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Suggested Solutions to Questions in Topic 6

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Suggested Solutions to Questions in Topic 6
1.
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Introduction of advertisement game between Firm A and B
The payoff matrix for an advertising game
a) Is there any dominant strategy
b) Find the Nash Equilibrium
Observe from this payoff matrix that if both firms decide to advertise Firm A makes a profit of
10 and Firm B will make a profit of 5. If Firm A advertises and Firm B does not advertise Firm
A will earn 15 and Firm B will earn zero.
First consider Firm A. Firm A should advertise no matter what Firm B does because Firm A does
best by advertising. If Firm B advertises A earns a profit of 10 if it advertises but only 6 if it
doesn’t. Thus Advertising is a dominant strategy for Firm A
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2
a) Find the Nash Equilibrium in the Introduction of Home Banking Services Games
Introduction of Home Banking Services (HB) Game Between Afriland First Bank and Atlantic
Bank
Atlantic Bank
Introduce
HB
Do
Not introduce HB
Introduce HB
24, 8
40, 6
Do Not
Introduce HB
30, 12
36, 10
Afriland First Bank
(payoffs are in millions of FCFA)
Solution
The Game Tree for Introduction of Home Banking Services (HB) Between Afriland First Bank
and Atlantic Bank
AFFB payoff AMB payoff
(payoffs are in Millions of FCFA)
HB
24
8
NHB
40
6
HB
30
12
NHB
36
10
HB
AM
AF
NHB
AM
In this game, African First Bank does not have a dominant strategy. It is better off not introducing
home banking if Atlantic Bank introduces one, but it prefers introducing home banking if Atlantic
bank does not introduce one. Despite the absence of dominant strategy for the Afriland First Bank,
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there is still a Nash equilibrium: Atlantic introduces home banking, and Afriland First bank
doesn’t.
It is interesting to see how Afriland First Bank might figure out which strategy to choose
in this game. If it envisions this payoff matrix, it should realise that while it does not have a
dominant strategy, Atlantic does “introduce home banking”.
3.
Table 1 gives the payoff matrix for two rival holiday companies, who intend to expand their
business by offering additional vacations to either Limbe or to Kribi.
Firm B
Limbe
Kribi
Limbe
100,
100
200,
200
Kribi
200,
200
100,
100
Firm A
(payoffs are in Thousand of FCFA)
Find the Nash equilibrium
Solution
Firm A payoff Firm B payoff
(Payoff is in 000 FCFA)
Limbe
100
100
Kribi
200
200
Limbe
200
200
Kribi
100
100
Limbe
B
A
Kribi
B
Here we have two Nash equilibrium (Limbe – Kribi) or (Kribi – Limbe)
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4.
Afriland First Bank and Atlantic Bank are competing in an oligopolistic industry. Afriland First
Bank, the larger of the two banks, is contemplating its capacity strategy, which we might broadly
characterise as “aggressive” and “passive”. The aggressive strategy involves a large increase in
capacity aimed at increasing the firm’s market share, while the passive strategy involves no change
in the Afriland First Bank’s capacity. Atlantic Bank, the smaller competitor, is also pondering its
capacity expansion strategy, it will also chose between an “aggressive strategy” or a “passive
strategy”. The table below shows the profits associated with each pair of choices made by the two
firms:
Atlantic Bank
Passive
Aggressive
Passive
0, 0
100, 300
Aggressive
300, 100
50, 50
Afriland First Bank
(payoffs are in millions of FCFA)
If Afriland First Bank could move first and Atlantic bank reacts what will be the best strategies?
Solution
AFFB payoff AMB payoff
(Payoffs are in Millions of FCFA)
Passive
0
0
Aggressive
100
300
Passive
300
100
Aggressive
50
50
Passive
AL
AFB
Aggressive
AL
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If AFB can choose first, then the best sequential move strategy will be “Aggressive” for AFB
and “passive” for AB. AFB will make the latter choice given that it can anticipate the decision
of its rival.
5.
Two banks, Afriland First Bank and Atlantic Bank, dominate the financial industry of
Cameroon, with Afriland First Bank producing a new product called Flash Cash and Atlantic
Bank producing Flash Cash Plus. Each bank has investigated the possibility of undertaking a
more aggressive advertising campaign in order to maintain its market share in a competitive
environment. The possible outcomes of such strategies are shown in the payoff matrix (Table
1) showing yearly profit figures (FCFA m).
