Review
We will spend up to 30 minutes reviewing Exam 1
• Know how your answers were graded.
• Know how to correct your mistakes. Your final exam is
cumulative, and may contain similar questions.
BA 210 Lesson II.1 Strategic Bargaining
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 you how to get the best deal
for yourself from bargaining.)
BA 210 Lesson II.1 Strategic Bargaining
2
Acknowledgements
Acknowledgements
Although much of the course content in
Part II is covered in the Krugman textbook,
the PowerPoint slides for the lessons are
self-contained, and are your primary text.
Thanks to the following students for
insights used in the Part II lesson slides:
Fabrizio Alliata
Karissa Garcia
BA 210 Lesson II.1 Strategic Bargaining
3
Overview
Overview
BA 210 Lesson II.1 Strategic Bargaining
4
Overview
Part II Overview
Supply and Demand Analysis from Part I helps consumers make
satisfying choices, helps managers make profitable decisions, and
helps governments make effective public policies when there are
impersonal competitive markets of large numbers of firms and
customers and workers. Game Theory in Part II now completes
supply and demand analysis when personal decisions interact.
For example, if Carlos tries to sell shoes and Aleisha tries to buy
shoes on eBay, each competes with many potential shoe sellers or
buyers, with price determined by aggregate supply and demand.
But if, instead, Aleisha happens to walk by Carlos’s trading stall
outside the Tijuana Wax Museum, they become tied to a few
potential shoe sellers or buyers in Tijuana, and they negotiate a
sales price separate from the worldwide competitive market.
BA 210 Lesson II.1 Strategic Bargaining
5
Overview
Each lesson in Part II is divided into examples. Each example
asks a Question and supplies an Answer. The questions are like
the questions on your exams: A game is described verbally and
you are asked questions about its solution. You must formulate
the game (identifying players, strategies, payoffs, …) then answer
those questions by solving the game.
Many examples also include a Comment about what general
lesson you learn from the example.
BA 210 Lesson II.1 Strategic Bargaining
6
Overview
Lesson II.1 Strategic Bargaining
Example 1: Bargaining verses Competition
Example 2: Rollback Solutions
Example 3: Take-it-or-Leave-it Offers
Example 4: Counter Offers
Example 5: Bargaining with Depreciation
Summary
Review Questions
Lesson II.2 Bargaining and Impatience
Lesson II.3 Sequential Quantity Competition
BA 210 Lesson II.1 Strategic Bargaining
7
Overview
Lesson 1 formulates and solves the following games:
Example 2: Century Mark Game. Has a simple solution, found
by starting at the end of the game and rolling back to the
beginning.
Examples 3, 4, 5: Sequential Bargaining Games. Have unique
rollback solutions. The solution favors the first mover, and so
favors aggressors.
BA 210 Lesson II.1 Strategic Bargaining
8
Example 1: Bargaining verses Competition
Example 1: Bargaining verses
Competition
BA 210 Lesson II.1 Strategic Bargaining
9
Example 1: Bargaining verses Competition
Question: Suppose
• Aleisha is willing to offer up to $59 to buy a pair of shoes.
• Brad, to pay $44; Claudia, $34.01; Darren, $24; Edwina, $10.
Suppose
• Andrew is willing to accept down to $8 to sell a pair of shoes.
• Betty, to sell $20; Carlos, $34; Donna, $48; Engelbert, $62.
Compute the competitive-equilibrium price of shoes if all 10
people trade shoes on eBay? (Ignore shipping costs.)
Alternatively, suppose Aleisha and Carlos do not use eBay, but
Aleisha walks by Carlos’s trading stall outside the Tijuana Wax
Museum. Compute the gains if they trade a pair of shoes.
Compute the price of shoes if they divide the gains 50-50.
BA 210 Lesson II.1 Strategic Bargaining
10
Example 1: Bargaining verses Competition
Answer: Competitive Markets have Many Independent Buyers ...
• Aleisha is willing to offer up to $59 to buy a pair of shoes.
• Brad, $44; Claudia, $34.01; Darren, $24; Edwina, $10.
