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 445 Lesson B.2 Sequential Bargaining and Impatience
1
Readings
Readings
Baye “Sequential Bargaining” (see the index)
Dixit Chapters 1, 2, 3
BA 445 Lesson B.2 Sequential Bargaining and Impatience
2
Overview
Overview
BA 445 Lesson B.2 Sequential Bargaining and Impatience
3
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
4
Best Alternative to No Agreement
Best Alternative to No Agreement
BA 445 Lesson B.2 Sequential Bargaining and Impatience
5
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
6
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 445 Lesson B.2 Sequential Bargaining and Impatience
7
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 445 Lesson B.2 Sequential Bargaining and Impatience
8
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 445 Lesson B.2 Sequential Bargaining and Impatience
9
Best Alternative to No Agreement
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
1
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sabitha's
Gain
Offered
Bob
100
100
0
BA 445 Lesson B.2 Sequential Bargaining and Impatience
10
Bilateral Alternatives to No Agreement
Bilateral Alternatives to No
Agreement
BA 445 Lesson B.2 Sequential Bargaining and Impatience
11
Mutual 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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
12
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 (nonunion) labor, and union labor earns $2,000 from outside employment.
What initial wage demand should the union make? Should
management accept that demand?
BA 445 Lesson B.2 Sequential Bargaining and Impatience
13
Mutual Alternatives to No Agreement
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.
Rounds to
End of
Game
Offer by
1
Management
25
?
?
2
Union
50
?
?
3
Management
75
?
?
4
Union
100
?
?
T otal Gain Union's Gain
to Divide
Offered
M.'s Gain
Offered
BA 445 Lesson B.2 Sequential Bargaining and Impatience
14
Mutual Alternatives to No Agreement
Starting 1 bargaining round
from the end of the game and
rolling back to the beginning
determines optimal offers and
counteroffers.
Rounds to
End of
Game
Offer by
1
Management
25
0
25
2
Union
50
25
25
3
Management
75
25
50
4
Union
100
50
50
T otal Gain Union's Gain
to Divide
Offered
M's Gain
Offered
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
15
Manipulating Alternatives to No Agreement
Manipulating Alternatives to No
Agreement
BA 445 Lesson B.2 Sequential Bargaining and Impatience
16
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
17
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 445 Lesson B.2 Sequential Bargaining and Impatience
18
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 445 Lesson B.2 Sequential Bargaining and Impatience
19
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
Management
25
0
25
2
Union
50
25
25
3
Management
75
25
50
4
Union
100
50
50
T otal Gain Union's Gain
to Divide
Offered
M's Gain
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 445 Lesson B.2 Sequential Bargaining and Impatience
20
Manipulating Alternatives to No 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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
21
Bargaining with Impatience
Bargaining with Impatience
BA 445 Lesson B.2 Sequential Bargaining and Impatience
22
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
23
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 445 Lesson B.2 Sequential Bargaining and Impatience
24
Bargaining with Impatience
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
25
Bargaining with Impatience
Answer: A bargaining payoff table for Bargainers A and B
making alternating offers over 5 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
?
?
4
B
100
?
?
5
A
100
?
?
BA 445 Lesson B.2 Sequential Bargaining and Impatience
26
Bargaining with Impatience
Starting 1 bargaining round from the end, B Rounds to
End of
should accept anything as being better than
Game
nothing. After deducing that, A’s best acceptable 1
offer to B is a pittance, leaving A with 100.
2
3
Offer by
T otal Gain
to Divide
A's Gain
Offered
(10% dis.)
B's Gain
Offered
(10% dis.)
A
100
100.00
0.00
B
100
90.00
10.00
A
100
91.00
9.00
Rolling back to 2 rounds from the end, A could
4
B
100
81.90
18.10
reject any offer and get 100 in the next round,
5
A
100
83.71
16.29
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 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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
27
Bargaining with Impatience
Rounds to
Comment: When both
End of
Game
bargainers discount the same 1
percent, the division of the gains 2
from trade is the same as if the 34
object itself were depreciating 5
by that percentage.
Offer by
T otal Gain
to Divide
(10% dep.)
A's Gain
Offered
B's Gain
Offered
A
65.61
65.61
0.00
B
72.90
65.61
7.29
A
81.00
73.71
7.29
B
90.00
73.71
16.29
A
100.00
83.71
16.29
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%.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
28
Relative Patience
Relative Patience
BA 445 Lesson B.2 Sequential Bargaining and Impatience
29
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
30
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
31
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
32
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.
