Economics of Strategy (ECON 4550) Maymester 2015 Foundations of Game Theory

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1 Economics of Strategy (ECON 4550) Maymester 05 Foundations of Game Theory Reading: Game Theory (ECON 4550 Courseak, Page 95) Definitions and Concets: Game Theory study of decision making settings in which the outcome for each decision maker deends uon not only their own actions, but also uon 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 Monooly and Perfect Cometition 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 otions or courses of action from which a layer is able to choose 3. Payoffs numerical measures of the desirability of every ossible outcome which could arise as a result of the strategies chosen by the layers Comlete Information an environment in which every layer knows all of the strategies available to all layers and the resulting ayoff for all layers at each ossible outcome Incomlete Information an environment in which at least one of the layers does not know all of the information that would be otentially relevant for making a decision at some oint in the game e.g., erhas the other layer can be of one of two different tyes, and I don t know for certain which tye he is Sequential Move Game a game in which layers make their decisions in sequence, with the choices made in the ast being observable by all layers when a resent decision is being made Simultaneous Move Game a game in which layers must each choose their strategies without being able to observe the strategy chosen by others (either the layers 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 layed between the same layers only one time Reeated Game a game that is layed between the same layers more than one time

2 Cooerative Game a game in which layers can enter into binding agreements before the start of the game Non-Cooerative Game a game in which layers cannot enter into binding agreements before the start of the game Credible romise/threat an announcement to behave in a certain way is credible if the stated action is in the best interest of the decision maker Non-Credible romise/threat an announcement to behave in a certain way is non-credible if the stated action is not in the best interest of the decision maker Nash Equilibrium a set of strategies, one for each layer, that are stable in the sense that no layer could increase his own ayoff by choosing a different strategy, given the strategies chosen by the other layers very often, a Nash Equilibrium will serve as a reasonable rediction of lay, since at the equilibrium strategies every layer 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 Rely for Player a strategy is a best rely to a chosen strategy of if the strategy in question gives Player a greater ayoff than any other available strategy Note: a set of strategies is a Nash Equilibrium if and only if each layer is choosing a strategy that is a best rely to the strategies being layed by others Dominant Strategy for Player a strategy is a dominant strategy if it is (strictly) a best rely to all available strategies of 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 ayoff than the strategy in question, for all available strategies of 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, exect her to use it Strategic Rule of Thumb #4 if your rival has a dominated strategy, exect her to never use it

3 Iterated Elimination of Dominated Strategies a rocess by which strategies that cannot be art of a Nash Equilibrium are eliminated from consideration, by. discarding any strategy that is dominated for a layer. examining the reduced game and discarding any strategy that is dominated for a layer in the reduced game 3. reeating Ste until no strategies for any layer can be discarded Any strategy that is discarded during the rocess of Iterated Elimination of Dominated Strategies can never be a strategy that is used by a layer at a Nash Equilibrium This rocess might lead to the identification of a unique Nash Equilibrium, even in seemingly comlex games (i.e., ones in which layers have several different available strategies) But, this rocess will not always whittle the game down to one unique strategy for each layer Prisoner s Dilemma a game in which every layer has a dominant strategy (so that the game has a unique equilibrium characterized by all layers using their dominant strategies), but in which there is some other outcome at which the ayoff of every single layer is higher than the equilibrium ayoff Collusion an effort by firms to coordinate their actions in an attemt to increase both total industry rofit and individual rofit of every firm (comared to the rofit levels which would result without any such coordination) Cartel a grou of firms who attemt to engage in collusion, either oenly/exlicitly or tacitly/imlicitly Reeated Prisoner s Dilemma a standard risoner s dilemma game which confronts the same layers reeatedly over time Mixed Extension of a Game an interretation of a game in which the strategy choice of a layer is allowed to be a robability distribution over their available ure strategies Nash s Existence Theorem for every game with any finite number of layers, each with a finite number of available ure strategies, there exists at least one Nash Equilibrium (otentially in mixed strategies ) Subgame a sequential move game of comlete information with n decision nodes has n subgames, each consisting of a decision node along with all subsequent branches of the game tree Subgame Perfect Nash Equilibrium an equilibrium in which every decision of every layer is otimal within all subgames of the larger game

