Prelim 1 Solutions

# Prelim 1 Solutions - 0:15435’ SGLU‘TIGN 516 HER...

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Unformatted text preview: 0%:15435’ SGLU‘TIGN 516'? HER sfrihtﬁ'Gé Psalm .1 1. (40 points: 12,5,1o,3,3,7)‘ Consider the following game. Three cards, an ace, a king, and a queen, are shufﬂed and one is dealt to player II. She looks at it (player I can’t see it). If it is the ace, she must say “Ace”; if the King, she can say “Ace” or “King”; if the queen, she says “Ace” or “Queen.” If she says “King,” the game is over and each player gets \$0; if “Queen,” player I gets-\$1 and II loses \$1. If player ’11 says “Ace,” player I can either believe her (B), in which case he loses \$1 and she gets \$1, or challenge her (C), in which case he loses \$4 and she gets 83 if she has the ace, and otherwise he gets \$3 and she loses \$4. Assume both players are risk- neutral and use von Neumann—Morgenstern utility functions with u(\$a:) = a; for a) Complete the extensive form of the game given below. Mark nodes and edges as appropriate, and all information sets. (If the edges from a node are marked B and C, mark the higher one B.) 'A ”t\ c) If AQ denotes “say “Ace” if have King, say “Queen” if have Queen,” etc. ., ‘ complete the strategic form of the game below. 903 A T \oa,\\K\TQS C“ A I Lek Q,\K.S- (“\, - \{J 2 [k ,WQ ' " a) . d) Suppose for the remaining parts that player I has marked the back of the cards, so he knows whatplayer II has. From the viewpoint of player I, how. does the extensive form of the game in (a) change? 1 .y '11‘5' 'Sgi'n‘j\g_ \\0\—@) Sex ' 1966.an ”home— . \er six I aux—t {gr-melt 09% make leg. . 1’th 0%; 525%. Sat’ [er—mi W (Ines: i’wd Brf—«f 'xe8;>rw5l\\$g 1 ‘ ' ' * e) How many pure strategies does I have now? Why? ‘ . _ . 1 A I Vixxox; 8 WC; siv‘oCkeﬁ/‘gﬂ :\ ‘ 3 \ecge MS} 2 I gReiﬁeeg a? 'ﬁmI/Q'x awe/a Dos it D. . £93 I BQQ, LLKQtvz Ai‘JWY \~ (0% °\)V\\ l .QKKKVLAV. at “MK ) f) Suppose now that II knows that I has marked the cards (and I knows ‘ ‘ that II knows, -etc.). Use Zermelo’s algorithm to determine strategies for the two players in this case (you can mark the diagram in '(a))., What kind of solution do these strategies form? What is the resulting payoﬁ‘? See wee-M 43% L." 5W. Veg." : ”Ipwiu' ECG ' '.. I (gag) Kg) "m _je'.£eh5m~ha€cgt Nfd" . , 2. (30 points; 4,4,6,5‘,5,6) Felix and Oscar are discussing how they. , . , l . decided they will go somewhere together, but have very different tastes. Felix’s 1 A ﬁrst choice is the theater, while Oscar’s is a hockey match; alternatively, they ' Felix is indiﬁerent‘betWeen the movie and a. . and 9.1/3 chance. of the theater. _ (a dangerous thing) and believe that their preferences satisfy the you Nonmem- 5/3 ”L. CT.) it ZWoQ“) i “okﬂlﬁb gust \ O ‘72.“ c) They discover that the f‘total utility’for the some higher than for either alternative. Should theybthereforego to the hockey motﬁIWhy or ' whynot? i 7 No‘ \t Makes no W47; 03) ﬁe {mew . ‘lim {Q’ce‘ewls .53,ka . . ' Maw-K xi;L- soésﬁ emf“ 1w: W 6) Now Oscar assigns a utility of 1 to the hockey and a utility of 0’ to the 4. _ theater. What is his new utility for the movie? What option now has the . , highest WWW? Let u}, «x 6.3%» new: yﬁh’f‘k'ﬁﬂ; 126099; ‘\ ' 9 CT) 4: 77g “0%) 2 “9“)“ 1/2 go to the theater, Felix’s top choice, . the hockey match, Oscar’s top choice. Is it ratio lottery? Why or why not ‘ ‘ . 1 . at on Mé‘m >‘ 9%4. T’ >3: ﬁr F . Sax“ 'HIE’GT”‘=Md°{—V;‘E1kzl >OKFW / 96o £3:th , M t» m, tang..- , ~ a) and(b) andassunuingfreedigpoealof f') Using the utility functions in utility, graph the cooperative payoff region and mark the strongly Pareto- . . quﬁmﬁga ,' 3. (20 points: 5,5,3,4,3) Consider the birmtrix game with 423 647 A: 623 B: 332 ' 634 ;.21 ‘ a) Write the game in strategic form. Label the strategies. 13) Find all Nash equilibria.‘ . 5‘93 CWQQQJ K‘oow waOSC-A ing~ bai _ e) Name all strongly dominated strategies of each player, and why they are ~ dominated. .,S‘ L: gkrbcjs dogmamax'eg (OJ 5': Q97? , 4 Fewer 1‘, 94w. A2,3)<< (ﬁg/q). Thig'tpﬁa . I ‘ d) Eliminate all strongly dominated strategies found m (c). In the resulting game, are any strategies of either player weakly or strongly dominated? _ . ". ~ Wen is! is JmenM . Let He. «VJ/(“q gawk ‘ 1: 3. \3 § WOU\J &0 M\ “AQQ (0} t1 ‘t3- \5‘ My \\ b3 ’ t\ t k _ \\ ‘\\ . ex 0 , t 2— ' 5 1. \\ ‘\ \\ K‘ 53. , _4. (10 points: 4,6) 3.) How does the fact that hath players Neumann—Morgenstem axioms simplify the an ’- \‘k Aka-en wfl‘o cussed/7%: WCW Q0 yNMthL\‘€~)5 {we Ntwag‘ 9d. j '&&-"{\W W Wot e .t\%*"? > have preferences Satisfying the von alysis of a game? I. . i) To see that ’D is the value of the game, you need to. consider the out- comes in the row corresponding to strategy 8 from player E’s point of view and the outcomes in the column corresponding tostrategy t . from player L’s point of view. 5 M g if. BeQV‘ch-CltWSOr (14:38— ' ' ' you need to consider the out- ..EE’S point T422: ,vS-w J l “f” I: was t ...
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