, one of these two possible strategies could be pursued: (1) PepsiCo could trigger a price war with Coca-cola in both the lemon-lime and cola markets. Each company earns $100 million in this case. (2) Coca-cola could acquiesce and each firm maintain it’s current 50/50 split of the cola market and split the lemon-lime market 30/70 (PepsiCo/Coca-cola). In this case, PepsiCo earns $227 million and Coca-cola earns $275million If you are the manager at PepsiCo, would you try to convince your colleagues that introducing the new soft drink is the most profitable strategy? Why or why not? The game could be represented by the following extensive form: Pepsi Not Introduce Introduce CocaCola 200 300 Price War 100 100 Accomodate 227 275 Notice that Coca-Cola’s best response if Pepsi introduces is to acquiesce to earn $275 million rather than to start a price war and earn $100. Thus, while Coca-Cola might threaten to start a price war in an attempt to keep you out of the market, this threat isn’t credible; your best option is to introduce. 6