4 A simle x game in which each layer has a dominant strategy Firm A Firm B High Price B Low Price B High Price A 0, 80 60, 96 Low Price A 44, 40 84, 56 In each cell the first number secifies the ayoff of layer (the layer whose strategies are secified by each distinct row) and the second number secifies the ayoff of layer (the layer whose strategies are secified by each distinct column) Each layer gets to choose their own strategy, but has no control over the strategy chosen by their rival (i.e., layer gets to choose the row, while layer gets to choose the column ) To identify a Nash Equilibrium, we must systematically address the question What should each layer 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 ossibly do Further (if necessary), Firm A could also try to figure out what Firm B will do (based uon what is best for Firm B to do ) From the ersective of Firm A If Firm B were to choose High Price B, then Firm A would want to choose Low Price A, since 44 0 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 rely to a choice of High Price by Firm B Low Price A is a best rely to a choice of Low Price by Firm B Low Price A is a best rely to anything B can do Note that Firm B also has a dominant strategy. From the ersective 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 Firm B High Price B Low Price B High Price A 0, 80 60, 96 Firm A Low Price A 44, 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 ayoff of (84) and Firm B realizes a ayoff of (56)

5 Solving a larger game by alying Iterated Elimination of Dominated strategies : a b c d e A 0, 5,, 0 3, 6 4, 4 Player B 8, 3 4, 8 3, 0, 4 7, 3 C 6, 4 5, 6 4, 6, 8, 7 D 4, 7 3, 8, 0 5, 9 5, 8 E, 4, 5, 4 0, 5, 0 Recognize that neither layer has a dominant strategy Further, for Player none of the strategies are dominated However, for strategy a is dominated by d => d is not always the best choice for, but it is always better than a (for anything that Player could choose) Eliminate a a b c d e A 0, 5,, 0 3, 6 4, 4 Player B 8, 3 4, 8 3, 0, 4 7, 3 C 6, 4 5, 6 4, 6, 8, 7 D 4, 7 3, 8, 0 5, 9 5, 8 E, 4, 5, 4 0, 5, 0 In this reduced game, for Player the strategy of A is now dominated by D => eliminate A a b c d e A 0, 5,, 0 3, 6 4, 4 Player B 8, 3 4, 8 3, 0, 4 7, 3 C 6, 4 5, 6 4, 6, 8, 7 D 4, 7 3, 8, 0 5, 9 5, 8 E, 4, 5, 4 0, 5, 0 In this reduced game, for the strategy of e is now dominated by d => eliminate e a b c d e A 0, 5,, 0 3, 6 4, 4 Player B 8, 3 4, 8 3, 0, 4 7, 3 C 6, 4 5, 6 4, 6, 8, 7 D 4, 7 3, 8, 0 5, 9 5, 8 E, 4, 5, 4 0, 5, 0

6 In this reduced game, for Player the strategies of B and D are dominated by C => eliminate B and D a b c d e A 0, 5,, 0 3, 6 4, 4 Player B 8, 3 4, 8 3, 0, 4 7, 3 C 6, 4 5, 6 4, 6, 8, 7 D 4, 7 3, 8, 0 5, 9 5, 8 E, 4, 5, 4 0, 5, 0 In this reduced game, for the strategies of b and c are dominated by d => eliminate b and c a b c d e A 0, 5,, 0 3, 6 4, 4 Player B 8, 3 4, 8 3, 0, 4 7, 3 C 6, 4 5, 6 4, 6, 8, 7 D 4, 7 3, 8, 0 5, 9 5, 8 E, 4, 5, 4 0, 5, 0 In this reduced game, for Player the strategy of E is dominated by C => eliminate E a b c d e A 0, 5,, 0 3, 6 4, 4 Player B 8, 3 4, 8 3, 0, 4 7, 3 C 6, 4 5, 6 4, 6, 8, 7 D 4, 7 3, 8, 0 5, 9 5, 8 E, 4, 5, 4 0, 5, 0 Via Iterated Elimination of Dominated Strategies, we have identified the strategy air of C for Player and d for as the unique Nash Equilibrium of the game

