Econ 221 (002 & 004) Problem Set 4

Problem Set 4This assignment is due Friday November 27 by 1:30pm (my o¢ce – BUTO 1011).Remember to write your NAME, STUDENT NUMBER, DISCUSSION GROUPand the name of your T.A. at the top of the front page of the problem set andto staple your problem set together before handing it in. Failure to do so mayresult in loss of marks on the assignment.All work must be shown. Answer ALL questions. At least one question will be graded.1. A small town has 100 cabs. 70 cabs are blue (Blue Cab Company) and 30 cabs aregreen (Green Cab Company). There is a hit-and-run accident at night (pedestrianinjury, not fatality) and an eyewitness present at the scene of the accident claims itwas a green cab that struck the pedestrian. No criminal charges are laid, but theinjured party does bring a civil lawsuit against the Green Cab Company to recovercosts associated with hospitalization and lost wages. The witness is called to testify(‘I saw a green cab strike the pedestrian’). The witness is also subjected to a visiontest and it is found that under conditions similar to those on the night of the accident,the witness can correctly identify a cab from other types of vehicles and the witnesscan correctly distinguish the colour of the cab (blue or green) X% of the time. Thejudge in the case will find in favour of the plainti§ if the beliefs of the judge assign aprobability of greater than 50% to a green cab being involved in the hit-and-run.(a) If X% = 100% (the witness always identifies the correct colour of the cab), howshould the judge rule in this case?(b) If X% = 75% (the witness correctly identifies the colour of the cab 75% of thetime), how should the judge rule in this case?(c) Suppose that evidence is presented that show that on the night in question therewere only 60 of the 70 blue cabs in service and 25 of the 30 green cabs in service.Would this lead to a change in the ruling of the judge (X% = 75%).2. You are a contestant in a game show. Three doors are presented (unopened). Behindeach door is a prize. Only one of the prizes is of any value, V > 0. Both of theother prizes are essentially useless and of no value. The arrangement of prizes behinddoors is essentially random. You must select one of the 3 doors. Before your door isopened the host will show you what is behind one of the two doors that have not beenselected. Two rules that are followed by the host: never open the door selected by thecontestant and always open a door with a useless (V = 0) prize, never open the doorwith the good (V > 0) prize. The host will now o§er you the option of sticking to youroriginal choice of door or switching to a di§erent door. You would never switch yourchoice to the door that was opened because you know it contains a useless (V = 0)prize. Let Di (i = 1, 2, 3) be the location of the good (V > 0) prize. Let Ri (i = 1, 2, 3)the door that is opened/revealed. You start by choosing door 2. After your selectionof door 2, the host opens door 1 to reveal a useless (V = 0) prize.1(a) What is the unconditional probability of the good (V > 0) prize being behindthe chosen door: P (D2)? A door that was not chosen? P (D1)? P (D3)? (Beforelearning that door 1 will be opened to reveal a useless (V = 0) prize).(b) What is the conditional/updated probability of the good (V > 0) prize beingbehind door 1? behind door 2? behind door 3?(c) What is the optimal strategy of the contestant with respect to sticking to theoriginal choice of door or switching to a di§erent door? How does the probabilityof winning the good (V > 0) prize change for the contestant?(d) Suppose that the same game is played with 4 doors, 1 good (V > 0) prize and 3useless (V = 0) prizes. Work through (a), (b), and (c) for this case.3. The U.N. (player U) is concerned with the weapons program of a Rogue dictator (playerR). The U.N. has sent weapons inspectors to the country, if R allows (a) the inspectorsto carry out their program then the game will come to an end. If R chooses to kick out(k) the inspectors, then U must choose how to respond (Escalate (e) or take a Passiverole (p)) and R will then choose whether to remain defiant (d) or Back down (b). Thegame tree can be shown as follows (payo§ to R, payo§ to U):(7,9)RakbddRebp(2,3)UR(5,6)(8,4)(4,8)(a) Find the subgame perfect Nash equilibrium to this game.(b) Suppose that the Rogue dictator may be one of two types. The Sane type, RShas payo§s as described in the game tree above. There is also a Crazy type, RC.The Crazy type likes confrontation and likes to be defiant. For the Crazy type,the game is as above except for two changes to the payo§s of RC:1. if RC allows the inspectors then RC will receive a payo§ of 3 (not 7)2. when R plays k, U plays e and R plays d the payo§ to RC is 9 instead of 2.The Rogue dictator knows his type. The U.N. is unable to observe the type but doesknow that the probability of the dictator being Sane is ? = 12 . Use Nature as anadditional player to draw the game tree.