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Review

Review. We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded. Know how to correct your mistakes. Your final exam is cumulative, and may contain similar questions. Deep Thought.

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Review

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  1. Review • We will spend up to 30 minutes reviewing Exam 1 • Know how your answers were graded. • Know how to correct your mistakes. Your final exam is cumulative, and may contain similar questions. BA 210 Lesson II.1 Strategic Bargaining

  2. Deep Thought I can picture in my mind a world without war, a world without hate. And I can picture us attacking that world, because they’d never expect it. ~ Jack Handey. (Translation: Today’s lesson teaches you how to get the best deal for yourself from bargaining.) BA 210 Lesson II.1 Strategic Bargaining

  3. Acknowledgements AcknowledgementsAlthough much of the course content in Part II is covered in the Krugman textbook, the PowerPoint slides for the lessons are self-contained, and are your primary text.Thanks to the following students for insights used in the Part II lesson slides:Fabrizio AlliataKarissa Garcia BA 210 Lesson II.1 Strategic Bargaining

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

  5. Overview Part II Overview Supply and Demand Analysis from Part I helps consumers make satisfying choices, helps managers make profitable decisions, and helps governments make effective public policies when there are impersonal competitive markets of large numbers of firms and customers and workers. Game Theory in Part II now completes supply and demand analysis when personal decisions interact. For example, if Carlos tries to sell shoes and Aleisha tries to buy shoes on eBay, each competes with many potential shoe sellers or buyers, with price determined by aggregate supply and demand. But if, instead, Aleisha happens to walk by Carlos’s trading stall outside the Tijuana Wax Museum, they become tied to a few potential shoe sellers or buyers in Tijuana, and they negotiate a sales price separate from the worldwide competitive market. BA 210 Lesson II.1 Strategic Bargaining

  6. Overview Each lesson in Part II is divided into examples. Each example asks a Question and supplies an Answer. The questions are like the questions on your exams: A game is described verbally and you are asked questions about its solution. You must formulate the game (identifying players, strategies, payoffs, …) then answer those questions by solving the game. Many examples also include a Comment about what general lesson you learn from the example. BA 210 Lesson II.1 Strategic Bargaining

  7. Overview Lesson II.1 Strategic Bargaining Example 1: Bargaining verses Competition Example 2: Rollback Solutions Example 3: Take-it-or-Leave-it Offers Example 4: Counter Offers Example 5: Bargaining with Depreciation Summary Review Questions Lesson II.2 Bargaining and Impatience Lesson II.3 Sequential Quantity Competition BA 210 Lesson II.1 Strategic Bargaining

  8. Overview Lesson 1 formulates and solves the following games: Example 2: Century Mark Game. Has a simple solution, found by starting at the end of the game and rolling back to the beginning. Examples 3, 4, 5: Sequential Bargaining Games. Have unique rollback solutions. The solution favors the first mover, and so favors aggressors. BA 210 Lesson II.1 Strategic Bargaining

  9. Example 1: Bargaining verses Competition Example 1: Bargaining verses Competition BA 210 Lesson II.1 Strategic Bargaining

  10. Example 1: Bargaining verses Competition Question: Suppose • 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. Suppose • Andrew is willing to accept down to $8 to sell a pair of shoes. • Betty, to sell $20; Carlos, $34; Donna, $48; Engelbert, $62. Compute the competitive-equilibrium price of shoes if all 10 people trade shoes on eBay? (Ignore shipping costs.) Alternatively, suppose Aleisha and Carlos do not use eBay, but Aleisha walks by Carlos’s trading stall outside the Tijuana Wax Museum. Compute the gains if they trade a pair of shoes. Compute the price of shoes if they divide the gains 50-50. BA 210 Lesson II.1 Strategic Bargaining

  11. Example 1: Bargaining verses Competition Answer: Competitive Markets have Many Independent Buyers ... • Aleisha is willing to offer up to $59 to buy a pair of shoes. • Brad, $44; Claudia, $34.01; Darren, $24; Edwina, $10. Aleisha Brad Claudia Darren Edwina BA 210 Lesson II.1 Strategic Bargaining

  12. Example 1: Bargaining verses Competition … and Many Independent Sellers • Andrew is willing to accept down to $8 to sell a pair of shoes. • Betty, $20; Carlos, $34; Donna, $48; Engelbert, $62. Aleisha Engelbert Brad Donna Claudia Carlos Darren Betty Edwina Andrew BA 210 Lesson II.1 Strategic Bargaining

