1 1 Deep Thought BA 445 Lesson B.2 Sequential Bargaining and Impatience Why can’t the ant and the...

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Deep Thought

BA 445 Lesson B.2 Sequential Bargaining and Impatience

Why can’t the ant and the caterpillar just get along? One eats grass, the other eats Caterpillars … Oh, I see now. ~ Jack Handey.

(Translation: Today is another lesson teaching you how to get the best deal for yourself from bargaining.)

2 2BA 445 Lesson B.2 Sequential Bargaining and Impatience

Readings

Readings

Baye “Sequential Bargaining” (see the index)Dixit Chapters 1, 2, 3


Overview

Overview

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Overview


Best Alternative to No Agreement is added to a bargainer’s share of the fixed positive gain from an agreement. — So, a seller can increase her price from a sale by finding alternative buyers.

Mutual Alternatives to No Agreement are added to both bargainers’ shares of the gains from an agreement. — So, a union seeks alternative employers and management alternative employees.

Manipulating Alternatives to No Agreement can be profitable even when alternatives decrease, depending on the relative decrease. — So, a union strikes as long as it costs management more.

Bargaining and Impatience interact like bargaining and depreciation. They should lead to an agreement before the gains from the agreement actually depreciate in the perception of the bargainers.

Relative Patience is profitable because bargaining is affected by the possibility of depreciation in the perceived gain from an agreement. — So, China gains when it is more patient than the US.


Best Alternative to No Agreement



Overview

Best Alternative to No Agreement is added to a bargainer’s share of the fixed positive gain from an agreement. — So, a seller can increase her price from a sale by finding alternative buyers.



Comment: When solving any Bargaining Game, a bargainer’s share of the fixed positive gain from an agreement is added to their Best Alternative To No Agreement (BATNA). In particular, a bargainer can increase their total payoff from an agreement by increasing their BATNA.



Question: Suppose Buyer Bob values the bag of fresh beans in front of him at 1 dollar and Seller Sabitha, in an effort to improve her bargaining position, has taken the time to find an alternative buyer would would pay $0.60. That drops the gains from a trade between Bob and Sabitha from $1.00 to $0.40.

As before, 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, and if Sabitha rejects then she can sell to the alternative buyer for $0.60. Will trade occur between Bob and Sabitha? If so, at what price to Bob?



Answer: The gain from Bob trading with Sabitha is $0.40. To consider all possible price offers (offers by Bob to buy or counteroffers by Sabitha to sell), consider a bargaining payoff table. The game lasts only one round, and gains are measured as a percentage of the $0.40 gain from Bob trading with Sabitha.

As before, Sabitha should accept anything as being better than nothing. So, Bob can get away with 100 percent of the gains minus a pittance. That leaves Bob with $0.40 gains from trade, and Sabitha with $0.00, meaning Bob pays $1.00-$0.40 = $0.60 to Sabitha, and Sabitha receives $0.60+$0.00 = $0.60 from Bob.

Rounds to Game End

Offer byTotal Gain to Divide

Bob's GainOffered

Sabitha's Gain

Offered

1 Bob 100 ? ?



Comment: In any Bargaining Game, a bargainers’ shares of the fixed positive gain from an agreement depends solely on the sequence of who makes offers and on any depreciation of the gains from an agreement over the bargaining rounds. In particular, the bargainers’ shares are independent of the level of the gain from an agreement, and of the BATNAs of the bargainers.

Rounds to Game End


Bob's GainOffered

Sabitha's Gain

Offered

1 Bob 100 100 0



Bilateral Alternatives to No Agreement

Bilateral Alternatives to No Agreement


Overview

Mutual Alternatives to No Agreement are added to both bargainers’ shares of the gains from an agreement. — So, a union seeks alternative employers and management alternative employees.

