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.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
1
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 how to get your best
deal from bargaining.)
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
2
Readings
Readings
Baye “Sequential Bargaining” (see the index)
Dixit Chapters 1, 2, 3
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
3
Overview
Overview
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
4
Overview
Part B Overview
Supply and Demand Analysis from Part A helps managers make
profitable decisions when there are impersonal competitive markets of
large numbers of firms and customers and workers.
Game Theory in Parts B and C now completes supply and demand
analysis when personal decisions interact.
For example, if Carlos tries to sell shoes or Aleisha tries to buy shoes
on eBay, they each compete 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 445 Lesson B.1 Sequential Bargaining and Depreciation
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Overview
Each lesson in Parts B and C 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 445 Lesson B.1 Sequential Bargaining and Depreciation
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Overview
Bargaining verses Competition depends on the number of alternative buyers and
sellers. Bargaining occurs when one buyer and one seller meet separate from a
competitive market.
Rollback Solutions find your optimal sequence of bargaining offers and
counteroffers, or any sequence of decisions, by making your last decision first.
— So, before your first cigarette, think about your last.
Take-it-or-Leave-it Offers capture the most gains from any agreement. Make the
other party an offer and do not let them make a counter offer; they must take
your first offer or leave it.
Counter Offers by the other party can eliminate the gains that could be captured
by your initial offer. — So, a buyer gains more from making an offer if he waits
until his tour bus schedule leaves no time for a counteroffer.
Bargaining and Depreciation should lead to an agreement before the gains from
the agreement actually depreciate. However, the possibility of depreciation
affects the bargaining agreement.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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Bargaining verses Competition
Bargaining verses Competition
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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Bargaining verses Competition
Overview
Bargaining verses Competition depends on the number of
alternative buyers and sellers. Bargaining occurs when
one buyer and one seller meet separate from a competitive
market.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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Rollback Solutions
Rollback Solutions
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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Rollback Solutions
Overview
Rollback Solutions find your optimal sequence of
bargaining offers and counteroffers, or any sequence of
decisions, by making your last decision first. — So, before
your first cigarette, think about your last.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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Take-it-or-leave-it Bargaining
Take-it-or-leave-it Bargaining
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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Take-it-or-leave-it Bargaining
Overview
Take-it-or-Leave-it Offers capture the most gains from any
agreement. Make the other party an offer and do not let
them make a counter offer; they must take your first offer or
leave it.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 Sabitha has 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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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 445 Lesson B.1 Sequential Bargaining and Depreciation
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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.
Proposer
I take 99%
of the
gain
from trade
I take 50%
of the
gain
from trade
I take 2%
of the
gain
from trade
Responder
Responder
Responder
Accept
Reject
Accept
Reject
Accept
Reject
99,1
0,0
50,50
0,0
2,98
0,0
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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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.
Proposer
I take 99%
of the
gain
from trade
I take 50%
of the
gain
from trade
I take 2%
of the
gain
from trade
Responder
Responder
Responder
Accept
Reject
Accept
Reject
Accept
Reject
99,1
0,0
50,50
0,0
2,98
0,0
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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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%.
Proposer
I take 99%
of the
gain
from trade
I take 50%
of the
gain
from trade
I take 2%
of the
gain
from trade
Responder
Responder
Responder
Accept
Reject
Accept
Reject
Accept
Reject
99,1
0,0
50,50
0,0
2,98
0,0
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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Counter Offers
Counter Offers
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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Counter Offers
Overview
Counter Offers by the other party can eliminate the gains
that could be captured by your initial offer. — So, a buyer
gains more from making an offer if he waits until his tour
bus schedule leaves no time for a counteroffer.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
32
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 445 Lesson B.1 Sequential Bargaining and Depreciation
33
Counter Offers
Question: Suppose Buyer Bob values the bag of fresh
beans in front of him at 1 dollar and Seller Sabitha has 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 445 Lesson B.1 Sequential Bargaining and Depreciation
34
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:
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sabitha's
Gain
Offered
1
Sabitha
100
?
