1 1Review BA 210 Lesson II.1 Strategic Bargaining We will spend up to 30 minutes reviewing Exam 1...

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ReviewReview

BA 210 Lesson II.1 Strategic BargainingBA 210 Lesson II.1 Strategic Bargaining

We will spend up to 30 minutesWe will spend up to 30 minutes reviewing Exam 1 reviewing Exam 1

• Know how your answers were graded.Know how your answers were graded.

• Know how to correct your mistakes. Your final exam is Know how to correct your mistakes. Your final exam is cumulative, and may contain similar questions.cumulative, and may contain similar questions.

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Deep ThoughtDeep Thought


I can picture in my mind I can picture in my mind a world a world without war, a world without hate. without war, a world without hate. And I can picture us attacking that And I can picture us attacking that world, because they’d never expect it. world, because they’d never expect it. ~ Jack Handey~ Jack Handey..

(Translation: Today’s lesson teaches you how to get the best deal (Translation: Today’s lesson teaches you how to get the best deal for yourself from bargaining.)for yourself from bargaining.)

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AcknowledgementsAcknowledgementsAlthough much of the course content in Although much of the course content in Part II is covered in the Krugman Part II is covered in the Krugman textbook, the PowerPoint slides for the textbook, the PowerPoint slides for the lessons are self-contained, and are your lessons are self-contained, and are your primary text.primary text.

Thanks to the following students for Thanks to the following students for insights used in the Part II lesson slides:insights used in the Part II lesson slides:Fabrizio AlliataFabrizio AlliataKarissa Garcia Karissa Garcia

AcknowledgementsAcknowledgements


4 4BA 210 Lesson II.1 Strategic BargainingBA 210 Lesson II.1 Strategic Bargaining

OverviewOverview

OverviewOverview

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Part II Overview Part II Overview Supply and Demand Analysis Supply and Demand Analysis from Part I helps consumers make from Part I helps consumers make satisfying choices, helps managers make profitable decisions, and satisfying choices, helps managers make profitable decisions, and helps governments make effective public policies when there are helps governments make effective public policies when there are impersonal competitive markets of large numbers of firms and impersonal competitive markets of large numbers of firms and customers and workers. customers and workers. Game Theory Game Theory in Part II now completes in Part II now completes supply and demand analysis when personal decisions interact. supply and demand analysis when personal decisions interact.

For example, if Carlos tries to sell shoes and Aleisha tries to buy For example, if Carlos tries to sell shoes and Aleisha tries to buy shoes on eBay, each competes with many potential shoe sellers or shoes on eBay, each competes with many potential shoe sellers or buyers, with price determined by aggregate supply and demand. buyers, with price determined by aggregate supply and demand. But if, instead, Aleisha happens to walk by Carlos’s trading stall But if, instead, Aleisha happens to walk by Carlos’s trading stall outside the Tijuana Wax Museum, they become tied to a few outside the Tijuana Wax Museum, they become tied to a few potential shoe sellers or buyers in Tijuana, and they negotiate a potential shoe sellers or buyers in Tijuana, and they negotiate a sales price separate from the worldwide competitive market.sales price separate from the worldwide competitive market.


OverviewOverview

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Each lesson in Part II Each lesson in Part II is divided into examples. Each example is divided into examples. Each example asks a asks a QuestionQuestion and supplies an and supplies an AnswerAnswer. The questions are like . The questions are like the questions on your exams: A game is described verbally and the questions on your exams: A game is described verbally and you are asked questions about its solution. You must formulate you are asked questions about its solution. You must formulate the game (identifying players, strategies, payoffs, …) then the game (identifying players, strategies, payoffs, …) then answer those questions by solving the game. answer those questions by solving the game.

Many examples also include a Many examples also include a CommentComment about what general about what general lesson you learn from the example. lesson you learn from the example.


OverviewOverview

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OverviewOverview

BA 210 Lesson II.1 Strategic Bargaining

Lesson II.1 Strategic BargainingLesson II.1 Strategic Bargaining

Example 1: Bargaining verses Competition Example 1: Bargaining verses Competition

Example 2: Rollback SolutionsExample 2: Rollback SolutionsExample 3: Take-it-or-Leave-it OffersExample 3: Take-it-or-Leave-it OffersExample 4: Counter OffersExample 4: Counter OffersExample 5: Bargaining with DepreciationExample 5: Bargaining with DepreciationSummarySummaryReview QuestionsReview Questions

Lesson II.2 Bargaining and ImpatienceLesson II.2 Bargaining and Impatience

Lesson II.3 Sequential Quantity CompetitionLesson II.3 Sequential Quantity Competition

8 8BA 210 Lesson II.1 Strategic Bargaining

Lesson 1 formulates and solves Lesson 1 formulates and solves the following games:the following games:Example 2: Example 2: Century Mark Game.Century Mark Game. Has a simple solution, found Has a simple solution, found

by starting at the end of the game and rolling back to the by starting at the end of the game and rolling back to the beginning.beginning.

Examples 3, 4, 5: Examples 3, 4, 5: Sequential Bargaining Games.Sequential Bargaining Games. Have unique Have unique rollback solutions. The solution favors the first mover, and so rollback solutions. The solution favors the first mover, and so favors aggressors.favors aggressors.

OverviewOverview



Example 1: Bargaining verses Example 1: Bargaining verses Competition Competition

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Question: Question: SupposeSuppose• Aleisha is willing to offer up to $59 to buy a pair of shoes.Aleisha is willing to offer up to $59 to buy a pair of shoes.• Brad, to pay $44; Claudia, $34.01; Darren, $24; Edwina, $10.Brad, to pay $44; Claudia, $34.01; Darren, $24; Edwina, $10.

SupposeSuppose• Andrew is willing to accept down to $8 to sell a pair of shoes.Andrew is willing to accept down to $8 to sell a pair of shoes.• Betty, to sell $20; Carlos, $34; Donna, $48; Engelbert, $62.Betty, to sell $20; Carlos, $34; Donna, $48; Engelbert, $62.

Compute the competitive-equilibrium price of shoes if all 10 Compute the competitive-equilibrium price of shoes if all 10 people trade shoes on eBay? (Ignore shipping costs.)people trade shoes on eBay? (Ignore shipping costs.)

