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Unformatted text preview: Midterm P830, Winter 2003 Total: 100 points NAME Read the questions over ﬁrst, and go back and check your work aﬂer. F eel free to doodle
or calculate on the backs of the pages. Be explicit about your answers if there ’3 a danger of lack of clarity — don ’t just put for example, “p=1/3 Write at the mixed strategy
equilibrium, Row will use Row 1 with probability 1/ . ” 1. (5 points) In the Bonk test, where two students had to declare which tire went ﬂat, how
many outcomes are Paretooptimal? Lt.
CH wheels, 0A. A “4% 2. Here is a game that indicates payoffs in a nuclear crisis situation. Each side decides
whether to launch its missiles against the other or not. X 5 move Stay at peace ‘ Strike (a) (4 points) Identify all pure strategy Nash equilibria. (Lip.ch t) a [wee (b) (6 points) Calculate the mixed strategy Nash equilibrium of the game. (Notice ttat
the game is symmetrical so if you calculate them for X, you don’t have to do it agaii. for Y, they will be the same.) ‘ WM of, Mia» lea, Ht
70(1—‘3 ’ "w" ﬂqgu—cw
—~ “(2 1 Co 5/04 The mixed strategy Nash equilibrium is: X WA (g/‘LI u/q)
\{ W (g/q, We) (c) (4 points) These payoffs are in the same order as one of the 2x2 games we discussed
in class. Was it the Prisoner‘s Dilemma, Chicken, the Stag Hunt or the Battle of the Sexes? HUNT. (We Uﬂppﬂ tr) (d) (4 points) Suppose that the game is changed so that first X decides whether to strike
or not. If X strikes the game ends, but if X does not strike Y can decide whether to strike
or not. If one country strikes, it gets 20 and the other gets 70. If neither strikes, both get 0. Draw the tree. .20 "70. 3. (6 points) In a gubernatorial election debate, ﬁrst candidate X speaks, then Y speaks,
then X speaks again, and that’s the end. At each speech a candidate can choose to attack
(A) or not (N). During the debate if they both attacked, the one who attacked ﬁrst gets —2
and the other gets 2. If only one attacked, the attacker gets 1 and the other 1. (It do esn’t
matter how many times that one attacked.) If neither attacked, both get 0. Construc1 the
tree ﬁrst, and then put in the payoffs. ' 4. (5 points) Solve the following tree by rolling back. (The ﬁrst payoff is A's, the second isB’sandthethirdisC’s.) A 3 2 ‘
I 5. (6 points) (a) Solve the game below using the iterated elimination of dominated
strategies. (Eliminate strategies that are either weakly or strongly dominated. Show your
answer by crossing out parts of the matrix, or if you want to redo it, copy the matrix below and cross out parts of it.) (b) (3 points) Which outcomes in the ﬁrst COLUMN are NOT Paretooptimal? “4,1 I f" ll,l Uf’» (I. (7 points) (a) Write down A's and B's strategies for the following tree. A """" 7 True or false (2 points each). Explain each answer in one short sentence. (a) Every matrix has at least one cell that is Pareto optimal. I
P91; w ermAM mm C‘UJ '1’ (so 1 4/ ﬂaw—ac (b) The winner’s curse is a problem only in an auction when you are unsure of your own value for the object. :1 r ARC}
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The/Le (9 We M’UM'A W 5 age‘s
see We f’ ‘
“tau VII/1:6 Uﬂ/Ue (c) Game the? analysis requires that each player know the other’s value for each outcome. I” ’ M NO 7—
C, {hi/lea 0% (McCarty/afar meme/meal»! _ (d) To say that my utilities for a portion of chocolate, strawberry, or vanilla ice cream are
0, 1/2 and 1 means by deﬁnition that I am indifferent between a plate with two portions of strawberry veﬁsus another plate with one portion of vanilla and one portion of
chocolate. Uhl‘hes [Le/)MS‘WT Ol‘dtcc’s H’NIBMO (4mié/é9/
Nb? Chate5 0* 6w!” ﬂecé’weﬂ rﬂ/ Cah.gz,tr/LF(o/\/ (e) Bridgebuming means acting ﬁrst in order to block some strategy that my adversary “ﬁghfgfgmgsfgzm WC 0% M1 tote/u emanate)" (f) For some games with no pure strategy Nash equilibrium if one applies the method of iterated dominance (success'Ely eliminating dominated strategies), one may end up
eliminating all strategies. To arms/MC A 9TW+T€O=1 we» N680 0M—
(Q'T'bG/L C/LC’I'YAMJI/Uw\ 0W8
(g) The Prisoner’s Dilemma game has no mixed strategy equilibrium. T
SHULQ 06.55691" L5 naAm/mvTJ NQT‘Hm‘a (ft1U LEFT A lama/t 'ro 06a tr We loam/(é, {my [€55 771m“ /'
(h) In Dutch auction the game ends after a single bid. ’1' We. Won/64 (WW: 0 06 seemo> “ELM/'1 m“: ’0’” 4"
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(i) I send a bid in for a ﬁrst price sealed bid auction, and the auctioneer sends me a message back saying that they changed the rules, that now it’s a second price sealed aid
auction, and I can submit another bid. I should raise my bid. I A ' i ~91} 
’Maw/p 81/) (£99 Thaw M4 uﬁwé ,A/ A/iﬂo Page
il‘t’u'i’ll‘O/V/ A'No ‘ZXALf/W NV UAJJE //V (4‘ 2" ﬂame (l) The game in question 2(b) is zerosum m ‘ pefice is (95 ﬂe/L 813/1. (pan! Mb lat/V17?— ‘3 ' . (4 points each) In the game below the players move in the order stated. Add in one or more information sets to show the following situations, concerning what they don’t
know. (a) Player 3 does not know 1’s move. (b) Player 3 does not know 1’s move and does not know 2’s move. 2.
‘l \e (0) Player 3 does not know 1’s move if 2 moves up but does know 2’s move if 1 moves r/‘<€< Rs
MM C[ o . (3 points each) Forty children come to a birthday party, twenty little boys and twenty
little girls. There are two big rooms, A and B. Each chooses at the same time to go into
room A or B and they have to stay there. For each situation below, state one pure
strategy Nash equilibrium, or say that there is none. (If one exists, just give one  don’t
try to list more than that.) (The questions are ordered by difﬁculty.) (a) Each child (boy or girl) likes a room better, the more children (boys plus girls) in it. [2.9 B} 1097] Zoe, 06;] (b ) Each child likes a room better the fewer children in it. _
E16: CHILD/10V“: £20 Q)“ {Nee/v j (0) Each boy likes a room better the fewer girls in it; each girl likes a room better the more boys in it.
Em a, we) 5:08.106») (d) Each child has only two levels of preference, liking and disliking. A child likes 2.
room if it has some children of the opposite sex but the proportion of the opposite sex in
the room is less than 50%. Otherwise the child dislikes the room. (So, for example, if
there are 50% or more girls, a boy dislikes it, but beyond that he doesn’t care exactly how
many there are.) PM ﬁnJLAUW'NC’VT 1% fmf Ear1 c?— + oM/Lr 15;
m1 {were} Arlew /Leom (9/2, A,“ gums mm m; Mort. EG. aoﬁ’acfl—ZOBWGJ; ,
9N8 E Z'O/blLtg't3,0C~3 ...
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This note was uploaded on 03/12/2008 for the course POL SCI 30 taught by Professor O' neill during the Spring '03 term at UCLA.
 Spring '03
 O' Neill

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