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.)

JF'S comments 2007 This was a pre-set question, yet still many bombed out; i gave 6 points out of 18 for just stating the exam questions. I used "weak" as a comment for your explanations identifications of key concepts....too many "weaks" and you may get less than a pass grade, 9/18. For those of you doing poorly go back and read Ch 2 in D&S -especially the explantions. Learn to use the language of game theory -new words, new concepts, new idea - with some precision. For a preset question I expected you to put in some effort organising your answer (sentences, not bullet points) as well as idtenifying key strategic concepts and issues.

Question 1 Question 2  Question 3  Question 4  Question 5 Question 6

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).

Explanation: Best response analysis (the x's and o's in the payoff matrix) yields one Nash Equilibria for this game, where both players quote the lowest possible price, $200, which yields a payoff of $200 to each. If you specify the highest amount, $400, and then think about what the other player would do if she also thought you were specifying this amount, she would have an incentive to undercut you by $100, leaving you with a penalty $300-$50 or 250. And if she undercuts you like this you're better matching her at $300 than holding out for $400. This argument basically uses the logic of best responses to say ($400,$400) can't be something each player would rationally expect the other players to do...there is always the incentive to undercut. Ditto for $300 each....It doesn't make strategic sense to claim $300 expecting the other to claim $300, and for you to expect the other to expect you to play $ 300. Becasue neither of you would be making a best response to what you believe the ooother player is doing...So we unravel down to $200. the only Nash Equilibrium in this game. ; ditto for the second highest amount, etc etc. This game is like the prisoner's dilemma in one respect - there are strategies which are better for both , eg quoting eiher $300 each or $400 each, with outcomes of $300 or $400 each, but these strategy combinations are not equilibria. That is, what appears to be individually rational behaviour leads to collectively poor outcomes. But the game is unlike the prisoner's dilemma in another respect in that neither player has a dominant strategy (note that quoting the highest price, $400, is a dominated strategy for each player).

Note: some people constructed the wrong payoff matrix by having each agent's reminbursement equal to what they claimed plus (minus) any reward / penalty . Eg if Blue claims $200 while Red claims $300, Red's payoff was calculated as reimbursement of thier claim , $300, less $50 penalty - $250 in total. Ditto for other situations where claims differ. The text says clearly that reimbursements for each player are the lower of the claimed amounts, not your own claimed amount...However, if you made this error and went on to analyze the rest of the probelm correctly, i typically gave a passing mark (9/17).

Question 1 Question 2  Question 3  Question 4  Question 5 Question 6

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 1 Question 2  Question 3  Question 4  Question 5 Question 6

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)

there is no Nash Equilibrium in this game - illustarting the circular reasoning so cleverly explained by the Sicilian ("if you chose that I would chose this, but you would expect me to chose this so wouldn't chose that....")

Version 2 (below)

Using best response analysis we can identify 4 nash equilibria, in cells2, 4, 3 and 6, as show on the graph below.This modified game may all seem too easy, since you might think that the Blue Player wants to do away with the red player anyhow. But in the game as we have set it up , the Blue player gets no extra benefit from the other player dieing....his only benefit is staying alive himself. With our binary payoffs, live or die, the blue player is completely indifferent to his own strategy and to the other player's strategy . There are two possible "live and let live" outcomes in cells 2 and 4, as well as the two "put the poison in both cups" strategy that lead to the death of the Sicilian.

Question 1 Question 2  Question 3  Question 4  Question 5 Question 6

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. The student has a dominant strategy to default. The Bank, expecting the student to play that dominant strategy should not (as a best response to her prediction) expect the loan to be repaid: don't make the loan is the Bank's best response to the student's dominant strategy (iterated elimination of dominated alternatives - "PB" for the student is dominated by "D") ).Notice that while this game is not the prisoner's dilemma (it is not the case that both parties have dominant strategies) there is a mutually benefical outcome that isn't an equilibrium outcome for this game, namely the outcome where the bank makes the loan and the student pays it back. Think of this mutually beneficial outcome that might have, but does not, occur as a lost opportunity. This lost opportunity is what we will bring up in class as the "cost of distrust".

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.The diagram above indicates that the rollback strategies for the Bank are to make the loan if the student pays back and otherwise not to make the loan, while the student should pay back the loan- in the rollback equilibrium prediction.

5.3 (4 points) Briefly (in a few sentences max) explain any differences or similarities between your predictions in the above two scenarios By changing the information structure of the game - in particular making it possible for the Student to "move first" - the student no longer has a dominant strategy (one that is best no matter what the other player does). This "first move" by the student is a type of committment to repaying the loan that the student can make, and as importantly, that the Bank can directly observe..before it decides to make the loan . Note that there is no possibility , in this simple game, for the student to default AFTER the bank has made the loan. The student has a first mover advantage (relative to the simultaneous move game - and also relative to the game where the bank moves first...but that's not part of this question). Another way to look at this is to note that the student's best response in this sequential game, to the Bank's conditional strategy (ML,DL) - ie making a loan if it is paid back and not making a loan if it isn't - is actually to pay back the loan. In the simultaneous game the bank doesn't have this type of conditional strategy available...it has only two strategies, make the loan or not . The student, while moving first (and not having a chance to move again later) isn't just reacting to the specifics of the bank's decision, to make a loan or not - but to the bank's strategy .

You can see the strategic issues a little more clearly, and deeply, by analysing the sequential game as a simultaneous game, as we did in the last lecture of term 1 (being able to do this sort of analysis was not necessary for the exam...but it might be helpful to you now as you try to understand the logic of this question) . Default is NOT a dominant strategy for the student viewed from this perspective...even though it was a dominant strategy in the original simultaneous move game.

JF comments: Notice one thing here - the student might be playing a 3-stage game where first, they send a message to the bank, like "I will pay back the loan" or "I won't pay back the loan" . Then the bank makes a choice as whether to make a loan or not after hearing this message, but without directly observing what the student will do, then the student has a choice to pay back or not, after observing whether the bank makes the loan or not. The words in this game are what is called "cheap talk messages" - actions which don't have any direct impact on outcomes (they are cheap to make): using game theory ask yourself whether the bank should believe cheap talk messages?? The student of course now has 4 strategies, which link cheap talk messages with behaviour. eg say you'll pay back at the beginning, but default later; say you'll pay back at the beginning and pay back later; say you won't pay the loan back, and don't, or say you won't pay it back, then do. Simply identifying these startegies makes you realise that a person's behaviour isn't necessarily the same as what they "say" that behaviour will be - it might be..but might not be. A good exercise (maybe for the final exam??) is to analyze the trust game with cheap talk by the student as a simultaneous game (4x4 payoff matirx) and then identify all the nash equilibria, as well as the sub game perfect Nash Equilibria.

Question 1 Question 2  Question 3  Question 4  Question 5 Question 6

Question 6 Multi Choice ANSWERS IN RED .

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 what 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) Click here to view a video clip on uctv (YouTube style or to download) on how to answer this question
(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 ____f, h _. (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 Z and second would eliminate strategy B_ (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

Question 1 Question 2  Question 3  Question 4  Question 5 Question 6