Aleisha
Brad
Claudia
Darren
Edwina
BA 210 Lesson II.1 Strategic Bargaining
11
Example 1: Bargaining verses Competition
… and Many Independent Sellers
• Andrew is willing to accept down to $8 to sell a pair of shoes.
• Betty, $20; Carlos, $34; Donna, $48; Engelbert, $62.
Aleisha
Engelbert
Brad
Donna
Claudia
Carlos
Darren
Betty
Edwina
Andrew
BA 210 Lesson II.1 Strategic Bargaining
12
Example 1: Bargaining verses Competition
Demand equals Supply determines Competitive Price
• At some price between $34 and $34.01, Aleisha, Brad and
Claudia each buy 1 pair, and Andrew, Betty and Carlos each
sell.
Aleisha
Engelbert
Brad
Donna
Claudia
Carlos
Darren
Betty
Edwina
Andrew
BA 210 Lesson II.1 Strategic Bargaining
13
Example 1: Bargaining verses Competition
Bargaining Occurs with One Buyer and One Seller
If Aleisha and Carlos meet separate from the competitive market,
then the gains if they were to trade a pair of shoes is the
difference between willingness to pay and willingness to sell.
• Aleisha is willing to offer $59 to buy a pair of shoes.
• Carlos is willing to accept $34 to sell a pair of shoes.
• The gain from trade is the difference, $25 = $59-$34.
If Aleisha and Carlos divided the gains from trade 50-50, then
each gets $12.50 gain, meaning Aleisha pays price $46.50 =
$59.00-$12.50, and Carlos receives price $46.50 =
$34.00+$12.50
BA 210 Lesson II.1 Strategic Bargaining
14
Example 1: Rollback Solutions
Example 2: Rollback Solutions
BA 210 Lesson II.1 Strategic Bargaining
15
Example 2: Rollback Solutions
Comment: Whether Aleisha and Carlos divide the gains from
trade 50-50 or 99-1 or some other division depends on the rules
for the sequence of bargaining offers and counteroffers. Consider
an instructive game to illustrate the best way to make any
sequence of decisions.
BA 210 Lesson II.1 Strategic Bargaining
16
Example 2: Rollback Solutions
Question: Consider the Century Mark Game
• Played by pairs of players taking turns.
• At each turn, each player chooses a number between 1 and 10
inclusive.
• This choice is added to sum of all previous choices (the initial
sum is 0).
• The first player to take the cumulative sum to 100 or more
wins.
How should you play the game if you were the first player?
BA 210 Lesson II.1 Strategic Bargaining
17
Example 2: Rollback Solutions
Answer: Start at the end of the game and work backward. This is
called backward induction and it defines a rollback solution, or
rollback equilibrium.
In the Century Mark Game, first ask: What number gets you to
100 next turn? If you bring the cumulative sum to 89, you can
take the cumulative sum to 100 on your next turn and win no
matter what your opponent does. (Whatever your opponent does,
you can make the sum of your two moves equal 11, and 89+11 =
100.)
BA 210 Lesson II.1 Strategic Bargaining
18
Example 2: Rollback Solutions
Complete Rollback Solution
• If you bring the cumulative sum to 89 on one turn, you can
take the cumulative sum to 100 on your next turn and win no
matter what your opponent does. (Whatever your opponent
does, you can make the sum of your two moves equal 11.)
• Hence, if you bring the cumulative sum to 78 on one turn, you
can take the cumulative sum to 89 on your next turn (and so
eventually win) no matter what your opponent does.
• And so on for sums 67, 56, 45, 34, 23, 12.
• Hence, if you play 1 on your first turn, you can bring the
cumulative sum to 12 on your next turn (and so eventually
win) no matter what your opponent does.
• That solution is a strategy --- a complete plan of actions no
matter what your opponent does.
BA 210 Lesson II.1 Strategic Bargaining
19
Example 3: Take-it-or-leave-it Bargaining
Example 3: Take-it-or-leave-it
Bargaining
BA 210 Lesson II.1 Strategic Bargaining
20
Example 3: Take-it-or-leave-it Bargaining
Comment 1: Bargaining Games have three parts.
