University of Canterbury: Economics Department

Midterm exam for Economics 223:March 29, 2007

You have 2 hours for this exam. You may leave any time after half an hour has elapsed. It is a closed book exam -no notes or books in written or electronic form are to be used. Please make sure all cellphones and electronic calculators are turned off and left at the front of the room - failure to do so will mean confiscation of your exam script and a zero mark. Please answer all questions in the exam booklet provided. Also, please make your hand writing legible - I won't mark what I can't read.

Question 1           18 marks        Dixit and Skeath game types
Question 2           17 marks        Traveller's Dilemma
Question 3           10 marks        Counting strategies
Question 4           10 marks        Battle Of Wits
Question 5           20 marks        Student Loans
Question 6           25 marks        5 x multi choice

Question 1  (18 Marks)  Dixit and Skeath (DS) classify games into a number of various "pure types" by asking and answering some interesting questions. Identify these types and briefly explain the key concepts/distinctions used to interpret and understand each type. (By "briefly" I mean no more than 1-2 sentences; you do not have to provide example games in your explanations.)

Question 2: (17 marks ) Two travellers , our old friends Red and Blue, had their luggage go missing while flying home on Air New Zealand after a cycling holiday in America . It turns out they each had used cash (no receipts) to buy a bicyle overseas and, indeed, had each purchased exactly the same kind of moderately priced touring bicyle, a Cannondale comfort bike. The Air NZ representative was a little suspicious about just accepting their self reported claims for reimbursement, so she directed them each to separate rooms. She explained to each traveller (now in separate rooms, so unable to communicate) that what she needed was a written and signed claim for reimbursement for one of three amounts: $200, $300 or $400. If the amounts written down by Red and Blue turned out to be the same, she explained, then the airline would reimburse that agreed amount to each traveller. If they differed however, the airline would only reimburse the lower reported amount to each traveller ,and, as an added incentive, would reward the traveller who wrote down the lower amount by an extra $150 but would penalize the traveller who claimed the higher amount by $50. Assume the travellers will never meet one another again and that each is only interested in their own monetary payoff.
2.1 (7 marks) Draw and clearly label the payoff matrix for this game.
2.2 (10 marks) Use game theory to analyze this game and predict the behaviour of the travellers. Clearly identify all Nash Equilibria in pure strategies…or explain why there are no nash Equilibria in pure strategies. Is this game related to a Prisoner's Dilemma game? (be brief in your answer).

Question 3 (10 points)The following game tree represents a 3 person sequential Game. How many strategies does each player {A,B,C } have in this game?  List the strategies for player's B and C. (you do NOT have to list the strategies for player A).

Question 4 (10 points) Two versions of the "Battle of Wits" game from the movie "The Princess Bride" are shown below. The Column Player is Wesley (the hero in the black mask), and the Row Player is the Sicilian (the talkative villain whose reasoning about which cup to choose is very clever, very witty, but ultimately of no avail) . Assume each player would rather live than die. Use game theory to answer  the following questions: In Version 1 of the game the Nash Equilibria in pure strategies are labelled by the cells _______ (write the label(s) of the relevant cells or put a "0" if no cells are Nash equilibria).  In version 2 of the game  the Nash Equilibria in pure strategies are labelled by the cells _______ (write the label(s) of the relevant cells or put a "0" if no cells are Nash equilibria)

Version 1 (below)

Version 2 (below)

Question 5: Student Loans (20 points) Banks and other credit agencies have a difficult time deciding whether to provide loans to students. They worry about whether or not the student will pay the loan back or whether the student will default on the loan. The student borrower, on the other hand, wants a loan, but, other things equal would rather not have to pay the loan back. The following 2x2 payoff matrix sets out a simple simultaneous game version of this strategic interaction between students and banks. payoffs in terms of preference ranks (higher numbers are better), with the row player the student, the column player the bank.

5.1 (5 points) Use the above payoff matrix and dominant strategy reasoning to predict the strategies and final payoffs of the game.

5.2 (11 points) Suppose that the game changes to a sequential game so that the student (row player) moves first and the Bank (column player) moves second . Draw and clearly label the game tree for this game and use rollback reasoning to predict the strategic behaviour of the student and the bank, as well as the final outcome of the game.

5.3 (4 points) Briefly (in a few sentences max) explain any differences or similarities between your predictions in the above two scenarios

Question 6 Multi Choice .

There are 5 multi-choice questions . Each question is worth 5 marks. Write your answer in your answer booklet.

