MIT-SCIENCE-Lectures-lectures14_15

14.122Final ExamAnswer all questions.You have3hours in which to complete the exam.1.(60Minutes–40Points)Answer each of the following subquestions briefly.Please show your calculations and provide rough explanations where you can’t give formal state-ments so I can give you partial credit.(a)Give a formal definition of what it means for a multistage game with observed actions to be continuous at infinity?Why do we care whether games are continuous at infinity?(b)Is the game below solvable by iterated strict dominance?Does it have a unique Nash equilibrium?(c)State Kakutani’s theorem.What correspondence is it applied to in the proof that anyfinite game has a Nash equilibrium?Where does the argument break down if you try to use Kakutani’s theorem in the same way to prove the existence of an equilibrium in the ”Name the Largest Number”game?(d)Is the following statement true or false:In genericfinite normal form games player 1’s equilibrium payoffis positive.(e)Find all of the subgames of the following extensive form game.(f)Given an example of a game in which you could argue that the subgame perfect equilibrium concept is too restrictive and rules out a reasonable outcome.Give an example of a game in which you could argue that the subgame perfect equilibrium concept is not restrictive enough and fails to rule out an unreasonable outcome.(Explain briefly what you would argue about each example.)(g)Find all of the Nash equilibria of the following game.(h)Find the Nash equilibrium of the simultaneous move game where player 1chooses a 1∈ ,player 2chooses a 2∈ ,and the payoffs area 2+2 2u 1(a 1,a 2)=−(a 1−1)2 a 1−−2and a 1+2 2u 2(a 1,a 2)=−(a 2−3)2 a 2−−2.(i)Suppose that in class I presented the following slight variant on Spence’s job market signalling model.Nature first chooses the ability θ∈{2,3}of player 1(with both choices being equally likely).Player 1observes θand chooses e ∈{0,1}.Player 2then observes e and chooses w .The players’utility functions are u 1(e,w ;θ)=w −ce/θ2and ∈ u 2(e,w ;θ)=−(w −θ)2For c =4.25this model has both a pooling equilibrium:(=2)=0∗θ1. (=0)=25∗e .2µ2(θ=2e =0)=0.5||s s (=3)=0∗θ1(=1)=2∗e 2µ2(θ=2e =1)=1s s and a separating equilibrium:(=2)=0∗θ1(=0)=2∗e 2µ2(θ=2e =0)=1||s s (=3)=1∗θ1(=1)=3∗e 2µ2(θ=2e =1)=0s s Suppose that after class two students come up to you in the hallway and ask you to settle an argument they are having about whether the equilibria fail the Cho-Kreps Intuitive Criterion.Assume that Irving arguesThe pooling equilibrium violates the intuitive criterion.The θ=3type couldmake a speech saying ‘I am choosing e =1.I know you are supposed to believethat anyone who gets an education is the low type,but this is crazy.The θ=2type would be worse offswitching to e =1even if you did choose w =3.I,on the other hand,being the θ=3type will be better from having switched toe =1if you choose w =3.Hence you should believe that I am the high typeand give me a high wage.’Freddy arguesThe separating equilibrium violates the intuitive criterion.It is inefficient.Be-fore he learns his type player 1could make a speech saying.‘This equilibrium iscation is of no value,yet with some probability I am going to have toincur substantial education costs.This is entirely due to your arbitrary beliefthat if I get no education I am the low type.If you instead believed that I wasthe low type with probability one-half in this case,then the return to educationwould be sufficiently low so as to allow the inefficiency to be avoided.Moreover,this more reasonable belief would turn out to be correct as I would then choosee =0regardless of my type.’What would you tell them?22.(40Minutes–20Points)Consider the following multistage game.Player1first has to choose how to divide $2between himself and player2(with only integer divisions being possible).Both players observe the division,and they then play the simultaneous move game with the dollar payoffs shown below.3Assume that each player is risk neutral and has utility equal to the sum of the number of dollars he or she receives in the divide the dollar game and the dollar payoffhe receives in the second stage game(a)Draw a tree diagram to represent the extensive form of this game.How many pure strategies does each player have in the normal form representation of this game?(b)Show that for any x the game has a Nash equilibrium in which player chooses to give both dollars to player2in the initial divide-the-two-dollars game.(c)For what values of x will the game have an unique subgame perfect equilibrium?