3.2.2. Dynamic Games of complete information: Backward Induction and Subgame perfection.

3.2.2. Dynamic Games of complete information: Backward Induction and Subgame perfection

Strategic Behavior in Business and Econ Outline 3.1. What is a Game ? 3.1.1. The elements of a Game 3.1.2 The Rules of the Game: Example 3.1.3. Examples of Game Situations 3.1.4 Types of Games 3.2. Solution Concepts 3.2.1. Static Games of complete information: Dominant Strategies and Nash Equilibrium in pure and mixed strategies 3.2.2. Dynamic Games of complete information: Backward Induction and Subgame perfection

Strategic Behavior in Business and Econ Dynamic Games of Complete Information Players choose their strategies sequentially, one after the other This means some players take actions knowing what others have done Because of this sequentiality they must represented using trees They are also “one-shot games”, that is, they are played only once Nevertheless, some special dynamic games consist of the repetition of a one-shot game played several times All the players have all the information regarding who are the other players, what are the own strategies and the strategies of the others, what are the own payoffs and the payoffs of the others, and what are the rules of the game Reminder

Strategic Behavior in Business and Econ Solution concepts for this type of games Nash Equilibrium (pure and mixed strategies) Works in any case (but sometimes it doesn't make sense) Equilibrium by Backward Induction When there is “perfect information” Subgame Perfect Equilibrium When there is “imperfect information”

Strategic Behavior in Business and Econ Reminder An equilibrium of the game is a choice of strategies by all the players that is stable, in the sense that Given what the other players are doing, nobody has any reason to change his or her own strategy

Strategic Behavior in Business and Econ Example: The Battle of the Sexes (dynamic version) Pat and Chris want to go out together after work They work on different places and before going to work they couldn't find any agreement on where to go The options were go to the Opera of go to the Football They both would like to go to a place together, but Pat prefers the Opera whereas Chris likes the Football better Thus, the situation is that after work (5 pm) each must decide where to go without knowing the choice of the other. Pat will choose first and will call Chris

Strategic Behavior in Business and Econ The environment of the game Players:Pat and Chris Strategies:(NOT ONLY) Opera or Football (more on this next) Payoffs:In this case we must “define” the payoffs in such a way that represent the game described (see the Table in the next slide) The Rules of the Game Timing of movesSimultaneous Sequential Nature of conflict and interactionCoordination Information conditionsSymmetric

Strategic Behavior in Business and Econ The game represented (Static version) Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 Look for the best replies

Strategic Behavior in Business and Econ The game represented (Static version) Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 There are two Nash Equilibria

Strategic Behavior in Business and Econ The game represented (Static version) Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 Pat and Chris going both to the Opera is stable

Strategic Behavior in Business and Econ The game represented (Static version) Football Opera Pat Chris 3, 1 Football Opera 0, 0 1, 3 -1, -1 Pat and Chris going both to the Football is stable

Strategic Behavior in Business and Econ The game represented (Dynamic version) Football Opera Pat The sequences of choices (moves) must be represented by a tree Chris Opera Football 3, 1 0,0 -1, -1 1, 3

Strategic Behavior in Business and Econ What are the strategies of the players ? For Pat (who moves first) Go to the Opera (and call Chris saying I am doing so) Go to the Football (and call Chris saying I am doing so) So, Pat has 2 strategies: Opera (O) Football (F)

Strategic Behavior in Business and Econ What are the strategies of the players ? For Chris (that moves second, after learning Pat's choice) Chris knows what Pat is doing, hence Chris' strategy must take into account the choice by Pat If Pat goes to the Opera I will go to the Opera, and if Pat goes to the Football I will go to the Football If Pat goes to the Opera I will go to the Opera, and if Pat goes to the Football I will go to the Opera If Pat goes to the Opera I will go to the Football, and if Pat goes to the Football I will go to the Opera If Pat goes to the Opera I will go to the Football, and if Pat goes to the Football I will go to the Football

