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Unformatted text preview: 093837 38/37 School of Economics and Political Science
University of Sydney ECON3012 — STRATEGIC BEHAVIOUR
(Semester 2, 2004) 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. Strong IEDS equilibria and Nash equilibria have the following relationship. a) A strong IEDS equilibrium must be a Nash equilibrium but the Nash equilibrium need
not be unique. b) A unique Nash equilibrium must be a strong IEDS equilibrium. c) A strong IEDS equilibrium must be a unique Nash equilibrium and a unique Nash
equilibrium must be a strong [EDS equilibrium.
d) None of the above. 1 38/37 Semester 2, 2004 Page 2 of 9 2. For games with a ﬁnite number of strategies and players a) an IEDS solution (possibly in mixed strategies) must exist.
b) a Nash equilibrium (possibly in mixed strategies) must exist.
c) a) and b). (:1) None of the above. 3. Sue and Alice are doing a joint assignment. If both work hard, the mark they get is worth 5
to each of them in utility terms. If one works hard, the mark they get is worth 2. If neither
works hard, the mark is worth 0 in utility terms. On the other side of the ledger, working hard
costs 3 in utility terms. They simultaneously choose whether to work hard (H) or slack off (S). In this game, a) slacking off is a strongly dominant strategy for each player.
b) working hard is a strongly dominant strategy for each player.
c) (H,H) and (8,5) are Nash equilibria. d) None of the above. 4. The game shown below a) is dominance solvable. b) does not have a Nash equilibrium.
0) has multiple Nash equilibria. d) None of the above. 5. The following game: 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.
c) has multiple pure strategy Nash equilibria.
d) None of the above. Player 2 T
Player 1 M L
B 3,4 R 38/37 Semester 2, 2004 Page 3 of 9 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: 1/3 andq: 1/4.
b) p=l/3 andq:2/3.
c) p:1/5andq=l/2.
d) None of the above. Player 2 T
Player 1
B 7. In the subgame perfect Nash equilibrium of the game in Figure l, the equilibrium payoffs
are: a) (3,6).
b) (4,4).
0) (5,5).
(1) None of the above. Player 1 Player 1 FIGURE 1. Extensive Form Game 1 8. Consider the entry game shown in Figure 2. In the backward induction solution to this game,
the payoffs are: a) (4,4). b) (2.8, 4.4). 0) (3,5). (1) None of the above. 38/37 Semester 2, 2004 Page 4 of 9 Firm 1 3,5 FIGURE 2. Extensive Form Game 2 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’s discount
factors are 51 and 52. In a subgame perfect Nash equilibrium: 3) player 1 gets 51M and player 2 gets (1 — 50M.
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  82)!” and player 2 gets 32M. 10. In the Stackelberg duopoly model, a) the firm that moves ﬁrst charges the higher price. b) the ﬁrm that moves second charges the higher price.
c) the ﬁrm that moves ﬁrst makes the most proﬁt. d) the ﬁrm that moves second makes the most proﬁt. 11. In a ﬁnitely repeated game, it is possible to have a subgame perfect Nash equilibrium in
which the choices in some period differ from the Nash equilibrium strategies of the stage game a) whenever the gains to cooperation are large enough. b) for every game in which the stage game that has no dominant strategies.
c) provided the stage game is played enough times. (1) only if the stage game has multiple Nash equilibria. 38/37 Semester 2, 2004 Page 5 of 9 12. Consider the stage game below. 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 z 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 (QC) or (HP). b) Any one of the 9. c) Only (QC), (HP) or (RF).
d) Only (QC), (HP) or (CF). 13. 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 1/2.
c) 5 2 3/4
(1) None of the above. 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 player—types use the same joint probability distribution of types. d) None of the above. 38/37 Semester 2, 2004 Page 6 of 9 15. Assume that in a game of incomplete information there are two players. Player 1 can be one
of two 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 BayeseNash equilibrium is a) 108.
b) 13.
c) 36.
