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Unformatted text preview: 38/37 School of Economics and Poiitica Science
University of Sydney ECON3012  STRATEGIC BEHAVIOUR
(Semester 2, 2005) W Duration: 2 hours (+ reading time) INSTRUCTIONS (1) The paper is in two sections. Section A consists of 20 multiple choice
questions and is worth 40% of the mark for the exam. Section B consists
of 3 problem questions and is worth 60% of the mark for the exam (20% per question). (2) Answer ALL questions. (3) When you submit your exam answers, place the multiple choice answer
sheet inside your answer booklet and return both. SECTION A: MULTIPLE CHOICE QUESTIONS Note. Unless the question speciﬁes otherwise, you should restrict attention to pure strategies
except in questions 5 and 6. 1. A strategy that is a best response to some opponent strategy a) must be a weakly dominant strategy. b) must be strongly dominant strategy. c) cannot be weakly dominated. d) can be weakly dominated but not strongly dominated. 2. The following is true: a) a strongly dominant strategy is a best response to all opponent strategies, but a weakly
dominant strategy may not be. h) a weakly dominant strategy is a best response to all opponent strategies. c) a weakly dominated strategy cannot strongly dominate another strategy.
d) None of the above. 1 38/37 Semester 2, 2005 Page 2 of 10 3. Assume that the row player can choose Top (T) or Bottom (B) and the column player can
choose Left (L) or Right (R). Assume that for the row player: ur(T,L) > ur(B,L)
ur(T,R) > ur(B,R)
Assume that for the column player:
uc(T,L) < uC(T,R)
uc(B,L) = uC(B,R)
Then:
a) (T,L) is a Nash equilibrium.
b) (B,L) is a Nash equilibrium. c) (ER) is a Nash equilibrium.
d) None of the above. 4. The game shown below a) has an IEDS solution.
b) does not have a Nash equilibrium. c) has multiple Nash equilibria.
(1) None of the above. L C R
T "m
B a) has an IEDS solution that may be reached using only pure strategies in the elimination process.
b) has an IEDS solution that may be reached if some mixed (nonpure) strategies are used in the elimination process.
0) has multiple pure strategy Nash equilibria.
(1) None of the above. 5. The following game: Player 2 L R T
Player 1 M B 38/37 Semester 2, 2005 Page 3 0f 10 6. Assume in the following game that player 1 plays T with probability p and that player 2
plays L with probability q. The mixed strategy solution is a) p: 1/4andq2 1/3.
b) p=l/2andq:2/3.
C) p: 1/3 andq=1/2.
d) None of the above. Player 1 7. In the subgame perfect Nash equilibrium of the game in Figure l, the equilibrium payoffs
are: a) (6,5).
b) (4,6). c) (5,5).
d) None of the above. Player 1 FIGURE 1. Extensive Form Game 1 38/37 Semester 2, 2005 Page 4 of 10 8. Consider the strategic form game shown below. a) This is consistent with an extensive form perfect information game in which player 1
moves ﬁrst. b) This is consistent with an extensive form perfect information game in which player 2
moves ﬁrst. c) This is not consistent with an extensive form perfect information game. d) Too little information is provided to draw a conclusion. Player 2 A F 9. Consider a two period ultimatum game in which there are M dollars to be distributed. In
period 1, player 1 makes an offer to player 2. If it is accepted, then distribution takes place in
accordance with the offer and the game ends. If the offer is rejected, then the game enters a
second period and this time player 2 gets to make an offer to player 1. If it is accepted, then
distribution takes place in accordance with the offer; if the offer is rejected, then both players
get nothing. Assume that the offer amount is continuously variable and that the player 1’s and
player 2’s discount factors are 51 and 82 respectively. In a subgame perfect Nash equilibrium: T
B Player 1 a) player 1 gets 51M and player 2 gets (1 — 51)M.
b) player 1 gets 52M and player 2 gets (1 — 52)M.
c) player 1 gets (1 — 51)M and player 2 gets 51M.
d) player 1 gets (1 — 52)M and player 2 gets 52M. 10. If a pure strategy game has a ﬁnite number of players and strategies a) it must have a Nash equilibrium. b) it must have a Nash equilibrium (derivable through backward induction) if it is a game
of perfect information. c) it can only have a Nash equilibrium if a player has a dominant strategy. d) None of the above. 38/37 Semester 2, 2005 Page 5 of 10 11. Consider the stage game below. Player 1 P Stage Game Plainly, there are 9 possible combinations of strategies for the stage game. Assume that the
stage game is played twice and that there is no discounting (5 = 1). Which of the 9 possible stage game strategy combinations could be played in the ﬁrst play of the stage game as part of
a subgame perfect Nash equilibrium for the repeated game? a) Only (CC) or (P,P). b) Any one of the 9. c) Only (C,C), (RP) or (RF).
