Resource Allocation and Decision Analysis (ECON 8010) Spring 2014 Foundations of Game Theory

Transcription

1 Resource Allocation and Decision Analysis (ECON 8) Spring 4 Foundations of Game Theory Reading: Game Theory (ECON 8 Coursepak, Page 95) Definitions and Concepts: Game Theory study of decision making settings in which the outcome for each decision maker depends upon not only their own actions, but also upon the actions of other decision makers Within economics, Game Theory is very useful for analyzing the behavior of firms in the intermediate market structures between Monopoly and Perfect Competition Pioneered by John von-neumann and Oskar Morgenstern s book Theory of Games and Economic Behavior (944) Three basic elements of any game:. Players decision makers whose behavior is to be analyzed. Strategies the different options or courses of action from which a player is able to choose 3. Payoffs numerical measures of the desirability of every possible outcome which could arise as a result of the strategies chosen by the players Complete Information an environment in which every player knows all of the strategies available to all players and the resulting payoff for all players at each possible outcome Incomplete Information an environment in which at least one of the players does not know all of the information that would be potentially relevant for making a decision at some point in the game e.g., perhaps the other player can be of one of two different types, and I don t know for certain which type he is Sequential Move Game a game in which players make their decisions in sequence, with the choices made in the past being observable by all players when a present decision is being made Simultaneous Move Game a game in which players must each choose their strategies without being able to observe the strategy chosen by others (either the players choose their strategies at the same time, or it is as if they choose them at the same time) One-Shot Game a game that is played between the same players only one time Repeated Game a game that is played between the same players more than one time

2 Cooperative Game a game in which players can enter into binding agreements before the start of the game Non-Cooperative Game a game in which players cannot enter into binding agreements before the start of the game Nash Equilibrium a set of strategies, one for each player, that are stable in the sense that no player could increase his own payoff by choosing a different strategy, given the strategies chosen by the other players very often, a Nash Equilibrium will serve as a reasonable prediction of play, since at the equilibrium strategies every player is behaving in a way that is in his own self-interest (given the behavior of others) Named after the game theorist John F. Nash, Jr. (994 Nobel Prize in Economics) Best Reply for Player a strategy is a best reply to a chosen strategy of Player if the strategy in question gives Player a greater payoff than any other available strategy Note: a set of strategies is a Nash Equilibrium if and only if each player is choosing a strategy that is a best reply to the strategies being played by others Dominant Strategy for Player a strategy is a dominant strategy if it is (strictly) a best reply to all available strategies of Player Strategic Rule of Thumb # if you have a dominant strategy, use it Dominated Strategy for Player a strategy is a dominated strategy there is some other available strategy that gives Player a (strictly) higher payoff than the strategy in question, for all available strategies of Player Strategic Rule of Thumb # if you have a dominated strategy, do not use it Strategic Rule of Thumb #3 if your rival has a dominant strategy, expect her to use it Strategic Rule of Thumb #4 if your rival has a dominated strategy, expect her to never use it 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 Mixed Extension of a Game an interpretation of a game in which the strategy choice of a player is allowed to be 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 A simple x game in which each player has a dominant strategy Firm A Firm B High Price B Low Price B High Price A, 8 6, 96 Low Price A 44, 4 84, 56 In each cell the first number specifies the payoff of player (the player whose strategies are specified by each distinct row) and the second number specifies the payoff of player (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 gets to choose the row, while player gets to choose the column ) To identify a Nash Equilibrium, we must systematically address the question What should each player do? Let us first examine the choice of Firm A 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 its 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 44 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 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 96 8 If Firm A were to choose Low Price A, then Firm B would want to choose Low Price B, since 56 4 Firm B High Price B Low Price B High Price A, 8 6, 96 Firm A Low Price A 44, 4 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)