?
2
Bob
100
?
?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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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, Sabitha can get
away with the whole gain from trade minus a pittance,
leaving Bob with the pittance (essentially, a zero gain to
Bob).
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sabitha's
Gain
Offered
1
Sabitha
100
0
100
2
Bob
100
?
?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
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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.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sabitha's
Gain
Offered
1
Sabitha
100
0
100
2
Bob
100
0
100
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
37
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.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sabitha's
Gain
Offered
1
Sabitha
100
0
100
2
Bob
100
0
100
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
38
Bargaining with Depreciation
Bargaining with Depreciation
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
39
Bargaining with Depreciation
Overview
Bargaining and Depreciation should lead to an agreement
before the gains from the agreement actually depreciate.
However, the possibility of depreciation affects the
bargaining agreement.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
40
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 445 Lesson B.1 Sequential Bargaining and Depreciation
41
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 20 cents. What
initial price should Buyer Bob offer? Should Seller Sam
accept that first offer?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
42
Bargaining 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.
Rounds to
Meltdown
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Bob's
Gain
Sam's
Gain
1
Bob
20
?
?
?
2
Sam
40
?
?
?
3
Bob
60
?
?
?
4
Sam
80
?
?
?
5
Bob
100
?
?
?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
43
Bargaining with Depreciation
Starting 1 bargaining round from the end of the game,
when the ice cream has 20 percent of its original value,
Sam should accept anything as being better than nothing.
So, Bob can get away with the whole 20 percent minus a
pittance, leaving Sam with the pittance.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Bob
20
20
0
2
Sam
40
?
?
3
Bob
60
?
?
4
Sam
80
?
?
5
Bob
100
?
?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
44
Bargaining with Depreciation
Rolling back to 2 steps to meltdown, when the ice cream
has 40 percent of its original value, Bob could reject any
offer and get 20 percent in the next round. So, Sam’s best
acceptable offer leaves Bob with 20 percent plus a
pittance, and Sam with 20 percent minus a pittance.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Bob
20
20
0
2
Sam
40
20
20
3
Bob
60
?
?
4
Sam
80
?
?
5
Bob
100
?
?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
45
Bargaining with Depreciation
Rolling back to 3 steps to meltdown, when the ice cream
has 60 percent of its original value, Sam could reject any
offer and get 20 percent in the next round. So, Bob’s best
acceptable offer leaves Sam with 20 percent plus a
pittance, and Bob with 40 percent minus a pittance.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Bob
20
20
0
2
Sam
40
20
20
3
Bob
60
40
20
4
Sam
80
?
?
5
Bob
100
?
?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
46
Bargaining with Depreciation
Rolling back to 4 steps to meltdown, when the ice cream
has 80 percent of its original value, Bob could reject any
offer and get 40 percent in the next round. So, Sam’s best
acceptable offer leaves Bob with 40 percent plus a
pittance, and Sam with 40 percent minus a pittance.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Bob
20
20
0
2
Sam
40
20
20
3
Bob
60
40
20
4
Sam
80
40
40
5
Bob
100
?
?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
47
Bargaining with Depreciation
Rolling back to the beginning of the game, when the ice
cream has 100% of its original value of $1.00, Sam could
reject any offer and get 40 percent in the next round. So,
Bob’s best acceptable offer leaves Sam with 40 percent
plus a pittance, and Bob with 60 percent minus a pittance.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Bob
20
20
0
2
Sam
40
20
20
3
Bob
60
40
20
4
Sam
80
40
40
5
Bob
100
60
40
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
48
Bargaining with Depreciation
Buyer Bob should thus initially offer price $0.40 for the ice
cream, and Seller Sam should accept that first offer. That
gives Bob 60 percent of the gain from trade and Sam 40
percent.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Bob
20
20
0
2
Sam
40
20
20
3
Bob
60
40
20
4
Sam
80
40
40
5
Bob
100
60
40
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
49
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.