7 Examle for which IEDS cannot solve the game: But, not all games can be solved by this aroach. Consider: Left Right To 8, 7 4, 5 Player Bottom, 6, 3 Neither layer has a dominant strategy or a dominated strategy => cannot whittle away any strategies by IEDS But, this game is simle enough that we can see (by drawing our best resonse arrows ) that there are actually multile Nash Equilibria Left Right To 8, 7 4, 5 Player Bottom, 6, 3 In a x game, what do the best resonse arrows reveal?. Whether a layer has or does not have a dominant strategy If all the arrows for a layer oint in the same direction, then the strategy associated with either the row or column to which they all oint is a dominant strategy (since this indicates that the best rely for the layer is the same, regardless of the strategy chosen by her rival) If all the arrows for a layer do NOT oint in the same direction, then the layer does not have a dominant strategy (since this indicates that the best rely for the layer deends uon the strategy chosen by her rival). Whether a air of strategies is or is not an equilibrium If for a articular cell all the arrows oint inward, then the air of strategies leading to this cell is a Nash Equilibrium (since this indicates that no layer could increase her own ayoff by choosing a different strategy) If for a articular cell any arrows oint outward, then the air of strategies leading to this cell is NOT a Nash Equilibrium (since this indicates that at least one layer could increase her ayoff by choosing a different strategy)

8 Ways to otentially maintain cooeration in a Prisoners Dilemma:. Resolving the dilemma by Reeated Interaction Consider the following Prisoners Dilemma: Cooerate Defect Cooerate 0, 0, Player Defect, 4, 4 Suose this game is reeated an uncertain number of times. After each eriod, the layers will meet again with robability Recognize that in a reeated game, strategies can be very comlex, since at any oint in time a layer can condition his choice on anything that has reviously been observed. Examles of strategies in this reeated Prisoners Dilemma: always choose cooerate in every eriod always choose defect in every eriod choose cooerate in every even eriod and defect in every odd eriod in each eriod lay whatever strategy my oonent layed last eriod (tit-for-tat) always choose cooerate so long as my oonent has chosen cooerate in every revious eriod, but once my oonent chooses defect then choose defect from in every single future eriod (grim trigger) If my oonent announces she will use grim trigger, what is my best rely? If I choose cooerate in every eriod, then my exected ayoff is: 3 4 C If I instead choose defect in every eriod, then my exected ayoff is: 3 4 D Always choosing cooerate is better if and only if: C D i 3 4 i Define... A Recognize that A... 3 Thus, A A (...) ( ) It follows that A( ) A For this game always choosing cooerate is a best rely to grim trigger (in which case reetition can sustain cooeration) so long as

9 . Resolving the dilemma by Legal Restrictions If the two layers can legally agree (ahead of time) to both cooerate, then clearly the dilemma can be resolved. But, such oen collusion between firms if often illegal in most countries However, U.S. cigarette manufacturers were able to resolve a dilemma by legal restriction in the early 970s Consider the following Prisoners Dilemma Phili Morris Advertise on TV No TV ads RJ Advertise on TV 0, 0 40, 5 Reynolds No TV ads 5, 50 35, 45 The simle fact that these firms were engaged in a reeated risoner s dilemma was not enough to resolve the dilemma rior to 97. However, on January, 97 they were able to resolve the dilemma and no longer advertise on TV Aroximate advertising exenditures by cigarette manufacturers: $300 million in 970; $40 in 97 Firms were able to realize larger rofits in 97 after resoling the dilemma => great news for cigarette manufacturers! How were they able to do this? Starting on //7, there was a legal agreement which made it that firms would no longer advertise on TV How could the Government let this haen!? The government was the one who made it haen => on //7 the Public Health Cigarette Smoking Act (which made it illegal to advertise cigarettes on TV and radio) went into effect