• Players or bargainers are managers, customers, or workers,
typically considered two at the time.
• Strategies are offers to divide a fixed positive gain from an
agreement, the rejection of such an offer, or the acceptance of
such an offer. The most common agreements is trading a good
at a particular price.
• Manager strategies include a price at which to sell a
consumption good or a wage to buy workers’ time.
• Customer strategies include a price at which to buy.
• Worker strategies include a wage at which to sell.
• Payoffs are percentage shares of the gain from an agreement, if
there is an agreement, or zero if there were no agreement.
BA 210 Lesson II.1 Strategic Bargaining
21
Example 3: Take-it-or-leave-it Bargaining
Comment 2: Bargaining Games in Part II assume Rationality and
Perfect Information are Common Knowledge.
• Rationality means players are perfect calculators and perfect
followers of their best strategies. Players have perfect
knowledge of their own interests.
• Perfect Information means players know the bargaining rules
and all offers that have taken place.
• Common Knowledge of rationality and perfect information
means all players assume all other players are rational and
have perfect information, and all players assume all other
players assume all other players are rational and have perfect
information, and so on.
• In contrast, poker professionals (sharks) do not assume bad
players (fish or suckers) are rational. Sharks try to trick fish.
BA 210 Lesson II.1 Strategic Bargaining
22
Example 3: Take-it-or-leave-it Bargaining
Question: Suppose Buyer Bob values the bag of fresh beans in
front of him at 1 dollar and Seller 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 210 Lesson II.1 Strategic Bargaining
23
Example 3: Take-it-or-leave-it Bargaining
Answer: Follow two steps to define any Bargaining Game:
 Determine the total gain from an agreement. When the
agreement means trading a good at a particular price, the gain is
the maximum price the buyer were willing to pay minus the
minimum price the seller were willing to receive. Here, total
gain is $1.
 Strategies are then percentage shares of the gain from trade.
Here, prices $0.01, $0.50, or $0.98 means Bob takes 99
percent, 50 percent, or 2 percent of the gain from trade.
 Determine the bargaining rules. Here, Bob makes an offer that
Sabitha accepts or rejects. Either way, the game is then over.
BA 210 Lesson II.1 Strategic Bargaining
24
Example 3: Take-it-or-leave-it Bargaining
Comment: Many Sequential-Move Games have complex rollback
solutions requiring a game tree to predict future responses to
current actions. Game trees or extensive forms consist of nodes
and branches. Nodes are connected to one another by the
branches, and come in two types. Some nodes are decision
nodes, where a player chooses an action branch to another node.
The other nodes are terminal nodes, where players receive the
outcomes of the actions taken by themselves and all other
players.
BA 210 Lesson II.1 Strategic Bargaining
25
Example 3: Take-it-or-leave-it Bargaining
Answer (continued): Here is a Game Tree where payoffs list the
gains from trade, with payoffs to the first player to act (Bob, the
proposer) listed first.
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 210 Lesson II.1 Strategic Bargaining
26
Example 3: Take-it-or-leave-it Offers
Starting at the end of the game, Bob predicts rational responses to
alternative proposals. Sabitha should accept anything as being
better than nothing.
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 210 Lesson II.1 Strategic Bargaining
27
Example 3: Take-it-or-leave-it Offers
Rolling back to the beginning of the game, determine Bob’s
rational proposal, where Bob assumes Sabitha is rational. Bob
should offer to take 99%. 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 210 Lesson II.1 Strategic Bargaining
28
Example 4: Counter Offers
Example 4: Counter Offers
BA 210 Lesson II.1 Strategic Bargaining
29
Example 4: Counter Offers
Comment: Bargainers can get most of the gains from trade by
making take-it-or-leave-it offers. So it is best for Buyer Bob to
make an offer on a bag of beans just as his tour bus is about to
leave.
To try to counter the take-it-or-leave-it strategy and get more of
the gains from trade, Seller Sabitha can try to make a (quick)
counter offer.