Alternatively, suppose Aleisha and Carlos do not use eBay, but Alternatively, suppose Aleisha and Carlos do not use eBay, but Aleisha walks by Carlos’s trading stall outside the Tijuana Aleisha walks by Carlos’s trading stall outside the Tijuana Wax Museum. Compute the gains if they trade a pair of shoes. Wax Museum. Compute the gains if they trade a pair of shoes. Compute the price of shoes if they divide the gains 50-50. Compute the price of shoes if they divide the gains 50-50.



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Answer: Competitive Markets have Many Independent Buyers ...Answer: Competitive Markets have Many Independent Buyers ...

• Aleisha is willing to offer up to $59 to buy a pair of shoes.Aleisha is willing to offer up to $59 to buy a pair of shoes.

• Brad, $44; Claudia, $34.01; Darren, $24; Edwina, $10.Brad, $44; Claudia, $34.01; Darren, $24; Edwina, $10.


Aleisha

Brad

Claudia

Darren

Edwina



Aleisha

Brad

Claudia

Darren

Edwina

… … and Many Independent Sellersand Many Independent Sellers

• Andrew is willing to accept down to $8 to sell a pair of shoes.Andrew is willing to accept down to $8 to sell a pair of shoes.

• Betty, $20; Carlos, $34; Donna, $48; Engelbert, $62.Betty, $20; Carlos, $34; Donna, $48; Engelbert, $62.

Engelbert

Donna

Carlos

Betty

Andrew



Aleisha

Brad

Claudia

Darren

Edwina

Demand equals Supply Demand equals Supply determines Competitive Pricedetermines Competitive Price

• At some price between $34 and $34.01, Aleisha, Brad and At some price between $34 and $34.01, Aleisha, Brad and Claudia each buy 1 pair, and Andrew, Betty and Carlos each Claudia each buy 1 pair, and Andrew, Betty and Carlos each sell.sell.

Engelbert

Donna

Carlos

Betty

Andrew



Bargaining Occurs with One Buyer and One SellerBargaining Occurs with One Buyer and One Seller

If Aleisha and Carlos meet separate from the competitive market, If Aleisha and Carlos meet separate from the competitive market, then the gains if they were to trade a pair of shoes is the then the gains if they were to trade a pair of shoes is the difference between willingness to pay and willingness to sell. difference between willingness to pay and willingness to sell.

• Aleisha is willing to offer $59 to buy a pair of shoes.Aleisha is willing to offer $59 to buy a pair of shoes.

• Carlos is willing to accept $34 to sell a pair of shoes.Carlos is willing to accept $34 to sell a pair of shoes.

• The gain from trade is the difference, $25 = $59-$34.The gain from trade is the difference, $25 = $59-$34.

If Aleisha and Carlos divided the gains from trade 50-50, then If Aleisha and Carlos divided the gains from trade 50-50, then each gets $12.50 gain, meaning Aleisha pays price each gets $12.50 gain, meaning Aleisha pays price $46.50 $46.50 = = $59.00-$12.50, and Carlos receives price $59.00-$12.50, and Carlos receives price $46.50 $46.50 = $34.00+= $34.00+$12.50$12.50



Example 1: Rollback SolutionsExample 1: Rollback Solutions


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Comment: Whether Aleisha and Carlos Comment: Whether Aleisha and Carlos divide the gains from divide the gains from trade 50-50 or 99-1 or some other division depends on the rules trade 50-50 or 99-1 or some other division depends on the rules for the sequence of bargaining offers and counteroffers. for the sequence of bargaining offers and counteroffers. Consider an instructive game to illustrate the best way to make Consider an instructive game to illustrate the best way to make any sequence of decisions.any sequence of decisions.



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Question: Consider the Century Mark GameQuestion: Consider the Century Mark Game

• Played by pairs of players taking turns.Played by pairs of players taking turns.

• At each turn, each player chooses a number between 1 and 10 At each turn, each player chooses a number between 1 and 10 inclusive.inclusive.

• This choice is added to sum of all previous choices (the initial This choice is added to sum of all previous choices (the initial sum is 0).sum is 0).

• The first player to take the cumulative sum to 100 or more The first player to take the cumulative sum to 100 or more wins.wins.

How should you play the game if you were the first player? How should you play the game if you were the first player?




Answer: Answer: Start at the end of the game and work backward. This is Start at the end of the game and work backward. This is called called backward induction backward induction and it defines a and it defines a rollback solution, rollback solution, oror rollback equilibrium. rollback equilibrium.

In the Century Mark Game, first ask: What number gets you to In the Century Mark Game, first ask: What number gets you to 100 next turn? If you bring the cumulative sum to 100 next turn? If you bring the cumulative sum to 8989, you can , you can take the cumulative sum to 100 on your next turn and win no take the cumulative sum to 100 on your next turn and win no matter what your opponent does. (Whatever your opponent does, matter what your opponent does. (Whatever your opponent does, you can make the sum of your two moves equal 11, and 89+11 = you can make the sum of your two moves equal 11, and 89+11 = 100.)100.)


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Complete Rollback SolutionComplete Rollback Solution

• If you bring the cumulative sum to If you bring the cumulative sum to 89 89 on one turn, you can on one turn, you can take the cumulative sum to 100 on your next turn and win no take the cumulative sum to 100 on your next turn and win no matter what your opponent does. (Whatever your opponent matter what your opponent does. (Whatever your opponent does, you can make the sum of your two moves equal 11.)does, you can make the sum of your two moves equal 11.)

• Hence, if you bring the cumulative sum to Hence, if you bring the cumulative sum to 7878 on one turn, you on one turn, you can take the cumulative sum to 89 on your next turn (and so can take the cumulative sum to 89 on your next turn (and so eventually win) no matter what your opponent does.eventually win) no matter what your opponent does.

• And so on for sums And so on for sums 67, 56, 45, 34, 23, 1267, 56, 45, 34, 23, 12..

• Hence, if you play 1 on your first turn, you can bring the Hence, if you play 1 on your first turn, you can bring the cumulative sum to 12 on your next turn (and so eventually cumulative sum to 12 on your next turn (and so eventually win) no matter what your opponent does.win) no matter what your opponent does.

• That solution is a That solution is a strategystrategy --- a complete plan of actions no --- a complete plan of actions no matter what your opponent does. matter what your opponent does.




Example 3: Take-it-or-leave-it BargainingExample 3: Take-it-or-leave-it Bargaining

Example 3: Take-it-or-leave-it Example 3: Take-it-or-leave-it BargainingBargaining


Comment 1: Bargaining GamesComment 1: Bargaining Games have three parts. have three parts.