Q6.a Consider the following two situations. (i) Two governments choose between two copyright protection policies: Policy (A) is very protective of the rights of their own pool of local talent - authors, writers, performers, etc - but only weak copyright protection for foreign authors, writers, etc   and Policy (B), signing an international treaty with weak protection for all authors, writers etc, no matter waht country they are located in. [You might be interested to know that policy A was used by the USA up until the twentieth century]. Regardless of what other governments are doing, imposing A type policies  always increases the political payoff (in votes) earned by the politicians in the home country by 10 but also always decreases the payoff earned by the other country’s politicians by 5. (ii) Fans at a sporting event choose between standing up and cheering for the home team and sitting down and not cheering. Regardless of what other fans are doing, standing up and cheering increases a fan’s enjoyment and thus always increases his or her payoff by 5, but the standing up makes it dificult for other fans to see and always decreases their payoff by 10. Which of these situations has the characteristics of a prisoners’ dilemma?Choose the answer from the list below and write the answer in your answer booklet.
(a) Only situation (i)
(b) Only situation (ii)
(c) Both situations (i) and (ii)
(d) Neither situation (i) nor situation (ii)
(e) there isn’t enough information to answer the question using Game theory

Q6.b Fishing Boat Owners (FBO) and the crews that work for them on the boats ( W) are playing an alternating offer bargaining game at the start of the fishing season. FBO and W must bargain together about how to share out the proceeds of the season's catch and they also have to work together to catch the fish , which the crew does, and process it for sale, which the FBO’s do. The fishing season is short and lasts only 3 weeks, with the total value of the catch being $50 in week 1, slightly less, $45, in week 2, and much less , $15, in week 3, with no fishing permitted after that. Getting late to the fishing grounds means less fish to catch. Workers and FBO’s each have to be paid more than $5 to make it worth their while to work. Due to circumstances beyond anyone’s control the boat owners and workers can meet to make offers and ratify agreements only once a week. The bargaining rules are that the FBO’s make an offer of a division of the catch’s value in week 1. The workers either accept or reject. If they accept , they go off to fish in week 1 and the catch value is split as agreed. If the workers reject the FBO’s offer they have to wait until week 2 to make a counter offer to the FBO’s. At that time (week 2) the FBO’s can either accept the workers offer and have the (smaller value) catch harvested and share the money as agreed, or reject , wait until week 3 then come back with a counter-counter-offer to the workers. This last counter offer is either accepted , the catch harvested and the (smaller still) money shared out as agreed, or it is rejected , no fish are caught and they each walk away with nothing. Assume offers and counter offers are made in units of $1 (ie no fractions), that each player values only their net receipts (ie $share of the catch less the $5 cost of their work efforts) , and that either party will accept a limiting offer rather than proceed to the next stage . Choose the answer from the list below that is closest to the prediction that game theory would make about the outcome of this bargaining game, and write the answer in your answer booklet.

1. The FBO’s will keep $35 for themselves in the first week and offer the workers $15, and the workers will accept, grudgingly.
2. The owners will will hold out as long as they can until week 3 and exploit the workers, dividing the $15 catch $10 for themselves and $5 for the worker, which the workers will grudgingly accept.
3. Workers reject low offers in week 1 and hold out till week 2 where they propose a $25, $20 split of the $45 proceeds ,keeping 25 for themselves, and the FBO’s will accept $20 rather than run the risk of not getting an agreement before the season runs out.
4. The Workers will keep $35 for themselves in the first week and the FBO's $15, as the FBO's grudgingly make them an offer they can't refuse in the first round.
5. The FBO’s will offer to split the first week's catch evenly $25, $25 each and the workers will accept rather than delay and run the risk of get a lower share of a smaller pie later.

Q6.cThe following payoff matrix describes a 3 person simultaneous game. The three players are Red, Blue, and Green. Each player has two moves: {T, B} for R, {Left, Right} for B, and {Left, Right} for G. Payoffs are preference ranks, higher numbers indicating more preferrd options, listed in order from left to right first for player R, then for player B then for player G. The Nash Equilibrium prediction(s) for the outcome of this game is/are _____. (Choose your answer(s) from the list a,b,c,d,e,f,g,h that labels each of the cells in the payoff matrix below and enter it into your answer booklet)

Q6.d The table below represents a 2 player simultaneous game with payoffs in dollars indicated in standard fashion:The first two steps in a process of iterated elimination of dominated alternatives would first eliminate  strategy _____and second would eliminate strategy_____??(Choose an answer from the list of strategies T,M,B,W,X,Y,Z to fill in the blanks in this sentence, writing your answer in the form “first X then Y” or  write “no strategy can be eliminated” and enter your answer in your answer booklet)

Q5.eConsider a variant of chicken in which each driver can go straight, swerve to his left, or swerve to his right. If one swerves to his left and the other swerves to his (own) right, then the cars will collide, just as they will if both drivers go straight. Thus the payoff table becomes:

This game has: (Choose an answer from the list below and enter it into your answer booklet)
(a) no pure strategy Nash equilibria but at least one mixed strategy Nash equilibrium
(b) 5 pure strategy Nash equilibria
(c) 4 pure strategy Nash equilibria
(d) 3 pure strategy Nash equilibria
(e) 2 pure strategy Nash equilibria
(f) 1 pure strategy Nash equilibrium