Lesson overview
Chapter 3 Games with Sequential Moves
Lesson I.3 Sequential Move Theory
Lesson I.4 Sequential Move Applications
Each Example Game introduces applications or techniques
• Example 1: Advertising
• Example 2: Entry Deterrence
• Example 3: Car Loan
• Example 4: Renegotiation
• Example 5: Compatible Web Browsers
• Example 6: Evicting Tenants
• Example 7: Strategic Voting
• Example 8: Nuisance Suits
• Review Problems
BA 592 Lesson I.4 Sequential Move Applications
1
Example 1: Advertising
Each example game in this “Sequential Move Applications”
lesson, or in any of the other “Applications” lessons in the course,
is presented in the same way as questions on an exam: I describe
a game verbally and ask questions about its solution. You must
formulate the game (identifying players, strategies, payoffs, …)
then answer those questions by solving the game.
BA 592 Lesson I.4 Sequential Move Applications
2
Example 1: Advertising
Question: Incumbant Senator Gray will run for reelection. The
challenger is Congresswoman Green. Senator Gray first decides
whether or not to run advertisements early on. The challenger
Green must then decide whether or not to enter the race. Issues to
think about in modeling the game:
 Players are Gray and Green. Gray moves first.
 Strategies for Gray are Ads, No Ads; for Green: In or Out.
 Ads are costly, so Gray would prefer not to run ads.
 Green will find it possible to win if Gray does not run ads,
but impossible if Gray does run ads.
Define a game tree for this Advertising Game, choosing payoff
numbers consistent with the issues above. Then, find the rollback
solution.
BA 592 Lesson I.4 Sequential Move Applications
3
Example 1: Advertising
Answer: Game tree
Grey
No Ads
Ads
Green
Green
Out
In
Out
In
4,2
2,4
3,3
1,1
BA 592 Lesson I.4 Sequential Move Applications
4
Example 1: Advertising
Answer: Rollback solution.
Grey plays Ads and gets payoff
3.
Green plays
{In if No Ads,
Out if Ads} and gets payoff 3.
Grey
No Ads
Ads
Green
Green
Out
In
Out
In
4,2
2,4
3,3
1,1
BA 592 Lesson I.4 Sequential Move Applications
5
Example 1: Advertising
Supplementary Question: Is there a first mover advantage in
the Advertising Game?
BA 592 Lesson I.4 Sequential Move Applications
6
Example 1: Advertising
Instructor’s Note:
• To see if the order matters in any game, rearrange the
sequence of moves.
• Examples in which order may matter:
 Adoption of new technology. Better to be first or last?
 Class presentation of a project. Better to be first or last?
• Sometimes order does not matter, such as the Prisoners’
dilemma with Confess the dominate strategy for both players.
• Is there such a thing as a second-mover advantage?
Sometimes, for example:



Sequential biding by two contractors.
Cake-cutting: One person cuts, the other gets to decide
how the two pieces are allocated.
Poker.
BA 592 Lesson I.4 Sequential Move Applications
7
Example 1: Advertising
Answer to Supplementary
Question: Suppose the
sequence of play in the
Advertising Game is changed
so that Green gets to first
commit to enter the race before
Grey commits to Ads. The
payoffs for the possible
outcomes are exactly the same
as before, except now, Green’s
payoff is listed first.
Green
Out
In
Grey
Grey
No Ads
Ads
No Ads
Ads
2,4
3,3
4,2
1,1
BA 592 Lesson I.4 Sequential Move Applications
8
Example 1: Advertising
Rollback Solution:
Green plays In and gets payoff
4.
Grey plays
{No Ads if Out,
No Ads if In} and gets payoff
2.
There is a first-mover
advantage.
Green
Out
In
Grey
Grey
No Ads
Ads
No Ads
Ads
2,4
3,3
4,2
1,1
BA 592 Lesson I.4 Sequential Move Applications
9
Example 2: Entry Deterrence
Question: A monopolist faces the prospect of loosing part of its
$10M profit by a potential competitor entering the industry. If the
competitor elects not to enter, it earns profits of $0 and the
monopolist maintains its profit of $10M. If the competitor enters,
the monopolist must either accommodate the entry or fight. If the
monopolist accommodates, both firms earn $4M. If the
monopolist fights, both firms lose $4M.
Should the competitor enter?
BA 592 Lesson I.4 Sequential Move Applications
10
Example 2: Entry Deterrence
Answer: Game tree and its
rollback solution.
Entrant plays In and gets payoff
4.
Monopolist plays
Accommodate if In and gets
payoff 4.
Entrant
Out
In
0,10
Monopolist
Accom.