10 Determining a Mixed Strategy Equilibrium: Consider the following game: Left Right Player To 4, 7 0, 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 robability with which chooses Left (so that q denotes the robability with which chooses Right ) denotes the robability with which Player chooses To (so that denotes the robability with which Player chooses Bottom ) Derive and grahically illustrate the Best Rely Corresondence for each layer Consider the choice by Player when chooses Left with robability q and Right with robability q T Player s exected ayoff from choosing To is: 4q 0( q) 0 6q B Player s exected ayoff from choosing Bottom is: 6q ( q) 4q Thus, Player s ayoff is strictly greater from choosing To as oosed to Bottom (in which case his best rely is ) if and only if T B 8 q Further, Player s ayoff is strictly greater from choosing Bottom as oosed to To (in which case his best rely is 0 ) if and only if q. 8 Finally, Player would realize the exact same exected ayoff from choosing To, Bottom, or any randomization between the two (in which case any value of 0 is a best rely ) if and only if q. 8 Visually, the Best Rely Corresondence of Player can be illustrated as:.8 q BR ( q) 0 0 Similarly, considering the choice by when Player chooses To with robability and Bottom with robability L s exected ayoff from choosing Left is: 7 3( ) 3 4 R 5( ) 5 s exected ayoff from choosing Right is: 4

11 Thus, s ayoff is strictly greater from choosing Left as oosed to Right (in which case his best rely is q ) if and only if L R Further, s ayoff is strictly greater from choosing Right as oosed to Left (in which case his best rely is q 0 ) if and only if. 5 Finally, would realize the exact same exected ayoff from choosing Left, Right, or any randomization between the two (in which case any value of 0 q is a best rely ) if and only if. 5 Visually, the Best Rely Corresondence of can be illustrated as: q BR ( ) Drawing the two corresondences in the same grah, we have: q BR ( ) BR ( q) If Player chooses *. 5 and chooses q *. 8, then each layer is choosing a mixed strategy that is a best rely to the strategy being layed by his rival => this air of mixed strategies is a Nash Equilibrium!

12 Sequential Move Games: Consider the following sequential move game: Firm Don t enter Market B Enter Market B Firm Firm Maintain Collusion Price War Maintain Collusion Price War Equilibrium strategies that are NOT subgame erfect => consider the following air of strategies: Firm chooses Maintain Collusion if Firm chooses Don t Enter Market B and chooses Price War if Firm chooses Enter Market B (i.e., Firm threatens a rice war if and only if Firm enters ) Firm chooses Don t Enter Market B Firm These air of strategies can be illustrated as: Don t enter Market B Enter Market B Firm Firm Maintain Collusion Price War Maintain Collusion Price War

13 ( Sequential Move Games continued) This air of strategies above does fit our definition of a Nash Equilibirium ( check all ossible unilateral deviations ), but it doesn t seem reasonable! Why not? The threat by Firm to engage in a rice war following entry is NOT CREDIBLE => does not seem reasonable to suort an equilibrium on such a threat that would clearly not be carried out if Player were to ever call the bluff of For sequential move games, a refinement of Nash Equilibrium has been develoed, in order to rule out such unreasonable equilibria => Subgame Perfect Nash Equilibrium So long as each layer has a different ayoff at each terminal node of the game, there will be exactly one SPNE for the game, which can be identified via backward induction First, consider the choice by Firm following each ossible choice by Firm Firm Don t enter Market B Enter Market B Firm Firm Maintain Collusion Price War Maintain Collusion Price War

14 ( Sequential Move Games continued) Next, considering the initial choice by Firm Firm Don t enter Market B Enter Market B Firm Firm Maintain Collusion Price War Maintain Collusion Price War Unique SPNE consists of the following air of strategies (note: we must secify the otimal choice at every decision node, even those that are not reached during the lay of the game) Firm chooses Maintain Collusion if Firm chooses Don t Enter Market B and chooses Maintain Collusion if Firm chooses Enter Market B Firm chooses Enter Market B This solution is illustrated most directly by drawing arrows along each relevant branch of the game tree (again, indicate exactly one choice after every decision node) and circling the ayoff vector at the realized outcome

