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

count

How many strategies does player C have in this game?(Choose an answer from below and enter it on your multi-choice answer sheet)

(a) 72 different strategies
(b) 13 different strategies
(c) 12 different strategies
(d) 6 different strategies
(e) 4 different strategies
(f) none of the above

A2 ( 5 marks) It is the year 2050. Two movie producers, one for Shrek 42 , the other for Pirates of the Caribbean 37, must choose between opening early in the year or opening later. The payoffs (in millions of dollars) for the movies are shown (in the standard way) in the accompanying table.

shrek

Choose one of the following answers that correctly fills in the blanks in the following statement: “The game illustrated here _____a prisoners’ dilemma because _____.”(Choose an answer from below and enter it on your multi-choice answer sheet)

(a)is; it has all the characteristics of a prisoners’ dilemma
(b) is not; neither Shrek 42 nor Pirates 37 has a dominant strategy
(c) is not; the game has no Nash equilibrium (in pure strategies)
(d)is not; using their dominant strategies gives the firms a mutually beneficial outcome
(e) is not; it is a battle of the sexes game with each producer prefer differring equilibria
(f) is; it is a simultaneous version of the entry deterrence game

A3 ( 5 marks) The penalty shoot in soccer is a dramatic game of strategic interaction. If we simplify we can think of two strategies for each player. The Goalie can decide to move to the left or the right. The Kicker can shoot to the (goalie's ) left or the (goalie's) right. Even if the goalie moves to the same side the Kicker kicks to, he doesn't always prevent the goal, and even if a kicker kicks to the opposite side to where the goalie has moved a goal isn't always scored. Sometimes the shot is just too powerful to block in relation to the goalie's body position. Sometimes the shooter blasts a shot over and above an open portion of the net. The following payoff matrix describes the moves and payoffs in a simultaneous 2x2 game. The payoffs for the goalie are success percentages in stopping goals: the chance (in %) that the ball doesn't go in the net and so there is no score score (higher such probs being better for the goalie) . Payoffs for the Kicker are preference ranks, higher numbers are better. His name is Beckham and he really doesn't care about scoring goals any more, only about outfoxing the goalie, and he really likes it when the goalie dives to the left and he does his famous "bend it like beckham" kick...whether or not it goes in. There is no Nash Equilibria in pure strategies in this game. However there is a mixed startegy Nash equilibrium. In that mixed strategy Nash equilibrium the kicker will be kicking to the left with what probability? On your multi choice answer sheet for A3 write down one of the integers 0,10,20,...through to ....90,100 .

A4 ( 5 marks) Examine following 2 player 4x4 payoff matrix. The first two steps of the method of iterated elimination of dominated alternatives can eliminate which two rows and colums? On your multi choice answer sheet for A4 write down the identifying letters of the rows or columns that are your answer to this question.

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. 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-repayent 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)

a) the unravelling of cooperation in a repeated prisoner's dilemma
b) an unfortunate signalling equilibrium with interest rates signalling high costs due to a large number of defaulters
c) moral hazard in the principal agent problem
d) adverse selection
e) a negative feedback cycle created by out-of-sync and unforgiving Tit for Tat strategies in in a repeated game between the bank and its customers
f)a bargaining game where customers fail to use rollback reasoning to calculate the consequences of defaulting on their loans
g) none of the above.

A6 ( 5 marks) Arts , Commerce and Education grads have a tough time getting jobs after they graduate. As a result, many of them enroll in research based graduate courses/programs to improve their chances of getting a job. The University that provides these grad courses gives out scarce scholarships, and it would prefer to give scholarships to students who are likely to complete their course program. The problem (for the University, and the students) is that the University can't tell who are the more able and who are the less able types. So the University decides to set up a screen based on the number of A's a student achieves in courses taken during the last year of their undergrad degree. Lets suppose that there are two types of grad students, those who are very productive (VP) in a research degree setting and those who are just average (AP). If a student gets m or more A-grades they receive a scholarship worth $25K, whereas if students get less than m A-grades they receive only $5K in scholarship money. Student payoffs are the level of their scholarship minus their personal costs of education in that final undergrad year. VP types have personal costs of $5K per A grade achieved whereas AP types have personal costs of $10K per A grade achieved. We're not talking about the level of fees or other living expenses, which are the same for either type, but rather just putting a money value on the personal time and effort to get an A-grade in an undergrad course . Both types can achieve A-grades.

