Econ 223 Examination:  June 5 2009
3 hours max: 6:30 pm-9:30 pm

This is a closed book exam: no notes, no books, no cell-phones, no calculators  just  you. Your script will be taken from you and you will receive a zero grade for this paper  should you be found with any of these items – so please deposit them at the front of the room. On the desk space in front of you you should have

Before you leave the exam room tonight. Make sure your name and student ID are clearly written on the A4 answer sheet and place it inside your answer book. Please leave your answer book at your assigned desk space when you leave (to preserve alphabetic arranging of scripts). No one will be admitted after 6:30 and you may leave any time after 6:30

There are 12 multi choice questions 5 marks each, 60 in total.
There are 3 "written answer" type questions in Part B for a total of 40 marks.

A1 ( 5 marks) The game tree below represents a stylized 3 player (A, B, C) sequential game (no payoffs are shown at the terminal nodes).

count strategies

How many strategies does player C have in this game?( ) 

A2 ( 5 marks) random breath checks The police have to decide whether to check for drinking drivers using roadblocks. Driver's have to decide whether to drink and drive or stay sober. Assume the game being played out by drivers and the police is a one shot simultaneous game with payoffs on a scale of 0 (bad) to 10 (very good) as show in the following 2x2 payoff matrix. You might not agree with these preferences - especially the preferences for the drivers - but assume for the sake of argument, as this payoff table does, that drivers do want to drink and drive...and not get checked by the police, and that if they do stay sober they would (mildly) prefer not to have the police stop and check them. The police on the other hand really do want to catch drinking drivers and they dislike not catching drinking drivers, while at the same time they would (mildly) prefer not to waste resources checking sober drivers. There is no Nash Equilibria in pure strategies in this game. However there is a mixed strategy Nash equilibrium. In that mixed strategy Nash equilibrium the Police will be checking on drinking drivers with what probability? On your multi choice answer sheet for A3 write down one of the integers 0,10,20,...through to ....90,100 that is closest to your answer.

A3( 5 marks) Supposing the payoffs for the Police in the previous question (A2) change in two small ways : (1) they slightly prefer ( 3 vs 2 now compared to 2 vs 2 before ) checking sober drivers to not checking sober drivers, since it gives them an opportunity to check for other things such as expired WOFS and registrations, underage drivers, criminal activity and (2) , their payoff from checking and catching drivers who drink drops from 10 down to 5. Payoffs to drivers remain unchanged. The effect of this change in the game on the Nash equilibrium is to:

YOU DIDN'T HAVE TO DRAW THESE DIAGRAMS...but they may be helpful for you in understanding the logic here. Just to check out the best response reasoning try calculating the expected payoffs for the polics for their two actions:

Now that we know the Police will always (against any pure or p-mixed strategy of drivers) check, the best response of (these) drivers is to stay sober.

A4 ( 5 marks) Examine following 2 player 4x4 payoff matrix. The first three steps of the method of iterated elimination of dominated alternatives can eliminate which rows and columns? On your multi choice answer sheet for A4 write down the identifying letters of the rows or columns in the order that they are eliminated with the first three steps in this iterative process.

 

Here's the final result: There are no columns that dominate any other (compare L with M, L with R then M with R....). But row B is dominated by row U. Then column L can be eliminated . After that rows T and U can be eliminated (in either order as they are both dominated by D...at this stage).

A5 ( 5 marks) Banks do not like to make loans to businesses (or students for that matter) who are likely to default on their loan payments, even more so in the current international climate. Of course even after background credit checks it is still very difficult to tell whether any particular customer will default on a loan. Imagine there is a wide range of types of customers of various default risks. When making a loan a bank charges an interest rate of say 14%, a rate that is expected to cover these losses from non-repayment plus make a normal profit. But it finds that at this interest many good risks decide this interest rate is too high and don't borrow. So the pool of customers borrowing money from the bank now includes a higher proportion of potential defaulters than the bank initially anticipated. The bank adjusts its interest rates up to reflect these extra expected costs of money lost on defaulted loans. But then more low risk types don't borrow at these new higher interest rates...and an upward spiral on interest rates continue, with a large number of bank loans defaulted on . This strategic situation is an example of (choose one item from the following list and write your answer on your multi choice answer sheet)