(d)For what values of x is there a subgame perfect equilibrium in which player1gives both dollars to player2in the initial divide-the-two-dollars game.(e)Can the game have a subgame perfect equilibrium in which player1’s total payoffis less than2?3.(40Minutes–20Points)Harvard and MIT are both considering whether to admit a particular student to their economics Ph.D.programs.Assume that MIT has read the student’s application carefully and knows the quality q of the student.Assume that Harvard faculty members are too busy to read applications carefully.Instead they must base their decisions on their prior about the student’s ability.Harvard’s prior is that q may be1,2or3and that each of these values is equally likely.Assume that each school must make one of two decisions on the student:admit with financial aid or reject(the student has no source of support and could not attend graduate school withoutfinancial aid).The schools make these decisions simultaneously.Assume that each school’s payoffin the game is0if they do not offer the student admission,-1if the student is offered admission and turns them down(this is costly both because the school loses prestige and because the slot could have been given to another student),and q−1.5if the student is offered admission and decides to come.Assume that if the student is admitted to both schools she chooses to come to MIT with probability0.65and to go to Harvard with probability0.35.In the following questions treat this as a two player game between Harvard(player1) and MIT(player2).(a)What type spacesΘ1andΘ2would you use to represent this situation as a static game of incomplete information?How many elements are in each set?Write down the val-ues of the utility functions u i(a1,a2;θ1,θ2)for a couple values of i,a1,a2,θ1,θ2to illustrate how to compute them.How many pure Bayesian strategies does each player have?(b)What actions are strictly(conditionally)dominated for each possible type of each player?(c)Find the Bayesian Nash equilibrium of this game.(d)Would Harvard be any better offif it could observe MIT’s admission decision before making its decision?44.(40Minutes–20Points)Consider the following model in which a worker’s choice of health play may signalhis health to his insurance company.Suppose naturefirst chooses the health of the workerchoosingθ∈{healthy,not healthy}.Assume that the probability that the worker is healthyis q.Player1observes whether he is healthy and then chooses a1∈{HMO,full insurance}. Player2,the competitive insurance market,observes a1and then chooses a price p for thechosen plan.Assume that the HMO plan will always pay half of the worker’s health care expenses,while the full insurance plan will pay all of the worker’s health care expenses.Assume thatthe worker’s expected health care expenses are$1000if the worker is healthy and$2000ifthe worker is not healthy.Assume that the worker is risk neutral and that his expectedutility from each health plan is equal to2000minus the sum of p and his unreimbursedhealth care expenditures,i.e.u1(HMO,p;healthy)=1500−pu1(full insurance,p;healthy)=2000−pu1(HMO,p;not healthy)=1000−pu1(full insurance,p;not healthy)=2000−pTo model a competitive insurance market assume that player2’s has a quadratic utilityfunction that makes it want to set p equal the expected payments that will be made undera plan,i.e.u2(HMO,p;healthy)=−(p−500)2u2(full insurance,p;healthy)=−(p−1000)2u2(HMO,p;not healthy)=−(p−1000)2u2(full insurance,p;not healthy)=−(p−2000)2(a)Does the model have a separating PBE where only the unhealthy workers buy fullinsurance?(b)For what values of q does this model have a pooling PBE where all types of player1buy full insurance?(c)Suppose that rather than there being just two types of player1there are a continuumof possible types.In particular,assume that player1observes his expected health careexpenditures for the year before making his health plan choice and that player2’s prior isthat these are uniformly distributed on[0,2000].What kinds of equilibria seem like theymight be possible in this model?Show that there is no PBE in which player1buys fullinsurance with positive probability.(d)What does this model suggest about the dangers of a free market in health insurance?What modifications to the model would be necessary if you wanted to think about theinefficiency of health insurance more seriously?5。

