(b) Is the game below solvable by iterated strict dominance? Does it have a unique Nash equilibrium?

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1 14.1 Final Exam Anwer all quetion. You have 3 hour in which to complete the exam. 1. (60 Minute 40 Point) Anwer each of the following ubquetion briefly. Pleae how your calculation and provide rough explanation where you can t give formal tatement o I can give you partial credit. (a) Give a formal definition of what it mean for a multitage game with oberved action to be continuou at infinity? Why do we care whether game are continuou at infinity? (b) I the game below olvable by iterated trict dominance? Doe it have a unique Nah equilibrium? (c) State Kakutani theorem. What correpondence i it applied to in the proof that any finite game ha a Nah equilibrium? Where doe the argument break down if you try to ue Kakutani theorem in the ame way to prove the exitence of an equilibrium in the Name the Larget Number game? (d) I the following tatement true or fale: In generic finite normal form game player 1 equilibrium payoff i poitive. (e) Find all of the ubgame of the following extenive form game. (f) Given an example of a game in which you could argue that the ubgame perfect equilibrium concept i too retrictive and rule out a reaonable outcome. Give an example of a game in which you could argue that the ubgame perfect equilibrium concept i not retrictive enough and fail to rule out an unreaonable outcome. (Explain briefly what you would argue about each example.) (g) Find all of the Nah equilibria of the following game.

2 (h) Find the Nah equilibrium of the imultaneou move game where player 1 chooe a 1 R, player chooe a R, and the payoff are u 1 (a 1,a )= (a 1 1) (a 1 a + ) and u (a 1,a )= (a 3) (a a 1 + ). (i) Suppoe that in cla I preented the following light variant on Spence job market ignalling model. Nature firt chooe the ability θ {, 3} of player 1 (with both choice being equally likely). Player 1 oberve θ and chooe e {0, 1}. Player then oberve e and chooe w R. The player utility function are u 1 (e, w; θ) = w ce/θ and u (e, w; θ) = (w θ). For c =4.5 thi model ha both a pooling equilibrium: 1( θ = ) = 0 ( e = 0) =. 5 µ (θ =e =0)=0.5 1 ( θ = 3) = 0 ( e = 1) = µ (θ =e =1)=1 and a eparating equilibrium: 1 ( θ = ) = 0 ( e = 0) = µ (θ =e =0)=1 1 ( θ = 3) = 1 ( e = 1) = 3 µ (θ =e =1)=0 Suppoe that after cla two tudent come up to you in the hallway and ak you to ettle an argument they are having about whether the equilibria fail the Cho-Krep Intuitive Criterion. Aume that Irving argue The pooling equilibrium violate the intuitive criterion. The θ = 3 type could make a peech aying I am chooing e = 1. I know you are uppoed to believe that anyone who get an education i the low type, but thi i crazy. The θ = type would be wore off witching to e = 1 even if you did chooe w =3. I, on the other hand, being the θ = 3 type will be better from having witched to e = 1 if you chooe w = 3. Hence you hould believe that I am the high type and give me a high wage. Freddy argue The eparating equilibrium violate the intuitive criterion. It i inefficient. Before he learn hi type player 1 could make a peech aying. Thi equilibrium i crazy. Education i of no value, yet with ome probability I am going to have to incur ubtantial education cot. Thi i entirely due to your arbitrary belief that if I get no education I am the low type. If you intead believed that I wa the low type with probability one-half in thi cae, then the return to education would be ufficiently low o a to allow the inefficiency to be avoided. Moreover, thi more reaonable belief would turn out to be correct a I would then chooe e = 0 regardle of my type. What would you tell them?

3 . (40 Minute 0 Point) Conider the following multitage game. Player 1 firt ha to chooe how to divide $ between himelf and player (with only integer diviion being poible). Both player oberve the diviion, and they then play the imultaneou move game with the dollar payoff hown below. 3 Aume that each player i rik neutral and ha utility equal to the um of the number of dollar he or he receive in the divide the dollar game and the dollar payoff he receive in the econd tage game (a) Draw a tree diagram to repreent the extenive form of thi game. How many pure trategie doe each player have in the normal form repreentation of thi game? (b) Show that for any x the game ha a Nah equilibrium in which player chooe to give both dollar to player in the initial divide-the-two-dollar game. (c) For what value of x will the game have an unique ubgame perfect equilibrium? (d) For what value of x i there a ubgame perfect equilibrium in which player 1 give both dollar to player in the initial divide-the-two-dollar game. (e) Can the game have a ubgame perfect equilibrium in which player 1 total payoff i le than?

