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# exam1-key - Eeo 122: Game Theory Summer Session II,...

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Unformatted text preview: Eeo 122: Game Theory Summer Session II, lllﬂll Professor David Lang Exam [ 20 August 2W3 NAME: k E I I. Multiple Choice Qaestiens For each ofthefot'lowlng. cheese the best tamer. Each question will be worth 3 points. Be sure to write the sewers in the boxes provided to the fell areas}: question. For questions 1-3, consider a game between a husband and a wif e in which the couple will choose to go to a baseball game or to the opera. The payoffs are listed in the table below, where the husband’s payoffs are listed ﬁrst. 1. Identity the pure strategy equilibrium: a. e” Husband plays Baseball; Wife plays Baseball b. Husband plays Baseball: Wife plays Opera E es" Husband plays ﬂpera; Wife plays Opera ii {El and [bl (£3 {at and be 2. In a mixed strategy equilibrium w “FE: FUE : t Uh a. Husband plays Baseball with probability lift P.- m- ; " ' ‘5 _ I b. Husband plays Baseball with probability iii 1:) H Pap (“l P) e. liasbaati plays Baseball with probability in {D F -~ Husband plays opera with probability is P —_~ 1/3 e. None of the above 3. In a mixed strategy equilibrium, .— a. Wife plays aassbaii with probability lt'ﬁ Husk-Mal '- 5: Up = E Up b. Wife plays Baseball with probability “3 _ _ e. Wife plays Baseball with probability is 21 0-13 - ' ‘11 + U a a) ti Wife plays IIlpera with probability EH (5 1 1 @ None of the above It as 2/5 For questions 4-5. Use the following table, in which Bush's payoffs are listed ﬁrst, New Mexico Iowa 63:- “In? Eli-31m .er - 4. Which of the following snatcgies represent a Nash Equilibrium? :1. Ohio, Colorado b. Florida, Colorado c. Pennsylvania, Colorado {at} Pennsylvania, New Mexico c. None of the above 5. Which of the strategies remain after using iterated elimination of dominated strategies? :1. Ghio, Colorado F: . E Fluﬁdlcﬂlumdﬂ g Chib‘iﬂ Aﬂoﬁl-‘lﬁj’ﬂ Haul-IAN“ ﬁ- o c. PennsyltAania, Colorado @J ? “in \$9me In}: Lyn-11 Pennsylvania. Newbiesico emu; Jwﬁm Hal; _.H A 115 E} - n. Home of the above [ED Um yMQmm 593w“. Lm’ NW“ For questions 15-3: Suppose that in the US, society can be represented as two people — yourself and an average person named “Either”. Each of you drives a car and can choose to put a pollution control device on your car, at a cost of\$1ﬂﬂ to the car owner. which would provide a beneﬁt in the form of cleaner air that is worth STD to each person. Thus, each of you has 2 possible actions — Install or Not Install. Suppose that this scenario is a simultaneous, one-time garner partially represented as the table below: m Not Install as a. When each person plays install1 what is the net beneﬁt in dollars to you? a. 100 if: 33 70+70 ~lco: HO a. o e. do T. When you ehoose blot install but Other ehooses Install, what is the net beneﬁt in dollars to you? a. [ﬁt] (t? to d. t} e. -3tl 3. Listing your strategy ﬁrst, determine the Nash equilibrium {equilibria) of this ante: a. Install, Install 3: h. Net Install. Install :2. Install, Not Install Not Install, Not Install :- wt and to 9. Suppose that you can model a simultaneous, Zero-sum game between 2 players by using a standard game tablefmatris. Then, any pure-strategy Nash equilibrium outcome must he the number in that column of the mania and the number in that row of the matrix. a. smallest: largest {ED largest; smallest e. smallest; smallest d. largest: largest e. {a} or {b} Fer questions ltl-l 1, use the following information: The sales for Firm t and Firm 2 depend on both its own priee and the competing ﬁtm's price, and can be represented as 01:2] - 'i' Q; = 27 - {1.7591 + DJ!" Where P. is the price charged by Firm 1, Q; are the sales by Firm 1, and P1 and Q1 are deﬁned similarly for Fin-n 2. The eost of produetion is constant at 6. “1 Given the priee from Firm 2, what is the best response ftmetion for Firm i'? l P|=tlill4jp}+ I? t? tests: TIT-Ci-QCE‘O'TSR+03?!) a. P1=(2JBJP2+21 311,-; E Noneot'theaboye “5:7; :: Zl~0.75ﬁ+8.5P1—ﬂ,'7§fl+t-[Egﬁ LSQ : 2‘35 +03% PF- igP, +17 1]. Listing the price efFirrn 1 ﬁrst, determine the Nash Equilibrium prices. a. 9. it] 1}. “3,9 l _ (a. 2323'] BIC?! + *" d. 3H. 32 . » e. 32,32 (Lia after ads-wars} i Mi Flap-1c) twee??