Strategic Behavior in Business and Econ What are the strategies of the players ? In dynamic (sequential moves) games, an strategy is a complete plan of action for the player. In this sense, a strategy must specify what action to take for each possible choice of an strategy by the other(s) player(s) In these games, strategies are of the form If... then I will..., and If... then I will..., and... An strategy implies knowing before hand (before the game starts) what to do in each possible contingency of the game

Strategic Behavior in Business and Econ What are the strategies of the players ? Think of a strategy in a dynamic game as a “guide” that you can write down in a piece of paper and give it to someone to play the strategy for you For instance, imagine that Chris is the CEO of the company and that there is a Limousine waiting to take Chris to either the Opera or the Football game. Chris is very busy and don't want to be bothered with phone calls. So, Chris tells Pat to call to the chauffeur after work telling him the chosen place. Once in the Limo, Chris can give the chauffeur instructions like: “If Pat is going to the Opera take me to the Opera, and if Pat is going to the Football take me to the Football”

Strategic Behavior in Business and Econ What are the strategies of the players ? Chris could also say to the chauffeur, “If Pat is going to the Opera take me to the Football, and if Pat is going to the Football take me to the Football” or any other combination of the type If... then I will..., and If... then I will... So, Chris has the 4 strategies we have seen before

Strategic Behavior in Business and Econ What are the strategies of the players ? Chris' strategy must be contingent on Pat's strategy If Pat goes to the Opera I will go to the Opera, and if Pat goes to the Football I will go to the Football If Pat goes to the Opera I will go to the Opera, and if Pat goes to the Football I will go to the Opera If Pat goes to the Opera I will go to the Football, and if Pat goes to the Football I will go to the Opera If Pat goes to the Opera I will go to the Football, and if Pat goes to the Football I will go to the Football

Strategic Behavior in Business and Econ What are the strategies of the players ? This strategies are often “represented” as If Pat goes to the Opera I will go to the Opera, and if Pat goes to the Football I will go to the Football If Pat goes to the Opera I will go to the Opera, and if Pat goes to the Football I will go to the Opera If Pat goes to the Opera I will go to the Football, and if Pat goes to the Football I will go to the Opera If Pat goes to the Opera I will go to the Football, and if Pat goes to the Football I will go to the Football O F O F O F What to do if Pat chooses O What to do if Pat chooses F

Strategic Behavior in Business and Econ The environment of the game Players:Pat and Chris Strategies:For Pat: O, F For Chris: O O, O F, F O, F F Payoffs:(see the Tree) The Rules of the Game Timing of movesSequential Nature of conflict and interactionCoordination Information conditionsSymmetric

Strategic Behavior in Business and Econ The environment of the game Players:Pat and Chris Strategies:For Pat: O, F For Chris: O O, O F, F O, F F Payoffs:(see the Tree) The Rules of the Game Timing of movesSequential Nature of conflict and interactionCoordination Information conditionsSymmetric What to do if Pat chooses O What to do if Pat chooses F

Strategic Behavior in Business and Econ Strategies in dynamic games In dynamic games (players move sequentially, some of them knowing the choices of other players) the strategies have to specify, as before, what to do in each possible contingency of the game. That is, what to do for each possible combination of the strategies chosen by the players that moved before. Think of a 3-person Battle of the Sexes... This becomes very complicated !!

Strategic Behavior in Business and Econ The game represented (Dynamic version) Football Opera Pat How to find the Nash Equilibria ? Chris Opera Football 3, 1 0,0 -1, -1 1, 3

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players Thus, this game Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players Can be represented in table form taking into account the strategies of the players For Pat: O, F For Chris: O O, O F, F O, F F

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players as this: Pat Chris

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players as this: Pat Chris Check that the payoffs in the table correspond to those in the tree !!