BA 210 Lesson II.1 Strategic Bargaining
30
Example 4: 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 210 Lesson II.1 Strategic Bargaining
31
Example 4: Counter Offers
Answer: To consider all possible price offers (offers by Bob to
buy or counteroffers by Sabitha to sell), consider a bargaining
payoff table:
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 210 Lesson II.1 Strategic Bargaining
32
Example 4: Counter Offers
Starting 1 bargaining round from the end of the game, Sabitha
either accepts Bob’s offer or makes a counteroffer that Bob must
either accept or reject. Bob should accept anything as being better
than nothing. So, 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 210 Lesson II.1 Strategic Bargaining
33
Example 4: Counter Offers
Rolling back to the beginning (2 bargaining rounds from the end
of the game), Sabitha could reject any offer and get 100 percent
in the next round. So, Bob’s best acceptable offer leaves Sabitha
with 100 percent, and Bob with 0 percent. That is, trade occurs,
and Bob pays $1.
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 210 Lesson II.1 Strategic Bargaining
34
Example 4: Counter Offers
Comment: Listening to a counteroffer thus eliminates all the
gains from trade that would otherwise have gone to the player
making a take-it-or-leave-it offer. Bob should have refused to
listen and, so, forced Sabitha to take or leave a first offer of a
pittance to Sabitha.
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 210 Lesson II.1 Strategic Bargaining
35
Example 5: Bargaining with Depreciation
Example 5: Bargaining with
Depreciation
BA 210 Lesson II.1 Strategic Bargaining
36
Example 5: Bargaining with Depreciation
Comment: Bargaining rules change if the value of the good
depreciates like ice cream in the time it takes to make
counteroffers to offers, and counter-counteroffers to counteroffers
and so on.
BA 210 Lesson II.1 Strategic Bargaining
37
Example 5: Bargaining with Depreciation
Question: Suppose Buyer Bob values an ice cream bar at 1 dollar
and Seller Sam has no alternative buyers and has no other value
for the ice cream. Suppose Bob can make the first offer to Sam,
who can accept or make a counter offer to Bob, who in turn can
accept or make a counter offer to Sam. Offers alternate thereafter.
But each counter offer takes valuable time which melts the ice
cream and, so, reduces the initial value of the ice cream by 50
cents. What initial price should Buyer Bob offer? Should Seller
Sam accept that first offer?
BA 210 Lesson II.1 Strategic Bargaining
38
Example 5: Counteroffers with Depreciation
Answer: To consider all possible price offers (offers by Bob to
buy or offers by Sam to sell), consider another bargaining payoff
table. The game ends if an offer is accepted or the ice cream is
completely melted and worthless, and gains are measured as a
percentage of the $1.00 gain from trading the ice cream before it
starts melting.
Rounds to
Meltdown
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Sam
50
?
?
2
Bob
100
?
?
BA 210 Lesson II.1 Strategic Bargaining
39
Example 5: Counteroffers with Depreciation
Starting 1 bargaining round from the end of the game, when the
ice cream has 50 percent of its original value, Bob should accept
anything as being better than nothing. So, Sam can get away
with the whole 50 percent minus a pittance, leaving Bob with the
pittance.
Rounds to
Meltdown
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Sam
50
0
50
2
Bob
100
?
?
BA 210 Lesson II.1 Strategic Bargaining
40
Example 5: Counteroffers with Depreciation
Rolling back to the beginning of the game, 2 steps to meltdown,
when the ice cream has 100 percent of its original value, Sam
could reject any offer and get 50 percent in the next round. So,
Bob’s best acceptable offer leaves Sam with 50 percent plus a
pittance, and Bob with 50 percent minus a pittance.
Rounds to
Meltdown
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Sam
50
0
50
2
Bob
100
50
50
BA 210 Lesson II.1 Strategic Bargaining
41
Example 5: Counteroffers with Depreciation
Buyer Bob should thus initially offer price $0.50 for the ice
cream, and Seller Sam should accept that first offer. That gives
Bob 50 percent of the gain from trade and Sam 50 percent.