  13. Example 1: Bargaining verses Competition Demand equals Supply determines Competitive Price • At some price between $34 and $34.01, Aleisha, Brad and Claudia each buy 1 pair, and Andrew, Betty and Carlos each sell. Aleisha Engelbert Brad Donna Claudia Carlos Darren Betty Edwina Andrew BA 210 Lesson II.1 Strategic Bargaining

  14. Example 1: Bargaining verses Competition Bargaining Occurs with One Buyer and One Seller If Aleisha and Carlos meet separate from the competitive market, then the gains if they were to trade a pair of shoes is the difference between willingness to pay and willingness to sell. • Aleisha is willing to offer $59 to buy 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. If Aleisha and Carlos divided the gains from trade 50-50, then each gets $12.50 gain, meaning Aleisha pays price $46.50 = $59.00-$12.50, and Carlos receives price $46.50 = $34.00+$12.50 BA 210 Lesson II.1 Strategic Bargaining

  15. Example 1: Rollback Solutions Example 2: Rollback Solutions BA 210 Lesson II.1 Strategic Bargaining

  16. Example 2: Rollback Solutions Comment: Whether Aleisha and Carlos divide the gains from trade 50-50 or 99-1 or some other division depends on the rules for the sequence of bargaining offers and counteroffers. Consider an instructive game to illustrate the best way to make any sequence of decisions. BA 210 Lesson II.1 Strategic Bargaining

  17. Example 2: Rollback Solutions Question: Consider the Century Mark Game • Played by pairs of players taking turns. • At each turn, each player chooses a number between 1 and 10 inclusive. • This choice is added to sum of all previous choices (the initial sum is 0). • The first player to take the cumulative sum to 100 or more wins. How should you play the game if you were the first player? BA 210 Lesson II.1 Strategic Bargaining

  18. Example 2: Rollback Solutions Answer: Start at the end of the game and work backward. This is called backward induction and it defines a rollback solution, or rollback equilibrium. In the Century Mark Game, first ask: What number gets you to 100 next turn? If you bring the cumulative sum to 89, you can take the cumulative sum to 100 on your next turn and win no matter what your opponent does. (Whatever your opponent does, you can make the sum of your two moves equal 11, and 89+11 = 100.) BA 210 Lesson II.1 Strategic Bargaining

  19. Example 2: Rollback Solutions Complete Rollback Solution • If you bring the cumulative sum to 89 on one turn, you can take the cumulative sum to 100 on your next turn and win no matter what your opponent does. (Whatever your opponent does, you can make the sum of your two moves equal 11.) • Hence, if you bring the cumulative sum to 78 on one turn, you can take the cumulative sum to 89 on your next turn (and so eventually win) no matter what your opponent does. • And so on for sums 67, 56, 45, 34, 23, 12. • Hence, if you play 1 on your first turn, you can bring the cumulative sum to 12 on your next turn (and so eventually win) no matter what your opponent does. • That solution is a strategy --- a complete plan of actions no matter what your opponent does. BA 210 Lesson II.1 Strategic Bargaining

  20. Example 3: Take-it-or-leave-it Bargaining Example 3: Take-it-or-leave-it Bargaining BA 210 Lesson II.1 Strategic Bargaining

  21. Example 3: Take-it-or-leave-it Bargaining • Comment 1: Bargaining Games have three parts. • Players or bargainers are managers, customers, or workers, typically considered two at the time. • Strategies are offers to divide a fixed positive gain from an agreement, the rejection of such an offer, or the acceptance of such an offer. The most common agreements is trading a good at a particular price. • Manager strategies include a price at which to sell a consumption good or a wage to buy workers’ time. • Customer strategies include a price at which to buy. • Worker strategies include a wage at which to sell. • Payoffs are percentage shares of the gain from an agreement, if there is an agreement, or zero if there were no agreement. BA 210 Lesson II.1 Strategic Bargaining

  22. Example 3: Take-it-or-leave-it Bargaining • Comment 2: Bargaining Games in Part II assume Rationality and Perfect Information are Common Knowledge. • Rationality means players are perfect calculators and perfect followers of their best strategies. Players have perfect knowledge of their own interests. • Perfect Information means players know the bargaining rules and all offers that have taken place. • Common Knowledge of rationality and perfect information means all players assume all other players are rational and have perfect information, and all players assume all other players assume all other players are rational and have perfect information, and so on. • In contrast, poker professionals (sharks) do not assume bad players (fish or suckers) are rational. Sharks try to trick fish. BA 210 Lesson II.1 Strategic Bargaining