Mutual Alternatives to No Agreement


Question: Consider union-management wage bargaining during Thanksgiving Weekend at Medieval Times Dinner Theatre. The extended weekend lasts Thursday through Sunday. On the Wednesday before the weekend, the employee’s union confronts the management over wages. The union presents its demand. The management either accepts or rejects it and returns the next day with a counteroffer. Offers alternate thereafter. Each day Medieval Times operates with union labor, Medieval Times makes a profit of $10,000 before paying union wages. Each day without an agreement with the union, Medieval Times makes a profit of $3,000 after paying for expensive scab (non-union) labor, and union labor earns $2,000 from outside employment.

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


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Rounds to End of Game


Union's GainOffered

M.'s GainOffered

1 Management 25 ? ?

2 Union 50 ? ?

3 Management 75 ? ?

4 Union 100 ? ?


Answer: To consider all possible wage offers (offers by the Union to sell labor or offers by Management to buy labor), consider another bargaining payoff table. The game ends if an offer is accepted or if the weekend ends without an accepted offer. Gains are measured as a percentage of the 4x(10-3-2) = 20 thousand gain from accepting the Union’s offer at the beginning.





Union's GainOffered

M's GainOffered

1 Management 25 0 25

2 Union 50 25 25


4 Union 100 50 50

Starting 1 bargaining round from the end of the game and rolling back to the beginning determines optimal offers and counteroffers.

The Union’s initial wage demand should give it 50% of the $20,000 gains from trade, or $10,000, and management should accept that demand. Since the BATNA for Medieval Times is $3,000x4 = $12,000 profit from scab labor and the BATNA for union labor is $2,000x4 = $8,000 from outside employment, the initial wage demand by the union should be $8,000+$10,000 = $18,000, and management should accept that demand, and so earn $12,000+$10,000 = $22,000 profit.



Manipulating Alternatives to No Agreement



Overview

Manipulating Alternatives to No Agreement can be profitable even when alternatives decrease, depending on the relative decrease. — So, a union strikes as long as it costs management more.



Comment: When a strategic bargainer observes that a better outside opportunity translates into a better share in a bargain, he will look for strategic moves that improve those outside opportunities. And he will notice that what matters is his outside opportunity relative to that of his rival. He will do better in the bargaining even if he makes a commitment or a threat that lowers both parties’ outside opportunities, as long as that of the rival is damaged more severely.



Question: Reconsider the outcome of the original union-management wage bargaining for Thanksgiving Weekend at Medieval Times Dinner Theatre, where each day Medieval Times operates with union labor, Medieval Times makes a profit of $10,000 before paying union wages, and each day without an agreement with the union, Medieval Times makes profit $3,000 from scab labor and union labor earns outside income $2,000.

Compute how outcomes change if the union gives up $500 per day of outside income to intensify their picketing and, thereby, reduces Medieval Times profit from scab labor by $1,000 per day.


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Union's GainOffered

M's GainOffered


2 Union 50 25 25


4 Union 100 50 50


Answer: The bargaining payoff table is not affected by changing the BATNA for each bargainer.

In particular, the Union and Management reach an agreement on the first round, and each get half of the gains from an agreement.



Originally, the Union’s initial wage demand should give it 50% of the $20,000 gains from trade, or $10,000, and management should accept that demand. Since the BATNA for Medieval Times is $3,000x4 = $12,000 profit from scab labor and the BATNA for union labor is $2,000x4 = $8,000 from outside employment, the initial wage demand by the union should be $8,000+$10,000 = $18,000, and management should accept that demand, and so earn $12,000+$10,000 = $22,000 profit.

But with intensified picketing, the BATNA for Medieval Times is $2,000x4 = $8,000 and the BATNA for union labor is $1,500x4 = $6,000 and the gains from trade is $26,000. So, the initial wage demand by the union should be $6,000+$13,000 = $19,000, and management should accept that demand, and so earn $8,000+$13,000 = $21,000 profit. The Union does better.



Bargaining with Impatience



Overview

Bargaining and Impatience interact like bargaining and depreciation. They should lead to an agreement before the gains from the agreement actually depreciate in the perception of the bargainers.



Comment: Now consider a different kind of cost of delay in reaching an agreement. Suppose the actual monetary value of the total gains from an agreement does not depreciate, but bargainers have a “time value of money” and therefore prefer early agreement to later agreement.



