Deep Thought
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.)
BA 210 Lesson II.2 Bargaining and Impatience
1
Overview
Overview
BA 210 Lesson II.2 Bargaining and Impatience
2
Overview
Lesson II.1 Strategic Bargaining
Lesson II.2 Bargaining and Impatience
Example 1: Best Alternative to No Agreement
Example 2: Mutual Alternatives to No Agreement
Example 3: Manipulating Alternatives to No Agreement
Example 4: Bargaining with Impatience
Example 5: Relative Impatience
Lesson II.3 Sequential Quantity Competition
Summary
Review Questions
BA 210 Lesson II.2 Bargaining and Impatience
3
Overview
Lesson II.2 formulates and solves the following games:
Examples 1, 2, 3, 4, 5: Sequential Bargaining Games. Have
unique rollback solutions. The solution favors the first mover,
and so favors aggressors; favors those with better outside
alternatives to an agreement; and favors those with more
patience.
BA 210 Lesson II.2 Bargaining and Impatience
4
Example 1: Best Alternative to No Agreement
Example 1: Best Alternative to No
Agreement
BA 210 Lesson II.2 Bargaining and Impatience
5
Example 1: Best Alternative to No Agreement
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.
BA 210 Lesson II.2 Bargaining and Impatience
6
Example 1: Best Alternative to No Agreement
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?
BA 210 Lesson II.2 Bargaining and Impatience
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Example 1: Best Alternative to No Agreement
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.
Rounds to
Game End
1
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sabitha's
Gain
Offered
Bob
100
?
?
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.
BA 210 Lesson II.2 Bargaining and Impatience
8
Example 1: Best Alternative to No Agreement
Comment: In any Bargaining Game, a bargainer’s 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
1
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sabitha's
Gain
Offered
Bob
100
100
0
BA 210 Lesson II.2 Bargaining and Impatience
9
Example 2: Bilateral Alternatives to No Agreement
Example 2: Bilateral Alternatives to
No Agreement
BA 210 Lesson II.2 Bargaining and Impatience
10
Review Questions
Question 1. Consider union-management negotiations over wages
for workers during Easter Weekend at Medieval Times Dinner
Theatre. The weekend lasts Saturday and Sunday. Each day
Medieval Times operates with union labor, it makes a profit of
$10,000. On the Friday night 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. 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?
BA 210 Lesson II.2 Bargaining and Impatience
11
Review Questions
Answer 1: 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 2x(10-3-2) = 10
thousand gain from accepting the Union’s offer at the beginning.
Rounds to
End of
Game
Offer by
1
Mgmt.
50
?
?
2
Union
100
?
?
T otal Gain Union's Gain Mgmt's Gain
to Divide
Offered
Offered
BA 210 Lesson II.2 Bargaining and Impatience
12
Review Questions
Rounds to
Gain Union's Gain Mgmt's Gain
Starting 1 bargaining round from End of Offer by TtootalDivide
Offered
Offered
Game
the end of the game, when a
1
Mgmt.
50
0
50
contract has 50 percent of its
2
Union
100
50
50
original value, the Union should accept anything as being better
than nothing. So, Management can get away with the whole 50
percent minus a pittance, leaving the Union with the pittance.
Rolling back to the beginning of the game, 2 bargaining rounds
from the end, when a contract has 100 percent of its original
value, Management could reject any offer and get 50 percent in
the next round. So, the Union’s best acceptable offer leaves
Management with 50 percent plus a pittance, and the union with
25 percent minus a pittance.
BA 210 Lesson II.2 Bargaining and Impatience
13
Review Questions
The Union’s initial wage demand should be for 50% of the
$10,000 profits, or $5,000, and management should accept that
demand. Since the BATNA for Medieval Times is $3,000x2 =
$6,000 profit from scab labor and the BATNA for union labor is
$2,000x2 = $4,000 from outside employment, the initial wage
demand by the union should be $4,000+$5,000 = $9,000, and
management should accept that demand, and so earn $20,000$9,000 = $11,000 profit.
Rounds to
End of
Game
Offer by
1
Mgmt.
50
0
50
2
Union
100
50
50
T otal Gain Union's Gain Mgmt's Gain
to Divide
Offered
Offered
BA 210 Lesson II.2 Bargaining and Impatience
14
Example 3: Manipulating Alternatives to No Agreement
Example 3: Manipulating Alternatives
to No Agreement
BA 210 Lesson II.2 Bargaining and Impatience
15
Example 3: Manipulating Alternatives to No Agreement
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.
BA 210 Lesson II.2 Bargaining and Impatience
16
Example 3: Manipulating Alternatives to No Agreement
Question: Reconsider the outcome of the original unionmanagement 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.
BA 210 Lesson II.2 Bargaining and Impatience
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Example 3: Manipulating Alternatives to No Agreement
Answer: The bargaining payoff table is not affected by changing
the BATNA for each bargainer.