- FEE-deﬁieh 12. Pepsi and Cake play a game where they must decide whether te advertise er net. When the game is played simultaneeusly, it can he represented in the table helew with Ceite's paveth listed ﬁrst. _E'_ Advertise Net Advertise [ﬁlm-ﬁli- t nae- If the game is really.r played sequentially, which ef the fellewing is true? Cake has a ﬁrst mever advantage if it moves ﬁrst Pepsi has a ﬁrst mever advantage if it meves ﬁrst Celte has a seeend mever advantage ifit nteves sewed a. h. e. 6.3;] {a} and {h} Jﬁuarddﬁrii e‘t‘t‘tcﬁi'l} deratmdt‘l— "Fur _‘ Nene ef the aheve . ardar 1H,: Flap M15 Mr hm Fer questiens 13‘ 14., use the tellewing interrrtatien: pizzeria 1 and pizzeria 2 play a simtdtaneeus game in ﬁnish they decide what prices to eharge fer their pizzas. The game is represented in the table heiew with the paveft‘s ef pizzeria 1 listed ﬁrst. —_ __ menstrual Pizzeria l “SEEM 13. Listing the strategy ef pizzeria 1 ﬁrst, determine the pure strategyr Nash equilibrium. High, High 3. h. Meditun. Medium C (E) Lew. Lew d. Medium, High E3. Nene ef the aheve 14. Listing the strategy of pizzeria 1 first, determine the mixed strategy equilibrium in order of the probability of playing Low, the probability of playing Medium, and the probability of playing High. a. b. a: (.525 15. Compare the following two games {13, 1!}, LG}; (US, US, U3] [HI-4, 132, “4): [ii-1, “2. U4] (“3, 1E3, 1H}; UM, U2, 1M} [HEEL 1325).;(13'5, 2193,15} None oftbeaboye “Higgit” 5+Fia‘H7‘ domt'aarimi by “Plasmas”! gym Fio‘fﬁﬁ Fﬂhbiirl‘ixf Ere—Fe? on “ﬂap,” Which of the following is true in mixed strategy equilibrium? Castor’s probability of playing W is higher in Game 1 relative to in Game 2 Castor’s probability of playing W is lower in Game 1 relative to in Game 2 Pollux's probability of playing "'1' is higher in Game 1 relative to in Game 2 (a) and [C] (bl and {Cl ﬁiiﬁ‘mlﬁ {is m'i’ gn- _A£L'+{ﬂr rial {:ﬁLﬁ-‘i'ﬂ't'xé a“? WeJrqlnﬂ MK (Vale-I5 ouci' assumes ﬂab}&nm 2)! Con lkﬁw Leia-relating. Faibyls 5%? in. Fatwa +6 coo-Pirm Cr?) 15, “Ha. Mﬁwef.’ ’EUw army as we?) = P* lit-F) s P“: s; l]. Free-Response Questions Far sash afrhefaﬂawing. be sure m 531014: :21! war}: Parrim' credit may be awarded Fur quesﬁnns 1—4. rsfer m the following simultaneous game batman Bruce and Ralph. 1. Find any and all pure and mixed strategy squsihris in this slmulmnenus game. Discuss yuur methodnlugy as you go. [20' points] ’2. F1 lm4wa asmgmsm 75-.) F q" W Y 4 l 1: i- PC m EON :3)? “3? 3p + (FF): '2;- + Hot?) 2. {In a single graph‘ shew the bust rcspnnse curves far bum Emma and Ralph and indicate the Equilibria. [15 paints] 3. Suppoac the game were played put Sﬂqmnﬂﬂlly rather than simultanenusly. What wuuld be the Suhgamt: Perfcct Equilibrium if Ralph plays ﬁrst?I How about iantce plays ﬁrst? [Iii paints] 3:4 Eaipin 5m Qté‘i‘ 1 Raipitiﬁ‘ grum‘j M mag Ralf-LIE w A 9?? @ﬁﬁ‘mws (Waxy) 4. mush-ate — but DO NUT SDLVE — the original simtﬂtaneeus game in extensive ferm ramer than in its payoff matrix. [I I] paints] Bruce. A E e. D _ ":_' __E@: “b “I *1 a "'4'" ,1 1% 3-. ‘f E " 7’ 1 '3 ‘2 5 '3’ I I G ‘5 I 1.11? i 7' O ‘5 Li 6 3 HGTE'i PM [35me my be agaﬂmdv 7113 +ng ...
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## This note was uploaded on 10/15/2009 for the course ECON 122 taught by Professor Bonanno,g during the Summer '08 term at UC Davis.

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exam1-key - Eeo 122: Game Theory Summer Session II,...

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