• Players or bargainersPlayers or bargainers are managers, customers, or workers, are managers, customers, or workers, typically considered two at the time.typically considered two at the time.

• Strategies Strategies are offers to divide a are offers to divide a fixed positive gain from an fixed positive gain from an agreementagreement, the rejection of such an offer, or the acceptance of , the rejection of such an offer, or the acceptance of such an offer. The most common agreements is trading a good such an offer. The most common agreements is trading a good at a particular price. at a particular price.

• Manager strategies include a price at which to sell a Manager strategies include a price at which to sell a consumption good or a wage to buy workers’ time.consumption good or a wage to buy workers’ time.

• Customer strategies include a price at which to buy. Customer strategies include a price at which to buy.

• Worker strategies include a wage at which to sell. Worker strategies include a wage at which to sell.

• Payoffs Payoffs are percentage shares of the gain from an agreement, are percentage shares of the gain from an agreement, if there is an agreement, or zero if there were no agreement.if there is an agreement, or zero if there were no agreement.



Comment 2: Bargaining Games in Part II Comment 2: Bargaining Games in Part II assume Rationality and assume Rationality and Perfect Information are Common Knowledge.Perfect Information are Common Knowledge.

•RationalityRationality means players are perfect calculators and perfect means players are perfect calculators and perfect followers of their best strategies. Players have perfect knowledge followers of their best strategies. Players have perfect knowledge of their own interests.of their own interests.

•Perfect InformationPerfect Information means players know the bargaining rules means players know the bargaining rules and all offers that have taken place.and all offers that have taken place.

•Common Knowledge Common Knowledge of of rationality and perfect information rationality and perfect information means all players assume all other players are rational and have means all players assume all other players are rational and have perfect information, and all players assume all other players perfect information, and all players assume all other players assume all other players are rational and have perfect assume all other players are rational and have perfect information, and so on. information, and so on.

•In contrast, poker professionals (sharks) do not assume bad In contrast, poker professionals (sharks) do not assume bad players (fish or suckers) are rational. Sharks try to trick fish. players (fish or suckers) are rational. Sharks try to trick fish.



Question: Suppose Buyer Bob Question: Suppose Buyer Bob values the bag of fresh beans in values the bag of fresh beans in front of him at 1 dollar and front of him at 1 dollar and Seller Sabitha Seller Sabitha has no alternative has no alternative buyers and has no other value for the beans. Suppose there is buyers and has no other value for the beans. Suppose there is only enough time before his tour bus leaves for Buyer Bob to only enough time before his tour bus leaves for Buyer Bob to make one offer to Seller Sabitha for the bag of beans. Seller make one offer to Seller Sabitha for the bag of beans. Seller Sabitha must either accept or reject that offer. Sabitha must either accept or reject that offer. Which of the Which of the prices $0.01, $0.50, or $0.98 should Buyer Bob offer? prices $0.01, $0.50, or $0.98 should Buyer Bob offer?



Answer: Answer: Follow two steps to define any Follow two steps to define any Bargaining GameBargaining Game: :

Determine the total Determine the total gain from an agreementgain from an agreement. When the . When the agreement means trading a good at a particular price, the gain is agreement means trading a good at a particular price, the gain is the maximum price the buyer were willing to pay minus the the maximum price the buyer were willing to pay minus the minimum price the seller were willing to receive. Here, total minimum price the seller were willing to receive. Here, total gain is $1.gain is $1.

Strategies are then percentage shares of the gain from trade. Strategies are then percentage shares of the gain from trade. Here, prices $0.01, $0.50, or $0.98 means Bob takes 99 Here, prices $0.01, $0.50, or $0.98 means Bob takes 99 percent, 50 percent, or 2 percent of the gain from trade.percent, 50 percent, or 2 percent of the gain from trade.

Determine the bargaining rules. Here, Bob makes an offer that Determine the bargaining rules. Here, Bob makes an offer that Sabitha accepts or rejects. Either way, the game is then over. Sabitha accepts or rejects. Either way, the game is then over.



Comment: Many Sequential-Move Games Comment: Many Sequential-Move Games have have complex rollback complex rollback solutions requiring a game tree to predict future responses to solutions requiring a game tree to predict future responses to current actions. current actions. Game trees Game trees or or extensive forms extensive forms consist of nodes consist of nodes and branchesand branches. Nodes are connected to one another by the . Nodes are connected to one another by the branches, and come in two types. Some nodes are decision branches, and come in two types. Some nodes are decision nodes, where a player chooses an action branch to another node. nodes, where a player chooses an action branch to another node. The other nodes are terminal nodes, where players receive the The other nodes are terminal nodes, where players receive the outcomes of the actions taken by themselves and all other outcomes of the actions taken by themselves and all other players.players.


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9 9 ,1

A cce p t

0 ,0

R e je ct

R e spo n d er

I take 9 9%o f theg a in

fro m tra de

5 0 ,50

A cce p t

0 ,0

R e je ct

R e spo n d er


fro m tra de

2 ,98

A cce p t

0 ,0

R e je ct

R e spo n d er

I ta ke 2%o f theg a in

fro m tra de

P ro po ser


Answer (continued): Answer (continued): Here is a Game Tree where payoffs list the Here is a Game Tree where payoffs list the gains from trade, with payoffs to the first player to act (Bob, the gains from trade, with payoffs to the first player to act (Bob, the proposer) listed first.proposer) listed first.



Starting at the end of the game, Starting at the end of the game, Bob predicts Bob predicts rational responses rational responses to to alternative proposals. Sabitha should accept anything as being alternative proposals. Sabitha should accept anything as being better than nothing. better than nothing.

Example 3: Take-it-or-leave-it OffersExample 3: Take-it-or-leave-it Offers

9 9 ,1

A cce p t

0 ,0

R e je ct

R e spo n d er


fro m tra de

5 0 ,50

A cce p t

0 ,0

R e je ct

R e spo n d er


fro m tra de

2 ,98

A cce p t

0 ,0

R e je ct

R e spo n d er


fro m tra de

P ro po ser

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9 9 ,1

A cce p t

0 ,0

R e je ct

R e spo n d er


fro m tra de

5 0 ,50

A cce p t

0 ,0

R e je ct

R e spo n d er


fro m tra de

2 ,98

A cce p t

0 ,0

R e je ct

R e spo n d er


fro m tra de

P ro po ser


Rolling back to the beginning of the game, Rolling back to the beginning of the game, determine Bob’s determine Bob’s rational proposalrational proposal, where , where Bob assumes Sabitha is rationalBob assumes Sabitha is rational. Bob . Bob should offer to take 99%.should offer to take 99%.