Fight
4,4
-4,-4
BA 592 Lesson I.4 Sequential Move Applications
11
Example 2: Entry Deterrence
Supplementary Question: Is there a first mover advantage in the
Advertising Game?
BA 592 Lesson I.4 Sequential Move Applications
12
Example 2: Entry Deterrence
Answer to Supplementary
Question: Suppose the
sequence of play is changed so
that Monopolist gets to first
commit to Accommodate or
Fight before Entrant commits to
Entry. The payoffs for the four
possible outcomes are exactly
the same as before, except now,
Monopolist’s payoff is listed
first.
Monopolist
Accom.
Fight
Entrant
Entrant
Out
In
Out
In
10,0
4,4
10,0
-4,-4
BA 592 Lesson I.4 Sequential Move Applications
13
Example 2: Entry Deterrence
Rollback Solution:
Monopolist plays Fight and gets
payoff 10.
Entrant plays {In if Accom.,
Out if Fight} and gets payoff 0.
There is a first-mover
advantage.
Monopolist
Accom.
Fight
Entrant
Entrant
Out
In
Out
In
10,0
4,4
10,0
-4,-4
BA 592 Lesson I.4 Sequential Move Applications
14
Example 3: Car Loan
Question: A bank must decide whether to offer Joe an auto loan
or not. The loan would be for $11,000, and the bank must decide
on a total amount L of principal and interest that he must repay at
the end of the year. If Joe accepts the loan, he can buy a car worth
$12,000 to him. He then decides whether to work hard, earning
$15,000, or loaf, earning $8,000. Joe would pay up to $5,000 to
be able to loaf. Joe has no other assets. If he decides not to repay
the loan, he loses the car and gets value $500 for using the car for
a year, and the bank collects $7,000 by reselling the car. If Joe
keeps the car, it retains its value of $12,000 to him even if it is a
year old. Consider an interest rate of 0%.
Should the bank offer a loan? If so, what should be the principal
and interest?
BA 592 Lesson I.4 Sequential Move Applications
15
Example 3: Car Loan
Answer: The Bank first decides whether to offer a loan, and at
what principal and interest L. Since L can be any real number,
we cannot draw a full game graph because it would have a
continuum of branches coming off it. So first fix L, then graph
the subgame. The Bank looses money if L < 11 (thousand), so
just consider L > 11.
The decision nodes in the subgame are in this order.
Bank: Offer loan or or don't offer one.
Joe: Accept or reject the offer.
Joe: Work or loaf. (If he loafs, he does not have the money to
pay back the loan.)
Joe: Repay or lose the car to the bank.
BA 592 Lesson I.4 Sequential Move Applications
16
Example 3: Car Loan
Subgame graph given L:
If the Bank does not offer a
loan, or if Joe turns down the
loan, its payoff is 0 and Joe’s
is 15 if he works or 13 (8+5)
if he loafs. If Joe works and
repays, the Bank gets L-11
and Joe gets 15+12 -L; if he
does not repay, the Bank gets
7-11 and Joe gets 15. If Joe
loafs and so does not repay,
the Bank gets 7-11 and Joe
gets 8+5.
Bank
No Offer
Offer Loan
Joe
Joe
Loaf
Work
Reject L
Accept L
0, 13
0, 15
Joe
Joe
Loaf
Work
Loaf
Work
0, 13
0, 15
Joe
Joe
Don't Pay
Don't Pay
Pay
-4, 13.5
-4, 15.5
L-11, 27-L
BA 592 Lesson I.4 Sequential Move Applications
17
Example 3: Car Loan
Some of Joe’s Choices
are independent of L.
Bank
No Offer
Offer Loan
Joe
Joe
Loaf
Work
Reject L
Accept L
0, 13
0, 15
Joe
Joe
Loaf
Work
Loaf
Work
0, 13
0, 15
Joe
Joe
Don't Pay
Don't Pay
Pay
-4, 13.5
-4, 15.5
L-11, 27-L
BA 592 Lesson I.4 Sequential Move Applications
18
Example 3: Car Loan
When L > 11.5:
The equilibrium path is
the Bank does not offer
the loan, and Joe works.
That path ends in payoff
0 for the bank.
Bank
No Offer
Offer Loan
Joe
Joe
Loaf
Work
Reject L
Accept L
0, 13
0, 15
Joe
Joe
Loaf
Work
Loaf
Work
0, 13
0, 15
Joe
Joe
Don't Pay
Don't Pay
Pay
-4, 13.5
-4, 15.5
L-11, 27-L
BA 592 Lesson I.4 Sequential Move Applications
19
Example 3: Car Loan
When L = 11.5:
There are multiple
rollback solutions
because if the Bank
Offers Loan, and Joe
Accepts Loan, and Joe
Works, then Joe is
indifferent between Not
Paying Back the Loan,
and Paying Back the
Loan.
