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ECON 20005: Competition and Strategy
创建日期：2024-05-15 15:22:45
所属分类：经济Economics
标签： ECON 20005
Competition and Strategy

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ECON 20005: Competition and Strategy

Tutorial Twelve (Sample Final) Solutions
Note: These are solution sketches that do not necessarily show all the work/intermediate
steps as required for full marks on the final. It is your task to clearly and succinctly
illustrate your knowledge on the final exam. When in doubt, show your work!
Question 1 (15 marks)
a. Find all pure strategy Nash Equilibria of the following two simultaneous- move games. If
there is no pure strategy Nash Equilibrium in one of these games, you should explain why
not.
(i)
P2
A B C
P1
a 9, 3 7, 4 3, 1
b 3, 6 11, 2 1, 4
c 6, 4 2, 3 8, 5
(ii)
P2
p q r
P1
m 2,−2 5,−5 13,−13
n 6,−6 5.6,−5.6 10.5,−10.5
o 10,−10 3,−3 −2, 2
(i) There is one pure strategy Nash Equilibrium: (c, C)
(ii) There is one pure strategy Nash Equilibrium: (n, q)
b. Consider the following sequential-move game:
BA
1
d
1, 1
c
3, 2
2
d
2, 3
c
1, 1
2
(i) Find all the pure-strategy Nash Equilibria to this game.
(iii) If a Nash Equilibrium is not Sub-game Perfect, explain why not.
(i) There pure-strategy NE to the game are (A, cc), (A, cd), (B, dd) (easiest to find these
using the normal form of the game)
(ii) The Nash Equilibira (A, cc) and (B, dd) are not sub-game perfect. They require player
2 to play either d after player 1 having played A, or c after player 1 having played B, both
of which would not be optimal for player 2 if the corresponding sub-game were reached.
Page 1 of 5
c. Consider the following game played between two players. Each player chooses 1, 3, 5 or
7 to add to a cumulative total. The cumulative total starts at 0. The players take it
in turns to choose one of these numbers, and they are able to view the cumulative total
at their turn. The player who causes the cumulative total to equal or exceed 66 is the
loser. Consider the player who starts. Explain whether she wins or loses in the Sub-game
Perfect Nash Equilibrium. Explain her equilibrium strategy.
(i) The first moving player wins in the sub-game perfect Nash Equilibrium.
(ii) There are various strategies that ensure her victory. One is to choose 1 at her first
move and then to ensure that the cumulative total is at 9, 17, 25, 33, 41, 49, 57 and 65
after her subsequent moves, such that her opponent has to cause the cumulative total to
equal or exceed 66. Another is to choose 1 at any move.
Question 2 (15 marks)
This is a penalty kick game in which the goalie has a third option (the option of remaining in the
middle and not reacting until the very last moment). We still assume that it is a simultaneous-
move game. The game table looks as follows:
Goalie
L M R
Kicker
L 30, 70 70, 30 90, 10
R 90, 10 70, 30 30, 70
a. Assume the kicker plays L with probability pL and R with probability 1 − pL , where
0 ≤ pL ≤ 1. Derive the goalies expected payoff for each of his pure strategies, and show
what the goalies best response is to any possible pL.
The goalies expected payoff is 70pL + 10(1 − pL) = 10 + 60pL when playing L, 30 when
playing M , and 10pL + 70(1 − pL) = 70− 60pL when playing R.
His best response is playing L if pL ≥ 0.5, and R if pL ≤ 0.5. (Playing M is never a best
response.)
b. Assume the goalie plays L with probability qL, R with probability qR and M with prob-
ability 1 − qL − qR,where qL ≥ 0,qR ≥ 0 and qL + qR ≤ 1. Derive the kickers expected
payoff for each of his pure strategies, and show what the kickers best response is to any
possible combination of qL and qR.
The kickers expected payoff is 30qL + 70(1 − qL − qR) + 90qR = 70 − 40qL + 20qR when
playing L,and 90qL + 70(1 − qL − qR) + 30qR = 70 + 20qL − 40qR when playing R.
His best response is L if qL ≤ qR,and R if qL ≥ qR.
c. Derive the Nash Equilibrium in mixed strategies.
It follows from parts (a) and (b) that the Nash Equilibrium in mixed strategies is (pL; qL, qR) =
(0.5; 0.5, 0.5)
Question 3 (25 marks)
Consider an industry with two firms Holden and Toyota that face an inverse demand function
P (Q) = 200−Q, where Q is the total quantity produced. Both firms produce at marginal costs
of 20 (and have no fixed costs).
Page 2 of 5
a. Assume that the two firms compete a la Cournot, i.e., that they simultaneously choose
the (continuous) quantity to produce. Derive the Nash equilibrium quantities that the
firms produce as well as their equilibrium profits. (Remember to show your work.)