A6 ( 5 marks) Tourist companies like to employ pleasant mannered students over the summer , especially those able to be pleasant under pressure when dealing with confused or unsettled tourist customers. It is difficult, if not impossible, for the employing agency to directly observe skills at being pleasant mannered under pressure. So they ask potential employees to participate in a series of time consuming workshops where they will be assessed on their demonstrated abilities to be pleasant under a variety of stressful conditions. The question is, how many of these workshops should they ask potential applicants to take? The plan is that those who know they don't have the required skills will choose NOT to take the workshops, while those who know they do have the required skills will take the workshops. The employers make a salary offer of $5000 for summer work as a front line representative to those who successfully complete n workshops, and an offer of $3300 for a summer job - but not in the frontline dealing directly with tourists - for those who don't complete the n workshops.

Lets suppose that there are only two types of applicants, those who have the required skills, the "haves" (H), and those who don't, the "have nots" (HN). Each applicant knows what type they are. Doing a course is hard work, but a little easier for the Haves than the Have Nots. The time and effort costs for a Have to successfully complete a course is $400 per course (assessed by the applicant themselves as a "money equivalent" for the time and effort per workshop). The time and effort costs for a Have Not to successfully complete a course is $600 per course. What is the smallest and what is the largest number n of courses that will satisfy the incentive compatability constraints for this problem of asymmetric information? Select your answer from the following list and write your selection(s) on your multi choice answer sheet. Note this question asks you to write down two responses unless you choose (a).

A7 ( 5 marks) Game theory argues that cooperative behaviour and mutually beneficial outcomes can be sustained in a repeated 2 player prisoner's dilemma game when players use tit-for-tat  (TFT) or grim (G) strategies in an indefinitely repeated game. These two trigger strategies have some (but not all) features in common when played in a Nash equilibrium that sustains cooperation. Select which features these two strategies have in common form the following list: (Choose your answer(s) from below and enter it on your multi-choice answer sheet - there may be more than one correct answer, marks are awarded only for a complete set of correct answers) 

A8   ( 5 marks) Shareholders (S) and Managers (M) in a small family publishing business are playing an alternating offer bargaining game about managerial compensation payments in the form of profit sharing options. However, the longer the bargaining goes on, the smaller the overall profits there are to share out. Profit opportunities are 100 this week, 70 next week, 30 in week 3 , 20 in week 4, and will be zero after that. Due to circumstances beyond anyone’s control S and M can only meet weekly to make offers and ratify agreements . The bargaining rules are that Shareholders S make an offer of a division of profits in week 1. Managers M either accept or reject that offer. If they accept , the profits avaliable at that time are split as agreed. If the offer is rejected they both have to wait until week 2 at which time M can make a counter offer to S. At that time (week 2) S can either accept and split the smaller amount of money as agreed, or reject , wait until week 3 then come back with a counter-counter-offer to M, and so on , alternating through to week 4 with M taking the initiative in week 4 . This last offer is either accepted by S, and the money available at that time shared out as agreed, or it is rejected , and they each take away nothing. Assume offers and counter offers are made in units of at least 1 (ie no fractions) and that either party has to be offered a positive inducement to not carry through to the next round.