Rounds to
Meltdown
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sam's Gain
Offered
1
Sam
50
0
50
2
Bob
100
50
50
BA 210 Lesson II.1 Strategic Bargaining
42
Example 6: Bargaining with Depreciation
Example 6: Bargaining with
Depreciation
BA 210 Lesson II.1 Strategic Bargaining
43
Example 6: Bargaining with Depreciation
Question: Suppose Buyer Bob values an ice cream bar at 1 dollar
and Seller Sam has no alternative buyers and has no other value
for the ice cream. Suppose Bob can make the first offer to Sam,
who can accept or make a counter offer to Bob, who in turn can
accept or make a counter offer to Sam. Offers alternate thereafter.
But each counter offer takes valuable time which melts the ice
cream and, so, reduces the initial value of the ice cream by 33 1/3
cents. What initial price should Buyer Bob offer? Should Seller
Sam accept that first offer?
BA 210 Lesson II.1 Strategic Bargaining
44
Example 6: Counteroffers with Depreciation
Answer: To consider all possible price offers (offers by Bob to
buy or offers by Sam to sell), consider another bargaining payoff
table. The game ends if an offer is accepted or the ice cream is
completely melted and worthless, and gains are measured as a
percentage of the $1.00 gain from trading the ice cream before it
starts melting.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
Sams's Gain
Offered
1
Bob
?
?
?
2
Sam
?
?
?
3
Bob
?
?
?
BA 210 Lesson II.1 Strategic Bargaining
45
Example 6: Counteroffers with Depreciation
Starting 1 bargaining round from the end of the game, when the
ice cream has 33 1/3 percent of its original value, Sam should
accept anything as being better than nothing. So, Bob can get
away with the whole 33 1/3 percent minus a pittance, leaving
Sam with the pittance.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
1
Bob
33 1/3
33 1/3
2
Sam
3
Bob
BA 210 Lesson II.1 Strategic Bargaining
Sams's Gain
Offered
0
46
Example 6: Counteroffers with Depreciation
Rolling back 2 steps to meltdown, when the ice cream has 66 2/3
percent of its original value, Bob could reject any offer and get 33
1/3 percent in the next round. So, Sam’s best acceptable offer
leaves Bob with 33 1/3 percent plus a pittance, and Sam with 33
1/3 percent minus a pittance.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
1
Bob
33 1/3
33 1/3
0
2
Sam
66 2/3
33 1/3
33 1/3
3
Bob
100
?
?
BA 210 Lesson II.1 Strategic Bargaining
Sams's Gain
Offered
47
Example 6: Counteroffers with Depreciation
Rolling back to the beginning of the game, 3 steps to meltdown,
when the ice cream has 100 percent of its original value, Sam
could reject any offer and get 33 1/3 percent in the next round.
So, Bob’s best acceptable offer leaves Sam with 33 1/3 percent
plus a pittance, and Bob with 66 2/3 percent minus a pittance.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
1
Bob
33 1/3
33 1/3
0
2
Sam
66 2/3
33 1/3
33 1/3
3
Bob
100
66 2/3
33 1/3
BA 210 Lesson II.1 Strategic Bargaining
Sams's Gain
Offered
48
Example 5: Counteroffers with Depreciation
Buyer Bob should thus initially offer price $1/3 for the ice cream,
and Seller Sam should accept that first offer. That gives Bob 66
2/3 percent of the gain from trade and Sam 33 1/3 percent.
Rounds to
End of
Game
Offer by
T otal Gain
to Divide
Bob's Gain
Offered
1
Bob
33 1/3
33 1/3
0
2
Sam
66 2/3
33 1/3
33 1/3
3
Bob
100
66 2/3
33 1/3
BA 210 Lesson II.1 Strategic Bargaining
Sams's Gain
Offered
49
Example 5: Counteroffers with Depreciation
Comment: The players in that counter offer game did not actually
make any counteroffers, but the possibility of counteroffers
affected the initial offer.