Rounds to
End of
Game
Offer by
1
Mgmt.
50
0
50
2
Union
100
50
50
T otal Gain Union's Gain Mgmt's Gain
to Divide
Offered
Offered
In particular, the Union and Management reach an agreement on
the first round, and each get half of the gains from an agreement.
BA 210 Lesson II.2 Bargaining and Impatience
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Example 3: Manipulating Alternatives to No Agreement
Originally, the Union’s initial wage demand should give it 50%
of the $10,000 gains from trade, or $5,000, and management
should accept that demand. Since the BATNA for Medieval
Times is $3,000x2 = $6,000 profit from scab labor and the
BATNA for union labor is $2,000x2 = $4,000 from outside
employment, the initial wage demand by the union should be
$4,000+$5,000 = $9,000, and management should accept that
demand, and so earn $20,000-$9,000 = $11,000 profit.
But with intensified picketing, the BATNA for Medieval Times is
$2,000x2 = $4,000 and the BATNA for union labor is $1,500x2 =
$3,000 and the gains from trade is $13,000. So, the initial wage
demand by the union should be $3,000+$6,500 = $9,500, and
management should accept that demand, and so earn $20,000$9,500 = $10,500 profit. The Union does better.
BA 210 Lesson II.2 Bargaining and Impatience
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Example 4: Bargaining with Impatience
Example 4: Bargaining with
Impatience
BA 210 Lesson II.2 Bargaining and Impatience
20
Example 4: Bargaining with Impatience
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.
BA 210 Lesson II.2 Bargaining and Impatience
21
Example 4: Bargaining with Impatience
Question: Compute a bargaining payoff table for Bargainers A
and B making alternating offers over 3 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.
BA 210 Lesson II.2 Bargaining and Impatience
22
Example 4: Bargaining with Impatience
Answer: A bargaining payoff table for Bargainers A and B
making alternating offers over 3 bargaining rounds, with
Bargainer A making the first offer:
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
(10% dis.)
B's Gain
Offered
(10% dis.)
1
A
100
?
?
2
B
100
?
?
3
A
100
?
?
BA 210 Lesson II.2 Bargaining and Impatience
23
Example 4: Bargaining with Impatience
Rounds to
A's Gain
Gain
Starting 1 bargaining round from End of Offer by TtootalDivide
Offered
Game
(10% dis.)
the end, B should accept
1
A
100
100
2
B
100
90
anything as being better than
A
100
91
nothing. After deducing that, A’s 3
best acceptable offer to B is a pittance, leaving A with 100.
B's Gain
Offered
(10% dis.)
0
10
9
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 the beginning of the game, 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.
BA 210 Lesson II.2 Bargaining and Impatience
24
Example 4: Bargaining with Impatience
Rounds to
Comment: When both bargainers End of
Game
discount the same percent, the
1
2
division of the gains from trade
3
is the same as if the object itself
were depreciating by that percentage.
Offer by
T otal Gain
to Divide
(10% dep.)
A's Gain
Offered
B's Gain
Offered
A
81
81
0
B
90
81
9
A
100
91
9
In the current problem, on the one hand, if there are 3 bargaining
rounds and if A and B each discount 10%, then A’s best
acceptable offer to B is 9 plus a pittance, leaving A with 91. On
the other hand, if there are 3 bargaining rounds and the object
itself depreciates 10% per round, then A’s best acceptable offer to
B is, again, 9 plus a pittance, leaving A with 91.
BA 210 Lesson II.2 Bargaining and Impatience
25
Example 5: Relative Impatience
Example 5: Relative Impatience
BA 210 Lesson II.2 Bargaining and Impatience
26
Example 5: Relative Impatience
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.
BA 210 Lesson II.2 Bargaining and Impatience
27
Example 5: Relative Impatience
Question: Compare outcomes from 3 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.
BA 210 Lesson II.2 Bargaining and Impatience
28
Example 5: Relative Impatience
Answer: A bargaining payoff table for Bargainers A and B, with
Bargainer A making the first offer, with 3 bargaining rounds,
with A discounting 10% between each period, and with B
discounting 20% between each period.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
(10% dis.)
B's Gain
Offered
(20% dis.)
1
A
100
?
?
2
B
100
?
?
3
A
100
?
?
BA 210 Lesson II.2 Bargaining and Impatience
29
Example 5: Relative Impatience
Rounds to
A's Gain
Gain
Starting 1 bargaining round from End of Offer by TtootalDivide
Offered
Game
(10% dis.)
the end, B should accept
1
A
100
100
2
B
100
90
anything as being better than
A
100
92
nothing. After deducing that, A’s 3
best acceptable offer to B is a pittance, leaving A with 100.
B's Gain
Offered
(20% dis.)
0
10
8
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 the beginning, 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.