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players as this: Pat Chris The “strange” appearance of the table proves that is it not suitable to represent Dynamic Games

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players as this: Pat Chris But it is very helpful (only) to find all the Nash Equilibria in Dynamic Games

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players as this: Pat Chris We can look for the players' Best Replies in the table (red circles) as usual

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players as this: Pat Chris And we find 3 Nash Equilibria

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players as this: Pat Chris And we find 3 Nash Equilibria Pat Chris (1) O OO (2) O OF (3) F FF

Strategic Behavior in Business and Econ Any Dynamic Game represented in tree form can also be represented in table form paying special attention to the strategies of the players as this: Pat Chris The 3 Nash Equilibria can be represented in the tree Pat Chris (1) O OO (2) O OF (3) F FF

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 The 3 Nash Equilibria can be represented in the tree Pat Chris (1) O OO

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 The 3 Nash Equilibria can be represented in the tree Pat Chris (1) O OO (2) O OF

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 The 3 Nash Equilibria can be represented in the tree Pat Chris (1) O OO (2) O OF (3) F FF

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat Chris (1) O OO (2) O OF (3) F FF

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 Thus, we basically have the same outcomes as in the static version, only that one of them (3, 1) appears twice

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 But, does the first equilibrium (1) make any sense ??

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 Equilibrium (1) says that if Pat goes to the Football Chris will go to the Opera !!

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 This won't be rational ! Hence, such equilibrium will never be carried out It is NOT sequentially rational

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 Equilibrium (1) is a “bad” equilibrium

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 Now, does the third equilibrium (3) make any sense ??

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 Equilibrium (3) says that if Pat goes to the Opera Chris will go to the Football !!

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 Again, this isn't rational ! Hence, such equilibrium will never be carried out It is NOT sequentially rational

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 Equilibrium (3) is a “bad” equilibrium as well

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 Only equilibrium (2) is a “good” equilibrium. It is sequentially rational

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Each Nash Equilibria produces a path that determines a final outcome and its corresponding payoff Pat ChrisPayoff (1) O OO3, 1 (2) O OF3, 1 (3) F FF1, 3 This equilibrium (2) is the one we would find by solving the game using Backward Induction

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Backward Induction If Path goes the Opera, the best choice by Chris is to go to the Opera as well

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Backward Induction If Path goes the Football, the best choice by Chris is to go to the Football as well

Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Backward Induction Then, knowing what the best replies of Chris are, the best choice by Pat is to go to the Opera

Although Chris seems to be in a better position (has the information), Pat knows that Chris has the information and can take advantage of !!! (First mover advantage) Strategic Behavior in Business and Econ Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 Backward Induction Then, knowing what the best replies of Chris are, the best choice by Pat is to go to the Opera

Strategic Behavior in Business and Econ Summary Dynamic games are games in which players move sequentially Dynamic games are better represented in tree form Special attention has to be placed in determining correctly the strategies of all the players Table form representation is useful to find all the Nash Equilibria But many of the Nash Equilibria found will not be “good” (sequentially rational) as the table misses the sequential nature of the game Backward Induction is the technique that finds only the “good” (sequentially rational) equilibria An equilibrium is sequentially rational if each time a player has to move, what the equilibrium says he or she has to do is a best reply to what has happened before in the game

Strategic Behavior in Business and Econ A word about Complete vs. Incomplete information and Perfect vs. Imperfect information When looking at the different types of games, we saw that there were games of Complete and of Incomplete information Now we will see that within the dynamic games (both of Complete and of Incomplete information) there are games of Perfect and of Imperfect information They do not refer to the same thing

Strategic Behavior in Business and Econ Complete vs. Incomplete information refers to what the players know about the game at the beginning

Strategic Behavior in Business and Econ Complete vs. Incomplete information refers to what the players know about the game at the beginning Perfect vs. Imperfect information refers to what the players know about what has happened in the game when they have to play

Strategic Behavior in Business and Econ The Battle of the Sexes (dynamic version) is a dynamic game of Perfect Information Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 When Pat has to play, Pat knows that it is the start of the game.