Use your knowledge of game theory to answer the following question: If bargaining reaches week 2 in this game, what division of the total profits of 70 in that week would be expected (according to game theory) to be going to the shareholders in that week? On your multi choice answer sheet in the space provided choose the correct answer from the list 0,1,2...up to .69,70. Remember, the number your write in your answer sheet should be the shareholders' payoff, not the managers payoff. This is a quickie verbal explanation - I'll get a video clip one soon. Start at the ultimatum game at the end in week 4 where $20 is available . M cab take 19 and offer 1 to S , and S will accept (rejecting gets them 0, and we assume they only care about their own payoffs ). Now back up to week 3. S has the initiative but only $30 in total is avaliable. They have to offer M $20 to prevent M from going to the next round and getting $19 , leaving $10 for themselves. Now keep backing up to week 2, $70 is avaliable it is M's initiative, and they expect that if S rejects then S can get $10 in next round. So offer them $11, keeping $59 for themsleves. The answer I asked for was the $11 at this stage. Note...we don't expect to get to this subgame , as we expect S at round 1 with $100 available to offer $50 to M and keep $40 for themsleves- and we expect M to accept this (ie we never actually get to any of the subgames - but everyone can figure out what would happen IF we did get there.).

A9   ( 5 marks) The 2x2 game tree below represents a game between a player now, and their future self. The idea (following Thomas Schelling) is that the player moving now ("you now") could train hard for a future event, eg the next Christchurch marathon, but unless the future self also is willing to train the player will not be able physically to run the marathon ; the self now will view that outcome as a failure rather than as success (successes are best, failures worst, and don't care is in between from the viewpoint of the player now, in the present ). The future self prefers to "relax and enjoy" as compared to the " hard work" of their own training and doesn't really care - positively or negatively - what the earlier self has done. The game being played here can be best described by saying that is it most like a : (Choose an answer from the list below and enter it on your multi-choice answer sheet)

(a) a prisoners’ dilemma game.
(b) a threat game.
(c) a battle of the sexes game.
(d) an assurance game.
(e) a trust game -why? because the future self can act opportunistically to take advantage of (harm) the current self while furthering their own interests, so the current self won't train, and won't trust any promises by the future self that it would train. Yes there is a slight difference to the Kreps style trust game that had as a default a status quo that was strictly worse for both than if the two players train (offer trust) and train (act fairly).
(f) a voluntary contributions (VCM) game


A10   ( 5 marks) A simple ultimatum game is being played out in the game tree below. The pairs of numbers at the terminal nodes (x, y) indicate the way the pie is split between the two players R and B . IF the players are only self interested then these numbers also indicate the players' payoffs.( In case of ties in outcomes for self interested players assume they prefer lower outcomes for the other player.) This question has two parts. First, work out the rollback equilibrium payoffs for this game played between two self interested players and make a note of it's identifying letter from the list a through j . Second imagine this is a game between two altruists: Blue only cares about the size of the pie going to red (more for Red is better for Blue ) and Red only cares about the size of the pie going to Blue (more for Blue is better for Red ). In case of ties in outcomes for the other player an altruistic prefer will take account of what their portion of the pie is and prefer more for himself/herself if that is possible. Now work out the rollback equilibrium payoffs for this game played between two altruistic players and make a note of it's identifying letter from the list a through jWrite the two letters from the multi options list for your two answers on your multi choice answer sheet in the order self interested players first, altruists second)

heres an answer in graphical form. I'll explain this in detail in a video clip soon....

 

A11  ( 5 marks) Two female friends, Alpha and Beta, are trying to decide what to wear to University today. They each choose between two different outfits, blue pants or red skirts, at the same time, in a simultaneous game with no communication. Alpha prefers wearing blue pants to red skirts, as long as she is not making the same choice as Beta , while Beta prefers wearing red skirts to blue pants, as long as she is not making the same choice as Alpha. The worst thing for both players is if they arrive at University wearing the same outfit - and it is just as bad whether that same outfit is blue pants or red skirts. This game is most like Choose an answer from the list below and enter it on your multi-choice answer sheet):

 