Example 3: Take-it-or-leave-it OffersExample 3: Take-it-or-leave-it Offers


Example 4: Counter OffersExample 4: Counter Offers



Comment: Bargainers Comment: Bargainers can get most of the gains from trade by can get most of the gains from trade by making take-it-or-leave-it offers. So it is best for Buyer Bob to making take-it-or-leave-it offers. So it is best for Buyer Bob to make an offer on a bag of beans just as his tour bus is about to make an offer on a bag of beans just as his tour bus is about to leave. leave.

To try to counter the take-it-or-leave-it strategy and get more of To try to counter the take-it-or-leave-it strategy and get more of the gains from trade, Seller Sabitha can try to make a (quick) the gains from trade, Seller Sabitha can try to make a (quick) counter offer. counter offer.



Question: Suppose Buyer Bob Question: Suppose Buyer Bob values the bag of fresh beans in values the bag of fresh beans in front of him at 1 dollar and front of him at 1 dollar and Seller Sabitha Seller Sabitha has no alternative has no alternative buyers and has no other value for the beans. But now suppose buyers and has no other value for the beans. But now suppose there is only enough time before his tour bus leaves for Bob to there is only enough time before his tour bus leaves for Bob to make one offer to Sabitha for the bag of beans and for Sabitha to make one offer to Sabitha for the bag of beans and for Sabitha to either to accept or to reject and to make a counteroffer that Bob either to accept or to reject and to make a counteroffer that Bob must either accept or reject. must either accept or reject. Will trade occur? If so, at what Will trade occur? If so, at what price? price?



Answer: Answer: To consider all possible price offers (offers by Bob to To consider all possible price offers (offers by Bob to buy or counteroffers by Sabitha to sell), consider a buy or counteroffers by Sabitha to sell), consider a bargaining bargaining payoff tablepayoff table::


Rounds to End of Game

Offer byTotal Gain to Divide

Bob's GainOffered

Sabitha's Gain

Offered

1 Sabitha 100 ? ?

2 Bob 100 ? ?


Starting 1 bargaining round from the end of the game,Starting 1 bargaining round from the end of the game, Sabitha Sabitha either accepts Bob’s offer or makes a counteroffer that Bob must either accepts Bob’s offer or makes a counteroffer that Bob must either accept or reject. Bob should accept anything as being better either accept or reject. Bob should accept anything as being better than nothing. So, Sabitha can get away with the whole gain from than nothing. So, Sabitha can get away with the whole gain from trade minus a pittance, leaving Bob with the pittance (essentially, trade minus a pittance, leaving Bob with the pittance (essentially, a zero gain to Bob). a zero gain to Bob).




Bob's GainOffered

Sabitha's Gain

Offered

1 Sabitha 100 0 100

2 Bob 100 ? ?


Rolling back to the beginning Rolling back to the beginning (2 bargaining rounds from the end (2 bargaining rounds from the end of the game), Sabitha could reject any offer and get 100 percent of the game), Sabitha could reject any offer and get 100 percent in the next round. So, Bob’s best acceptable offer leaves Sabitha in the next round. So, Bob’s best acceptable offer leaves Sabitha with 100 percent, and Bob with 0 percent. That is, trade occurs, with 100 percent, and Bob with 0 percent. That is, trade occurs, and Bob pays $1. and Bob pays $1.




Bob's GainOffered

Sabitha's Gain

Offered

1 Sabitha 100 0 100

2 Bob 100 0 100


Comment: Comment: Listening to a counteroffer thus eliminates all the Listening to a counteroffer thus eliminates all the gains from trade that would otherwise have gone to the player gains from trade that would otherwise have gone to the player making a take-it-or-leave-it offer. Bob should have refused to making a take-it-or-leave-it offer. Bob should have refused to listen and, so, forced Sabitha to take or leave a first offer of a listen and, so, forced Sabitha to take or leave a first offer of a pittance to Sabitha.pittance to Sabitha.




Bob's GainOffered

Sabitha's Gain

Offered

1 Sabitha 100 0 100

2 Bob 100 0 100


Example 5: Bargaining with DepreciationExample 5: Bargaining with Depreciation

Example 5: Bargaining with Example 5: Bargaining with DepreciationDepreciation


Comment: Bargaining rules change Comment: Bargaining rules change if the value of the good if the value of the good depreciatesdepreciates like ice cream in the time it takes to make like ice cream in the time it takes to make counteroffers to offers, and counter-counteroffers to counteroffers to offers, and counter-counteroffers to counteroffers and so on.counteroffers and so on.



Question: Question: Suppose Buyer Bob Suppose Buyer Bob values an ice cream bar at 1 dollar values an ice cream bar at 1 dollar and and Seller Sam Seller Sam has no alternative buyers and has no other value has no alternative buyers and has no other value for the ice cream. Suppose Bob can make the first offer to Sam, for the ice cream. Suppose Bob can make the first offer to Sam, who can accept or make a counter offer to Bob, who in turn can who can accept or make a counter offer to Bob, who in turn can accept or make a counter offer to Sam. Offers alternate thereafter. accept or make a counter offer to Sam. Offers alternate thereafter. But each counter offer takes valuable time which melts the ice But each counter offer takes valuable time which melts the ice cream and, so, reduces the initial value of the ice cream by 50 cream and, so, reduces the initial value of the ice cream by 50 cents. cents. What initial price should Buyer Bob offer? Should Seller What initial price should Buyer Bob offer? Should Seller Sam accept that first offer? Sam accept that first offer?



Answer: Answer: To consider all possible price offers (offers by Bob to To consider all possible price offers (offers by Bob to buy or offers by Sam to sell), consider another buy or offers by Sam to sell), consider another bargaining payoff bargaining payoff tabletable. The game ends if an offer is accepted or the ice cream is . The game ends if an offer is accepted or the ice cream is completely melted and worthless, and gains are measured as a completely melted and worthless, and gains are measured as a percentage of the $1.00 gain from trading the ice cream before it percentage of the $1.00 gain from trading the ice cream before it starts melting.starts melting.