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
?
?
4
B
100
?
?
5
A
100
?
?
BA 445 Lesson B.2 Sequential Bargaining and Impatience
33
Relative Patience
A's Gain
B's Gain
Starting 1 bargaining round from the end, B Rounds to
T otal Gain
End of
Offer by
Offered
Offered
to Divide
should accept anything as being better than
Game
(10% dis.)
(20% dis.)
nothing. After deducing that, A’s best
1
A
100
100.00
0.00
acceptable offer to B is a pittance, leaving A
2
B
100
90.00
10.00
with 100.
3
A
100
92.00
8.00
Rolling back to 2 rounds from the end, A
4
B
100
82.80
17.20
could reject any offer and get 100 in the
5
A
100
86.24
13.76
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.
Rolling back to 4 rounds from the end, A could reject any offer and get 92 in the next
round,
which after A’s 10% discount is worth 82.8 in the current round. After deducing that, B’s
best acceptable offer to A is 82.8 plus a pittance, leaving B with 17.2.
Rolling back to the beginning, B could reject any offer and get 17.2 in the next round,
which after B’s 20% discount is worth 13.76 in the current round. . After deducing that,
A’s best acceptable offer to B is 13.76 plus a pittance, leaving A with 86.24.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
34
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.
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
?
?
4
B
100
?
?
5
A
100
?
?
BA 445 Lesson B.2 Sequential Bargaining and Impatience
35
Relative Patience
A's Gain
B's Gain
Starting 1 bargaining round from the end, B Rounds to
T otal Gain
End of
Offer by
Offered
Offered
to Divide
should accept anything as being better than
Game
(20% dis.)
(10% dis.)
nothing. After deducing that, A’s best
1
A
100
100.00
0.00
acceptable offer to B is a pittance, leaving
2
B
100
80.00
20.00
A with 100.
3
A
100
82.00
18.00
Rolling back to 2 rounds from the end, A
4
B
100
65.60
34.40
could reject any offer and get 100 in the
5
A
100
69.04
30.96
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.
Rolling back to 4 rounds from the end, A could reject any offer and get 82 in the next
round,
which after A’s 20% discount is worth 65.6 in the current round. After deducing that, B’s
best acceptable offer to A is 65.6 plus a pittance, leaving B with 34.4.
Rolling back to the beginning, B could reject any offer and get 34.4 in the next round,
which after B’s 10% discount is worth 30.96 in the current round. . After deducing that,
A’s best acceptable offer to B is 30.96 plus a pittance, leaving A with 69.04.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
36
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
37
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
38
Summary
Summary
BA 445 Lesson B.2 Sequential Bargaining and Impatience
39
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
40
Summary
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.
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
100.00
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
BA 445 Lesson B.2 Sequential Bargaining and Impatience
41
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
Offer by
T otal Gain
to Divide
A's Gain
Offered
(40% dis.)
B's Gain
Offered
(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
BA 445 Lesson B.2 Sequential Bargaining and Impatience
42
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
43
Review 1: Relative Impatience
Review 1: Relative Impatience
BA 445 Lesson B.2 Sequential Bargaining and Impatience
44
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
45
Review 1: Relative Impatience
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).
BA 445 Lesson B.2 Sequential Bargaining and Impatience
46
Review 1: Relative Impatience
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.
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
100.00
0.00
2
S
100
90.00
10.00
3
B
100
91.20
9.80
4
S
100
82.08
17.92
BA 445 Lesson B.2 Sequential Bargaining and Impatience
47
Review 1: Relative Impatience
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.
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
100.0000
0.0000
2
S
100
99.0000
1.0000
3
B
100
99.0200
0.9800
4
S
100
98.0298
1.9702
BA 445 Lesson B.2 Sequential Bargaining and Impatience
48
Review 1: Relative Impatience
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.
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
3
B
100
2
98
4
S
100
0
100
BA 445 Lesson B.2 Sequential Bargaining and Impatience
49
Review 1: Relative Impatience
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.
BA 445 Lesson B.2 Sequential Bargaining and Impatience
50
BA 445
Managerial Economics
End of Lesson B.2
BA 445 Lesson B.2 Sequential Bargaining and Impatience
51