15 Multile Choice Questions:. Consider a two layer game between Player and. Player has two available strategies: Strategy A and Strategy B. has three available strategies: Strategy c, Strategy d, and Strategy e. If Strategy A of Player is a Best Rely to a choice of Strategy c by, 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 (erhas all) of the above answers is correct.. Collusion refers to A. a game in which all layers have a dominant strategy. B. a game in which all layers comletely disregard the imact of their own actions on their own ayoff. C. an effort by firms to coordinate their actions in an attemt to increase both total industry rofit and individual rofit of every firm (comared to the rofit levels which would result without any such coordination). D. a market structure in which firms are able to engage in First Degree Price Discrimination. 3. Which of the following statements corresonds 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, exect her to never use it. C. If you have a dominant strategy, recognize that your rival will exect that you will never use it. D. More than one (erhas all) of the above answers is correct. 4. Nash s Existence Theorem states that A. individuals can only every overcome a Prisoners Dilemma by reeated 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 layers, each with a finite number of available ure strategies, has at least one Nash Equilibrium (otentially in Mixed Strategies ). 5. 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 ).

16 6. A game that is layed between the same layers more than one time is called a A. One-Shot Game. B. Reeated Game. C. Game of Incomlete Information. D. Game of Comlete Information. 7. The ioneering work Theory of Games and Economic Behavior was written by A. Adam Smith. B. John F. Nash, Jr. and Lloyd Shaley. C. John Maynard Keynes. D. John von-neumann and Oskar Morgenstern. 8. 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 layer is allowed to be a robability distribution over his available ure strategies. C. there can never be any Nash Equilibria. D. layers are never allowed to lay Pure Strategies. 9. In a simultaneous move game with two layers, it must always be the case that A. the sum of the ayoffs of both layers is maximized at an outcome that is a Nash Equilibrium. B. the sum of the ayoffs of both layers is minimized at an outcome that is a Nash Equilibrium. C. there is at least one Nash Equilibrium (otentially in mixed strategies ). D. More than one (erhas all) of the above answers is correct. Problem Solving or Short Answer Questions:. Consider the layer simultaneous move game below: Left Right Player To 80, 65 50, 45 Bottom 70, 75 35, 55 A. Does this game fit the definition of a Prisoner s Dilemma? Clearly exlain why or why not. B. Determine all Nash Equilibria of this game.

17 . Consider the following sequential move game: Player Left Right left right left right Identify a air of strategies that is a Nash Equilibrium but is not a Subgame Perfect Nash Equilibrium. 3. For each of the following games, determine if either layer has a Dominant Strategy and identify all Pure Strategy Nash Equilbria. 3A. Left Right To 0, 5, 3 Player Bottom 8, 7 6, 3B. 3C. 3D. Player Player Player Left Right To, 5 6, 9 Bottom 8, 4, 3 Left Right To 0, 5 4, 0 Bottom 6, 0, 5 Left Right To 30, 5 0, 35 Bottom 0, 45 40, 5

18 4. Determine all Nash Equilibria of each of the following games. 4A. Left Right To 3,, 0 Player Bottom, 4, 5 4B. 4C. 4D. Player Player Player Left Right To 4, 9, 5 Bottom 0, 7 0, 3 Left Center Right To 6, 0 5, 0, 9 Middle, 6 3, 8 5, 5 Bottom 4, 8 9, 0 0, 7 Left Center Right To, 4 5, 0 8, 3 Middle 0, 7, 3 4, 0 Bottom, 5 3, 6, 9 5. Consider the following simultaneous move game: Left Center Right To 8, 8 0, 0 0, 0 Player Middle 0, 0 4, 4 0, 0 Bottom 0, 0 0, 0, For the Mixed Extension of this game, let denote the robability that Player chooses To, denote the robability that Player chooses Middle, let 3 denote the robability that Player chooses Bottom, q denote the robability that chooses Left, q denote the robability that chooses Center, and q3 q q denote the robability that chooses Right. 5A. Identify all Pure Strategy Nash Equilibria. 5B. Is the strategy air,, 3,, and q 3, q, q3,, a Nash 3 5C. Equilibrium? Exlain why or why not. Is the strategy air,, 3,,0 and q,,,,0 q q3 a Nash Equilibrium? Exlain why or why not. 5D. Based uon your answers to arts (5B) and (5C), identify a different air of strategies that is a Mixed strategy Nash Equilibrum.