Example 5: Counteroffers with DepreciationExample 5: Counteroffers with Depreciation

Rounds to Meltdown


Bob's GainOffered

Sam's GainOffered

1 Sam 50 ? ?

2 Bob 100 ? ?


Starting 1 bargaining round from the end of the game,Starting 1 bargaining round from the end of the game, when the when the ice cream has 50 percent of its original value, Bob should accept ice cream has 50 percent of its original value, Bob should accept anything as being better than nothing. So, Sam can get away anything as being better than nothing. So, Sam can get away with the whole 50 percent minus a pittance, leaving Bob with the with the whole 50 percent minus a pittance, leaving Bob with the pittance. pittance.


Rounds to Meltdown


Bob's GainOffered

Sam's GainOffered

1 Sam 50 0 50

2 Bob 100 ? ?


Rolling back to the beginning of the game, 2 steps to meltdown, Rolling back to the beginning of the game, 2 steps to meltdown, when the ice cream has 100 percent of its original value, Sam when the ice cream has 100 percent of its original value, Sam could reject any offer and get 50 percent in the next round. So, could reject any offer and get 50 percent in the next round. So, Bob’s best acceptable offer leaves Sam with 50 percent plus a Bob’s best acceptable offer leaves Sam with 50 percent plus a pittance, and Bob with 50 percent minus a pittance.pittance, and Bob with 50 percent minus a pittance.


Rounds to Meltdown


Bob's GainOffered

Sam's GainOffered

1 Sam 50 0 50

2 Bob 100 50 50


Buyer Bob Buyer Bob should thus initially offer price $0.50 for the ice should thus initially offer price $0.50 for the ice cream, and Seller Sam should accept that first offer. That gives cream, and Seller Sam should accept that first offer. That gives Bob 50 percent of the gain from trade and Sam 50 percent. Bob 50 percent of the gain from trade and Sam 50 percent.


Rounds to Meltdown


Bob's GainOffered

Sam's GainOffered

1 Sam 50 0 50

2 Bob 100 50 50



Example 6: Bargaining with Example 6: Bargaining with DepreciationDepreciation


Question: Question: Suppose Buyer Bob Suppose Buyer Bob values an ice cream bar at 1 dollar values an ice cream bar at 1 dollar and and Seller Sam Seller Sam has no alternative buyers and has no other value has no alternative buyers and has no other value for the ice cream. Suppose Bob can make the first offer to Sam, for the ice cream. Suppose Bob can make the first offer to Sam, who can accept or make a counter offer to Bob, who in turn can who can accept or make a counter offer to Bob, who in turn can accept or make a counter offer to Sam. Offers alternate thereafter. accept or make a counter offer to Sam. Offers alternate thereafter. But each counter offer takes valuable time which melts the ice But each counter offer takes valuable time which melts the ice cream and, so, reduces the initial value of the ice cream by 33 1/3 cream and, so, reduces the initial value of the ice cream by 33 1/3 cents. cents. What initial price should Buyer Bob offer? Should Seller What initial price should Buyer Bob offer? Should Seller Sam accept that first offer? Sam accept that first offer?



Answer: Answer: To consider all possible price offers (offers by Bob to To consider all possible price offers (offers by Bob to buy or offers by Sam to sell), consider another buy or offers by Sam to sell), consider another bargaining payoff bargaining payoff tabletable. The game ends if an offer is accepted or the ice cream is . The game ends if an offer is accepted or the ice cream is completely melted and worthless, and gains are measured as a completely melted and worthless, and gains are measured as a percentage of the $1.00 gain from trading the ice cream before it percentage of the $1.00 gain from trading the ice cream before it starts melting.starts melting.




Bob's GainOffered

Sams's GainOffered

1 Bob ? ? ?

2 Sam ? ? ?

3 Bob ? ? ?


Starting 1 bargaining round from the end of the game,Starting 1 bargaining round from the end of the game, when the when the ice cream has 33 1/3 percent of its original value, Sam should ice cream has 33 1/3 percent of its original value, Sam should accept anything as being better than nothing. So, Bob can get accept anything as being better than nothing. So, Bob can get away with the whole 33 1/3 percent minus a pittance, leaving away with the whole 33 1/3 percent minus a pittance, leaving Sam with the pittance. Sam with the pittance.




Bob's GainOffered

Sams's GainOffered

1 Bob 33 1/3 33 1/3 0

2 Sam

3 Bob


Rolling back 2 steps to meltdown, Rolling back 2 steps to meltdown, when the ice cream has 66 2/3 when the ice cream has 66 2/3 percent of its original value, Bob could reject any offer and get percent of its original value, Bob could reject any offer and get 33 1/3 percent in the next round. So, Sam’s best acceptable offer 33 1/3 percent in the next round. So, Sam’s best acceptable offer leaves Bob with 33 1/3 percent plus a pittance, and Sam with 33 leaves Bob with 33 1/3 percent plus a pittance, and Sam with 33 1/3 percent minus a pittance.1/3 percent minus a pittance.




Bob's GainOffered

Sams's GainOffered

1 Bob 33 1/3 33 1/3 0

2 Sam 66 2/3 33 1/3 33 1/3

3 Bob 100 ? ?


Rolling back to the beginning of the game, 3 steps to meltdown, Rolling back to the beginning of the game, 3 steps to meltdown, when the ice cream has 100 percent of its original value, Sam when the ice cream has 100 percent of its original value, Sam could reject any offer and get 33 1/3 percent in the next round. could reject any offer and get 33 1/3 percent in the next round. So, Bob’s best acceptable offer leaves Sam with 33 1/3 percent So, Bob’s best acceptable offer leaves Sam with 33 1/3 percent plus a pittance, and Bob with 66 2/3 percent minus a pittance.plus a pittance, and Bob with 66 2/3 percent minus a pittance.




Bob's GainOffered

Sams's GainOffered

1 Bob 33 1/3 33 1/3 0

2 Sam 66 2/3 33 1/3 33 1/3

3 Bob 100 66 2/3 33 1/3


Buyer Bob Buyer Bob should thus initially offer price $1/3 for the ice cream, should thus initially offer price $1/3 for the ice cream, and Seller Sam should accept that first offer. That gives Bob 66 and Seller Sam should accept that first offer. That gives Bob 66 2/3 percent of the gain from trade and Sam 33 1/3 percent. 2/3 percent of the gain from trade and Sam 33 1/3 percent.