Atlantic Bank proctor
lever
Advertise
No Advertise
Advertise
11, 6
16, 0
No Advertise
7, 9
11, 3
Afriland First Bank
22, 2
(payoffs are in millions of FCFA)
a) Use the payoff matrix to explain what it meant by the term “dominant strategy” for the
respective banks.
b) Suppose the payoff values in the bottom-right square now change from 11, 3 respectively
to 22, 2, how will this influence the dominant strategy of each bank? Will the Nash
equilibrium change?
Solution
a) A dominant strategy for Afriland First Bank would be one that is optimal for that company
irrespective of Atlantic bank strategy. Similarly, the dominant strategy for Atlantic Bank
would be the one that is optimal for it, irrespective of Afriland First Bank strategy. On the
other hand, the Nash equilibrium is where Afriland First Bank is carrying out its optimal
strategy giving what Atlantic bank is doing likewise. The dominant strategy for Afriland
First Bank is to advertise, because whatever Atlantic Bank does, advertising still proves to
be the most profitable strategy for Afriland First Bank. The best strategy of Atlantic Bank
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is also to advertise. So (advertise, advertise is the dominant strategy for both firms.
b) In this case Afriland First Bank does not have a dominant strategy because advertising is
not the best course of action taking both Atlantic Bank’s strategies into consideration.
Therefore, the Nash equilibrium will not change since Afriland First Bank will know that
Atlantic Bank will have a dominant strategy of advertising whatever it does.
6.
Find a Nash equilibrium: UNICS Finance Versus Western Union
Commission charges per money transfer of 5000 FCFA or less
UNICS
Western
Union
10.25
7.25
8.25
6.25
10.5
11.5
12.5
110.5
66; 190
79; 201
82; 212
75; 223
68; 199
82; 211
86; 224
80; 237
70; 198
85; 214
90; 229
85; 244
73; 191
89; 208
95; 225
81; 205
Solution
With four strategies for each firm, this looks like a complicated game. But we can greatly simplify
it by searching for dominated strategy. To see why, compare Western Union’s payoffs among the
four rows in the table. You will see that Western Union’s payoff is always higher in row 1 – a
money transfer commission charge of 8.25 FCFA – than in any other row. Thus, a commission
charge of 8.25 FCFA for Western Union is a dominant strategy and the other three prices are
dominated strategies.
If UNICS assumes that Western Union will follow its dominant strategy, UNICS should
expect Western Union to set a commission charge per money transfer of 8.25 FCFA. Given this
commission charge, UNICS best response is to set a price of 12.5 FCFA.
The Nash equilibrium in this game is for Western Union to set a commission charge of 8.25 FCFA
and UNICS to set a commission charge of 12.5 FCFA.
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If Western Union could move first and UNICS reacts find the Nash equilibrium.
Commission charges per money transfer
UNICS
6
Western
Union
6.25
7.25
8.25
9.25
10.5
11.5
12.5
110.5
85; 190
79; 201
82; 212
75; 223
87; 199
82; 211
86; 224
80; 237
93; 198
85; 214
90; 229
85; 244
97; 191
89; 208
95; 225
81; 205
Solution
With four strategies for each firm, this looks like a complicated game. But we can greatly simplify
it by searching for dominated strategy. To see why, compare Western Union’s payoffs among the
four rows in the table. You will see that Western Union’s payoff is always higher in row 1 – a
money transfer commission charge of 6.25 FCFA – than in any other row. Thus, a commission
charge of 6.25 FCFA for Western Union is a dominant strategy and the other three prices are
dominated strategies.
If UNICS assumes that Western Union will follow its dominant strategy, UNICS should
expect Western Union to set a commission charge per money transfer of 6.25 FCFA. Given this
commission charge, UNICS best response is to set a price of 11.5 FCFA.
The Nash equilibrium in this game is for Western Union to set a commission charge of 6.25 FCFA
and UNICS to set a commission charge of 11.5 FCFA.
8.
Afriland First Bank and Amity Bank are the most reliable banks in Cameroon. These two banks
compete head-to-head for expansion in Equatorial Guinea. The following table shows the profit in
millions of FCFA that each firm earns when it makes the expansion decision in Equatorial Guinea.
Find the Nash Equilibrium in the Modified Capacity Expansion Game between Afriland First Bank
(AFB) and Amity Bank (AB).
AB
Large
Small
Do not Build
AFB
Large
Small
Do not Build
0; 0
80; 120
90; 180
120; 80
160; 160
150; 200
150; 90
170; 180
180; 180
Payoffs are in millions of FCFA.
