ECON 2100 Principles of Microeconomics (Summer 2016) Game Theory and Oligopoly

Transcription

1 ECON 2100 Principles of Microeconomics (Summer 2016) Game Theory and Oligopoly Relevant readings from the textbook: Mankiw, Ch. 17 Oligopoly Suggested problems from the textbook: Chapter 17 Questions for Review (Page 366): 1 and 5 Chapter 17 Quick Check Multiple Choice (Pages ): 1 Chapter 17 Problems and Applications (Pages ): 2, 4, 5, 6, 7, 8, and 9 Definitions and Concepts: Collusion an effort by firms to coordinate their actions in an attempt to increase both total industry profit and individual profit of every firm (compared to the profit levels which would result without any such coordination) Cartel a group of firms who attempt to engage in collusion, either openly/explicitly or tacitly/implicitly Game Theory the study of decision making environments in which the outcome for any one decision maker depends upon not only their own actions, but also upon the actions of other decision makers Three basic elements of any game: 1. Players the decision makers whose behavior is to be analyzed 2. Strategies the different options or courses of action available to a player 3. Payoffs numerical measures of the desirability of every possible outcome which could arise as a result of the strategies chosen by the players Simultaneous Move Game a game in which each player must choose their own strategy without being able to observe the strategies chosen by other players (in such games it is as if each player is deciding what to do at exactly the same time ) Best Reply for Player 1 a strategy is a best reply to a chosen strategy of Player 2 if the strategy gives Player 1 a greater payoff than any other available strategy Dominant Strategy for Player 1 a strategy is a dominant strategy if it is a best reply to every available strategy of Player 2 Strategic Rule of Thumb #1 if you have a dominant strategy, use it

2 Strategic Rule of Thumb #2 if your rival has a dominant strategy, expect her to use it; thus, you should use your strategy which is the best reply to the dominant strategy of your rival Nash Equilibrium a pair of strategies is a Nash Equilibrium if every player is choosing a strategy that is a best reply to the strategies which are chosen by her rivals Prisoner s Dilemma a game in which every player has a dominant strategy (so that the game has a unique equilibrium characterized by all players using their dominant strategies), but in which there is some other outcome at which the payoff of every single player is higher than the equilibrium payoff Repeated Prisoner s Dilemma a standard prisoner s dilemma game which confronts the same players repeatedly over time Mixed Extension of a Game an interpretation of a game in which the strategy of a player is a probability distribution over their available pure strategies Nash s Existence Theorem for every game with any finite number of players, each with a finite number of available pure strategies, there exists at least one Nash Equilibrium (potentially in mixed strategies )

3 2 by 2 simultaneous move game Example 1: Consider two firms operating in a market, each of which must choose either a high or low price Suppose the payoffs for the two firms at the four possible outcomes are summarized by the payoff matrix below (want to figure out what the players will do, assuming this information is given and known by both players ) Firm B High Price B Low Price B Firm A High Price A 120, 80 60, 96 Low Price A 144, 40 84, 56 First number in cell specifies the payoff of player 1 (the player whose strategies are specified by each distinct row) and the second number specifies the payoff of player 2 (the player whose strategies are specified by each distinct column) Each player gets to choose their own strategy, but has no control over the strategy chosen by their rival (i.e., player 1 gets to choose the row, while player 2 gets to choose the column ) What should each player do? Firm A has no control over the choice of Firm B, and further Firm A does not know what Firm B will necessarily do, but However, Firm A could determine what it s own best choice would be for each of the things Firm B could possibly do Further (if necessary), Firm A could also try to figure out what Firm B will do (based upon what is best for Firm B to do ) From the perspective of Firm A If Firm B were to choose High Price B, then Firm A would want to choose Low Price A, since If Firm B were to choose Low Price B, then Firm A would want to choose Low Price A, since In this game, for Firm A : Low Price A is a best reply to a choice of High Price by Firm B Low Price A is a best reply to a choice of Low Price by Firm B Low Price A is a best reply to anything B can do Note that in this example, Firm B also has a dominant strategy. From the perspective of Firm B If Firm A were to choose High Price A, then Firm B would want to choose Low Price B, since If Firm A were to choose Low Price A, then Firm B would want to choose Low Price B, since 56 40