Strategic Behavior in Business and Econ The Battle of the Sexes (dynamic version) is a dynamic game of Perfect Information Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 When Pat has to play, Pat knows that it is the start of the game. When is Chris' turn to move, Chris knows whether Pat has gone to the Opera or to the Football

Strategic Behavior in Business and Econ The Battle of the Sexes (dynamic version) is a dynamic game of Perfect Information Football Opera Pat Chris Opera Football 3, 1 0,0 -1, -1 1, 3 When Pat has to play, Pat knows that it is the start of the game. When is Chris' turn to move, Chris knows whether Pat has gone to the Opera or to the Football Perfect information means that every time a player has to move, she or he knows the “history” of all the previous moves by other players in the game

Strategic Behavior in Business and Econ The Invest and buy game is a dynamic game of Imperfect Information A seller and a buyer want to trade an item. Before the trade takes place, the buyer can do an investment that would increase the value of the item. Without knowing if the buyer has done the investment or not, the seller can offer the item at either $1 or $2 (million) Once the buyer observes the offer by the seller, she can Accept or Reject the offer. If the offer is accepted the trade is carried out and both players collect their payoffs whereas if the offer is rejected both get $0 unless the buyer has done the investment (-$0.2 for the buyer in such case). The item has a value of $1.8 for the buyer, and the investment would increase such value in $0.5 but has a cost of $0.2

Strategic Behavior in Business and Econ The environment of the game Players:Buyer and Seller Strategies:For the Buyer: (too complex to describe here) (Has 32 strategies !) For the Seller: Offer at $2 or Offer at $1 Payoffs:(Next slide) The Rules of the Game Timing of movesSequential Nature of conflict and interactionCompetition Information conditionsComplete but Imperfect

Strategic Behavior in Business and Econ Payoff computation for the Buyer Value $1.8 Investment added value$0.5 Investment cost$0.2 The Payoff for the Seller is just the selling price

Strategic Behavior in Business and Econ Buyer Seller The game represented Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 The sequence of moves is represented in the game tree

Strategic Behavior in Business and Econ Buyer Seller The game represented Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 The sequence of moves is represented in the game tree But there is nothing in the tree that shows that the Seller doesn't know if the Buyer has invested or not

Strategic Behavior in Business and Econ Buyer Seller The game represented Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 The sequence of moves is represented in the game tree The fact of not having Perfect Information is represented with a dotted line that joins the nodes that contain the same information

Strategic Behavior in Business and Econ Buyer Seller The game represented Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 The sequence of moves is represented in the game tree This dotted line means that the Seller doesn't know in which of the two nodes the game is

Strategic Behavior in Business and Econ Buyer Seller The game represented Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 What would this mean for the Buyer ?

Strategic Behavior in Business and Econ Buyer Seller The game represented Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 What would this mean for the Buyer ? Means that the Buyer doesn't know the offer by the Seller (doesn't make any sense here)

Strategic Behavior in Business and Econ Buyer Seller The game represented Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 And what would this mean for the Buyer ?

Strategic Behavior in Business and Econ Buyer Seller The game represented Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 And what would this mean for the Buyer ? The Buyer doesn't remember if she invested or not ! (again, doesn't make any sense)

Strategic Behavior in Business and Econ Summary A game has Perfect Information if any player, every time he or she has to move, has all the information about all the moves of all the players that have played earlier If there is at least one player that at some point doesn't have all the information about the past moves in the game, then the game has Imperfect information

Strategic Behavior in Business and Econ Nash equilibrium in Dynamic Games The concept of Nash equilibrium in Dynamic Games is the Same as in Static games: every player is playing a best reply to what the other players are doing Looking for Nash equilibria in a tree representation is (usually) a difficult task To fin the Nash equilibria in a Dynamic game, the game must be represented in table form. Then, special attention must be paid to the strategies of the players Nash equilibria can be found in both, games with Perfect Information and games with Imperfect Information

Strategic Behavior in Business and Econ Example: The Centipede Game Two players take turns choosing either to take a slightly larger share of a slowly increasing pot, or to pass the pot to the other player. The payoffs are arranged so that if one passes the pot to the opponent and the opponent takes the pot on the next round, one receives slightly less than if one had taken the pot on this round.