19 6. Consider the following three layers sequential move game. In each ayoff vector, the to number indicates the ayoff of Player, the middle number indicates the ayoff of, and the bottom number indicates the ayoff of layer 3. Determine the unique Subgame Perfect Nash Equilibrium of this game. Player A B C D E F G H Player 3 Player 3 Player 3 Player 3 Player 3 Player 3 I J K L M N O P Q R S T U V Player W X Y Z A Player B Consider the following two layer simultaneous move game: Player Left Right To a, b, Bottom, c, d Secify values of a, b, c, and d for which: 7A. To is a dominant strategy for Player and Left is a dominant strategy for. 7B. neither layer has a dominant strategy and there are no Pure Strategy Equilibria. 7C. neither layer has a dominant strategy and there are two Pure Strategy Equilibria. 7D. there are no Pure Strategy Equilibria and the unique Mixed Strategy Equilibrium is characterized by Player choosing To with robability 5 and choosing Left with robability 3.

20 8. Consider the following two layer simultaneous move game: Player Left Right To,, 0 Bottom 0, 8, 8 8A. Does this game fit the definition of a Prisoners Dilemma? Clearly exlain. 8B. If this game is layer only once, what outcome would you exect to observe? 8C. For the remained of this question, suose that this game will be reeated an uncertain number of times. After each eriod, the game will be layed exactly one more time with robability. In this reeated game, if were to follow a strategy of always choose Right in every single eriod, what is the best rely of Player? Determine the exected value of Player s ayoff when he uses this best rely against s strategy of always choose Right in every single eriod. 8D. Continuing to suose that this game will be reeated as described above, suose that is using the grim trigger strategy (i.e., choosing Left initially, and continuing to choose Left in each future eriod if and only if Player has chosen To in every single revious eriod). Determine Player s exected ayoff from choosing To in every eriod against this strategy of. Determine Player s exected ayoff from choosing Bottom in every eriod against this strategy of. For what values of can cooeration be maintained in this reeated game? Answers to Multile Choice Questions:. B. C 3. A 4. D 5. B 6. B 7. D 8. B 9. C Answers to Problem Solving or Short Answer Questions: A. The definition of a Prisoner s Dilemma was: a game in which every layer has a dominant strategy (so that the game has a unique equilibrium characterized by all layers using their dominant strategies), but in which there is some other outcome at which the ayoff of every single layer is higher than the equilibrium ayoff. For the game under consideration, both layers do have dominant strategies ( To is the dominant strategy for Player and Left is the dominant strategy for ). However, when Player lays To and lays Left, the realized

21 ayoffs are (80) for Player and (65) for. That is, each layer is realizing the highest ayoff that she could ossibly realize. From here, it is clear that this is not a Prisoner s Dilemma (since there is not some other outcome at which the ayoff of every single layer is higher than the equilibrium ayoff ). B. Since each layer has a dominant strategy, the unique Nash Equilibrium is for each layer to follow her dominant strategy. That is, the unique Nash Equilibrium is for Player to lay To and for to lay Left.. By backward induction, the unique Subgame Perfect Nash Equilibrium is: Player Left Right left right left right Recognize that at this outcome does not realize her highest ossible ayoff of (00). Any Nash Equilibrium that is NOT Subgame Perfect must involve a layer making a non-credible threat/romise. Suose that attemts to make such a non-credible commitment in order to realize her highest ossible ayoff of (00). More recisely, uses the strategy of choosing left following a choice of Left by Player and choosing left following a choice of Right by Player. If were to in fact lay according to this strategy, then the best rely of Player would be to choose Left. This air of strategies is illustrated as: Player Left Right left right left right This air of strategies is an equilibrium, since neither layer can increase her own ayoff by changing only her choice at any of the decision nodes. But again, it is not a Subgame