  23. Example 3: Take-it-or-leave-it Bargaining Question: Suppose Buyer Bob values the bag of fresh beans in front of him at 1 dollar and Seller Sabithahas no alternative buyers and has no other value for the beans. Suppose there is only enough time before his tour bus leaves for Buyer Bob to make one offer to Seller Sabitha for the bag of beans. Seller Sabitha must either accept or reject that offer. Which of the prices $0.01, $0.50, or $0.98 should Buyer Bob offer? BA 210 Lesson II.1 Strategic Bargaining

  24. Example 3: Take-it-or-leave-it Bargaining • Answer: Follow two steps to define any Bargaining Game: • Determine the total gain from an agreement. When the agreement means trading a good at a particular price, the gain is the maximum price the buyer were willing to pay minus the minimum price the seller were willing to receive. Here, total gain is $1. • Strategies are then percentage shares of the gain from trade. Here, prices $0.01, $0.50, or $0.98 means Bob takes 99 percent, 50 percent, or 2 percent of the gain from trade. • Determine the bargaining rules. Here, Bob makes an offer that Sabitha accepts or rejects. Either way, the game is then over. BA 210 Lesson II.1 Strategic Bargaining

  25. Example 3: Take-it-or-leave-it Bargaining Comment: Many Sequential-Move Games have complex rollback solutions requiring a game tree to predict future responses to current actions. Game trees or extensive forms consist of nodes and branches. Nodes are connected to one another by the branches, and come in two types. Some nodes are decision nodes, where a player chooses an action branch to another node. The other nodes are terminal nodes, where players receive the outcomes of the actions taken by themselves and all other players. BA 210 Lesson II.1 Strategic Bargaining

  26. Example 3: Take-it-or-leave-it Bargaining Answer (continued): Here is a Game Tree where payoffs list the gains from trade, with payoffs to the first player to act (Bob, the proposer) listed first. BA 210 Lesson II.1 Strategic Bargaining

  27. Example 3: Take-it-or-leave-it Offers Starting at the end of the game, Bob predicts rational responses to alternative proposals. Sabitha should accept anything as being better than nothing. BA 210 Lesson II.1 Strategic Bargaining

  28. Example 3: Take-it-or-leave-it Offers Rolling back to the beginning of the game, determine Bob’s rational proposal, where Bob assumes Sabitha is rational. Bob should offer to take 99%. BA 210 Lesson II.1 Strategic Bargaining

  29. Example 4: Counter Offers Example 4: Counter Offers BA 210 Lesson II.1 Strategic Bargaining

  30. Example 4: Counter Offers Comment: Bargainers can get most of the gains from trade by 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 leave. 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) counter offer. BA 210 Lesson II.1 Strategic Bargaining

  31. Example 4: Counter Offers Question: Suppose Buyer Bob values the bag of fresh beans in front of him at 1 dollar and Seller Sabithahas no alternative 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 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 must either accept or reject. Will trade occur? If so, at what price? BA 210 Lesson II.1 Strategic Bargaining

  32. Example 4: Counter Offers Answer: To consider all possible price offers (offers by Bob to buy or counteroffers by Sabitha to sell), consider a bargaining payoff table: BA 210 Lesson II.1 Strategic Bargaining

  33. Example 4: Counter Offers Starting 1 bargaining round from the end of the game,Sabitha either accepts Bob’s offer or makes a counteroffer that Bob must either accept or reject. Bob should accept anything as being better than nothing. So, Sabithacan get away with the whole gain from trade minus a pittance, leaving Bob with the pittance (essentially, a zero gain to Bob). BA 210 Lesson II.1 Strategic Bargaining

  34. Example 4: Counter Offers Rolling back to the beginning (2 bargaining rounds from the end of the game), Sabitha could reject any offer and get 100 percent in the next round. So, Bob’s best acceptable offer leaves Sabitha with 100 percent, and Bob with 0 percent. That is, trade occurs, and Bob pays $1. BA 210 Lesson II.1 Strategic Bargaining

  35. Example 4: Counter Offers Comment: Listening to a counteroffer thus eliminates all the gains from trade that would otherwise have gone to the player 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 pittance to Sabitha. BA 210 Lesson II.1 Strategic Bargaining