Question: Compute a bargaining payoff table for Bargainers A and B making alternating offers over 5 bargaining rounds, with Bargainer A making the first offer, A discounting 10% between each period, and B also discounting 10% between each period.That is, both bargainers believe that having only 90 cents right now is as good as having $1 one round later.





A's GainOffered

(10% dis.)

B's GainOffered

(10% dis.)

1 A 100 ? ?

2 B 100 ? ?

3 A 100 ? ?

4 B 100 ? ?

5 A 100 ? ?


Answer: A bargaining payoff table for Bargainers A and B making alternating offers over 5 bargaining rounds, with Bargainer A making the first offer:




A's GainOffered

(10% dis.)

B's GainOffered

(10% dis.)

1 A 100 100.00 0.00

2 B 100 90.00 10.00

3 A 100 91.00 9.00

4 B 100 81.90 18.10

5 A 100 83.71 16.29

Starting 1 bargaining round from the end, B should accept anything as being better than nothing. After deducing that, A’s best acceptable offer to B is a pittance, leaving A with 100.

Rolling back to 2 rounds from the end, A could reject any offer and get 100 in the next round, which after A’s 10% discount is worth 90 in thecurrent round. After deducing that, B’s best acceptable offer to A is 90 plus a pittance, leaving B with 10.

Rolling back to 3 rounds from the end, B could reject any offer and get 10 in the next round, which after B’s 10% discount is worth 9 in the current round. . After deducing that, A’s best acceptable offer to B is 9 plus a pittance, leaving A with 91.

Rolling back to 4 rounds from the end, A could reject any offer and get 91 in the next round, which after A’s 10% discount is worth 81.9 in the current round. After deducing that, B’s best acceptable offer to A is 81.9 plus a pittance, leaving B with 18.1.

Rolling back to the beginning, B could reject any offer and get 18.1 in the next round, which after B’s 10% discount is worth 16.29 in the current round. . After deducing that, A’s best acceptable offer to B is 16.29 plus a pittance, leaving A with 83.71.



Comment: When both bargainers discount the same percent, the division of the gains from trade is the same as if the object itself were depreciating by that percentage.

In the current problem, on the one hand, if there are 5 bargaining rounds and if A and B each discount 10%, then A’s best acceptable offer to B is 83.71% plus a pittance, leaving A with 16.29%. On the other hand, if there are 5 bargaining rounds and the object itself depreciates 10% per round, then A’s best acceptable offer to B is, again, 83.71% plus a pittance, leaving A with 16.29%.




(10% dep.)

A's GainOffered

B's GainOffered

1 A 65.61 65.61 0.00

2 B 72.90 65.61 7.29

3 A 81.00 73.71 7.29

4 B 90.00 73.71 16.29

5 A 100.00 83.71 16.29


Relative Patience

Relative Patience


Overview

Relative Patience is profitable because bargaining is affected by the possibility of depreciation in the perceived gain from an agreement. — So, China gains when it is more patient than the US.

Relative Patience


Comment: We now see that more patience (less discounting) translates into a better share in a bargain. So a strategic bargainer will learn patience.

The Chinese symbol for patience can also mean restraint and control. It is formed by 2 different characters with the image of blade on the heart.

The word connotes how difficult it is to practice patience.

Relative Patience


Question: Compare outcomes from 5 rounds of bargaining in three different scenarios:

1) Both bargainers discount the money payoff from an agreement 10 percent each round.

2) The first mover (the first to make an offer) discounts the money payoff from an agreement 10 percent each round, but the second mover discounts 20 percent.

3) The first mover (the first to make an offer) discounts the money payoff from an agreement 20 percent each round, but the second mover discounts 10 percent.

Relative Patience


Answer: A bargaining payoff table for Bargainers A and B, with Bargainer A making the first offer, with 5 bargaining rounds, with A discounting 10% between each period, and with B discounting 20% between each period.



A's GainOffered

(10% dis.)