Solution
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Figure 2
Modified Capacity Expansion Game between Afriland First Bank (AFB) and Amity Bank
AB)
AFB payoff
AB payoff
Payoffs are in millions of FCFA
Large
Large
FCFA0
FCFA0
Small
FCFA120
FCFA80
Do Not Build
FCFA150
FCFA90
Large
FCFA80
FCFA120
Small
FCFA160
FCFA160
Do Not Build
FCFA170
FCFA180
Large
FCFA90
FCFA180
Small
FCFA150
FCFA200
Do Not Build
FCFA180
FCFA180
AB
Small
AFB
AB
Do Not Build
AB
Neither player has a dominant strategy, and with three strategies rather than two, the task of finding
a Nash equilibrium is daunting. No matter what Amity bank does, Afriland First Bank is always
better off by choosing “Small” and the same conclusion holds the other way round. In this
simultaneous-move game, Amity Bank cannot observe Afriland First Bank’s decision beforehand,
and therefore Afriland First Bank Cannot force Amity Bank hand.
9.
Afriland First Bank and Amity Bank are competing in an oligopolistic industry. Afriland First
Bank, the larger of the two banks, is contemplating its capacity strategy, which we might broadly
characterise as “aggressive” and “passive”. The aggressive strategy involves a large increase in
capacity aimed at increasing the firm’s market share, while the passive strategy involves no change
in the Afriland First Bank’s capacity. Amity Bank, the smaller competitor, is also pondering its
capacity expansion strategy, it will also chose between an “aggressive strategy” or a “passive
strategy”. The table below shows the profits associated with each pair of choices made by the two
firms:
Amity Bank
8
Aggressive
Passive
Aggressive
25, 9
33, 10
Passive
30, 13
36, 12
Afriland First Bank
(payoffs are in millions of FCFA)
a) If both banks decide their strategies simultaneously, what is the Nash equilibrium?
b) If Afriland First Bank could move first and Amity bank reacts what will be the best
strategies?
Solution
AFFB payoff AMB payoff
(payoffs are in Millions of FCFA)
Aggressive
25
9
Passive
33
10
Aggressive
30
13
Passive
36
12
Aggressive
AM
AFB
Passive
AM
a) If both banks choose simultaneously, then Afriland First bank will choose its dominant
strategy, “Passive”. Amity bank knowing the Afriland First Bank has a dominant strategy,
will assume AFB will play this strategy and choose “aggressive”. The Nash equilibrium,
therefore, has AFB selecting Passive and AB selecting “Aggressive”.
b) If AFB can choose first, then the new best sequential move strategy will be “Aggressive”
for AFB and “passive” for AB.
9
10
This game is set in the South Pacific in 1943. Admiral Imamura must transport Japanese troops
from the port of Rabaul in New Britain, across the Bismarck Sea to New Guinea. The Japanese
fleet could either travel north of New Britain, where it is likely to be foggy, or south of New
Britain, where the weather is likely to be clear. U.S. Admiral Kenney hopes to bomb the troop
ships. Kenney has to choose whether to concentrate his reconnaissance aircraft on the Northern or
the Southern route. Once he finds the convoy, he can bomb it until its arrival in New Guinea.
Kenney’s staff has estimated the number of days of bombing time for each of the outcomes. The
payoffs to Kenney and Imamura from each outcome are shown in the box below. The game is
modeled as a“zero-sum game.” For each outcome, Imamura’s payoff is the negative of Kenney’s
payoff.
a)
b)
c)
d)
Is there any dominant strategy for Kenney?
Is there any dominant strategy for Imamura?
Is there any equilibrium,
Nash equilibrium? Explain your result.
Solution
a) This game does not have a dominant strategy equilibrium, since there is no dominant
strategy for Kenney. His best choice depends on what Imamura does.
b) No
c) The only Nash equilibrium for this game is where Imamura chooses the northern route
and Kenney concentrates his search on the northern route. To check this, notice that if
Imamura goes North, then Kenney gets an expected two days of bombing if he (Kenney)
chooses North and only one day if he (Kenney) chooses South. Furthermore, if Kenney
concentrates on the north, Imamura is indifferent between going north or south, since he
can be expected to be bombed for two days either way. Therefore if both choose “North,”
then neither has an incentive to act differently. You can verify that for any other
combination of choices, the admiral or Kenney would want to change. As things actually
worked out, Imamura chose the Northern route and Kenney concentrated his search on
the North. After about a day’s search the Americans found the Japanese
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