Bank
No Offer
Offer Loan
Joe
Joe
Loaf
Work
Reject L
Accept L
0, 13
0, 15
Joe
Joe
Loaf
Work
Loaf
Work
0, 13
0, 15
Joe
Joe
Don't Pay
Don't Pay
Pay
-4, 13.5
-4, 15.5
L-11, 27-L
Let us avoid this case.
BA 592 Lesson I.4 Sequential Move Applications
20
Example 3: Car Loan
When 11 < L < 11.5:
The Bank’s strategy
is offer the loan, and
it gets payoff L-11.
Bank
No Offer
Offer Loan
Joe
Joe
Loaf
Work
Reject L
Accept L
0, 13
0, 15
Joe
Joe
Loaf
Work
Loaf
Work
0, 13
0, 15
Joe
Joe
Don't Pay
Don't Pay
Pay
-4, 13.5
-4, 15.5
L-11, 27-L
BA 592 Lesson I.4 Sequential Move Applications
21
Example 3: Car Loan
The Bank’s rollback equilibrium payoffs thus depend on L:
• If L > 11.5, payoff = 0.
• If L = 11.5, there are multiple rollback equilibria.
• If 11 < L < 11.5, payoff = L-11.
Should the bank offer a loan? If so, what should be the principal
and interest? Yes, offer the loan at principal and interest L =
11.5-e, and so earn payoff .5-e. Joe earns payoff 15.5+e, which
is .5+e more than if he did not get the loan.
BA 592 Lesson I.4 Sequential Move Applications
22
Example 4: Renegotiation
Supplementary Question: Now suppose Joe has another option: if
Joe loafs and so cannot repay the full amount of the loan, he can
offer the bank some smaller amount S not to foreclose on the loan
and repossess the car.
What would that amount S be? How does the outcome change if
both Joe and the bank know that this kind of renegotiation is
possible?
BA 592 Lesson I.4 Sequential Move Applications
23
Example 4: Renegotiation
The bank collects
$7,000 by reselling the
car, so if Joe loafs and
offers any S > $7,000,
then the Bank would
accept. So offer S =
$7,000+e. Hence, the
game tree changes:
Bank
No Offer
Offer Loan
Joe
Joe
Loaf
Work
Reject L
Accept L
0, 13
0, 15
Joe
Joe
Loaf
Work
Loaf
Work
0, 13
0, 15
Joe
Joe
Renegotiate
Don't Pay
Pay
-4+e, 18-e
-4, 15.5
L-11, 27-L
BA 592 Lesson I.4 Sequential Move Applications
24
Example 4: Renegotiation
When L > 11.5:
The equilibrium path is
the Bank does not offer
the loan, and Joe works.
That path ends in payoff
0 for the bank.
Bank
No Offer
Offer Loan
Joe
Joe
Loaf
Work
Reject L
Accept L
0, 13
0, 15
Joe
Joe
Loaf
Work
Loaf
Work
0, 13
0, 15
Joe
Joe
Renegotiate
Don't Pay
Pay
-4+e, 18-e
-4, 15.5
L-11, 27-L
BA 592 Lesson I.4 Sequential Move Applications
25
Example 4: Renegotiation
When 11 < L < 11.5:
The Bank’s strategy
is offer the loan, and
it gets payoff L-11.
Bank
No Offer
Offer Loan
Joe
Joe
Loaf
Work
Reject L
Accept L
0, 13
0, 15
Joe
Joe
Loaf
Work
Loaf
Work
0, 13
0, 15
Joe
Joe
Renegotiate
Don't Pay
Pay
-4+e, 18-e
-4, 15.5
L-11, 27-L
BA 592 Lesson I.4 Sequential Move Applications
26
Example 4: Renegotiation
Summary: When Joe has the option to loaf and offer the bank
some smaller amount S not to foreclose on the loan and repossess
the car, he would offer S = $7,000+e. But when both Joe and the
Bank know that this kind of renegotiation is possible, the Bank
no longer offers the loan, which eliminates the previous gains of
payoff .5-e to the Bank and .5+e to Joe.
BA 592 Lesson I.4 Sequential Move Applications
27
Example 5: Compatible Web Browsers
Question: Microsoft and Google are planning to introduce a new
type of Web browser. They must choose between two platforms,
ActiveX and Java. If they introduce different platforms, their
profits are zero. If they introduce the same platform, their profits
are 1, plus Microsoft gets 1 more if the platform is ActiveX and
Google gets 1 more if the platform is Java.
Which platform should Microsoft choose if it can choose first?
Which platform should Google choose if it can choose first?
Is there a first mover advantage?
Now suppose Microsoft can choose first unless Google rushes
development of their browser. Rushing development cost Google
0.5, and it allows Google to choose their platform first.