An important and easy-to-neglect issue in building the frame is the weight of the water it will need to support.One cubic foot of water contains about 71/2 gallons and weighs approximately 63 pounds [2]. For a pool of any reasonable size, this quickly adds up; for large volumes of water, careful structural analysis will be important to ensure that the frame will be able to hold the weight, especially if the pool is being recessed into a platform. One simple way to minimize the amount of water used and thus the amount of weight that must be supported is to make the pool less deep; this is often a very reasonable option since the actual depth of the pool will always not be obvious to the audience.The frame should then be covered with plywood (given the loads it will be sup-porting, something substantial like 3/4 ′′ plywood should be used. It then needs to be covered with a pond liner so that it can be filled with water. These are readily available from garden stores as waterproof plastic sheets, which can be cut to size and fit into the pool.The pool can then be filled with water. Once filled, it should be emptied or changed regularly — the last thing anyone needs on the stage is a place for bacteria and other unpleasant things to grow, and a stagnant pool of water fits that bill all too well. A pool treatment can also be useful here, though ones containing chlorine are not ideal due to their distinctive smell.3 Building the waterfallMany of the same techniques from building a pool of water apply to building a waterfall. Again, we will need to use a waterproof pond liner around a frame and support structure. The difference, of course, is that the waterfall will be a stream (albeit one that sometimes flows horizontally and sometimes flows vertically), so it will not have as much depth.This allows the support structure to be more lightweight.A common means for building streams is to construct a set of curved ribs, then lay a thin layer of plywood or lauan over it that conforms to the curve of the ribs. This should then be covered with the pond liner.The water should flow through the stream and down the waterfall into the pool, which serves as a reservoir. It then needs to be returned to the top of the waterfall; for this, a pump is required. A household sump pump, available from the Home Depot or similar hardware store in the $100 range, is probably the easiest option.ℵ0-2For especially large installations, a common pump may not suffice. One option is to obtain a larger pump; McMaster-Carr carries sump pumps up to 3/4 hp and effluent pumps up to 2 hp, though the latter may be a bit expensive for this purpose at over $700[3]. A simple alternative is to use multiple smaller pumps. Submersible pumps are best for this application: other types of pumps may allow the pump to be placed further away from the pool, reducing noise, but adding a drain to the pool requires creating and sealing a hole in the liner, which only adds complexity. Sump pumps may pump into a standard garden hose, which is certainly easy to find and easy to work with. However, the pump will be able to operate more efficiently if PVC pipe (usually or 11/2 ′′ , though it depends on the pump) is used.11/4 ′′The pressure created by a pump, especially a large one, may be rather high. This could lead to the unusual effect of the water spewing forcefully upward or in a powerful stream — which, while useful for a fountain, isn’t very realistic in a stream effect.The solution here is to increase the diameter of the pipe by using couplers with successively larger diameters. This causes the same amount of water to flow through a larger area, and thus with less force.Waterfalls and streams are generally rather noisy. The sound of water flowing down the stream and into the pool can be a rather pleasant one, and can be a helpful part of the sound design for creating the outdoor atmosphere. If, however, that doesn’t fit with the sound designer’s vision, it becomes the technical designer’s problem to fix.Reducing the velocity of the flowing water is an obvious solution with an obvious drawback.Adding rocks and similar objects to the stream will help reduce the volume of water that needs to flow, reducing the noise created (and also allowing a smaller, quieter pump to be used). It is also quieter to not have water fall directly from the waterfall into the pool; instead, it should flow onto a surface that slopes into the pool.References[1] M. Powers, “Pools, waterfalls, etc.” available at /howto/water/pools.htm.[2] P. Carter, Backstage Handbook, 3rd ed. Louisville, Kentucky: Broadway Press,1994.ℵ0-3[3] McMaster-Carr, Inc. /.ℵ0-4。