What is the smallest and what is the largest number m of A grades 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).

a) there are no levels of A grades that works to satisfy both incentive compatability constraints
b) 1 A-grade
c) 2 A-grades
d) 3 A-grades
e) 4 A-grades
f) 5 A-grades
g) 6 A-grades

A7 ( 5 marks) The following game tree diagram describes a 2 player sequential game between player A and player B, each having two moves. The payoff matrix associated with the analysis of this sequential game is shown below the tree may have several Nash equilibria. Some, but not all, of the entries in that payoff matrix have been filled in for you. Select as your answer to this multi-choice question A7 the labelled cell or cells a,b,c,d,e,f,g,or h from that payoff matrix that are a Nash Equilibrium for this game that are also NOT sub-game perfect equilibrium, or write “none of the cells in the payoff matrix”. (Write your answer on your multi choice answer sheet )

trust

Part B 4 Questions

Question B1: elementary 2x2 games (18 marks)

The question for this part - and an answer template -is on the opposite side of your multi choice answer sheet. Write your answers on that answer sheet . Do not write your answers in your answer booklet.

Question B2: inverse probability (10 marks) (write answers in answer book)

Context: Sex between consenting persons over 16 (or some other approved age) has become an accepted part of secular western society, at least for a significant number of the population. This change in behaviour, or at least wider acceptability of sexual behaviours outside of marriage, started in the 60's. But the advent of AIDS in the 80's and its subsequent lethal and explosive spread around the world in both heterosexual and homosexual populations made sexual "games" between consenting adults much riskier. Sexually transmitted diseases like AIDS are a classic example of imperfect private information in strategic interaction. Just as in the market for used cars where buyers tend to be ignorant about quality and sellers tend to know more about quality, the lemons problem of adverse selection emerges in sexual transaction games. If both players know their own and their partner's sexual health (or lack thereof) they play a much different "game" than if they don't, and an especialy different game if they know that each knows more about their own sexual health than their partner's sexual health. (Would you have sex with someone who you know has had an AIDS test and who knows you know that (s)he has had an AIDS test, but is not showing you the test results?) So a variety of signalling and screening devices have arisen to test for the presence or absence, of sexually transmitted diseases like AIDS. But how should those tests be interpreted? How would you or any player "know" privately that they were likely or not to have AIDS on the basis of a test result? [this is not a question for you to answer here...it is just part of the context for the inverse probability question comig up).

One biological test for AIDS, called the ELISA test, involves detecting antibodies to the HIV virus in the blood stream . It is an indirect test for AIDS, but not a perfect test. If someone actually has AIDS this test will pick it up (ie show a positive test result for AIDS) most, but not all, of the time, say with an 80% chance . If someone doesn't have AIDS the test will typically indicate a negative result, but every now and then, 10% of the time, an ELISA test will give a positive test result even though the person tested does not actually have AIDS. So ELISA is an imperfect test for AIDS, with both false positives and false negatives.

Now the questions: Part A (5 marks): Suppose that among a certain class of "at risk" young men and women in Christchurch - called MISFITS - the prevalence of AIDS has been found to be 1 in 100. That is, out of 1000 of MISFITS , 10 have been found to have AIDS. Now, a young man James comes to you for advice - James being an "at risk" type just like one of the MISFITS. James has just had a positive ELISA test result and wants you to advise him on his chances of having AIDS. What do you say to him, and, as important...why (ie explain your reasoning)?

Part B (5 marks): In South Africa, a typical black male has a 30% chance of having AIDS (and in some areas even higher) - ie 300 out of 1000 South African black males actually have aids (the tragic story of how this occurred can be viewed on a PBS documentary -if you are interested, contact me after the exam). Suppose that James -obviously a young black male - now tells you that he is a recent immigrant from South Africa, having lived there almost all of his life. Does your advice to him on how to interpret his T+ test change...and why?

Note: you do not need to use a calculator -if you do compute a fraction in your advice to James- just leave it as a fraction, no need to reduce it down or give a decimal equivalent.