CD Only (C,C), (RP) or (CF). 12. Consider the following stage game. Player 2 Cooperate Defect Cooperate
Player 1
Defect Stage Game If this game is repeated an inﬁnite number of times and a grim trigger strategy is employed,
then (Cooperate, Cooperate) can be sustained in a subgame perfect Nash equilibrium if and only if a) 5 21/5. b) 5 2 7/12. c) 6 2 3/4 d) None of the above. 38/37 Semester 2, 2005 Page 6 of 10 13. If we compare the grim trigger strategy and the forgiving trigger strategy in terms of their
ability to sustain as subgame perfect Nash equilibria the playing of choices that are not Nash
equilibria of the stage game, then the use of the forgiving trigger strategy: a) requires smaller minimum values of 5 in order to compensate for the fact that punish
ment is less severe. b) requires larger minimum values of 5 in order to compensate for the fact that punishment
is less severe. c) may require either smaller or larger minimum values of 5 depending on whether or not
cyclical behaviour is involved. d) has no effect on the required minimum values of 5. 14. In games of incomplete information, the “common priors” assumption means that a) all playertypes have the same probability of meeting a particular opponent type. b) all types of a given player have the same probability of meeting a particular opponent
type. c) all playertypes use the same joint probability distribution of types. d) None of the above. 15. Assume that in a game of incomplete information there are two players. Player 1 can be one
of three types, each of which has two possible strategies. Player two can be one of three types,
each of which has three possible strategies. The number of possible strategy combinations to be considered in identifying a BayesNash equilibrium is a) 54.
b) 15.
c) 108.
d) None of the above. 16. The following table gives the joint probability distribution for different type combinations.
Based on this table: Player 2 Type X Type Y Type 2 T eA
Playerl yp Type B a) the unconditional probability of a Type Y is 2/5, while p(Y J A) : 1/3.
b) p(B  Z) = 2/3 and p(Z  B) = 1/5.
c) p(A 12) = 2/3 and p(B I Y) = 3/4.
(1) the unconditional probability of a Type A is 1/2, while p(A  X) = 3/5. 38/37 Semester 2, 2005 Page 7 of 10 17. Consider the following game:
Player 2A Player 2B L R L R
T Player 2 is of type A with probability p and type B with probability 1 — p. In a BayesNash
equilibrium T Player 1 Player 1 a) player 2A plays R and player 2B plays L for all p, whereas player 1 plays T if and only
if p 2 3/4. b) player 2A plays R and player 2B plays L for all p, whereas player 1 plays T if and only
if p 2 1 / 3. c) player 2A plays L and player 2B plays R for all p, whereas player 1 plays T if and only
if p 2 3 / 4. CD player 2A plays L and player 2B plays R for all p, whereas player 1 plays T if and only
if p 2 1 / 3. 18. Suppose that each of two Cournot duopolists with a linear industry demand curve may be
either a low cost or a high cost type (a low cost ﬁrm 1 has the same constant marginal cost as a
low cost ﬁrm 2 and similarly for a high cost ﬁrm 1 and ﬁrm 2). If the probability that ﬁrm 1’s
opponent is low cost is 0.25 and the probability that ﬁrm 2’s opponent is low cost is 0.5, then
in a BayesNash equilibrium involving positive outputs: a) a high cost ﬁrm l’s output will be greater than a high cost ﬁrm 2’s output.
b) a high cost ﬁrm 2’s output will be greater than a high cost ﬁrm 1’s output.