BA 210 Lesson II.1 Strategic Bargaining
50
Summary
Summary
BA 210 Lesson II.1 Strategic Bargaining
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 210 Lesson II.1 Strategic Bargaining
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 210 Lesson II.1 Strategic Bargaining
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 210 Lesson II.1 Strategic Bargaining
54
Summary
Rounds to
Gain
A's Gain
B's Gain
Starting 1 bargaining round from End of Offer by TtootalDivide
Offered
Offered
Game
the end, when the gain has
1
B
50
0
50
33 1/3 percent of its original
2
A
100
50
50
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 210 Lesson II.1 Strategic Bargaining
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
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 210 Lesson II.1 Strategic Bargaining
B's Gain
Offered
56
Summary
Rounds to
Starting 1 bargaining round from the
End of
Offer by
Game
end, when the gain has 33 1/3 percent
1
A
of its original value, B should accept
2
B
anything as being better than nothing.
3
A
After deducing that, A’s best acceptable
offer to B is a pittance, leaving A with 25 percent.
T otal Gain
to Divide
A's Gain
Offered
B's Gain
Offered
33 1/3
33 1/3
0
66 2/3
33 1/3
33 1/3
100
66 2/3
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 210 Lesson II.1 Strategic Bargaining
57
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.1 Strategic Bargaining
58
Review Questions
Review Question 1
BA 210 Lesson II.1 Strategic Bargaining
59
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, 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. Medieval Times can open only on those
days after an agreement is reached the night before.
What initial wage demand should the union make? Should
management accept that demand?
BA 210 Lesson II.1 Strategic Bargaining
60
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 $20,000 gain
from accepting the Union’s initial offer.
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.1 Strategic Bargaining
61
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.1 Strategic Bargaining
62
Review Questions
The Union’s initial wage demand should be for 50% of the
$20,000 profits, or $10,000, and management should accept that
demand.
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.1 Strategic Bargaining
63
Review Questions
Review Question 2
BA 210 Lesson II.1 Strategic Bargaining
64
Review Questions
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 210 Lesson II.1 Strategic Bargaining
65
Review Questions
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 210 Lesson II.1 Strategic Bargaining
66
Review Questions
Rounds to
Pepperdine's Candidate's
Gain
Starting 1 bargaining round from End of Offer by TtootalDivide
Gain
Gain
Game
Offered
Offered
the end of the game, when a
1
Pepperdine 33 1/3
33 1/3
0
contract has 33 1/3 percent of its 2 Candidate 66 2/3 33 1/3 33 1/3
3
Pepperdine 100
?
?
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 210 Lesson II.1 Strategic Bargaining
67
Review Questions
Rounds to
Pepperdine's Candidate's
T otal Gain
Rolling back to 3 bargaining
End of
Offer by
Gain
Gain
to Divide
Game
Offered
Offered
rounds before the end, when a
1
Pepperdine 33 1/3
33 1/3
0
2
Candidate
66 2/3
33 1/3
33 1/3
contract has 100 percent of its
3
Pepperdine 100
66 2/3
33 1/3
original value, the Candidate
could reject any offer and get 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 210 Lesson II.1 Strategic Bargaining
68
Review Questions
Review Question 3
BA 210 Lesson II.1 Strategic Bargaining
69
Review Questions
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 210 Lesson II.1 Strategic Bargaining
70
Review Questions
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
BA 210 Lesson II.1 Strategic Bargaining
LA's Gain
Offered
71
Review Questions
Rounds to
Gain Dallas's Gain LA's Gain
Starting 1 bargaining round from End of Offer by TtootalDivide
Offered
Offered
Game
the end of the game, when an
1
Dallas
33 1/3
33 1/3
0
2
LA
66 2/3
33 1/3
33 1/3
agreement has 33 1/3 percent of
3
Dallas
100
?
?
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 210 Lesson II.1 Strategic Bargaining
72
Review Questions
Rounds to
T otal Gain Dallas's Gain LA's Gain
Rolling back to 3 bargaining
End of
Offer by
to Divide
Offered
Offered
Game
rounds before the end, when an
1
Dallas
33 1/3
33 1/3
0
2
LA
66 2/3
33 1/3
33 1/3
agreement has 100 percent of its
3
Dallas
100
66 2/3
33 1/3
original value, LA could reject
any offer and get 33 1/3 percent in the 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, BostonDallas 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 210 Lesson II.1 Strategic Bargaining
73
BA 210
Introduction to Microeconomics
End of Lesson II.1
BA 210 Lesson II.1 Strategic Bargaining
74