In more advanced games (covered in Part III),
counteroffers can actually occur, and ice cream can start
melting, if players lack perfect information about the payoffs
of other players. Bob’s value of the ice cream, for example,
can be unknown to Sam, so Bob can try to signal a lower
value of ice cream by offering a low initial price. Sam might
then conclude Bob has almost zero value for the ice cream
and, so, might counteroffer the ice cream at a price near
zero.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
50
Summary
Summary
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
51
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 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.1 Sequential Bargaining and Depreciation
52
Summary
A bargaining payoff table for Bargainers A and B, with
Bargainer A making the only offer: 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 percent.
Rounds to
End of
Game
1
Offer by
T otal Gain
to Divide
A's Gain
Offered
B's Gain
Offered
A
100
100
0
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
53
Summary
A bargaining payoff table for Bargainers A and B, with
Bargainer A making the first offer, when the potential gains
from bargaining fully depreciate after 2 Rounds:
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
B's Gain
Offered
1
B
50
0
50
2
A
100
50
50
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
54
Summary
Rounds to
T otal Gain
A's Gain
B's Gain
Starting 1 bargaining round
End of
Offer by
to Divide
Offered
Offered
Game
from the end, when the gain
1
B
50
0
50
2
A
100
50
50
has 33 1/3 percent of its
original value, A should accept anything as being better
than nothing. After deducing that, B’s best acceptable offer
to A is a pittance, leaving B with 25 percent.
Rolling back to the beginning, when the gain has 100
percent of its original value, B could reject any offer and get
50 percent in the next round. After deducing that, A’s best
acceptable offer to B is 50 percent plus a pittance, leaving
A with 50 percent.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
55
Summary
A bargaining payoff table for Bargainers A and B, with
Bargainer A making the first offer, when the potential gains
from bargaining fully depreciate after 3 Rounds:
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
B's Gain
Offered
1
A
33 1/3
33 1/3
0
2
B
66 2/3
33 1/3
33 1/3
3
A
100
66 2/3
33 1/3
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
56
Summary
Rounds to
T otal Gain
A's Gain
Starting 1 bargaining round from
End of
Offer by
to Divide
Offered
Game
the end, when the gain has 33 1/3
1
A
33 1/3
33 1/3
percent of its original value, B
2
B
66 2/3
33 1/3
should accept anything as being
3
A
100
66 2/3
better than nothing. After deducing
that, A’s best acceptable offer to B is a pittance, leaving A with 25
percent.
B's Gain
Offered
0
33 1/3
33 1/3
Rolling back to 2 rounds from the end, when the gain has 66 2/3
percent of its original value, A could reject any offer and get 33 1/3
percent in the next round. After deducing that, B’s best acceptable offer
to A is 33 1/3 percent plus a pittance, leaving B with 33 1/3 percent.
Rolling back to the beginning, when the gain has 100 percent of its
original value, B could reject any offer and get 33 1/3 percent in the
next round. After deducing that, A’s best acceptable offer to B is 33 1/3
percent plus a pittance, leaving A with 66 2/3 percent.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
57
Summary
A bargaining payoff table for Bargainers A and B, with
Bargainer A making the first offer, when the potential gains
from bargaining fully depreciate after 4 Rounds:
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
B's Gain
Offered
1
B
25
25
0
2
A
50
25
25
3
B
75
50
25
4
A
100
50
50
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
58
Summary
Starting 1 bargaining round from the end,
when the gain has 25 percent of its original
value, A should accept anything as being
better than nothing. After deducing that, B’s
best acceptable offer to A is a pittance,
leaving B with 25 percent.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
B's Gain
Offered
1
B
25
25
0
2
A
50
25
25
3
B
75
50
25
4
A
100
50
50
Rolling back to 2 rounds from the end, when the gain has 50 percent of its original value,
B could reject any offer and get 25 percent in the next round. After deducing that, A’s
best acceptable offer to B is 25 percent plus a pittance, leaving A with 25 percent.