EXPLANATION: Le'ts say the strategy profiles are (Red skirt, Red skirt) , (Red skirt, Blue pants), (Blue pants, Red skirt), (Blue pants, Blue pants) where the first component the list is Alpha's (row player, green, in the table below) strategy and the second component" is Beta's (column player,bBlack) strategy. (Red skirt, Red skirt) and (Bluepants, Blue pants) whwre both are wearing the same item , are at the bottom of each player's ranking, and equally so. This follows from the statement " The worst thing for both players is if they arrive at University wearing the same outfit - and it is just as bad whether that same outfit is blue pants or red skirts." In the payoff matrix below I've indicated that by a zero in the corresponding cells (any low number that is the same in these two cells is fine) . How does Alpha compare the other two situations (Redskirt, Blue pants), (Bluepants, Red skirt), the two situations where the two of them are making different choices? According to the description "Alpha prefers wearing blue pants to red skirts, as long as she is not making the same choice as Beta ," this is when she prefers blue to red, ie (Bluepants, Red skirt) to (Red skirt, Blue pants).In the table below this is the comparison along the diagonal running from bottom left to top right. i Have indicated that with the payoff numbers 2 and 1 forAlpha . How does Beta compare the other two situations where the two of them are making different choices? according to the description this is when she prefers red to blue, ie (Redskirt, Blue pants), (Blue pants, Red skirt) - i've indicated this with a number 2 greater than 1 , in that off diagonal...

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A12 ( 5 marks) Consider the following game of international relations between two countries. One country, Black, moves first , the other, Red, moves second so can observe and react to what Black does. Each country can choose to act aggressively (A) or to be friendly (F). Aggression can be in the form of occasional and sudden armed attacks, or trade restrictions/sanctions, or restrictive border crossing policies, or grabbing disputed land for settlements, etc. Payoffs are index numbers reflecting gains or losses in property values and civilian well-being , as well as political payoffs, relative to status quo payoffs of (0,0) when both countries act in a friendly manner towards one another. Negative numbers are overall losses, positive numbers are gains. As you can see, both losses and gainsrelative to the staus quo are higher for the black country than for the red. Unfortunately rollback analysis of this game leads to a predictable outcome of mutual aggression - with both countries justifying their behaviour in terms of "resolve" to fight aggression ( ie if you act- or are expected to act -aggressively towards me you can expect me to act aggressively towards you).

myerson game

Nobel prize winner Roger Myerson analyzed a similar game, and used the argument that the mutual aggression outcome of this game could be changed if the country moving second in this game could somehow be punished for failing to exercise "restraint". In Myerson's argument the punishment is a reputational loss in repeated play of games of this type, imposed (perhaps from international sanctions) just when a country was observed to act in an aggressive manner against another country who was first acting in a friendly manner . So if payoffs without the reputation loss is X then payoffs with a reputation loss R is X-R. Which of the following values of R, lost reputation, will NOT change the expected outcome of this game? In your multi-choice answer sheet for this question write down ALL of the cells from the list below {a,b,c,d,e,f} that will NOT lead to cooperation being sustained in the repeated game. (There may be more than one answer, or select "none of the above"; Enter your answer(s) on your multi-choice answer sheet- as noted, there may be more than one correct answer) It might have helped had you looked at a similar version of this question last year...P S Does this remind youa bit of the Palestine (Black) and Israeli (Red) conflcit? - I say "a bit" becasue that is a (terrible) repeated game ....but much of Myerson's argument about resolve and restraint applies to that coflcit as wella s his critique of American unilateralism.

 

Part B 3 Questions

B1 Strategic moves and Bullying (20 marks) (write answers in answer book)

Bullying in schools is a problem. Imagine (as we did in the midterm) two players Betty Blue and Rachel Red. Betty can act like a bully (B) or be nice (N). Similarly Rachel can fight back (F) or be nice (N). Rachel's most preferred combination is where everybody is nice and her worst is where she gets bullied and has to fight back. Betty on the other hand likes it best when she acts like a bully and Rachel is nice , passively accepting that, while her worst is being nice and having Rachel fight back. In between preference ranks are indicated as in the following game tree - where higher numbers reflect higher preferences, and payoffs are in the conventional order (B's first, then R's).. You might think that bullies and nice people have different preferences than this...but for the sake of argument accept these preferences, and assume that both players know these preferences and that they know each knows, etc..