4 A simple x game in which only one player has a dominant strategy Firm A Firm B High Price B Low Price B High Price A 5, 5 96, 64 Low Price A 5, 38 5, 35 Drawing the Best Reply arrows : Firm B High Price B Low Price B High Price A 5, 5 96, 64 Firm A Low Price A 5, 38 5, 35 By either Strategic Rule of Thumb # or # we see that Firm A should choose Low Price. But, what should Firm B do? Strategic Rule of Thumb #3 if your rival has a dominant strategy, expect her to use it Strategic Rule of Thumb #4 if your rival has a dominated strategy, expect her to never use it Thus, Firm B should choose High Price => the unique Nash Equilibrium is: Firm A Firm B High Price B Low Price B High Price A 5, 5 96, 64 Low Price A 5, 38 5, 35

5 Two examples of product development decisions: Example 3: ( anti-coordination game ) Volkswagen and Ford must each decide to either develop or not develop a new compact hybrid car Suppose the payoffs are: Ford Develop Hybrid Not Develop Develop Hybrid, 95, VW Not Develop 5, 5 85, 9 Start by recognizing that, for the given payoffs, neither player has a dominant strategy Ford Develop Hybrid Not Develop Develop Hybrid, 95, VW Not Develop 5, 5 85, 9 There are two Pure Strategy Nash Equilibria (one in which VW develops the hybrid and Ford does not, and another in which Ford develops the hybrid and VW does not) Mixed Extension: Let q denote the probability with which VW chooses Develop (so that q denotes the probability with which VW chooses Not Develop ) Let p denote the probability with which Ford chooses Develop (so that p denotes the probability with which Ford chooses Not Develop ) Deriving the Best Response Correspondence for VW, see that Develop (i.e., q ) is the strictly better choice if and only if ( p) ( p) 5( p) 85( p) 35( p) 5 p 35 7 p Similarly, deriving the Best Reply Correspondence for Ford, recognize that Develop (i.e., p ) is the strictly better choice if and only if 95( q) 5( q) ( q) 9( q) 5( q) 5q 5 3 q. 75 4

6 Visually q BR T ( p).75 p BR F (q).875 The graph above illustrates all three equilibria of this game: i. Pure Strategy Nash Equilibrium in which VW chooses Develop and Ford chooses Not Develop ( q and p ) ii. Pure Strategy Nash Equilibrium in which VW chooses Not Develop and Ford chooses Develop ( q and p ) iii. Mixed Strategy Nash Equilibrium in which VW chooses Develop with probability q.75 and Ford chooses Develop with probability p. 875 Example 4: ( cat and mouse game ) Suppose Lexus and Hyundai must choose a design for their model, either sleek or boxy Hyundai wants their car to look like the Lexus, while Lexus wants their car to look different than the Hyundai Hyundai Sleek Hyundai Boxy Hyundai Sleek Lexus 6, 4, 9 Lexus Boxy Lexus 7, 3, No Pure Strategy Nash Equilibrium => consider the Mixed Extension of the game Let q denote the probability with which Hyundai chooses Sleek (so that q denotes the probability with which Hyundai chooses Boxy ) Let p denote the probability with which Lexus chooses Sleek (so that p denotes the probability with which Lexus chooses Boxy ) Deriving the Best Response Correspondence for Hyundai, recognize that Sleek (i.e., q ) is the strictly better choice if and only if 4( p) ( p) 9( p) ( p) 5 p ( p) p.6 6 6

7 Similarly, deriving the Best Reply Correspondence for Lexus, recognize that Sleek (i.e., p ) is the strictly better choice if and only if 6( q) ( q) 7( q) 3( q) 7( q) q 7 7 q. 875 Visually q 8 8 BR H ( p) p BR L (q) The graph above illustrates the unique equilibrium of this game => a Mixed Strategy Nash 7 Equilibrium in which Hyundai chooses Sleek with probability q 8 and Lexus chooses Sleek with probability p 6