Rolling back to 3 rounds from the end, when the gain has 75 percent of its original value,
A could reject any offer and get 25 percent in the next round. After deducing that, B’s
best acceptable offer to A is 25 percent plus a pittance, leaving B with 50 percent.
Rolling back to the beginning, when the gain has 100 percent of its original value, B could
reject any offer and get 50 percent in the next round. After deducing that, A’s best
acceptable offer to B is 50 percent plus a pittance, leaving A with 50 percent.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
59
Summary
A bargaining payoff table for Bargainers A and B, with
Bargainer A making the first offer, when the potential gains
from bargaining fully depreciate after 5 Rounds:
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
B's Gain
Offered
1
A
20
20
0
2
B
40
20
20
3
A
60
40
20
4
B
80
40
40
5
A
100
60
40
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
60
Summary
Starting 1 bargaining round from the end, when the
gain has 20 percent of its original value, 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 20 percent.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
A's Gain
Offered
B's Gain
Offered
1
A
20
20
0
2
B
40
20
20
3
A
60
40
20
Rolling back to 2 rounds from the end, when the
4
B
80
40
40
gain has 40 percent of its original value, A could
5
A
100
60
40
reject any offer and get 20 percent in the next
round. After deducing that, B’s best acceptable offer to A is 20 percent plus a pittance, leaving B with
20 percent.
Rolling back to 3 rounds from the end, when the gain has 60 percent of its original value, B could
reject any offer and get 20 percent in the next round. After deducing that, A’s best acceptable
offer to B is 20 percent plus a pittance, leaving A with 40 percent.
Rolling back to 4 rounds from the end, when the gain has 80 percent of its original value, A could
reject any offer and get 40 percent in the next round. After deducing that, B’s best acceptable
offer to A is 40 percent plus a pittance, leaving B with 40 percent.
Rolling back to the beginning, when the gain has 100 percent of its original value, B could
reject any offer and get 40 percent in the next round. After deducing that, A’s best acceptable
offer to B is 40 percent plus a pittance, leaving A with 60 percent.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
61
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.1 Sequential Bargaining and Depreciation
62
Review 1: Bargaining with Depreciation
Review 1: Bargaining with
Depreciation
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
63
Review 1: Bargaining with Depreciation
Question 1. Consider union-management negotiations over
wages for workers during Thanksgiving Weekend at
Medieval Times Dinner Theatre. The extended weekend
lasts Thursday through Sunday. Each day Medieval Times
operates, it makes a profit of $10,000. On the Wednesday
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. Offers alternate
thereafter. Medieval Times can open only on those days
after an agreement is reached.
What initial wage demand should the union make? Should
management accept that demand?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
64
Review 1: Bargaining with Depreciation
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 $40,000 gain from accepting the Union’s initial offer.
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
Offered
to Divide
M.'s Gain
Offered
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
65
Review 1: Bargaining with Depreciation
Rounds to
T otal Gain Union's Gain M's Gain
Starting 1 bargaining round
End of
Offer by
to Divide
Offered
Offered
Game
from the end of the game,
Management
1
25
0
25
Union
2
50
25
25
when a contract has 25
Management
3
75
?
?
percent of its original value,
Union
4
100
?
?
the Union should accept
anything as being better than nothing. So, Management
can get away with the whole 25 percent minus a pittance,
leaving the Union with the pittance.
Rolling back to 2 bargaining rounds from the end, when a
contract has 50 percent of its original value, Management
could reject any offer and get 25 percent in the next round.
So, the Union’s best acceptable offer leaves Management
with 25 percent plus a pittance, and the union with 25
percent minus a pittance.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
66
Review 1: Bargaining with Depreciation
Rounds to
Rolling back to 3 bargaining rounds End of Offer by T otal Gain Union's Gain
to Divide
Offered
before the end, when a contract has Game
Management
1
25
0
75 percent of its original value, the
Union
2
50
25
Union could reject any offer and get
Management
3
75
25
25 percent in the next round. So,
Union
4
100
50
Management’s best acceptable
offer leaves the Union with 25 percent plus a pittance, and
Management with 50 percent minus a pittance.