Holden chooses the quantity qH that maximizes its profit
πH(qH) = (200 − qH − qT − 20)qH = (180 − qH − qT )qH
given qT . Differentiating with respect to qH gives the first order condition
0 = 180− 2qH − qT .
Rearranging gives Holdens best response qH =
180−qT
2
. Similarly, Toyotas best response
is qT =
180−qH
2
.
Inserting Holdens best response into Toyotas best response implies
qT =
180− 180−qT
2
2
⇒ q∗T = 60
Consequently, q∗H = 180− q
∗
T = 180− 60 = 60.
Equilibrium profits are π∗H = π
∗
T = (200 − 2× 60 − 20) × 60 = 60
2 or 3600.
b. Assume that the two firms compete a la Bertrand, i.e., that they simultaneously choose
their (continuous) prices, that consumers buy from the firm that sets the lower price, and
that consumers buy the same quantity from both firms if both firms set the same price.
Find the unique Nash equilibrium in pure strategies and the corresponding equilibrium
profits. Also prove that there cannot exist any other Nash equilibrium.
In the unique Nash equilibrium (NE) prices are p∗H = p
∗
T = 20.
Equilibrium profits are π∗H = π
∗
T = 0.
There cannot exist a NE with min{pH , pT } < 20, as firms making negative profits would
have an incentive to deviate. There cannot exist a NE in which min{pH , pT } > 20, as at
least one firm would have an incentive to deviate and to set a slightly lower price than its
competitor. There can also not exist a NE in which min{pH , pT } = 20 < max{pH , pT },
as the firm setting the lower price would have an incentive to deviate and to increase its
price. Hence the above NE must be unique.
c. Now assume that the government subsidizes Holden with 30 per car that they sell. (In
other words, Holden receives from the government 30 per car that they sell.) Again
assuming Cournot competition, derive the equilibrium quantities and profits of the two
firms.
Now Holden chooses the quantity qH that maximizes its profit
πH(qH) = (200 − qH − qT − 20 + 30)qH = (210 − qH − qT )qH
which leads to its best response qH =
210−qT
2
. Toyotas best response is still qT =
180−qH
2
.
Inserting Holdens best response into Toyotas best response implies
qT =
180− 210−qT
2
2
⇒ q∗T = 50
Consequently, q∗H =
210−q∗
T
2
= 210−50
2
= 80
Equilibrium profits are π∗H = (200 − 80 − 50 − 20 + 30) × 80 = 80
2 = 6400 and π∗T =
(200 − 80− 50− 20) × 50 = 502 = 2500.
Page 3 of 5
d. Now assume again Bertrand competition, and describe one Nash equilibrium in pure
strategies for the case in which the government subsidizes Holden with 30 per car that
they sell. Also derive the firms equilibrium profits.
There are two NE: p∗H = 20 and p
∗
T = 20 + ǫ,or p
∗
H = 20 − ǫ and p
∗
T = 20, with ǫ being
arbitrarily small.
To derive the equilibrium profit of Holden, note that P (Q) = 200 − Q implies Q(P ) =
200 − P . Hence π∗H(200 − 20)(20 − 20 + 30) = 5400, while π
∗
T = 0.
Question 4 (25 marks)
Suppose that a society consists of only two players and that both of them drive a car. Each
player must decide whether to put a pollution control device on their car. Installing such a
device on a car costs $100 and provides a benefit that is worth $70 to each player (both players
benefit since they both breathe cleaner air). If both players install the device, the air is twice
as clean, providing a benefit of $140.
Each player has two possible actions: “Install” or “Not Install.” Assume for the moment
that this game is a simultaneousmove oneshot game.
a. Construct the game table and find all pure strategy Nash Equilibria.
Game table:
Player 2
Install Not
Player 1
Install 40, 40 −30, 70
Not 70,−30 0, 0
The unique Nash equilibrium is (Not,Not).
b. Is this game a Prisoners Dilemma Game? Why/Why not?
It is a Prisoners Dilemma Game as each player has a dominant strategy, and the dominance
solution is worse for both players than one of the non-equilibrium outcomes.
c. Assume for the remainder of the question that the game is played infinitely often, that
both players have a discount factor of δ < 1, and that the pollution device only lasts for
one year so that the two players must decide at the start of each year whether to install
the device.
What infinite payoff will each player receive if they both play a grim strategy and co-
operate forever (by installing the device)?
If they both play a grim strategy and cooperate forever, their infinite payoff is 40
1−δ
.
d. What infinite payoff will a player receive by cheating in this period (and not installing the
device) if the opponent plays a grim strategy?