B's GainOffered

(20% dis.)

1 A 100 ? ?

2 B 100 ? ?

3 A 100 ? ?

4 B 100 ? ?

5 A 100 ? ?

Relative Patience

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Starting 1 bargaining round from the end, B should accept anything as being better than nothing. After deducing that, A’s best acceptable offer to B is a pittance, leaving A with 100.Rolling back to 2 rounds from the end, A could reject any offer and get 100 in the next round, which after A’s 10% discount is worth 90 in the current round. After deducing that, B’s best acceptable offer to A is 90 plus a pittance, leaving B with 10.Rolling back to 3 rounds from the end, B could reject any offer and get 10 in the next round, which after B’s 20% discount is worth 8 in the current round. . After deducing that, A’s best acceptable offer to B is 8 plus a pittance, leaving A with 92.






A's GainOffered

(10% dis.)

B's GainOffered

(20% dis.)

1 A 100 100.00 0.00

2 B 100 90.00 10.00

3 A 100 92.00 8.00

4 B 100 82.80 17.20

5 A 100 86.24 13.76

Relative Patience


A bargaining payoff table for Bargainers A and B, with Bargainer A making the first offer, with 5 bargaining rounds, with A discounting 20% between each period, and with B discounting 10% between each period.



A's GainOffered

(20% dis.)

B's GainOffered

(10% dis.)

1 A 100 ? ?

2 B 100 ? ?

3 A 100 ? ?

4 B 100 ? ?

5 A 100 ? ?

Relative Patience




A's GainOffered

(20% dis.)

B's GainOffered

(10% dis.)

1 A 100 100.00 0.00

2 B 100 80.00 20.00

3 A 100 82.00 18.00

4 B 100 65.60 34.40

5 A 100 69.04 30.96

Starting 1 bargaining round from the end, B should accept anything as being better than nothing. After deducing that, A’s best acceptable offer to B is a pittance, leaving A with 100.Rolling back to 2 rounds from the end, A could reject any offer and get 100 in the next round, which after A’s 20% discount is worth 80 in the current round. After deducing that, B’s best acceptable offer to A is 80 plus a pittance, leaving B with 20.Rolling back to 3 rounds from the end, B could reject any offer and get 20 in the next round, which after B’s 10% discount is worth 18 in the current round. . After deducing that, A’s best acceptable offer to B is 18 plus a pittance, leaving A with 82.



Relative Patience


Summary: Compare outcomes from 5 rounds of bargaining in three different scenarios. 1) Both bargainers discount the money payoff from an agreement 10

percent each round. Outcome: First mover A’s best acceptable offer to B is 16.29%, leaving A with 83.71%.

2) The first mover (the first to make an offer) discounts the money payoff from an agreement 10 percent each round, but the second mover discounts 20 percent. Outcome: A’s best acceptable offer to B is 13.76%, leaving A with 86.24%.

3) The first mover (the first to make an offer) discounts the money payoff from an agreement 20 percent each round, but the second mover discounts 10 percent. Outcome: A’s best acceptable offer to B is 30.96 plus a pittance, leaving A with 69.04.

A does best when his is more patient (discounts less) than B.

Relative Patience


Comment: When a strategic bargainer observes that more patience (less discounting) translates into a better share in a bargain, he will seek partners less patient than himself and avoid those more patient.

Relative Patience


Summary

Summary


Summary

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


A bargaining payoff table for Bargainers A and B, with Bargainer A making the first offer, with 5 bargaining rounds, with A discounting 30% between each period, and with B discounting 40% between each period.



A's GainOffered

(30% dis.)

B's GainOffered

(40% dis.)

1 A 100 100.00 0.00

2 B 100 70.00 30.00

3 A 100 82.00 18.00

4 B 100 57.40 42.60

5 A 100 74.44 25.56

Summary


A bargaining payoff table for Bargainers A and B, with Bargainer A making the first offer, with 5 bargaining rounds, with A discounting 40% between each period, and with B not discounting. (If A did not discount, he would have 100% of the gains from an agreement.) Rounds to

End of Game


A's GainOffered

(40% dis.)