M's Gain
Offered
25
25
50
50
Rolling back to the beginning, 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 50
percent minus a pittance.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
67
Review 1: Bargaining with Depreciation
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 be for 50% of the
$40,000 profits, or $20,000, and management should
accept that demand.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
68
Review 2: Bargaining with Depreciation
Review 2: Bargaining with
Depreciation
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
69
Review 2: Bargaining with Depreciation
Question 2. Consider negotiations over wages for adjunct Management
Science Professors at the Business Administration Division of Seaver
College of Pepperdine University. Pepperdine seeks a professor for
the next 3 semesters. The only candidate is willing to work for $15,000
per semester and Pepperdine is willing to pay up to $25,000 per
semester, so the potential gains from trade are $10,000 per semester.
Before the first semester, Pepperdine confronts the candidate over
wages. Pepperdine presents its offer. The candidate either accepts it
or rejects it and returns the next semester with a counteroffer. Offers
alternate thereafter. Employment only occurs in those semesters after
an agreement is reached.
What percentage of the gains from trade should Pepperdine offer?
Should the candidate accept that initial offer?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
70
Review 2: Bargaining with Depreciation
Answer 2: To consider all possible wage offers (offers by
Pepperdine to buy labor by the candidate to sell labor),
consider another bargaining payoff table. The game ends
if an offer is accepted or if the three semesters end without
an accepted offer, and gains are measured as a
percentage of the $30,000 gain from accepting
Pepperdine’s offer before the first semester.
Rounds to
End of
Game
Pepperdine's Candidate's
T otal Gain
Gain
Gain
to Divide
Offered
Offered
Offer by
1
Pepperdine
33 1/3
33 1/3
0
2
Candidate
66 2/3
33 1/3
33 1/3
3
Pepperdine
100
66 2/3
33 1/3
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
71
Review 2: Bargaining with Depreciation
Rounds to
Pepperdine's Candidate's
T otal Gain
Starting 1 bargaining round
End of
Offer by
Gain
Gain
to Divide
Game
Offered
Offered
from the end of the game,
1
Pepperdine 33 1/3
33 1/3
0
2
Candidate
66 2/3
33 1/3
33 1/3
when a contract has 33 1/3
3
Pepperdine 100
?
?
percent of its original value,
the Candidate should accept anything as being better than
nothing. So, Pepperdine can get away with the whole 33
1/3 percent minus a pittance, leaving the Candidate with
the pittance.
Rolling back to 2 bargaining rounds from the end, when a
contract has 66 2/3 percent of its original value, Pepperdine
could reject any offer and get 33 1/3 percent in the next
round. So, the Candidate’s best acceptable offer leaves
Pepperdine with 33 1/3 percent plus a pittance, and the
Candidate with 33 1/3 percent minus a pittance.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
72
Review 2: Bargaining with Depreciation
Rounds to
Pepperdine's Candidate's
Twhen
otal Gain a
Rolling back to 3 bargaining rounds before
End of the
Offerend,
by
Gain
Gain
to Divide
Game
Offered
Offered
contract has 100 percent of its
1
Pepperdine 33 1/3
33 1/3
0
original value, the Candidate could
2
Candidate
66 2/3
33 1/3
33 1/3
reject any offer and get 33 1/3
3
Pepperdine 100
66 2/3
33 1/3
percent in the next round. So,
Pepperdine’s best acceptable offer leaves the Candidate with 33 1/3
percent plus a pittance, and Pepperdine with 66 2/3 percent minus a
pittance.