4 2 by 2 simultaneous move game Example 1 (continued) Firm B High Price B Low Price B Firm A High Price A 120, 80 60, 96 Low Price A 144, 40 84, 56 The unique Nash Equilibrium is for Firm A to choose Low Price A and for Firm B to choose Low Price B. As a result, Firm A realizes a payoff of (84) and Firm B realizes a payoff of (56) 2 by 2 simultaneous move game Example 3 (as numbered in lecture): Suppose Toyota and Ford must each decide to either develop or not develop a new compact hybrid car Ford Develop Hybrid Not Develop Toyota Develop Hybrid 100, , 90 Not Develop 60, , 85 Consider first the choice of Ford If Toyota chooses Develop, then Ford prefers Not Develop (since 90>65) If Toyota chooses Not Develop, then Ford prefers Develop (since 110>85) Ford does not have a dominant strategy since there is no one strategy available to Ford which is a best reply to everything Toyota could do => best choice for Ford is not immediately clear Think about the choice of Toyota If Ford chooses Develop, then Toyota prefers Develop (since 100>60) If Ford chooses Not Develop, then Toyota prefers Develop (since 125>70) So, Toyota does have a dominant strategy. Implications: o Toyota should use it s dominant strategy (i.e., Toyota should Develop Hybrid ) o Further, Ford should be able to place itself in the shoes of Toyota and correctly recognize that Toyota will choose to Develop Hybrid (since doing so is a dominant strategy for Toyota) o The only reasonable action for Ford to take is to Not Develop (i.e., choose the best reply to the dominant strategy of its rival) Ford Develop Hybrid Not Develop Toyota Develop Hybrid 100, , 90 Not Develop 60, , 85

5 What exactly do the best response arrows show us? 1. Whether a player has or does not have a dominant strategy If all the arrows for a player point in the same direction, then the strategy associated with either the row or column to which they all point is a dominant strategy (since this indicates that the best reply for the player is the same, regardless of the strategy chosen by her rival) If all the arrows for a player do NOT point in the same direction, then the player does not have a dominant strategy (since this indicates that the best reply for the player depends upon the strategy chosen by her rival) 2. Whether a pair of strategies is or is not an equilibrium If for a particular cell all the arrows point inward, then the pair of strategies leading to this cell is a Nash Equilibrium (since this indicates that no player could increase her own payoff by choosing a different strategy) If for a particular cell any arrows point outward, then the pair of strategies leading to this cell is NOT a Nash Equilibrium (since this indicates that at least one player could increase her payoff by choosing a different strategy) 2 by 2 simultaneous move game Example 4 (as numbered in lecture): (illustrates that there can be multiple equilibria ) What if the payoffs in Example 3 had instead been Ford Develop Hybrid Not Develop Toyota Develop Hybrid 110, , 100 Not Develop 115, , 90 Now neither player has a dominant strategy Further, we have two equilibria (one in which Toyota develops the hybrid and Ford does not, and another in which Ford develops the hybrid and Toyota does not) Ford Develop Hybrid Not Develop Toyota Develop Hybrid 110, , 100 Not Develop 115, , 90 2 by 2 simultaneous move game Example 5 (as numbered in lecture): ( cat and mouse game ) Suppose Lexus and Kia must choose a design for their 2011 model: sleek or boxy Kia wants their car to look like the Lexus, while Lexus wants their car to look different than the Kia Kia Sleek Kia Boxy Kia Lexus Sleek Lexus 160, , 90 Boxy Lexus 170, , 120 No Pure strategy Nash Equilibruim