If the opponent plays a grim strategy, the infinite payoff when cheating in this period is
70.
e. What values of δ sustain the collusive outcome of this game? Provide the intuition for
your solution.
Page 4 of 5
The collusive outcome is sustained if and only if
40δ
1− δ
≥ 70⇒ 40 ≥ 70(1 − δ)⇒ 70δ ≥ 30⇒ δ ≥
3
7
f. Now assume that in each period one of the two players is involved in a deadly car accident
with probability 25%. For what values of δ can the collusive outcome be sustained in this
case?
The effective discount factor becomes 3δ
4
. Hence the collusive outcome is sustained if and
only if
3δ
4
≥
3
7
⇒ δ ≥
4
7
Question 5 (20 marks)
Consider a charitable donations game simultaneously played by N = 100 citizens. Each citizen
must decide whether to donate to $11 charity or not. If n citizens donate to the charity, then
each of the N citizens receives a private benefit worth $n2 regardless of whether they donated
or not (since they feel good about charity donations in general). If an individual donates, they
suffer a private financial cost of $11. Throughout, n must be a whole number from 0 to 100.
a. Find all Nash Equilibria number of donors, nne.
Given n donors, if citizen n + 1 switches from shirking to participating they receive a
payoff of P (n + 1) = (n+ 1)2 − 11 = n2 + 2n+ 1− 11 = n2 + 2n− 10
Given n donors, each citizen’s private benefit to not donating (shirking) is: S(n) = n2.
Solving for the interior Nash Equilibrium we have:
P (n+ 1) = S(n)⇔ n2 + 2n− 10 = n2 ⇒ nne = 5
As P (n+ 1) intersects S(n) from below, this is an Assurance collective action game with
two additional Nash equilibria at nne = 0 and nne = N = 100.
b. Find the socially optimal number of donors, n∗.
To find the socially optimal number of donors, we must first write down the total surplus
function for society for a given number of donors n and non-donors 100 − n:
T (n) = n(n2 − 11) + (100 − n)n2 = 100n2 − 11n
The first-order derivative of T (n) is:
∂T (n)
∂n
= 200n − 11
Which is strictly positive for all n ≥ 1 implying T (n) is always increasing in n. Hence,
n∗ = N = 100 is the socially optimal number of donors.
c. What type of Collective-Action problem is this and why?
This is an Assurance collective action problem as the P (n + 1) function intersects the
S(n) function from below. When no people are donating, S(n) is much higher than
P (n + 1) meaning it’s privately optimal for no-one to donate. When there are lots of
people donating, P (n + 1) is much larger than S(n) meaning it’s privately optimal for
everyone to donate.
Page 5 of 5
d. Suppose n is currently equal to 0 and that the government can simply convince individuals
to donate (say through other means such as tax breaks). How many people does the
government have to convince to donate in order for n = N in a Nash Equilibrium?
Recall the (unstable) interior Nash Equilibrium to this Assurance game is at nne = 5.
Thus if the government could convince n = 6 people to donate, we would reach the Nash
Equilibrium of nne = N since P (n+ 1)− S(n) grows as n grows for all n > 6.
Question 6 (20 marks)
There are four bidders interested to buy the the DVD “Lord of the Rings: Return of the King”
in an auction. Their valuations V are $5, $10, $20 and $25. Each bidder only knows her
own valuation, but all bidders know the distribution function from which all the valuations are
drawn. Assume that the bids are continuous variables (from $0 to $100).
a. Describe the optimal strategy of a bidder with valuation V in an first-price sealed-bid
auction. What bidder wins in equilibrium and at what price?
The bidder chooses his bid b to maximize his expected payoff P (b)× (V − b), where P (b)
is the probability that his bid b exceeds the largest bid r of the other bidders.
Hence, bidders face a trade-off: The higher b, the larger the probability P (b) that they
get the book, but the smaller the payoff V − b when they get the book. From the first
order condition that governs a bidders optimal bid we have:
∂P (b)
∂b
(
V − b
)
− P (b) = 0⇒ b = V − P (b)×
(∂P (b)
∂b
)
−1
< V
It directly follows that the bidder optimally shades his bid and bid b < V . Moreover, his
bid b must increase in his valuation V . As all bidders play this strategy, the bidder with
valuation V = $25 gets the book at some price b < $25.
b. Describe the optimal strategy of a bidder with valuation V in a second-price sealed-bid
auction. What bidder wins in equilibrium and at what price?
It is the bidders dominant strategy to bid b = V .
Any higher bid may lead to a negative payoff, and any lower bid may lead to the bidder
not getting the book while she could have gotten it for a price below V .
As all bidders play their dominant strategy, the bidder with valuation V = $25 gets the
book at the price b = $20.
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MOEC0021 STA 106 ENGG 1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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