A1 ( 6 marks) The game tree below represents a stylised 3 player (A, B, C) sequential game (no payoffs are shown at the terminal nodes).

How many strategies does player B have in this game?(Write your answer in the space provided for this question on the multi choice answer sheet) Player B appears in 3 situations in the game, at nodes b.1,b.2,b.3. A strategy needs to specify precisely what the player will do in each of those situations - so one specific startegy is uut, another is uum etc, where we list what B could possibly do in order of the nodes. there are 2x2x3=12 different strategies.

A2 ( 6 marks) Dixit and Skeath present a number of stories about strategic interaction that are intended to be motivating examples. Each story has a central strategic point. The main strategic point of the "Why are Professor's so mean?" story is: (Choose one answer from the list below and enter it on your multi-choice answer sheet )

(a) to be unpredictable: don't play a strategy that systematically allows you to be exploited
(b) to show that rollback reasoning , "look forward and reason back", implies that players discount the future highly
(c) to show that when each player has a temptation to break a collective agreement if no one else breaks the agreement, but also wants to avoid being a "sucker" if everyone else breaks the agreement, rational strategic play may mean that everyone is worse off than they might otherwise have been
(d) to explain how inflexible and bureaucratic rules may be strategically useful when some players cannot be expected to carry through with their threats
(e)to illustrate that it is not whether a player think that some particular action "makes sense" for intelligent rational play in a coordination game but that payers need focal points in order to avoid poor outcomes in such games
(f) to show that game theory reasoning involves circluar arguments about how one player is thinking another player will be thinking about what they are thinking about what the other player is thinking about......
(g) to show that rational intelligent players believe in a golden rule for the modern age: he who has the gold makes the rules

A3 ( 6 marks) Some workers like leaving early to beat the rush hour (5 pm) traffic, but if everyone leaves early it just shifts the congestion problem in time. Of course, people are different and some abhor congestion less than others...if they can get home early! Suppose preferences for Row (red) and Column (Blue) for leaving early or at 5pm and for the associated congestion are as indicated in the following payoff table for a one-shot simultaneous game. Preferences ranks are in letter grades (A is better than B is better than C is better than D, etc). On your Multi Choice answer sheet write the letter(s), a,b,c,or d, of all the cells corresponding to the Nash equilibria of this game. Answer : use a best response analysis (x's for Red, o's for Blue) to find the mutual best responses - Nash equilibria- in cells c and b. The slightly tricky part is to understand how to work with letters indicating preference ranks rather than numbers.

A4 ( 6 marks) Two friends, Ralph and Bob, are trying to decide what to wear to a Star Wars fancy dress party. They each can choose between two different outfits, a C3PO outfit and a Yoda outfit, at the same time, in a simultaneous game, but in different places and with no means of communication. The main thing that is important to each of them is that they arrive at the party with different outfits, although in this case Bob has a preference to show up as Yoda rather than C3PO while Ralph has a preference to show up as C3PO rather than as Yoda. And while both showing up in the same outfit is bad, both showing up as Yoda is MUCH worse than both showing up as C3PO - other party goers will make ruthless fun of them both dressed as Yoda!. This game is most like (Choose an answer from the list below and enter it on your multi-choice answer sheet):

(a) a prisoner's dilemma game
(b) An assurance game (I initially had "c" as the answer becasue I was thinking of a different question - several students picked up on the error; see the post on the discussion forum)
(c) A chicken game- they disagree in preference for the two NE and if they don't cooridnate one non equilibrium outcome is very bad for both.
(d) a battle of the sexes game
(e) a game of pure coordination
(f) a stag hunt game
(g) none of the above