6 Problems: 1. Consider the 2 player simultaneous move game below: Player 2 Left Right Player 1 Top 80, 65 50, 45 Bottom 70, 75 35, 55 1A. Does this game fit the definition of a Prisoner s Dilemma? Clearly explain why or why not. 1B. Determine all Nash Equilibria of this game. Multiple Choice Questions: 1. In a simultaneous move game with two players, it must always be the case that A. the sum of the payoffs of both players is maximized at an outcome that it not a Nash Equilibrium. B. each player has a dominant strategy. C. there is at least one Nash Equilibrium (potentially in mixed strategies ). D. More than one of the above answers are correct. 2. Collusion refers to A. a game in which no player has a dominant strategy. B. the amount by which the output of a firm is below the efficient scale of production. C. an effort by firms to coordinate their actions in an attempt to increase both total industry profit and individual profit of every firm (compared to the profit levels which would result without any such coordination). D. a market structure in which there are no substantial barriers to entry, but in which competing firms sell differentiated products. 3. Consider a two player game between Player 1 and Player 2. Player 1 has two available strategies: Strategy A and Strategy B. Player 2 has three available strategies: Strategy c, Strategy d, and Strategy e. If Strategy A of Player 1 is a Best Reply to a choice of Strategy c by Player 2, then A. Strategy A must be a dominant strategy for Player 1. B. Strategy A cannot be a dominant strategy for Player 1. C. Strategy B cannot be a dominant strategy for Player 1. D. More than one of the above answers is correct. 4. Which of the following is NOT one of the three basic elements of a game? A. Players. B. Rules of the Game. C. Strategies. D. Payoffs.

7 5. Consider a two player game in which Player 1 has four available strategies ( Strategy A, Strategy B, Strategy C, and Strategy D ) and Player 2 has two available strategies ( Strategy e and Strategy f ). Suppose that for Player 1 : Strategy B is a best reply to a choice of Strategy e by Player 2 ; and Strategy D is a best reply to a choice of Strategy f by Player 2. Further, Strategy e is a dominant strategy for Player 2. It follows that this game A. has exactly two Nash Equilibria. B. has exactly one Nash Equilibrium. C. does not have any Nash Equilibria (even in mixed strategies ). D. None of the above answers are correct (since it would be necessary to know the exact values of all payoffs in order to determine the number of Nash Equilibria of the game). For questions 6 through 9, consider the 2 player simultaneous move game below: Player 2 C D Player 1 A 20, 25 40, 15 B 8, 35 60, For this game A. neither Player 1 nor Player 2 has a dominant strategy. B. both Player 1 and Player 2 each have a dominant strategy. C. Player 1 has a dominant strategy, but Player 2 does not. D. Player 2 has a dominant strategy, but Player 1 does not. 7. If Player 1 were to choose Strategy A, then the best reply for Player 2 would be to choose A. Strategy A. B. Strategy B. C. Strategy C. D. Strategy D. 8. Which of the following pairs of strategies is a Nash Equilibrium? A. Player 1 chooses Strategy A ; Player 2 chooses Strategy C. B. Player 1 chooses Strategy B ; Player 2 chooses Strategy C. C. Player 1 chooses Strategy B ; Player 2 chooses Strategy D. D. More than one of the above answers is correct If Player 2 were to randomize and choose Strategy C with probability 4 and choose Strategy D with probability 43, then Player 1 A. would maximize her own payoff by choosing Strategy A. B. would maximize her own payoff by choosing Strategy B. C. would maximize her own payoff by choosing Strategy A with 1 probability 3 and choosing Strategy B with probability 32. D. has no control over her own payoff (since the value of her expected payoff is the same for any randomization over her two available strategies).

8 Answer to Problem: 1A. The definition of a Prisoner s Dilemma was: a game in which every player has a dominant strategy (so that the game has a unique equilibrium characterized by all players using their dominant strategies), but in which there is some other outcome at which the payoff of every single player is higher than the equilibrium payoff. For the game under consideration, both players do have dominant strategies ( Top is the dominant strategy for Player 1 and Left is the dominant strategy for Player 2). However, when Player 1 plays Top and Player 2 plays Left, the realized payoffs are (80) for Player 1 and (65) for Player 2. That is, each player is realizing the highest payoff that she could possible realize. From here, it is clear that this is not a Prisoner s Dilemma (since there is not some other outcome at which the payoff of every single player is higher than the equilibrium payoff ). 1B. Since each player has a dominant strategy, the unique Nash Equilibrium is for each player to follow their dominant strategy. That is, the unique Nash Equilibrium is for Player 1 to play Top and for Player 2 to play Left. Answers to Multiple Choice Questions: 1. C 2. C 3. C 4. B 5. B 6. A 7. C 8. D 9. B