Should Google rush their development?
BA 592 Lesson I.4 Sequential Move Applications
28
Example 5: Compatible Web Browsers
Answer: Which platform should
Microsoft choose if it can
choose first? ActiveX, giving
Microsoft 2 and Google 1.
Which platform should Google
choose if it can choose first?
Java, giving Microsoft 1 and
Google 2.
Is there a first mover
advantage? Yes, either firm
earns 1 more moving first
rather than second.
Should Google rush? Yes, they
earn 1 more at a cost of 0.5
Microsoft
ActiveX
Java
Google
Google
ActiveX
Java
ActiveX
Java
2,1
0,0
0,0
1,2
Google
ActiveX
Java
Microsoft
Microsoft
ActiveX
Java
ActiveX
Java
1,2
0,0
0,0
2,1
BA 592 Lesson I.4 Sequential Move Applications
29
Example 6: Evicting Tenants
Question: A landlord has three tenants, Alfred, Betty, and Charlie,
in a rent-controlled apartment building in New York City. A new
law says that the landlord has the right to evict one tenant per
building. The landlord calculates that the value of a vacant
apartment is $15,000, both to the tenant and to her. She sends the
following letter to each of her tenants: “Tomorrow I will be
visiting your building. I will offer A $1,000 if he agrees to vacate
his apartment voluntarily; otherwise, I will evict him. If A agrees
to vacate voluntarily, I will then offer B $1,000 if she agrees to
vacate his apartment voluntarily; otherwise, I will evict her. If B
agrees to vacate voluntarily, I will evict C.”
How many vacant apartments will the landlord have? What
profit will the landlord have?
BA 592 Lesson I.4 Sequential Move Applications
30
Example 6: Evicting Tenants
Answer: This is a game
between tenants A and B. (The
Landlord has committed to the
Get Evicted
letter, and C has no choices.) A
and B both vacate, and C gets
-15, 0
evicted.
The landlord thus gains 3
vacant apartments and pays
$2,000. So the landlord gains
$45,000 = $43,000 - $2,000.
A
Vacate
B
Get Evicted
Vacate
-14, -15
-14,-14
BA 592 Lesson I.4 Sequential Move Applications
31
Example 7: Strategic Voting
Question: Three legislators, Alfred, Betty, and Charlie, are voting
on whether to give themselves a pay raise. The raise is worth b,
but each legislator who votes for the raise incurs a cost of voter
resentment equal to a < b. The outcome is decided by majority
rule. A votes first, then B sees A’s choice and votes, then C sees
A’s and B’s choice and votes.
How should A vote?
BA 592 Lesson I.4 Sequential Move Applications
32
Example 7: Strategic Voting
A
Answer: A should vote
Against, believing that
B and C will then vote
For the raise.
For
Against
B
B
For
Against
For
Against
C
C
C
C
For
Against
For
Against
For
Against
For
Against
b-a,b-a,b-a
b-a,b-a,b
b-a,b,b-a
-a,0,0
b,b-a,b-a
0,-a,0
0,0,-a
0,0,0
BA 592 Lesson I.4 Sequential Move Applications
33
Example 8: Nuisance Suits
Question: Alfred contemplates suing Betty over the purported
damage done to Alfred’s son sustained while using the bike jump
in Betty’s yard. Suppose A’s court cost for initiating a suit is a,
his legal costs of going to trial are b, and Betty’s cost of
defending herself is c. Suppose both sides know these costs and
also share the knowledge that the probability that A will win the
suit is p and the expected amount of the settlement is x. Assume
px < b. Finally, suppose that before the case goes to trial (but
after the suit is initiated), the parties can settle out of court for the
amount s.
Should A file suit?
BA 592 Lesson I.4 Sequential Move Applications
34
Example 8: Nuisance Suits
Answer: There are two
rollback solutions, depending
on whether A choose Offer to
Settle or No Offer to Settle.
But in either rollback solution,
A does not file suit.
A
No Suit
File Suit
0,0
A
No Offer
Offer Settle
A
B
Pursue Suit
Drop S.
Reject Offer
Accept O.
px-a-b,-px-c
-a,0
A
s-a,-s
Pursue Suit
Drop S.
px-a-b,-px-c
-a,0
BA 592 Lesson I.4 Sequential Move Applications
35
Review Problems
Mall Location Game
Page 83, Question S8.
Answer
BA 592 Lesson I.4 Sequential Move Applications
36
Review Problems
Political Game
Page 85, Question U4.
Answer
BA 592 Lesson I.4 Sequential Move Applications
37
BA 592
Game Theory
End of Lesson I.4
BA 592 Lesson I.4 Sequential Move Applications
38