8 Another example of Determining a Mixed Strategy Equilibrium: Consider the following game: Player Left Right Player Top 4, 7, Bottom 6, 3, 5 This game does not have any Pure Strategy Nash Equilibria Consider the Mixed Extension of the game, in which q denotes the probability with which Player chooses Left (so that q denotes the probability with which Player chooses Right ) p denotes the probability with which Player chooses Top (so that p denotes the probability with which Player chooses Bottom ) Derive and graphically illustrate the Best Reply Correspondence for each player Consider the choice by Player when Player chooses Left with probability q and Right with probability q T Player s expected payoff from choosing Top is: 4q ( q) 6q B Player s expected payoff from choosing Bottom is: 6q ( q) 4q Thus, Player s payoff is strictly greater from choosing Top as opposed to Bottom (in which case his best reply is p ) if and only if 8 q. 8 T B Further, Player s payoff is strictly greater from choosing Bottom as opposed to Top (in which case his best reply is p ) if and only if q. 8 Finally, Player would realize the exact same expected payoff from choosing Top, Bottom, or any randomization between the two (in which case any value of p is a best reply ) if and only if q. 8 Visually, the Best Reply Correspondence of Player can be illustrated as:.8 q 5 4 BR ( q ) Similarly, considering the choice by Player when Player chooses Top with probability p and Bottom with probability p L Player s expected payoff from choosing Left is: 7 p 3( p) 3 4 p R p 5( p) 5 Player s expected payoff from choosing Right is: 4 p p

9 Thus, Player s payoff is strictly greater from choosing Left as opposed to Right (in which case his best reply is q ) if and only if L R p Further, Player s payoff is strictly greater from choosing Right as opposed to Left (in which case his best reply is q ) if and only if p. 5 Finally, Player would realize the exact same expected payoff from choosing Left, Right, or any randomization between the two (in which case any value of q is a best reply ) if and only if p. 5 Visually, the Best Reply Correspondence of Player can be illustrated as: q BR ( p).5 Drawing the two correspondences in the same graph, we have: p q BR ( p).8.5 p BR ( q ) If Player chooses p *. 5 and Player chooses q *. 8, then each player is choosing a mixed strategy that is a best reply to the strategy being played by his rival => this pair of mixed strategies is a Nash Equilibrium!

10 Multiple Choice Questions:. Consider a two player game between Player and Player. Player has two available strategies: Strategy A and Strategy B. Player has two available strategies: Strategy c and Strategy d. If Strategy A of Player is a Best Reply to a choice of Strategy c by Player, then A. Strategy A cannot be a dominant strategy for Player. B. Strategy B cannot be a dominant strategy for Player. C. Player must use Strategy A at any Nash Equilibrium of the game. D. More than one (perhaps all) of the above answers is correct.. Which of the following statements corresponds to one of the decision making rules of thumb discussed in lecture? A. If you have a dominant strategy, use it. B. If your rival has a dominant strategy, expect her to never use it. C. If you have a dominant strategy, recognize that your rival will expect that you will never use it. D. More than one (perhaps all) of the above answers is correct. 3. 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. More than one of the above answers is correct (since more than one of the above is NOT one of the three basic elements of a game ). 4. 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 is a Nash Equilibrium. B. the sum of the payoffs of both players is minimized at an outcome that is a Nash Equilibrium. C. there is at least one Nash Equilibrium (potentially in mixed strategies ). D. More than one (perhaps all) of the above answers is correct. 5. A game that is played between the same players more than one time is called a A. One-Shot Game. B. Repeated Game. C. Game of Incomplete Information. D. Game of Complete Information. 6. In the Mixed Extension of a game, A. the strategy chosen by each of your rivals should be viewed as entirely random. B. the strategy choice of each player is allowed to be a probability distribution over his available pure strategies. C. there can never be any Nash Equilibria. D. players are never allowed to play Pure Strategies.

11 7. Nash s Existence Theorem states that A. individuals can only every overcome a Prisoners Dilemma by repeated interaction. B. the unique equilibrium of any simultaneous move game can be determined by Iterated Elimination of Dominated Strategies. C. the unique Subgame Perfect Nash Equilibrium of any sequential move game can be determined by backward induction. D. every game with a finite number of players, each with a finite number of available pure strategies, has at least one Nash Equilibrium (potentially in Mixed Strategies ). 8. The pioneering work Theory of Games and Economic Behavior was written by A. Adam Smith. B. John F. Nash, Jr. and Lloyd Shapley. C. John Maynard Keynes. D. John von-neumann and Oskar Morgenstern. 9. A refers to 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 their equilibrium payoff. A. Cat and Mouse Game B. Prisoners Dilemma C. repeated game D. Best Reply Correspondence Problem Solving or Short Answer Questions:. Determine all Nash Equilibria of each of the following games. A. Player Top 3,, Player Bottom, 4, 5 B. Player Player Top 4, 9, 5 Bottom, 7, 3