Question B3: Trust (12 marks) (write answers in answer book)

Trust is an important part of any relationship or transaction. Without trust there are many valuable opportunities - valuable for all players - that will go wasting. Many people would simply rather not be involved in a game where they are going to be ripped off, even though if they could trust the other player (and perhaps vice versa) the game would be worth playing. The following payoff table describes payoffs and actions in a 2 player, one-shot trust game. The column player Rachel can choose either to Trust (T) or distrust (DT) the row player Bob. Bob, the row player can choose to honour that trust (HT) by acting in an honest sincere hardworking manner or he can act opportunistically (OPP), taking advantage of the situation. Bob's payoffs are his preference ranks 1 to 4 for various combinations of actions, while Rachel's payoffs are 0 for a benchmark (she doesn't trust and Bob acts opportunistically) , and other cells are valued as gains or losses relative to that benchmark, higher numbers being more preferred.

trust game

B3(a) (5 marks) Draw and label the game tree for the sequential game where Rachel moves first and Bob second, Bob observing whether Rachel is acting to trust him or not before Bob makes his choice (Please use the convention for payoffs that the first player's payoffs come first in the payoff list) . Use rollback reasoning to show that the payoff to each player is the same in this sequential game as in the simultaneous game. Then analyze this sequential game as a simultaneous game and verify that in both the original simultaneous game and in the sequential game Bob has a dominant strategy to always act opportunistically.

B3(b) (7 marks) Bob's dominant strategy to always act opportunistically in the previous two versions of this game (B3(a)) creates a problem that both players would like to solve. Suppose we change the order of play yet again. Draw and label the game tree for the sequential game where Bob moves first and Rachel second (Again, please use the convention for payoffs that the first player's payoffs come first in the payoff list) . Then analyze this sequential game as a simultaneous game, identifying the Nash Equilibrium predction(s). Does Bob still have a dominant strategy to act opportunistically in this new game? Briefly explain how changing the order of play in this one-shot trust game with Bob "moving first" turns what appears to be a dominant strategy (always act opportunistically) into a strategy rational intelligent players would no longer expect. .

B4 Credibility and drink driving (25 marks)

Consider the following stylized game that tries to use game theory to analyze the strategic problems of self control with some addictive or compulsive behaviours. A young male goes to a party, and in his sober state (as player M1 in the game tree below), he hands over his car keys to his host. Sure enough, our young male drinks too much alcohol and gets a little "tipsy", a polite way of saying "drunk as a skunk". As the party ends and he is about to leave he has a choice of whether to ask for his car keys back or not (as player M2 in the game tree2) . His host (player H) can give (G) the car keys back or withold (WH) them. Payoff numbers representing preference ranks of all 3 players are represented in the payoff matrix. If you study the game tree you'll see that M2's most preferred option is to obtain his car keys back when he (M2) asks for them, and his worst (as M2) is to be given his car keys back when he hasn't asked for them, with the in between outcomes ranked as indicated. The host is a polite person, and her preferences are to agree with what her guest wants to do with his own property, at least with what her guest asks for and wants at the time the guest leaves (perhaps the host has been drinking too) . Unfortunately, as you see if you look closely, there is some conflict of interest in this game between the preferences of the sober young man as M1 and the "tipsy" young man in the case where the "tipsy" young man asks for his keys back.

1) (5 marks) use game theory to analyze this game and predict behaviour and payoffs.

2) (20 marks) As Player M1, the sober young man, the predicted behaviour and outcome is not to your liking. After a little bit of thinking, and talking to yourself, including that part of you (M2) that likes drinking too much and driving home, you (M1) decide to try to change the game by making a strategic move. You explain to the host that being one person, you really know, quite intimately, this other self who drinks too much at a party , and in effect this other self has made a promise to you not to ask for the keys back, and you are relaying this message to the host, reminding her that it is in her interest not to give you (or M2) the keys if you (M2) doesn't ask for them. Identify the credibility problem with this approach. What can you as M1 (the sober young male) do to solve the credibility problem and credibly gain some self control. (A good answer will briefly list all the methods for achieving credibility in a strategic game that Dixit and Skeath identify and check which might be relevant in this game)