22 Perfect Equilibrium, since it rests uon making a sub-otimal choice in the Subgame following a choice of Right by Player. 3A. Player has a dominant strategy of To (it is a best rely for each available strategy of ). has a dominant strategy of Left (it is a best rely for each available strategy of Player ). Thus, the unique equilibrium of the game is a Pure Strategy Nash Equilibrium in which Player lays To and lays Left. 3B. Player does not have a dominant strategy ( Bottom is the best rely to Left, while To is the best rely to Right ). has a dominant strategy of Right (it is a best rely for each available strategy of Player ). Thus, the unique equilibrium of the game is a Pure Strategy Nash Equilibrium in which lays Right (his dominant strategy) and Player lays To (her best rely to the dominant strategy of ). 3C. Neither layer has a dominant strategy. For Player, To is the best rely to Left, while Bottom is the best rely to Right. For, Left is the best rely to To, while Right is the best rely to Bottom. Thus, the game has two Pure Strategy Nash Equilibria one in which Player lays To and lays Left, and another in which Player lays Bottom and lays Right. 3D. Neither layer has a dominant strategy. For Player, To is the best rely to Left, while Bottom is the best rely to Right. For, Right is the best rely to To, while Left is the best rely to Bottom. Thus, the game has no Pure Strategy Nash Equilibria. 4A. This game has two Pure Strategy Nash Equilibria as follows: Left Right To 3,, 0 Player Bottom, 4, 5 Additionally, letting denote the robability with which Player lays To and letting q denote the robability with which lays Left, a Mixed Strategy Nash Equilibrium can be determined as: T B 3q ( q) q ( q) q q * and L R 4( ) 5( ) * 4B. This game does not have any Pure Strategy Nash Equilibria: Left Right To 4, 9, 5 Player Bottom 0, 7 0, 3

23 However, by Nash s Existence Theorem we know that there must exist a Mixed Strategy Equilibrium. Letting denote the robability with which Player lays To and letting q denote the robability with which lays Left, a Mixed Strategy Nash Equilibrium can be determined as: T B 4q ( q) 0( q) q 0 0q q * 3 and L R 9 7( ) 5 3( ) 7 3 * 5 4C. Start by recognizing that for, Right is dominated by Left. Once we eliminate Right from consideration, we are left with the reduced game: Left Center Right To 6, 0 5, 0, 9 Player Middle, 6 3, 8 5, 5 Bottom 4, 8 9, 0 0, 7 In this reduced game, Bottom is dominated by To, giving us: Left Center Right To 6, 0 5, 0, 9 Player Middle, 6 3, 8 5, 5 Bottom 4, 8 9, 0 0, 7 From here, recognize that in this reduced game Left is dominated by Center, so that we have: Left Center Right To 6, 0 5, 0, 9 Player Middle, 6 3, 8 5, 5 Bottom 4, 8 9, 0 0, 7 In this reduced game, Middle is dominated by To. Thus, the unique Nash Equilibrium of this game is for Player to choose To and to choose Center, as illustrated below: Left Center Right To 6, 0 5, 0, 9 Player Middle, 6 3, 8 5, 5 Bottom 4, 8 9, 0 0, 7

24 4D. Start by recognizing that for Player, Bottom is dominated by To. Further, if we consider the reduced game in which Player does not consider Bottom, then for Player Right is dominated by Left. Eliminating Bottom for Player and Right for leaves us with the following reduced game: Left Center Right To, 4 5, 0 8, 3 Player Middle 0, 7, 3 4, 0 Bottom, 5 3, 6, 9 First recognize that in this reduced game there are two Pure Strategy Nash Equilibria as illustrated below: Left Center Right To, 4 5, 0 8, 3 Player Middle 0, 7, 3 4, 0 Bottom, 5 3, 6, 9 Further, considering the mixed extension of this reduced game (in which denotes the robability with which Player lays To and q denotes the robability with which lays Left ), there is a Mixed Strategy Nash Equilibrium which can be identified as follows: T M q 5( q) 7( q) 5 3q 7 7q q * and L C 4 ( ) 3( ) * 3 5A. There are three Pure Strategy Nash Equilibria as follows: one in which Player chooses To and chooses Left ; a second in which Player chooses Middle and chooses Center ; and a third in which Player chooses Bottom and chooses Right. 5B. The strategy air,, 3,, and q 3, q, q3,, is not a Nash Equilibrium. 3 To see this, recognize that if were to use q, q, q3,,, then Player s 3 8 ayoff from laying To is 8 0 0, which is strictly greater than his ayoff of laying either Middle (of ) or Bottom (of 0 0 ). Thus, Player s best rely would be,, 3,0,0, not,, 3,,. This 3 alone imlies that the given strategy air is not a Nash Equilibrium. 5C. The strategy air,, 3,,0 and q,,,,0 q q3 is a Nash Equilibrium. To see this, we must verify that each layer is choosing a strategy that is a best rely to the q q, q,,0, then Player strategy being layed by his rival. When uses, 3