12 . For each of the following games, determine if either player has a Dominant Strategy and identify all Pure Strategy Nash Equilbria. A. Player Top, 5, 3 Player Bottom 8, 7 6, B. C. D. Player Player Player Player Top, 5 6, 9 Bottom 8, 4, 3 Player Top, 5 4, Bottom 6,, 5 Player Top 3, 5, 35 Bottom, 45 4, 5 3. Consider the player simultaneous move game below: Player Player Top 8, 65 5, 45 Bottom 7, 75 35, 55 3A. Does this game fit the definition of a Prisoner s Dilemma? Clearly explain why or why not. 3B. Determine all Nash Equilibria of this game.

13 4. Golden Fleece and JonShawn are two firms that produce men s clothing. They must simultaneously choose their product lines for next year. They each broadly have a choice of either a traditional line or a trendy line. Historically, the clothing of Golden Fleece has tended to appeal to consumers with conservative tastes, while the clothing of JonShawn has tended to appeal to those consumers who want to be on the cutting edge of fashion. If the two firms choose similar lines, then their products will be less differentiated from one another. As a result, they would be competing for essentially the same segment of consumers, making joint profits lower. More precisely, the profits of the firms will be: GF 95 and JS 7, if both introduce a traditional line ; GF 85 and JS, if both introduce a trendy line ; GF 55 and JS, if Golden Fleece introduces a Trendy line and JonShawn introduces a traditional line ; and GF 85 and JS, if Golden Fleece introduces a traditional line and JonShawn introduces a trendy line. 4A. Illustrate the interaction between these two firms by way of a payoff matrix. 4B. Identify all Pure Strategy Nash Equilibria of this game. 4C. Considering the Mixed Extension of the game (in which Golden Fleece chooses traditional line with probability p and JonShawn chooses traditional line with probability q ), graphically illustrate the Best Reply Correspondence of each player. Based upon this graph, identify any Mixed Strategy Nash Equilibria of this game. 5. Consider the following two player simultaneous move game: Player Player Top a, b, Bottom, c, d Specify values of a, b, c, and d for which: 5A. Top is a dominant strategy for Player and Left is a dominant strategy for Player. 5B. neither player has a dominant strategy and there are no Pure Strategy Equilibria. 5C. neither player has a dominant strategy and there are two Pure Strategy Equilibria. 5D. there are no Pure Strategy Equilibria and the unique Mixed Strategy Equilibrium is characterized by Player choosing Top with probability 5 and Player choosing Left with probability 3.

14 Answers to Multiple Choice Questions:. B. A 3. B 4. C 5. B 6. B 7. D 8. D 9. B Answers to Problem Solving or Short Answer Questions: A. This game has two Pure Strategy Nash Equilibria as follows: Player Top 3,, Player Bottom, 4, 5 Additionally, letting p denote the probability with which Player plays Top and letting q denote the probability with which Player plays Left, a Mixed Strategy Nash Equilibrium can be determined as: T B 3q ( q) q ( q) q q * and L R p 4( p) 5( p) 4 3p 5 5p p * B. This game does not have any Pure Strategy Nash Equilibria: Player Top 4, 9, 5 Player Bottom, 7, 3 However, by Nash s Existence Theorem we know that there must exist a Mixed Strategy Equilibrium. Letting p denote the probability with which Player plays Top and letting q denote the probability with which Player plays Left, a Mixed Strategy Nash Equilibrium can be determined as: T B 4q ( q) ( q) q q q * 3 and