Bob's GainOffered

Sams's GainOffered

1 Bob 33 1/3 33 1/3 0

2 Sam 66 2/3 33 1/3 33 1/3

3 Bob 100 66 2/3 33 1/3


Comment: Comment: The players in that counter offer game did not actually The players in that counter offer game did not actually make any counteroffers, but the possibility of counteroffers make any counteroffers, but the possibility of counteroffers affected the initial offer. affected the initial offer.



SummarySummary

SummarySummary


SummarySummary

A bargainers’ shares A bargainers’ shares of the fof the fixed positive gain from an agreement ixed positive gain from an agreement depends solely on the sequence of who makes offers and on any depends solely on the sequence of who makes offers and on any depreciation of the gains from an agreement over the bargaining depreciation of the gains from an agreement over the bargaining rounds. In particular, the rounds. In particular, the bargainers’ shares are independent of bargainers’ shares are independent of the level of the gain from an agreement. the level of the gain from an agreement.


SummarySummary

A bargaining payoff table A bargaining payoff table for Bargainers A and B, with Bargainer for Bargainers A and B, with Bargainer A making the only offer: B should accept anything as being A making the only offer: B should accept anything as being better than nothing. After deducing that, A’s best acceptable offer better than nothing. After deducing that, A’s best acceptable offer to B is a pittance, leaving A with 100 percent.to B is a pittance, leaving A with 100 percent.Rounds to

End of Game


A's GainOffered

B's GainOffered

1 A 100 100 0


SummarySummary

A bargaining payoff table A bargaining payoff table for Bargainers A and B, with Bargainer for Bargainers A and B, with Bargainer A making the first offer, when the potential gains from A making the first offer, when the potential gains from bargaining fully depreciate after 2 rounds:bargaining fully depreciate after 2 rounds:



A's GainOffered

B's GainOffered

1 B 50 0 50

2 A 100 50 50


SummarySummary



A's GainOffered

B's GainOffered

1 B 50 0 50

2 A 100 50 50

Starting 1 bargaining round from Starting 1 bargaining round from the end,the end, when the gain has when the gain has 33 1/3 percent of its original 33 1/3 percent of its original value, A should accept anything as being better than nothing. value, A should accept anything as being better than nothing. After deducing that, B’s best acceptable offer to A is a pittance, After deducing that, B’s best acceptable offer to A is a pittance, leaving B with 25 percent.leaving B with 25 percent.

Rolling back to the beginning, Rolling back to the beginning, when the gain has 100 percent of when the gain has 100 percent of its original value, B could reject any offer and get 50 percent in its original value, B could reject any offer and get 50 percent in the next round. After deducing that, A’s best acceptable offer to the next round. After deducing that, A’s best acceptable offer to B is 50 percent plus a pittance, leaving A with 50 percent.B is 50 percent plus a pittance, leaving A with 50 percent.


SummarySummary

A bargaining payoff table A bargaining payoff table for Bargainers A and B, with Bargainer for Bargainers A and B, with Bargainer A making the first offer, when the potential gains from A making the first offer, when the potential gains from bargaining fully depreciate after 3 rounds:bargaining fully depreciate after 3 rounds:



A's GainOffered

B's GainOffered

1 A 33 1/3 33 1/3 0

2 B 66 2/3 33 1/3 33 1/3

3 A 100 66 2/3 33 1/3


SummarySummary

Starting 1 bargaining round from the Starting 1 bargaining round from the end,end, when the gain has 33 1/3 percent when the gain has 33 1/3 percent of its original value, B should accept of its original value, B should accept anything as being better than nothing. anything as being better than nothing. After deducing that, A’s best acceptable After deducing that, A’s best acceptable offer to B is a pittance, leaving A with 25 percent.offer to B is a pittance, leaving A with 25 percent.

Rolling back to 2 rounds from the end, Rolling back to 2 rounds from the end, when the gain has 66 2/3 percent of its when the gain has 66 2/3 percent of its original value, A could reject any offer and get 33 1/3 percent in the next original value, A could reject any offer and get 33 1/3 percent in the next round. After deducing that, B’s best acceptable offer to A is 33 1/3 percent round. After deducing that, B’s best acceptable offer to A is 33 1/3 percent plus a pittance, leaving B with 33 1/3 percent.plus a pittance, leaving B with 33 1/3 percent.

Rolling back to the beginning, Rolling back to the beginning, when the gain has 100 percent of its original when the gain has 100 percent of its original value, B could reject any offer and get 33 1/3 percent in the next round. After value, B could reject any offer and get 33 1/3 percent in the next round. After deducing that, A’s best acceptable offer to B is 33 1/3 percent plus a pittance, deducing that, A’s best acceptable offer to B is 33 1/3 percent plus a pittance, leaving A with 66 2/3 percent.leaving A with 66 2/3 percent.



A's GainOffered

B's GainOffered

1 A 33 1/3 33 1/3 0

2 B 66 2/3 33 1/3 33 1/3

3 A 100 66 2/3 33 1/3

58 58

Review QuestionsReview Questions


Review QuestionsReview Questions You should try to answer some of the following questions You should try to answer some of the following questions before the next class. before the next class. You will not turn in your answers, but students may request You will not turn in your answers, but students may request to discuss their answers to begin the next class. to discuss their answers to begin the next class. Your upcoming Exam 1 and cumulative Final Exam will Your upcoming Exam 1 and cumulative Final Exam will contain some similar questions, so you should eventually contain some similar questions, so you should eventually consider every review question before taking your exams.consider every review question before taking your exams.



Review Question 1Review Question 1


Question 1. Question 1. Consider union-management negotiations over wages Consider union-management negotiations over wages for workers during Easter Weekend at Medieval Times Dinner for workers during Easter Weekend at Medieval Times Dinner Theatre. The weekend lasts Saturday and Sunday. Each day Theatre. The weekend lasts Saturday and Sunday. Each day Medieval Times operates, it makes a profit of $10,000. On the Medieval Times operates, it makes a profit of $10,000. On the Friday night before the weekend, the employee’s union confronts Friday night before the weekend, the employee’s union confronts the management over wages. The union presents its demand. the management over wages. The union presents its demand. The management either accepts or rejects it and returns the next The management either accepts or rejects it and returns the next day with a counteroffer. Medieval Times can open only on those day with a counteroffer. Medieval Times can open only on those days after an agreement is reached the night before.days after an agreement is reached the night before.

What initial wage demand should the union make? Should What initial wage demand should the union make? Should management accept that demand?management accept that demand?