25 8 would realize a ayoff of from laying Middle, and a ayoff of from laying To, a ayoff of from laying Bottom. Thus, any strategy that lays Bottom with robability zero is a best rely for Player. The strategy,, 3,,0 satisfies this criterion. Similarly, when Player uses,, 3,,0, then would realize a ayoff of from laying Left, a ayoff of from laying 3 3 Center, and a ayoff of from laying Right. Thus, any strategy that lays Right with robability zero is a best rely for. The strategy q q, q,,0 satisfies this criterion., 3 5D. Beyond the Mixed Strategy Nash Equilibrium given in art (5C), three other Mixed Strategy Nash Equilibria exist. Two of these are similar to that given in (5C), in that for each layer they lace ositive weight on two of the three available strategies. These two equilibria are: Player lays,, 3 0,, and lays, q, q3 0,, 4 4 and Player lays,,,0 and lays q, q, q,0 q ; 3, 5 5 3, 5 5. The final mixed strategy equilibrium involves each layer randomizing over all three of his available strategies. More recisely, this equilibrium is: Player lays,,, 4 q, q, q, 4. and lays 3, , The unique Subgame Perfect Nash Equilibrium can be identified via backward induction as: Player A B C D E F G H Player 3 Player 3 Player 3 Player 3 Player 3 Player 3 I J K L M N O P Q R S T U V Player W X Y Z Player A B

26 7A. For To to be a dominant strategy for Player, we need a and c. For Left to be a dominant strategy for, we need b and d. 7B. For neither layer to have a dominant strategy and for no Pure Strategy Equilibria to exist, we need the best resonse arrows to result in a cycle. One way to have this is for a, b, c, and d. 7C. For neither layer to have a dominant strategy and for two Pure Strategy Equilibria to exist, we need the best resonse arrows to be such that either: To/Left and Bottom/Right each have both arrows ointing inward; or To/Right and Bottom/Left each have both arrows ointing inward. To have the former, we would need a, b, c, and d. 7D. Again, to have no Pure Strategy Equilibria we need the best resonse 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 otimally choose a mixed strategy when is laying Left with robability 3, we need: a ( ) ( ) a c 3 a c Similarly, in order for to otimally choose a mixed strategy when Player is laying To with robability 5, we need: 4 b 4 ( ) ( ) d b d b 6 A set of values satisfying these restrictions is: a 3, c 4, b 0, and d 3. 8A. The definition of a Prisoner s Dilemma was: a game in which every layer has a dominant strategy (so that the game has a unique equilibrium characterized by all layers using their dominant strategies), but in which there is some other outcome at which the ayoff of every single layer is higher than the equilibrium ayoff. For Player, Bottom is a dominant strategy, and for, Right is a dominant strategy. When this air is used, each layer gets a ayoff of (8). If instead Player chose To and chose Left, then each layer would get a ayoff of (). So, yes, this game does fit the definition of a Prisoners Dilemma. 8B. If the game is layed only once, then each layer should use her dominant strategy. We should exect Player to lay Bottom, to lay Right, and the layers to each realize a ayoff of (8). 8C. If were to always choose Right in every single eriod (regardless of the choices made by Player ), then Player should choose Bottom in every eriod. If this air of strategies is chosen, then Player will realize a ayoff of (0) in each and every eriod. Given the robability of having the reeated game layed one more time, this outcome gives Player an exected ayoff of: i i c d

27 i Recall that. Thus, the ayoff of Player in this case is i D. Suose uses the grim trigger strategy. If Player chooses To in every eriod, then she realizes a ayoff of () in each eriod, giving her an exected ayoff of i If instead Player chooses Bottom in every eriod, then she realizes a ayoff of (0) in the first eriod and a ayoff of (8) in any subsequent eriod. This choice gives her an exected ayoff of: i 0 8 i 0 8 Cooeration (i.e., chooses Left and Player chooses To in every eriod) can be maintained so long as: 0 8 i