c) a high cost ﬁrm 1 and a high cost ﬁrm 2 will produce the same output. d) Too little information is provided to draw a conclusion. 19. If we compare a two—player game in which there is complete information about player 1
with an otherwise identical game in which there is incomplete information about player 1, then a) player 1 may be better or worse off in the complete information case but player 2 cannot
be worse off, since player 2 can always choose to ignore the information. b) if one player is better off in the complete information game, then the other player must
be worse off. c) both players are equally well off in both games since the incomplete information game
works in terms of expected values. d) None of the above. 38/37 Semester 2, 2005 Page 8 of 10 20. Firm H is considering taking over ﬁrm P. Firm P is worth either 0, 5 or 10 with equal
probability when under its own management. Under the management of ﬁrm H, it is worth
5/3 times as much as under its own management. Finn H makes a takeover offer of y, where
y is either 0, 5 or 10, and ﬁrm P of type x (where x = O, 5 or 10) speciﬁes the minimum
amount it will accept as a takeover offer, m(x). Payoffs are zero to both parties if the takeover offer is rejected. If accepted, ﬁrm H gets x (5/3) — y and ﬁrm P gets y —x. In a Bayes—Nash
equilibrium: a) ﬁrm H successfully takes over ﬁrm P due to the greater value that the ﬁrm has under
ﬁrm H’s management. b) ﬁrm H only takes over ﬁrm P if ﬁrm P’s type is at least 5. c) ﬁrm H makes a takeover offer of 0 and the takeover succeeds, if at all, only when
ﬁrm P’s type is 0. d) None of the above. 38/37 Semester 2, 2005 Page 9 of 10 SECTION B: PROBLEM QUESTIONS Q1. Do the following: (i) For the following game, identify an elimination path that leads to a pure strategy IEDS
equilibrium. If there is a choice of path, ﬁnd a second path that either leads to a different
pure strategy IEDS equilibrium or does not lead to a pure strategy IEDS equilibrium. Player2
L C R
,1 T
B Player 1 (ii) Now consider a second game. Player 2
L R Player 1 B Find the pure strategy Nash equilibria (if any) in the game. Also ﬁnd any additional
' mixed strategy equilibria. Q2. Consider the following Cournot duopoly game with nonconstant marginal costs. Demand
is given by P 2 a — Q, where a is a positive constant and Q = Q1 + Q2. Firms 1 and 2 each
have the total cost function: TCi =4Qi+Qi2 (i: 172) Consequently, the proﬁt of ﬁrm i is given by: n, = TR, — TC,
= For —4Q.— Q?
= (a _ Q1 — Qlei —4Qi — Qiz
Thus
7t1=(a — Q1  Q2)Q14Q1~ Q12
= (a~Qz—4)Qi 4912.
Similarly, R2 = (a  Q1 —4)Q2  2Q22 (i) Using the proﬁt functions above, ﬁnd the best response (reaction) function of each ﬁrm.
Solve for the Cournot equilibrium quantities and proﬁts for the two ﬁrms (these will be
functions of a). Simplify these as far as possible. Note. You may assume without proof that the solution quantities are positive, so you
do not have to worry about nonnegativity restrictions. 38/37 Semester 2, 2005 Page 10 of 10 (ii) Now suppose that the Cournot game is played in the second period of a two period
game. In period 1, the two ﬁrms simultaneously choose whether or not to promote the industry product through advertising. The value of a in the industry demand function is
given by the following: 6 if neither ﬁrm advertises
a x 8 if one ﬁrm advertises
10 if both ﬁrms advertise The cost to a ﬁrm of advertising is 915. In period 2, the ﬁrms play the Cournot game with
the value of a in the industry demand function determined by the choices in period 1.
Solve for the subgame perfect Nash equilibrium of this game assuming that each ﬁrm
seeks to maximise proﬁt net of advertising cost (assume no discounting). (iii) Is the equilibrium level of advertising in the interests of the two ﬁrms (or are there
choices that would give a better outcome for both)? Explain. If the level of advertising
is not in the interests of the two ﬁrms, give an intuitive explanation for their sub—optimal
choices. QB. Players 1 and 2 must choose between c00perating C and ﬁghting F. Player 1 is of a
single type. Player 2 can be either a Bully or a Reciprocator. The Bully player 2 likes to take
advantage of cooperative behaviour and hence gets the highest payoff by playing F against C.
The Reciprocator player 2 likes to treat others as they treat him. His highest payoffs are from
playing C against C and F against F. Player 1 is of a single type but her payoffs are nevertheless
inﬂuenced by the type of her opponent. The probability that player 2 is a Bully is p and hence
the probability that player 2 is a Reciprocator is 1 — p. The payoff tables are given below: Player 2 Bully Player 2 Reciprocator Player 1 Solve for all (pure strategy) Bayes—Nash equilibria, indicating their dependence on p if appro
priate. Hint. While no playertype has a dominant strategy, if you think about the best responses of
the playertypes, you can narrow down the range of possible equilibrium strategy combinations substantially. End Of Paper ...
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