  36. Example 5: Bargaining with Depreciation Example 5: Bargaining with Depreciation BA 210 Lesson II.1 Strategic Bargaining

  37. Example 5: Bargaining with Depreciation Comment: Bargaining rules change if the value of the good depreciates like ice cream in the time it takes to make counteroffers to offers, and counter-counteroffers to counteroffers and so on. BA 210 Lesson II.1 Strategic Bargaining

  38. Example 5: Bargaining with Depreciation Question: Suppose Buyer Bob values an ice cream bar at 1 dollar and Seller Sam has no alternative buyers and has no other value 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 accept or make a counter offer to Sam. Offers alternate thereafter. But each counter offer takes valuable time which melts the ice cream and, so, reduces the initial value of the ice cream by 50 cents. What initial price should Buyer Bob offer? Should Seller Sam accept that first offer? BA 210 Lesson II.1 Strategic Bargaining

  39. Example 5: Counteroffers with Depreciation Answer: To consider all possible price offers (offers by Bob to buy or offers by Sam to sell), consider another bargaining payoff table. The game ends if an offer is accepted or the ice cream is completely melted and worthless, and gains are measured as a percentage of the $1.00 gain from trading the ice cream before it starts melting. BA 210 Lesson II.1 Strategic Bargaining

  40. Example 5: Counteroffers with Depreciation Starting 1 bargaining round from the end of the game, when the ice cream has 50 percent of its original value, Bob should accept anything as being better than nothing. So, Sam can get away with the whole 50 percent minus a pittance, leaving Bob with the pittance. BA 210 Lesson II.1 Strategic Bargaining

  41. Example 5: Counteroffers with Depreciation Rolling back to the beginning of the game, 2 steps to meltdown, 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, Bob’s best acceptable offer leaves Sam with 50 percent plus a pittance, and Bob with 50 percent minus a pittance. BA 210 Lesson II.1 Strategic Bargaining

  42. Example 5: Counteroffers with Depreciation Buyer Bob should thus initially offer price $0.50 for the ice cream, and Seller Sam should accept that first offer. That gives Bob 50 percent of the gain from trade and Sam 50 percent. BA 210 Lesson II.1 Strategic Bargaining

  43. Example 6: Bargaining with Depreciation Example 6: Bargaining with Depreciation BA 210 Lesson II.1 Strategic Bargaining

  44. Example 6: Bargaining with Depreciation Question: Suppose Buyer Bob values an ice cream bar at 1 dollar and Seller Sam has no alternative buyers and has no other value 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 accept or make a counter offer to Sam. Offers alternate thereafter. 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 cents. What initial price should Buyer Bob offer? Should Seller Sam accept that first offer? BA 210 Lesson II.1 Strategic Bargaining

  45. Example 6: Counteroffers with Depreciation Answer: To consider all possible price offers (offers by Bob to buy or offers by Sam to sell), consider another bargaining payoff table. The game ends if an offer is accepted or the ice cream is completely melted and worthless, and gains are measured as a percentage of the $1.00 gain from trading the ice cream before it starts melting. BA 210 Lesson II.1 Strategic Bargaining

  46. Example 6: Counteroffers with Depreciation Starting 1 bargaining round from the end of the game, when the ice cream has 33 1/3 percent of its original value, Sam should accept anything as being better than nothing. So, Bob can get away with the whole 33 1/3 percent minus a pittance, leaving Sam with the pittance. BA 210 Lesson II.1 Strategic Bargaining

  47. Example 6: Counteroffers with Depreciation Rolling back 2 steps to meltdown, when the ice cream has 66 2/3 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 leaves Bob with 33 1/3 percent plus a pittance, and Sam with 33 1/3 percent minus a pittance. BA 210 Lesson II.1 Strategic Bargaining

  48. Example 6: Counteroffers with Depreciation Rolling back to the beginning of the game, 3 steps to meltdown, 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. 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. BA 210 Lesson II.1 Strategic Bargaining

  49. Example 5: Counteroffers with Depreciation Buyer Bob should thus initially offer price $1/3 for the ice cream, 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. BA 210 Lesson II.1 Strategic Bargaining

  50. Example 5: Counteroffers with Depreciation Comment: The players in that counter offer game did not actually make any counteroffers, but the possibility of counteroffers affected the initial offer. BA 210 Lesson II.1 Strategic Bargaining

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