Pepperdine’s initial wage offer should be for Pepperdine to keep 66 2/3
percent of the $30,000 gain from employment for all three semesters,
and the Candidate should accept that initial offer and, so, keep 33 1/3
percent of the $30,000 gain. In particular, since the Candidate is willing
to work for $45,000 for all three semesters, he should be paid $45,000
+ $10,000 = $55,000.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
73
Review 3: Bargaining with Depreciation
Review 3: Bargaining with
Depreciation
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
74
Review 3: Bargaining with Depreciation
Question 3. Two companies, one in Dallas and one in LA, are using the same
Boston-based lawyer, William Alan Shatner (aka, Captain Kirk). Each firm
needs Shatner’s services in January, February, and March. As a result of
coordinating schedules, Shatner could each month make a triangle route
(Boston-Dallas-LA) rather than making two separate trips. The airfares each
month are:
Triangle $1000, Boston-Dallas roundtrip $400, Boston-LA roundtrip $800
Before the January trip, Dallas (the Dallas company) confronts LA (the LA
company) over its share of the triangle airfare. Dallas presents its offer. LA
either accepts it for each of the three trips, or rejects it and returns after the
January trip with a counteroffer. After the January trip, Dallas either accepts
that counteroffer for each of the remaining two trips, or rejects it and returns
after the February trip with a counteroffer. After the February trip, LA either
accepts that counteroffer for the remaining trip, or rejects it and ends the game.
What percentage of the gains from trade should Dallas offer? Should LA
accept that initial offer?
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
75
Review 3: Bargaining with Depreciation
Answer 3: To consider all possible offers to split the gains
from an agreement on splitting the cost of the triangle
route, consider another bargaining payoff table. The game
ends if an offer is accepted or if the three months end
without an accepted agreement, and gains are measured
as a percentage of the gain from using the triangle route for
all three months.
Rounds to
End of
Game
Offer by
1
Dallas
33 1/3
?
?
2
LA
66 2/3
?
?
3
Dallas
100
?
?
T otal Gain Dallas's Gain
to Divide
Offered
LA's Gain
Offered
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
76
Review 3: Bargaining with Depreciation
Rounds to
T otal Gain Dallas's Gain LA's Gain
Starting 1 bargaining round
End of
Offer by
to Divide
Offered
Offered
Game
from the end of the game,
1
Dallas
33 1/3
33 1/3
0
2
LA
66 2/3
33 1/3
33 1/3
when an agreement has
3
Dallas
100
?
?
33 1/3 percent of its original
value, LA should accept anything as being better than
nothing. So, Dallas can get away with the whole 33 1/3
percent minus a pittance, leaving the LA with the pittance.
Rolling back to 2 bargaining rounds from the end, when an
agreement has 66 2/3 percent of its original value, Dallas
could reject any offer and get 33 1/3 percent in the next
round. So, the LA’s best acceptable offer leaves Dallas
with 33 1/3 percent plus a pittance, and LA with 33 1/3
percent minus a pittance.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
77
Review 3: Bargaining with Depreciation
Rounds to
Twhen
otal Gain an
Dallas's Gain LA's Gain
Rolling back to 3 bargaining rounds before
End of the
Offerend,
by
to Divide
Offered
Offered
Game
agreement has 100 percent of its
1
Dallas
33 1/3
33 1/3
0
original value, LA could reject any
2
LA
66 2/3
33 1/3
33 1/3
offer and get 33 1/3 percent in the
3
Dallas
100
66 2/3
33 1/3
next round. So, Dallas’s best
acceptable offer leaves LA with 33 1/3 percent plus a pittance, and
Dallas with 66 2/3 percent minus a pittance.
Recall, the airfares each month are: Triangle $1000, Boston-Dallas
roundtrip $400, Boston-LA roundtrip $800. In particular, the gain from
three triangle trips is 3x(1200-1000) = 600.
Dallas’s initial offer should be for Dallas to keep 66 2/3 percent of the
$600 gain, or $400. So Dallas pays 3x400-400 = $800. Likewise, LA
gains $200 and pays 3x800-200 = $2200.
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
78
BA 445
Managerial Economics
End of Lesson B.1
BA 445 Lesson B.1 Sequential Bargaining and Depreciation
79