Answer 1: Answer 1: To consider all possible wage offers (offers by the To consider all possible wage offers (offers by the Union to sell labor or offers by Management to buy labor), Union to sell labor or offers by Management to buy labor), consider another consider another bargaining payoff tablebargaining payoff table. The game ends if an . The game ends if an offer is accepted or if the weekend ends without an accepted offer is accepted or if the weekend ends without an accepted offer. Gains are measured as a percentage of the $20,000 gain offer. Gains are measured as a percentage of the $20,000 gain from accepting the Union’s initial offer.from accepting the Union’s initial offer.



Union's GainOffered

Mgmt's GainOffered

1 Mgmt. 50 ? ?

2 Union 100 ? ?

62 62

Starting 1 bargaining round from Starting 1 bargaining round from the end of the game,the end of the game, when a when a contract has 50 percent of its contract has 50 percent of its original value, the Union should accept anything as being better original value, the Union should accept anything as being better than nothing. So, Management can get away with the whole 50 than nothing. So, Management can get away with the whole 50 percent minus a pittance, leaving the Union with the pittance. percent minus a pittance, leaving the Union with the pittance.

Rolling back to the beginning of the game, 2 bargaining rounds Rolling back to the beginning of the game, 2 bargaining rounds from the end, when a contract has 100 percent of its original from the end, when a contract has 100 percent of its original value, Management could reject any offer and get 50 percent in value, Management could reject any offer and get 50 percent in the next round. So, the Union’s best acceptable offer leaves the next round. So, the Union’s best acceptable offer leaves Management with 50 percent plus a pittance, and the union with Management with 50 percent plus a pittance, and the union with 25 percent minus a pittance.25 percent minus a pittance.





Union's GainOffered

Mgmt's GainOffered

1 Mgmt. 50 0 50

2 Union 100 50 50



The Union’s initial wage demand should be for 50% of the The Union’s initial wage demand should be for 50% of the $20,000 profits, or $10,000, and management should accept that $20,000 profits, or $10,000, and management should accept that demand.demand.



Union's GainOffered

Mgmt's GainOffered

1 Mgmt. 50 0 50

2 Union 100 50 50


Question 2. Question 2. Consider negotiations over wages for adjunct Consider negotiations over wages for adjunct Management Science Professors at the Business Administration Management Science Professors at the Business Administration Division of Seaver College of Pepperdine University. Division of Seaver College of Pepperdine University. Pepperdine seeks a professor for the next 3 semesters. The only Pepperdine seeks a professor for the next 3 semesters. The only candidate is willing to work for $15,000 per semester and candidate is willing to work for $15,000 per semester and Pepperdine is willing to pay up to $25,000 per semester, so the Pepperdine is willing to pay up to $25,000 per semester, so the potential gains from trade are $10,000 per semester. Before the potential gains from trade are $10,000 per semester. Before the first semester, Pepperdine confronts the candidate over wages. first semester, Pepperdine confronts the candidate over wages. Pepperdine presents its offer. The candidate either accepts it or Pepperdine presents its offer. The candidate either accepts it or rejects it and returns the next semester with a counteroffer. Offers rejects it and returns the next semester with a counteroffer. Offers alternate thereafter. Employment only occurs in those semesters alternate thereafter. Employment only occurs in those semesters after an agreement is reached.after an agreement is reached.

What percentage of the gains from trade should Pepperdine What percentage of the gains from trade should Pepperdine offer? Should the candidate accept that initial offer?offer? Should the candidate accept that initial offer?




Answer 2: Answer 2: To consider all possible wage offers (offers by To consider all possible wage offers (offers by Pepperdine to buy labor by the candidate to sell labor), consider Pepperdine to buy labor by the candidate to sell labor), consider another another bargaining payoff tablebargaining payoff table. The game ends if an offer is . The game ends if an offer is accepted or if the three semesters end without an accepted offer, accepted or if the three semesters end without an accepted offer, and gains are measured as a percentage of the $30,000 gain from and gains are measured as a percentage of the $30,000 gain from accepting Pepperdine’s offer before the first semester.accepting Pepperdine’s offer before the first semester.Rounds to

End of Game


Pepperdine's Gain

Offered

Candidate'sGain

Offered

1 Pepperdine 33 1/3 33 1/3 0

2 Candidate 66 2/3 33 1/3 33 1/3

3 Pepperdine 100 66 2/3 33 1/3



Starting 1 bargaining round from Starting 1 bargaining round from the end of the game,the end of the game, when a when a contract has 33 1/3 percent of its contract has 33 1/3 percent of its original value, the Candidate original value, the Candidate should accept anything as being better than nothing. So, should accept anything as being better than nothing. So, Pepperdine can get away with the whole 33 1/3 percent minus a Pepperdine can get away with the whole 33 1/3 percent minus a pittance, leaving the Candidate with the pittance. pittance, leaving the Candidate with the pittance.

Rolling back to 2 bargaining rounds from the end, when a Rolling back to 2 bargaining rounds from the end, when a contract has 66 2/3 percent of its original value, Pepperdine could contract has 66 2/3 percent of its original value, Pepperdine could reject any offer and get 33 1/3 percent in the next round. So, the reject any offer and get 33 1/3 percent in the next round. So, the Candidate’s best acceptable offer leaves Pepperdine with 33 1/3 Candidate’s best acceptable offer leaves Pepperdine with 33 1/3 percent plus a pittance, and the Candidate with 33 1/3 percent percent plus a pittance, and the Candidate with 33 1/3 percent minus a pittance.minus a pittance.



Pepperdine's Gain

Offered

Candidate'sGain

Offered

1 Pepperdine 33 1/3 33 1/3 0

2 Candidate 66 2/3 33 1/3 33 1/3

3 Pepperdine 100 ? ?



Rolling back to 3 bargaining Rolling back to 3 bargaining rounds before the end, when a rounds before the end, when a contract has 100 percent of its contract has 100 percent of its original value, the Candidate original value, the Candidate could reject any offer and get 33 1/3 percent in the next round. could reject any offer and get 33 1/3 percent in the next round. So, Pepperdine’s best acceptable offer leaves the Candidate with So, Pepperdine’s best acceptable offer leaves the Candidate with 33 1/3 percent plus a pittance, and Pepperdine with 66 2/3 33 1/3 percent plus a pittance, and Pepperdine with 66 2/3 percent minus a pittance.percent minus a pittance.