B's GainOffered

(0% dis.)

1 A 100 100.00 0.00

2 B 100 60.00 40.00

3 A 100 60.00 40.00

4 B 100 36.00 64.00

5 A 100 36.00 64.00

Summary

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Review Questions


Review Questions You should try to answer some of the following

questions before the next class. You will not turn in your answers, but students may

request to discuss their answers to begin the next class. Your upcoming Exam 2 and cumulative Final Exam will

contain some similar questions, so you should eventually consider every review question before taking your exams.


Review 1: Relative Impatience



Comment: TheFasterTimes.Com posts an account of bargaining for a tiny glass bottle covered in a landscape painting at Shanghai’s Yuyuan market.

The games began at a whopping 380RMB, or about $56US. Over the course of 20 minutes, the buyer brought the price down by pointing out design flaws and pretending to be distracted by a porcelain doll in a red silk dress. When the shopkeeper refused to go below 80RMB, the buyer walked away, slowly. “Come back,” the saleswoman cried, racing after her. “You can have it for 60.” Deal, at just under $9US, or about 1/6 of the asking price.

Bargaining in China is an art form that requires a lot of patience, and a willingness of the buyer to signal their patience by spending time.


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Question 1. Consider negotiations over the price of a tiny glass bottle covered in a landscape painting at Shanghai’s Yuyuan market. Buyer Betty values the bottle at $65, and Seller Shen’s cost of supplying the bottle is $5. S makes the first offer, with 4 bargaining rounds of alternating offers, and with S discounting 2% between each period.

Compare bargaining outcomes when B discounts 10% between each period with B discounting only 1% between each period, and with B discounting 100% between each period (which is absolute impatience).

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Answer 1: A bargaining payoff table for Bargainers B and S, with Bargainer S making the first offer, with 4 bargaining rounds, with B discounting 10% between each period, and with S discounting 2% between each period.



B's GainOffered

(10% dis.)

S's GainOffered

(2% dis.)

1 B 100 100.00 0.00

2 S 100 90.00 10.00

3 B 100 91.20 9.80

4 S 100 82.08 17.92

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A bargaining payoff table for Bargainers B and S, with Bargainer B making the first offer, with 4 bargaining rounds, with B discounting only 1% between each period, and with S discounting 2% between each period.



B's GainOffered

(1% dis.)

S's GainOffered

(2% dis.)

1 B 100 100.0000 0.0000

2 S 100 99.0000 1.0000

3 B 100 99.0200 0.9800

4 S 100 98.0298 1.9702

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A bargaining payoff table for Bargainers B and S, with Bargainer B making the first offer, with 4 bargaining rounds, with S discounting 100% between each period, and with S discounting 2% between each period.



B's GainOffered

(100% dis.)

S's GainOffered

(2% dis.)

1 B 100 100 0

2 S 100 0 100

3 B 100 2 98

4 S 100 0 100

50 50



Summary: In the first case, when B discounts 10% between each period, B captures 82.08% of the $60 gains from trade, or about $49.25, leaving S with about $10.75. Buyer Betty thus makes an acceptable offer of $65-$49.25 = $15.75, leaving Seller Shen with gain $15.75-$5 = $10.75.

On one extreme, when B discounts only 1% between each period, B captures 98.0298% of the $60 gains from trade, or about $58.82, leaving S with about $1.18. Buyer Betty thus makes an acceptable offer of $65-$58.82 = $6.18, leaving Seller Shen with gain $6.18-$5 = $1.18.

On the other extreme, when B discounts 100% between each period, B captures 0% of the $60 gains from trade, or $0, leaving S with $60. Buyer Betty thus makes an acceptable offer of $65, leaving Seller Shen with gain $65-$5 = $60. It is just as if Seller Shen made a take-it-or-leave-it offer.

Buyer Betty does better when she is more patient. In an extreme, she gets no gains when she has no patience.

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End of Lesson B.2


BA 445 Managerial Economics

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