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Reminder: An strategy is a complete plan of action. Should specify what to do in each possible contingency of the game

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Thus, the strategies of the players in this game should have 2 components: What to do the first time I have to move What to do the second time I have to move

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Hence, each player has 4 strategies: P P P T T P T T

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Clearly, T P (or TT) doesn't make any sense ! But it must be considered for completeness

Strategic Behavior in Business and Econ The Centipede Game (in table form) Player 1 Player 2 Once again, the table representation doesn't show the dynamics of the game. It's only useful to find the Nash Equilibria

Strategic Behavior in Business and Econ The Centipede Game (in table form) Player 1 Player 2 Looking for the Nash Equilibria (by finding the Best Replies), we have 4 equilibria that lead to the same outcome but thru different combinations of strategies

Strategic Behavior in Business and Econ The Centipede Game (in table form) Player 1 Player 2 The explanation is that since the final outcome is reached in the first step of the game, what the strategies say about what to do in later steps is not important at all

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game The 4 Nash Equilibria represented

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Clearly, not all 4 Nash Equilibria make sense (consider, for instance, the purple one). That's why “bare” Nash Equilibrium is not the RIGHT concept in dynamic games

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Dynamic games must be solved by Backward Induction (if the game is of Perfect Information) or by Subgame Perfection (if the game is of Imperfect Information)

Strategic Behavior in Business and Econ Equilibrium by Backward Induction The equilibrium of a game by Backward Induction is found by inspecting the tree representation of the game Backward Induction selects only those Nash equilibria that are sequentially rational Backward Induction can be used only in games with Perfect Information

Strategic Behavior in Business and Econ Solving the game by Backward Induction consists of: Start at the top-rightmost end of the tree For each set of choice branches of a player that has to move, find the corresponding best choice at that point Draw an “arrow” marking the path (branch) that corresponds to that best choice Proceed left wise to find another set of choice branches Again, find the corresponding best choice at that point according to the path (arrows) already found and mark it with an arrow Proceed in the same way until a best choice has been found for all the players along the sequence of the game Equilibrium by Backward Induction

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game The right-most end of the tree corresponds to the last choice by Player 2

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Clearly, the best choice by Player 2 is to take the pot (Take)

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Clearly, the best choice by Player 2 is to take the pot (Take) We draw an arrow indicating the path produced by this choice

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Now we know (and all the players as well) what would be the outcome of the game if the sequence of the game reaches this point

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game We now move left to find one set of decision branches for Player 1

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Clearly, the best choice for Player 1 at this point is to take the pot (Take) as all the players know what would happen if Player 1 chooses Pass at this point

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Clearly, the best choice for Player 1 at this point is to take the pot (Take) as all the players know what would happen if Player 1 chooses Pass at this point We draw an arrow indicating the path produced by this choice

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Again, we know (and all the players as well) what would be the outcome of the game if the sequence of the game reaches this point

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game We now move left to find another set of decision branches for Player 2

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Again, we know (and all the players as well) what would be the outcome of the game if the sequence of the game reaches this point

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game We now move left to find another set of decision branches for Player 1 (which corresponds to the start of the game)

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game Thus, the equilibrium found by Backward Induction is the one in which all the players take the pot as soon as they can !! (Was the “green” equilibrium when we found all the Nash equilibria)

Strategic Behavior in Business and Econ 1212 1, 00, 23, 12, 4 4, 3 Take Pass The Centipede Game The equilibrium by Backward Induction is: Player 1 chooses: T T Player 2 chooses:T T