PepperdinePepperdine’s initial wage offer should be for Pepperdine to keep ’s initial wage offer should be for Pepperdine to keep 66 2/3 percent of the $30,000 gain from employment for all three 66 2/3 percent of the $30,000 gain from employment for all three semesters, and the Candidate should accept that initial offer and, semesters, and the Candidate should accept that initial offer and, so, keep 33 1/3 percent of the $30,000 gain. In particular, since so, keep 33 1/3 percent of the $30,000 gain. In particular, since the Candidate is willing to work for $45,000 for all three the Candidate is willing to work for $45,000 for all three semesters, he should be paid $45,000 + $10,000 = $55,000.semesters, he should be paid $45,000 + $10,000 = $55,000.



Pepperdine's Gain

Offered

Candidate'sGain

Offered

1 Pepperdine 33 1/3 33 1/3 0

2 Candidate 66 2/3 33 1/3 33 1/3

3 Pepperdine 100 66 2/3 33 1/3


Question 3. Question 3. Two companies, one in Dallas and one in LA, are using the same Two companies, one in Dallas and one in LA, are using the same Boston-based lawyer, William Alan Shatner (aka, Captain Kirk). Each firm Boston-based lawyer, William Alan Shatner (aka, Captain Kirk). Each firm needs Shatner’s services in January, February, and March. As a result of needs Shatner’s services in January, February, and March. As a result of coordinating schedules, Shatner could each month make a triangle route coordinating schedules, Shatner could each month make a triangle route (Boston-Dallas-LA) rather than making two separate trips. The airfares each (Boston-Dallas-LA) rather than making two separate trips. The airfares each month are: month are: Triangle $1000, Boston-Dallas roundtrip $400, Boston-LA roundtrip $800Triangle $1000, Boston-Dallas roundtrip $400, Boston-LA roundtrip $800

Before the January trip, Dallas (the Dallas company) confronts LA (the LA Before the January trip, Dallas (the Dallas company) confronts LA (the LA company) over its share of the triangle airfare. Dallas presents its offer. LA company) over its share of the triangle airfare. Dallas presents its offer. LA either accepts it for each of the three trips, or rejects it and returns after the either accepts it for each of the three trips, or rejects it and returns after the January trip with a counteroffer. After the January trip, Dallas either accepts January trip with a counteroffer. After the January trip, Dallas either accepts that counteroffer for each of the remaining two trips, or rejects it and returns that counteroffer for each of the remaining two trips, or rejects it and returns after the February trip with a counteroffer. After the February trip, LA either after the February trip with a counteroffer. After the February trip, LA either accepts that counteroffer for the remaining trip, or rejects it and ends the accepts that counteroffer for the remaining trip, or rejects it and ends the game. game.

What percentage of the gains from trade should Dallas offer? Should LA What percentage of the gains from trade should Dallas offer? Should LA accept that initial offer?accept that initial offer?




Answer 3: Answer 3: To consider all possible offers to split the gains from To consider all possible offers to split the gains from an agreement on splitting the cost of the triangle route, consider an agreement on splitting the cost of the triangle route, consider another another bargaining payoff tablebargaining payoff table. The game ends if an offer is . The game ends if an offer is accepted or if the three months end without an accepted accepted or if the three months end without an accepted agreement, and gains are measured as a percentage of the gain agreement, and gains are measured as a percentage of the gain from using the triangle route for all three months.from using the triangle route for all three months.Rounds to

End of Game


Dallas's GainOffered

LA's GainOffered

1 Dallas 33 1/3 ? ?

2 LA 66 2/3 ? ?

3 Dallas 100 ? ?



Starting 1 bargaining round from Starting 1 bargaining round from the end of the game,the end of the game, when an when an agreement has 33 1/3 percent of agreement has 33 1/3 percent of its original value, LA should its original value, LA should accept anything as being better than nothing. So, Dallas can get accept anything as being better than nothing. So, Dallas can get away with the whole 33 1/3 percent minus a pittance, leaving the away with the whole 33 1/3 percent minus a pittance, leaving the LA with the pittance. LA with the pittance.

Rolling back to 2 bargaining rounds from the end, when an Rolling back to 2 bargaining rounds from the end, when an agreement has 66 2/3 percent of its original value, Dallas could agreement has 66 2/3 percent of its original value, Dallas could reject any offer and get 33 1/3 percent in the next round. So, the reject any offer and get 33 1/3 percent in the next round. So, the LA’s best acceptable offer leaves Dallas with 33 1/3 percent plus LA’s best acceptable offer leaves Dallas with 33 1/3 percent plus a pittance, and LA with 33 1/3 percent minus a pittance.a pittance, and LA with 33 1/3 percent minus a pittance.




LA's GainOffered

1 Dallas 33 1/3 33 1/3 0

2 LA 66 2/3 33 1/3 33 1/3

3 Dallas 100 ? ?



Rolling back to 3 bargaining Rolling back to 3 bargaining rounds before the end, when an rounds before the end, when an agreement has 100 percent of its agreement has 100 percent of its original value, LA could reject original value, LA could reject any offer and get 33 1/3 percent in the next round. So, Dallas’s any offer and get 33 1/3 percent in the next round. So, Dallas’s best acceptable offer leaves LA with 33 1/3 percent plus a best acceptable offer leaves LA with 33 1/3 percent plus a pittance, and Dallas with 66 2/3 percent minus a pittance.pittance, and Dallas with 66 2/3 percent minus a pittance.

Recall, the airfares each month are: Triangle $1000, Boston-Recall, the airfares each month are: Triangle $1000, Boston-Dallas roundtrip $400, Boston-LA roundtrip $800. In particular, Dallas roundtrip $400, Boston-LA roundtrip $800. In particular, the gain from three triangle trips is 3the gain from three triangle trips is 3xx(1200-1000) = 600.(1200-1000) = 600.

DallasDallas’s initial offer should be for Dallas to keep 66 2/3 percent ’s initial offer should be for Dallas to keep 66 2/3 percent of the $600 gain, or $400. So Dallas pays 3of the $600 gain, or $400. So Dallas pays 3xx400-400 = $800. 400-400 = $800. Likewise, LA gains $200 and pays 3Likewise, LA gains $200 and pays 3xx800-200 = $2200.800-200 = $2200.




LA's GainOffered

1 Dallas 33 1/3 33 1/3 0

2 LA 66 2/3 33 1/3 33 1/3

3 Dallas 100 66 2/3 33 1/3

74 74

End of Lesson II.1End of Lesson II.1


BA 210 Introduction to BA 210 Introduction to MicroeconomicsMicroeconomics

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