4 3. (40 Minute 0 Point) Harvard and MIT are both conidering whether to admit a particular tudent to their economic Ph.D. program. Aume that MIT ha read the tudent application carefully and know the quality q of the tudent. Aume that Harvard faculty member are too buy to read application carefully. Intead they mut bae their deciion on their prior about the tudent ability. Harvard prior i that q may be 1, or 3 and that each of thee value i equally likely. Aume that each chool mut make one of two deciion on the tudent: admit with financial aid or reject (the tudent ha no ource of upport and could not attend graduate chool without financial aid). The chool make thee deciion imultaneouly. Aume that each chool payoff in the game i 0 if they do not offer the tudent admiion, -1 if the tudent i offered admiion and turn them down (thi i cotly both becaue the chool loe pretige and becaue the lot could have been given to another tudent), and q 1.5 if the tudent i offered admiion and decide to come. Aume that if the tudent i admitted to both chool he chooe to come to MIT with probability 0.65 and to go to Harvard with probability In the following quetion treat thi a a two player game between Harvard (player 1) and MIT (player ). (a) What type pace Θ 1 and Θ would you ue to repreent thi ituation a a tatic game of incomplete information? How many element are in each et? Write down the value of the utility function u i (a 1,a ; θ 1,θ ) for a couple value of i, a 1,a,θ 1,θ to illutrate how to compute them. How many pure Bayeian trategie doe each player have? (b) What action are trictly (conditionally) dominated for each poible type of each player? (c) Find the Bayeian Nah equilibrium of thi game. (d) Would Harvard be any better off if it could oberve MIT admiion deciion before making it deciion? 4

5 4. (40 Minute 0 Point) Conider the following model in which a worker choice of health play may ignal hi health to hi inurance company. Suppoe nature firt chooe the health of the worker chooing θ {healthy, not healthy}. Aume that the probability that the worker i healthy i q. Player 1 oberve whether he i healthy and then chooe a 1 {HMO, full inurance}. Player, the competitive inurance market, oberve a 1 and then chooe a price p for the choen plan. Aume that the HMO plan will alway pay half of the worker health care expene, while the full inurance plan will pay all of the worker health care expene. Aume that the worker expected health care expene are $1000 if the worker i healthy and $000 if the worker i not healthy. Aume that the worker i rik neutral and that hi expected utility from each health plan i equal to 000 minu the um of p and hi unreimbured health care expenditure, i.e. u 1 (HMO,p; healthy) = 1500 p u 1 (full inurance,p; healthy) = 000 p u 1 (HMO,p; not healthy) = 1000 p u 1 (full inurance,p; not healthy) = 000 p To model a competitive inurance market aume that player ha a quadratic utility function that make it want to et p equal the expected payment that will be made under a plan, i.e. u (HMO,p; healthy) = (p 500) u (full inurance,p; healthy) = (p 1000) u (HMO,p; not healthy) = (p 1000) u (full inurance,p; not healthy) = (p 000) (a) Doe the model have a eparating PBE where only the unhealthy worker buy full inurance? (b) For what value of q doe thi model have a pooling PBE where all type of player 1 buy full inurance? (c) Suppoe that rather than there being jut two type of player 1 there are a continuum of poible type. In particular, aume that player 1 oberve hi expected health care expenditure for the year before making hi health plan choice and that player prior i that thee are uniformly ditributed on [0, 000]. What kind of equilibria eem like they might be poible in thi model? Show that there i no PBE in which player 1 buy full inurance with poitive probability. (d) What doe thi model ugget about the danger of a free market in health inurance? What modification to the model would be neceary if you wanted to think about the inefficiency of health inurance more eriouly? 5