Strategic Behavior in Business and Econ The Centipede Game The real version of the game has 100 “legs” (hence the name of Centipede) It doesn't matter how long the game is, the equilibrium by Backward Induction is always the same: TTT · · · T for each player In experiments played by people, though, they wait some time to take the pot Nevertheless, it's a good example to show how Backward Induction works

Strategic Behavior in Business and Econ Market Entry A firm wants to enter a new market, and its main concern is about the reaction of an incumbent company that currently is making a profit $100,000 If the reaction is aggressive, the challenger will suffer a loss of $10,000, and the incumbent's profits will be reduced to only $20,000 (because of the costs of the fight). On the other hand, if the incumbent chooses to accommodate to the new market scenario, then the two companies will share the $100,000 profit ($50,000 each) Obviously, the challenger can always choose to stay out if that seems to be the best choice (with a profit of $0)

Strategic Behavior in Business and Econ The environment of the game Players:Firm 1 (challenger) and Firm 2 (incumbent) Strategies:Firm 1: Enter or Stay out Firm 2: Aggressive or Accommodate Payoffs:as described The Rules of the Game Timing of movesSequential Nature of conflict and interactionConflict (market) Information conditionsSymmetric

Strategic Behavior in Business and Econ The game represented Firm 1 Firm 2 Enter Aggressive Stay out Accommodate (0, 100,000) (- 10,000, 20,000) (50,000, 50,000)

Strategic Behavior in Business and Econ This is a credibility (reputation) game Firm 1 Firm 2 Enter Aggressive Stay out Accommodate (0, 100,000) (- 10,000, 20,000) (50,000, 50,000) Backward Induction Firm 2, if called to play, will choose to Accommodate

Strategic Behavior in Business and Econ This is a credibility (reputation) game Firm 1 Firm 2 Enter Aggressive Stay out Accommodate (0, 100,000) (- 10,000, 20,000) (50,000, 50,000) Backward Induction Firm 2, if called to play, will choose to Accommodate Being Aggressive is not a “rational” choice as it would harm itself

Strategic Behavior in Business and Econ This is a credibility (reputation) game Firm 1 Firm 2 Enter Aggressive Stay out Accommodate (0, 100,000) (- 10,000, 20,000) (50,000, 50,000) Backward Induction: Firm 1, foreseeing the choice of Firm 2, will Enter

Strategic Behavior in Business and Econ This is a credibility (reputation) game Firm 1 Firm 2 Enter Aggressive Stay out Accommodate (0, 100,000) (- 10,000, 20,000) (50,000, 50,000) Backward Induction Firm 1, foreseeing the choice of Firm 2, will Enter The final outcome is that the two firms share the market

Strategic Behavior in Business and Econ This is a credibility (reputation) game Firm 1 Firm 2 Enter Aggressive Stay out Accommodate (0, 100,000) (- 10,000, 20,000) (50,000, 50,000) Backward Induction Firm 1, foreseeing the choice of Firm 2, will Enter Being “aggressive” could be used by Firm 2 as a threat to prevent entry by Firm 1

Strategic Behavior in Business and Econ This is a credibility (reputation) game Firm 1 Firm 2 Enter Aggressive Stay out Accommodate (0, 100,000) (- 10,000, 20,000) (50,000, 50,000) Backward Induction Firm 1, foreseeing the choice of Firm 2, will Enter But it is an “incredible threat”. If Firm 1 enters the market, being aggressive is against Firm 2's own interest

Strategic Behavior in Business and Econ Summary In dynamic games, the players should look forward (to understand the game) and think backwards (to solve the game) Backward Induction shows that “incredible threats” (or “incredible promises”) should not be taken into account What a player knows about the choices of other players is as important as what the other players know about what you know about their choices

Strategic Behavior in Business and Econ Equilibrium by Subgame Perfection (Subgame Perfect Equilibrium) The equilibrium of a game by Subgame Perfection is found again by inspecting the tree representation of the game Subgame Perfection selects only those Nash equilibria that are sequentially rational Subgame Perfection is the technique to use in games with Imperfect Information.