BA 210 Lesson II.2 Bargaining and Impatience
30
Example 5: Relative Impatience
A bargaining payoff table for Bargainers A and B, with Bargainer
A making the first offer, with 3 bargaining rounds, with A
discounting 20% between each period, and with B discounting
10% between each period.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
(20% dis.)
B's Gain
Offered
(10% dis.)
1
A
100
?
?
2
B
100
?
?
3
A
100
?
?
BA 210 Lesson II.2 Bargaining and Impatience
31
Example 5: Relative Impatience
Rounds to
A's Gain
Gain
Starting 1 bargaining round from End of Offer by TtootalDivide
Offered
Game
(20% dis.)
the end, B should accept
1
A
100
100
2
B
100
80
anything as being better than
A
100
82
nothing. After deducing that, A’s 3
best acceptable offer to B is a pittance, leaving A with 100.
B's Gain
Offered
(10% dis.)
0
20
18
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 the beginning, 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.
BA 210 Lesson II.2 Bargaining and Impatience
32
Example 5: Relative Impatience
Summary: Compare outcomes from 3 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 9%, leaving A with 91%.
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 8%, leaving A with 92%.
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 18 plus a pittance, leaving A with 82.
A does best when his is more patient (discounts less) than B.
BA 210 Lesson II.2 Bargaining and Impatience
33
Example 5: Relative Impatience
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.
BA 210 Lesson II.2 Bargaining and Impatience
34
Summary
Summary
BA 210 Lesson II.2 Bargaining and Impatience
35
Summary
A bargainer’s 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.
BA 210 Lesson II.2 Bargaining and Impatience
36
Example 5: Relative Impatience
A bargaining payoff table for Bargainers A and B, with Bargainer
A making the first offer, with 3 bargaining rounds, with A
discounting 30% between each period, and with B discounting
40% between each period.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
(30% dis.)
B's Gain
Offered
(40% dis.)
1
A
100
100
0
2
B
100
70
30
3
A
100
82
18
BA 210 Lesson II.2 Bargaining and Impatience
37
Example 5: Relative Impatience
A bargaining payoff table for Bargainers A and B, with Bargainer
A making the first offer, with 3 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
Offer by
T otal Gain
to Divide
A's Gain
Offered
(40% dis.)
B's Gain
Offered
(0% dis.)
1
A
100
100
0
2
B
100
60
40
3
A
100
60
40
BA 210 Lesson II.2 Bargaining and Impatience
38
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 1 and cumulative Final Exam will
contain some similar questions, so you should eventually
consider every review question before taking your exams.
BA 210 Lesson II.2 Bargaining and Impatience
39
Review Questions
Review Question 1
BA 210 Lesson II.2 Bargaining and Impatience
40
Review Questions
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.
BA 210 Lesson II.2 Bargaining and Impatience
41
Review Questions
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 2
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).
BA 210 Lesson II.2 Bargaining and Impatience
42
Review Questions
Answer 1: A bargaining payoff table for Bargainers B and S, with
Bargainer S making the first offer, with 2 bargaining rounds, with
B discounting 10% between each period, and with S discounting
2% between each period.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
B's Gain
Offered
(10% dis.)
S's Gain
Offered
(2% dis.)
1
B
100
100
0
2
S
100
90
10
BA 210 Lesson II.2 Bargaining and Impatience
43
Review Questions
A bargaining payoff table for Bargainers B and S, with Bargainer
S making the first offer, with 2 bargaining rounds, with B
discounting only 1% between each period, and with S
discounting 2% between each period.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
B's Gain
Offered
(1% dis.)
S's Gain
Offered
(2% dis.)
1
B
100
100
0
2
S
100
99
1
BA 210 Lesson II.2 Bargaining and Impatience
44
Review Questions
A bargaining payoff table for Bargainers B and S, with Bargainer
S making the first offer, with 2 bargaining rounds, with B
discounting 100% between each period, and with S discounting
2% between each period.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
B's Gain
Offered
(100% dis.)
S's Gain
Offered
(2% dis.)
1
B
100
100
0
2
S
100
0
100
BA 210 Lesson II.2 Bargaining and Impatience
45
Review Questions
Summary: In the first case, when B discounts 10% between each period, B
captures 90% of the $60 gains from trade, or $54, leaving S with $6. Buyer
Betty thus makes an acceptable offer of $65-$54 = $11, leaving Seller Shen
with gain $11-$5 = $6.
On one extreme, when B discounts only 1% between each period, B captures
99% of the $60 gains from trade, or $59.40, leaving S with $0.60. Buyer Betty
thus makes an acceptable offer of $65-$59.40 = $5.60, leaving Seller Shen
with gain $5.60-$5 = $0.60.
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.
BA 210 Lesson II.2 Bargaining and Impatience
46
BA 210
Introduction to Microeconomics
End of Lesson II.2
BA 210 Lesson II.2 Bargaining and Impatience
47