bullying

Rachel's parents are concerned. They don't want their daughter to be bullied so they approach the school, and you, for advice. Your job is to act as a consultant to Rachel, her parents, and the school, suggesting strategic moves that Rachel might make to improve the game for her. Use game theory to explain the credibility problem involved in using a strategic move to change this game. Then identify and briefly explain (in one or two sentences) all the methods for solving credibility problems for strategic moves that Dixit and Skeath identify in Chapter 10 , briefly explaining how each might or might not be relevant to solving the bullying problem that the parents and Rachel are concerned about.

Question B2: inverse probability (10 marks) (write answers in answer book) click here for a suggested answer in pdf

Part B2.1 (5 marks):Billy Bob (BB) is about to buy a used car ,but is uncertain about it's quality. To keep things simple assume there are only two possible qualities: the car is in very good shape (a plum) or it is in very bad shape (a lemon). BB can't tell from looking, and in the town where he lives and shops for cars , most used cars are lemons - indeed everyone knows that 80 out of 100 cars in this town are lemons and only 20 out of 100 are plums. The AA (Automobile Association) can run a diagnostic test for BB. It is not a perfect test, but it is pretty accurate. To keep things simple lets say that the AA can only give BB two reports "a Lemon" or "a Plum". If you took 10 cars that really are plums the AA test will correctly report that 9 times, but the other 1 time it will make an error and report the car as a lemon. If you took any 10 cars that really are lemons the AA test will correctly report that 8 times out of the 10 , but the other 2 times it will make an error and report the car as a plum. So the AA is a bit more accurate at detecting plums (good cars) than it is at detecting lemons (bad cars). Now just before BB asks the AA for a test on a particular car he doesn't know whether the AA is going to say that this car is "a Lemon" or "a Plum", but he can work out a strategy for what to believe , depending on which AA repor he getst. What should BB, as an intelligent rational player of games, believe about the chances of the car really being a plum when it is reported as "a plum" and what should BB believe about the chances of it being a lemon when it is reported as "a lemon"? Briefly explain your answer - in the answer booklet provided. (fractions OK, no need to reduce to decimals...)

 

Part B2.2 (5 marks): Employee absenteeism (for reasons other than legitimate illness) in the construction industries is a problem in in New Zealand. Not dramatically serious - about 10 out of 100 employees will be absent from work for more than a week during a year . But employers are loathe to hire people who are going to be absent from the workplace that often. The question arises as to whether there is a good diagnostic test for employee absenteeism. A consultant argues on the basis of statistics about absentee workers that most absentees are young males, under 25. When asked to "quantify" this assertion he says statistics show that of every 10 employees who are absent for more than 1 week in a year , 7 will be males under 25 . But when you look at statistics for employees who are not absent during the year it's spread fairly evenly, 50/50, ie about half being young men under 25and half being other types of employees. So, argues the consultant, young males are significantly "over represented" in the population of absentee employees, with a 70% chance of absenteeism . Do you agree with this assessment or not? How would you, as an intelligent rational player of games, using the same statistics as the conslutant, advise employers about the chances of a young male under 25 being absent from work in this industry? Briefly explain your answer. (fractions OK, no need to reduce them to decimals...)

 

Question B3: (10 marks) (mark the payoff table on the reverse side of A4 multi choice answer sheet and write explanations in answer book) click here for a pdf answer

The payoff table on the reverse side of the A4 multi answer sheet describes a 4 player simultaneous game where each player , Red, Blue, Green and Black has 2 moves, x or y, indicating one of two travel routes. The payoffs to each player for various strategy profiles are travel time in minutes. It is assumed that every player prefers lower travel times and that each player is only concerned about his or her own travel time. Clearly identify with the letter N all strategy profiles in the table that are Nash Equilibria of this driving game (put the letter N beside the relevant rows in the space provided). In your answer booklet briefly explain your reasoning. Answer - this is basically the first half of the last question on last years final -