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 The fact of not having Perfect Information is represented with a dotted line that joins the nodes that contain the same information

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction The top-right most end of the three corresponds to a choice for the Buyer

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction The best choice for the Buyer at this point would be to Accept the offer

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction We mark this best choice with an arrow

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction The next right most end of the three corresponds to another choice for the Buyer

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction Again, the best choice for the Buyer at this point would be to Accept the offer

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction But now the best choice for the Buyer at this point would be to Reject the offer

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction Now we should move left wise in the tree

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction And then we find the decision branches of the Seller

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction But remember !! The Seller doesn't know whether she is at the upper or at the lower part of the tree !!

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction If she were at the upper part (if the Buyer choses to Invest), the best choice would be to offer the item at $2 (to get a payoff of 2)

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction But if she were at the lower part (if the Buyer choses to Not to Invest), the best choice would be to offer the item at $1 (to get a payoff of 1)

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction But since the Seller doesn't know the choice by the Buyer (Invest or Don't Invest), we can not find what will be the best choice at this point

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction Backward induction can not solve this game !

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction What happens is that this part of the game is a “static game” The Seller has to choose without knowing the choice of the Buyer

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction It's like a game of simultaneous moves

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction That must be solved using the table representation

Strategic Behavior in Business and Econ Example: The Invest and Buy game Let's try to solve this game by Backward Induction That must be solved using the table representation Buyer Seller

Strategic Behavior in Business and Econ Example: The Invest and Buy game Let's try to solve this game by Backward Induction We look for the Best Replies Buyer Seller

Strategic Behavior in Business and Econ Example: The Invest and Buy game Let's try to solve this game by Backward Induction To find that the (partial) equilibrium of this part of the game is (Invest, $2) Buyer Seller

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction Thus, we finally have that the Buyer choosing Invest and the Seller choosing $2 are the best choices in this part of the game.

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction We also mark them with arrows

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction Thus, the final equilibrium is: The Buyer Invests and later will Accept any offer except the $2 offer if he doesn't invest The Seller offers the item at $2

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction This is a Subgame Perfect Equilibrium

Strategic Behavior in Business and Econ Buyer Seller Example: The Invest and Buy game Buyer $1 $2 $1 $2 Accept Reject Invest Don't Invest -0.2, 0 0, 0 1.1, 1 0.1, 2 0.8, 1 -0.2, 2 Let's try to solve this game by Backward Induction The equilibrium produces a path towards 0.1, 2

Strategic Behavior in Business and Econ Solving the game by Subgame Perfection consists of: Proceed as in Backward Induction as far left as possible When we reach a point in which a player has imperfect information (has to choose without knowing the choices before), create the corresponding table and solve (best replies) the “static game” that corresponds to that part of the tree Draw “arrows” marking the path (branches) that correspond to the equilibrium found in the table representation Proceed in the same way until a best choice has been found for all the players along the sequence of the game Equilibrium by Subgame Perfection

Strategic Behavior in Business and Econ Summary Dynamic games are games in which players move sequentially In dynamic games, the players should look forward (to understand the game) and think backwards (to solve the game) Dynamic games are better represented in tree form Backward Induction is the technique that finds only the “good” (sequentially rational) equilibria in games with Perfect Information Subgame Perfection is the technique that finds only the “good” (sequentially rational) equilibria in games with Imperfect Information Backward Induction and Subgame Perfection show that “incredible threats” (or “incredible promises”) should not be taken into account What a player knows about the choices of other players is as important as what the other players know about what you know about their choices

3.2.2. Dynamic Games of complete information: Backward Induction and Subgame perfection.

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3.2.2. Dynamic Games of complete information: Backward Induction and Subgame perfection.

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