(c) Consider an equilibrium where RC plays a single action, but both RS and U followmixed strategies. RS plays k with probability ? (and a with probability 1 ? ?)2and U plays e with probability ? (p with probability 1??). Find the equilibrium(pure) strategy of type RC and find the equilibrium (mixed) strategy of U (? and1 ? ?). (recall ? = 12 )(d) To complete the description of the equilibrium from (c), look for the mixed strategythat is followed by RS. It will be necessary to derive expressions for the beliefsof U if they observe k, i.e. P (RS|k) and P (RC|k). Find the equilibrium mixedstrategy of RS (? and 1 ? ?) and the beliefs of U regarding the type of playerthey face when they observe k being played.4. Consider the following game of incomplete information. Player S is unable to observethe type of player T (type T1 or T2). The probability distribution over types is commonknowledge. Note that payo§s in this game are ordered player S followed by player T :(?S, ?T ).prob = 3/8pN(4, 3)T2GET2LqM(4, 6)S (8, 4)prob = 5/8(6,7)T1GEonLMmST1(5, 3)(10, 4)onT1(12, 2)(5, 8)qpT2(3, 10)(2, 6)(?S, ?T)(a) What is a separating equilibrium in a Bayesian game? What is a Pooling equilibriumin a Bayesian game. What are the beliefs of a player in a Bayesian game?What is the relation between beliefs of a player and the type of equilibrium (Poolingvs. Separating)?(b) Consider a Bayesian perfect Nash equilibrium involving mixed strategies. PlayerS plays L with probability ? and M with probability 1 ? ?. Type T1 plays Ewith probability ? and G with probability 1 ? ?. There is no other mixing in thegame. Find the mixed strategy of player S and the strategy of type T2.(c) Complete the description of the Bayesian perfect Nash equilibrium from (b) byfinding the equilibrium strategy of type T1 and the beliefs of player S.(d) Describe a pooling equilibrium to this game if one exists. If there is no poolingequilibrium clearly demonstrate that no pooling equilibrium exists.35. There are two types of cars sold in the used car market. Sellers of high-quality cars havea reservation price (the minimum price they will accept for a car) of RH = $1800, whilelow-quality (Lemons) sellers have a reservation price of RL = $1000. The proportionof High-quality cars available in the market is ?H = 23 . Risk-neutral buyers are willingto-payWTP H = $2400 for a high-quality car and WTP L = $1200 for a low-qualitycar. For a car of indeterminate quality (the buyer is unable to verify the quality ofthe car) the buyer is willing-to-pay E{WTP}. Assume that due to limited supply,the prices in the high-quality and low-quality (if separated) or used-car (if pooled)markets are set equal to the relevant WTP. (eg. if there is a market for high-quality,P H = WTP H = $2400).(a) Determine the amount the buyer will be willing to pay (E{WTP}) for a car whenthey are unable to verify its quality. Explain whether both types of car, or only‘lemons’ will be sold in the used-car market.(b) Starting from the situation in (a), what is the maximum cost, cH, that the highqualityseller would be willing to incur in order to reliably signal ‘high-quality’ tothe buyer?(c) In order for the signal in (b) to be reliable, it must not be worthwhile for thelow-quality seller to send the same signal. What is the minimum cost, cL, of thesignal for low-quality that will insure this does not happen?(d) The cost to a seller of providing warranty coverage on a used vehicle is a functionof the type of vehicle and the number of kilometers of coverage o§ered in thewarranty with:cH(d) = .01dcL(d) = .03dExplain whether there is a level of warranty coverage (a number of kilometers d?)such that there will be a separating equilibrium.6. Consider the following game between players S and R. Player S may be either type s4or type t with P(Ss) = 34 .aaabax yx ySsStNSsStbR RbR Rb(3,2)(5,0)(2,3)(0,1)(1,1)(6,4)(4,2)(2,3)(a) Under a separating equlibrium, player R will have beliefs that assign probability1 to facing a particular type of player given observation of x or y. Describe thebeliefs of player R under a separating equilibriumPR(Ss|x)PR(Ss|y)PR(St|x)PR(St|y)(b) Find a separating Bayesian-perfect Nash equilibrium to the game if one exists.(c) Under a pooling equilibrium, player R will not be able to infer the type of playerS from the action, what are the beliefs of player R in such an equilibrium?(d) Find a pooling perfect Bayesian-perfect Nash equilibrium to the game if one exists.

 

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