14.122 Final ExamAnswer all questions. You have 3 hours in which to complete the exam.1. (60 Minutes – 40 Points) Answer each of the following subquestions briefly. Pleaseshow 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 observedactions to be continuous at infinity? Why do we care whether games are continuous atinfinity?(b) Is the game below solvable by iterated strict dominance? Does it have a uniqueNash equilibrium?(c) State Kakutani’s theorem. What correspondence is it applied to in the proof thatany finite game has a Nash equilibrium? Where does the argument break down if you tryto 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 generic finite normal form games player1’s equilibrium payoff is 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 perfectequilibrium concept is too restrictive and rules out a reasonable outcome. Give an exampleof a game in which you could argue that the subgame perfect equilibrium concept is notrestrictive enough and fails to rule out an unreasonable outcome. (Explain briefly whatyou 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 1 choosesa1∈, player 2 chooses a2∈, and the payoffs area2+22u1(a1,a2)=−(a1− 1)2a1−−2anda1+22u2(a1,a2)=−(a2− 3)2a2−−2.(i) Suppose that in class I presented the following slight variant on Spence’s job marketsignalling model. Nature first chooses the ability θ ∈{2, 3} of player 1 (with both choicesbeing equally likely). Player 1 observes θ and chooses e ∈{0, 1}. Player 2 then observese and chooses w . The players’ utility functions are u1(e, w; θ)=w − ce/θ2and∈u2(e, w; θ)=−(w − θ)2For c =4.25 this model has both a pooling equilibrium:(=2)=0∗θ1. ( = 0) = 2 5∗e.2µ2(θ =2e =0)=0.5||ss(=3)=0∗θ1(=1)=2∗e2µ2(θ =2e =1)=1ssand a separating equilibrium:(=2)=0∗θ1(=0)=2∗e2µ2(θ =2e =0)=1||ss(=3)=1∗θ1(=1)=3∗e2µ2(θ =2e =1)=0ssSuppose that after class two students come up to you in the hallway and ask you tosettle an argument they are having about whether the equilibria fail the Cho-Kreps IntuitiveCriterion. Assume that Irving arguesThe pooling equilibrium violates the intuitive criterion. The θ = 3 type 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 off switching to e = 1 even if you did choose w =3. I,on the other hand, being the θ = 3 type will be better from having switched toe = 1 if 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 1 could make a speech saying. ‘This equilibrium iscrazy. Education 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 = 0 regardless of my type.’What would you tell them?232. (40 Minutes – 20 Points)Consider the following multistage game. Player 1 first has to choose how to divide$2 between himself and player 2 (with only integer divisions being possible). Both playersobserve the division, and they then play the simultaneous move game with the dollar payoffsshown below.Assume that each player is risk neutral and has utility equal to the sum of the numberof dollars he or she receives in the divide the dollar game and the dollar payoff he receivesin the second stage game(a) Draw a tree diagram to represent the extensive form of this game. How many purestrategies 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 togive both dollars to player 2 in 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 player 1 givesboth dollars to player 2 in the initial divide-the-two-dollars game.(e) Can the game have a subgame perfect equilibrium in which player 1’s total payoffis less than 2?3. (40 Minutes – 20 Points)Harvard and MIT are both considering whether to admit a particular student to theireconomics Ph.D. programs. Assume that MIT has read the student’s application carefullyand knows the quality q of the student. Assume that Harvard faculty members are toobusy to read applications carefully. Instead they must base their decisions on their priorabout the student’s ability. Harvard’s prior is that q may be 1, 2 or 3 and that each ofthese values is equally likely.Assume that each school must make one of two decisions on the student: admit withfinancial aid or reject (the student has no source of support and could not attend graduateschool without financial aid). The schools make these decisions simultaneously.Assume that each school’s payoff in the game is 0 if they do not offer the studentadmission, -1 if the student is offered admission and turns them down (this is costly bothbecause the school loses prestige and because the slot could have been given to anotherstudent), and q − 1.5 if the student is offered admission and decides to come.Assume that if the student is admitted to both schools she chooses to come to MITwith probability 0.65 and to go to Harvard with probability 0.35.In the following questions treat this as a two player game between Harvard (player 1)and MIT (player 2).(a) What type spaces Θ1and Θ2would you use to represent this situation as a staticgame of incomplete information? How many elements are in each set? Write down the val-ues