(1) 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 Z T e A
Player 1 yp
Type B a) the unconditional probability of a Type Y is 2/5, while p(Y I A) = 1/3.
b) p(B I2) = 2/3 andp(Z  B) : 2/5.
c) p(A I Z) = 2/3 and p(B [ Y) = 3/4.
d) the unconditional probability of a Type A is 1/2, while p(A  X) = 2/3. 17. Consider the following game:
Player 2A Player 2B L R L R
T
B Player 2 is of type A with probability p and type B with probability 1 — p. In a Bayes‘Nash
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 2/3. b) player 2A plays R and player 2B plays L for all p, whereas player 1 plays T if and only
if p 2 1/2. c) player 2A plays L and player 2B plays R for all p, whereas player 1 plays T if and only
if p 2 2/3. d) player 2A plays L and player 2B plays R for all p, whereas player 1 plays T if and only
if p 2 1/2. 38/37 Semester 2, 2004 Page 7 of 9 18. Suppose that each of two Cournot duopolists may be either a low cost or a high cost type
(a low cost ﬁrm I has the same 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: 3) a high cost ﬁrm 1’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 piayer is worse off in the complete information game, then the other player must
be better off. c) both players are equally well off in both games since the incomplete information game
works in terms of expected values.
(1) None of the above. 20. Firm H is considering taking over ﬁrm P. Firm P is worth either 10, 15 or 20 with equal
probability when under its own management. Under the management of ﬁrm H, it is worth
5/3 times as much. Firm H makes a takeover offer of y, where y is either 10, 15 or 20 and ﬁrm P accepts the offer if and only if y is at least as large as the value of the ﬁrm under ﬁrm P’s
management. The expected payoff to ﬁrm H from an offer of y is: melee—y]
b>’%“"l(3’%19>%yl (1) None of the above. 38/37 Semester 2, 2004 Page 8 of 9 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 [EDS equilibrium or does not lead to a pure strategy [EDS equilibrium. Player 2
L C
T
Player 1 M
B
(ii) Now consider a second game.
Player 2
L R
T
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. Demand is given by P = 8 m Q, where
Q = Q1 + Q2. Firms 1 and 2 have constant marginal costs of Cl and C2, respectively. (i) Solve for the Cournot equilibrium quantities and proﬁts for the two ﬁrms (these will be
functions of cl and Cg).
Note. You may assume without proof that the solution quantities are positive, so you
do not have to worry about nonnegativity restrictions. (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 invest in cost
reduction. If a ﬁrm invests in cost reduction, its marginal cost c, equals 1. If it does
not invest in cost reduction, its marginal cost c, equals 2. The investment costs 23/9. In
period 2, the ﬁrms play the Coumot game with their costs determined by the choices in
period 1. Solve for the subgame perfect Nash equilibrium of this game assuming that
ﬁrms seek to maximise proﬁts net of investment (assume no discounting). (iii) For any ﬁrm that undertakes the investment, identify the relationship between the in
vestment in cost reduction and the cost savings achieved at its equilibrium output. Com—
ment on this relationship. 38/37 Semester 2, 2004 Page 9 of 9 Q3. There are two members in a team. They simultaneously choose to either work (H)ard or to
(S)hirk. If a member 1' chooses H, that member incurs a personal cost of effort Ci. If a member
1' chooses S, then the cost of effort is zero. If both members choose H, the total proﬁts are 12, if
only one of them chooses H, then the total proﬁts are 6, and if neither chooses H, then the total
proﬁts are 4. The realised proﬁts are shared equally. This leads to the following payoff table: Player 2 The value of c, (i = 1,2) depends on a player’s type (each player knows her own type but not
the type of the other team member). For a low cost type, c, = 2. For a high cost type, c, = 4.
The probability that the other team member is low cost is p and hence the probability that the
other team member is high cost is (l — p). Solve for all (pure strategy) BayesN ash equilibria, indicating their dependence on p if appro
priate. Comment on the intuition behind the equilibria.
Hint. Does either type have a dominant strategy? End 01‘ Paper ...
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