15 L R p 7( p) 5 p 3( p) 9 p 7 p 3 p * 5 A. Player has a dominant strategy of Top (it is a best reply for each available strategy of Player ). Player has a dominant strategy of Left (it is a best reply for each available strategy of Player ). Thus, the unique equilibrium of the game is a Pure Strategy Nash Equilibrium in which Player plays Top and Player plays Left. B. Player does not have a dominant strategy ( Bottom is the best reply to Left, while Top is the best reply to Right ). Player has a dominant strategy of Right (it is a best reply for each available strategy of Player ). Thus, the unique equilibrium of the game is a Pure Strategy Nash Equilibrium in which Player plays Right (his dominant strategy) and Player plays Top (her best reply to the dominant strategy of Player ). C. Neither player has a dominant strategy. For Player, Top is the best reply to Left, while Bottom is the best reply to Right. For Player, Left is the best reply to Top, while Right is the best reply to Bottom. Thus, the game has two Pure Strategy Nash Equilibria one in which Player plays Top and Player plays Left, and another in which Player plays Bottom and Player plays Right. D. Neither player has a dominant strategy. For Player, Top is the best reply to Left, while Bottom is the best reply to Right. For Player, Right is the best reply to Top, while Left is the best reply to Bottom. Thus, the game has no Pure Strategy Nash Equilibria. 3A. 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 and Left is the dominant strategy for Player ). However, when Player plays Top and Player plays Left, the realized payoffs are (8) for Player and (65) for Player. That is, each player is realizing the highest payoff that she could possibly 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 ). 3B. Since each player has a dominant strategy, the unique Nash Equilibrium is for each player to follow her dominant strategy. That is, the unique Nash Equilibrium is for Player to play Top and for Player to play Left. 4A. Given the description of the game, the resulting payoff matrix is: Golden Fleece JonShawn Traditional Trendy Traditional 95, 7 85, Trendy 55, 85,

16 4B. Drawing the Best Reply Arrows as below: Golden Fleece JonShawn Traditional Trendy Traditional 95, 7 85, Trendy 55, 85, We see that there are two Pure Strategy Nash Equilibrum, one in which Golden Fleece chooses Trendy and JonShawn chooses Traditional, and one in which Golden Fleece chooses Traditional and JonShawn chooses Trendy. 4C. The Best Reply Correspondence for Golden Fleece can be derived by first recognizing when Golden Fleece would have a strict preference for choosing p (i.e., always choosing a traditional line). This will be the case so long as: Trad Trend GF 95 q 85( q) 55q 85( q) GF ( q) 6q 6q 5 q Thus, choosing p (i.e., always choosing trendy) is the best reply if q Finally, if q 5. 65, then any value of p between zero and one is a best reply. 8 Similarly, for JonShawn, we have that q (i.e., always choosing a traditional line) is the unique best reply if: Trad Trend JS 7 p ( p) p ( p) JS 8( p) 5p 3p p It follows that choosing q (i.e., always choosing a trendy line) is the best reply if p Finally, if p , then any value of q between zero and one is a 3 3 best reply. Graphically: q q BR ( p) p (q) p BR For the mixed extension of the game, we have a Nash Equilibrium wherever the two Best Reply Correspondences intersect. Thus, the graph above identifies not just the two aforementioned Pure Strategy Nash Equilibria (of ( p, q) (, ) and ( p, q) (,) ), but 8 5 ( p, q),. also reveals the existence of a Mixed Strategy Nash Equilibrium with 3 8

17 5A. For Top to be a dominant strategy for Player, we need a and c. For Left to be a dominant strategy for Player, we need b and d. 5B. For neither player to have a dominant strategy and for no Pure Strategy Equilibria to exist, we need the best response arrows to result in a cycle. One way to have this is for a, b, c, and d. 5C. For neither player to have a dominant strategy and for two Pure Strategy Equilibria to exist, we need the best response arrows to be such that either: Top/Left and Bottom/Right each have both arrows pointing inward; or Top/Right and Bottom/Left each have both arrows pointing inward. To have the former, we would need a, b, c, and d. 5D. Again, to have no Pure Strategy Equilibria we need the best response arrows to result in a cycle. As noted in (7B), one way to have this is with a, b, c, and d. In order for Player to optimally choose a mixed strategy when Player is playing Left with probability 3, we need: a ( ) ( ) a c a c Similarly, in order for Player to optimally choose a mixed strategy when Player is playing Top with probability 5, we need: 4 b 4 ( ) ( ) d b d b 6 A set of values satisfying these restrictions is: a 3, c 4, b, and d 3. c d