A5 (6 marks) In South Africa there are nearly 1 million children with HIV becasue their mothers had HIV. Almost all of these children are expected to die of AIDS in the next 8-10 years. In most developed western countries mother to child transmission of HIV is now almost (but not quite) non existent, even though in the beginning of the AIDS epidemic (1980's) the rate of mother to child transmission of HIV was as high in western countries as it is now in South Africa. Why is this so? The short answer is the availability of modern drug treatments for AIDs . Technically the drugs are also "available" in South Africa - but often only at the high prices that they are available at in wealthier western countries. But how can someone in South Africa with an income of $25 a year afford a drug that is priced at $10,000 a year? Multinational pharmaceutical companies (BigPharma) say they would gladly lower prices for AIDS drugs in South Africa if South Africans would introduce and enforce restrictions on the resale of those drugs back to wealthy countries like America or Europe. But if BigPharma did actually price AIDS drug low enough to make them affordable to South Africa, people in the South Africa would benefit by NOT enforcing restrictions on resale of these drugs (why not buy enough for your own medical uses and then buy some more to sell back in America or Europe at high prices there?).

Imagine that Big Pharma and South Africans are engaged two possible one-shot sequential games, as outlined in the game trees below. In game 1 South Africa (BLUE) moves first, in game 2 BigPharma (RED) moves first. Big Pharma has only two prices they can offer, PH, price high , or PL, price low, and South Africans have only two actions, ER, enforce restrictions on resale back to America/Europe, or not, NER. Let payoffs be described by the numbers in payoff matrices below (higher numbers indicate more preferred outcomes for that player; "0" indicates status quo and positive and negative numbers indicate gains and losses relative to that status quo; the payoffs for the player who moves first are recorded first in the list.):

Using rollback reasoning to predict the outcomes and payoffs of each game , choose ONE item from the following list that most accurately describes the predicted strategic behaviour in these games

(a) Both RED (Big Pharma) and BLUE (South Africa) have first mover advantages
(b) Neither RED (Big Pharma) nor BLUE (South Africa) have first mover advantages
(c) RED (Big Pharma) has a first mover advantage BLUE (South Africa) does not
(d) BLUE (South Africa) has a first mover advantage but RED (Big Pharma) does not
(e) the concept of first mover advantage does not apply in these games

A6 (6 marks) The 2x2 game tree below represents a game between a player now, and their future self, trying to stay on a diet. The red player (the dieter now) has definite preferences: succeed is better than don't care is better than fail over the different paths of play, but the future self is pretty impulsive, preferring to enjoy food rather than starve herself irrespective of what the current self wants or does. 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) I was looking for answer e, a trust game, following the logic in the graph, but i accepted d an entry dterrence game becasue (it can be agrued) that the pattern of conflict and cooperation there is very similar to that in the entry dterrence game, and the equilibrium in that game (3rd highest for both) is close to what we see here. The key idea here , once we're focussing on sequential games, is to look for similarities in patterns of conflcit and cooperation and in the predicted outcome.

(a) a prisoners’ dilemma game.
(b) a threat game.
(c) a stop-go game.
(d) an entry deterrence game.
(e) a trust game
(f) a voluntary contributions (VCM) game
(g) a short term centipede game

A7 ( 6 marks) Examine following 2 player 4x4 payoff matrix. The first two 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 two steps in this iterative process. Answer: eliminate row T then eliminate column L

A8 (6 marks) Two travelers , 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 bicycle overseas and, indeed, each had purchased exactly the same kind of cheap touring bicycle. 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 traveler (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 traveler. If they differed however, the airline would only reimburse the lower reported amount to each traveler ,and, as an added incentive, would reward the traveler who wrote down the lower amount by an extra $150 but would penalize the traveler who claimed the higher amount by $50. Assume the travelers will never meet one another again and that each is only interested in their own monetary payoff. Which strategies would be predicted using the Nash Equilibrium concept of game theory . (Choose the label or labels from the cells in the table below to identify your answer(s) and enter your answer(s) on your multi-choice answer sheet) Answer: Cell a- see best response and Nash equilibrium below.

A9 ( 6 marks) A simple ultimatum bargaining game is being played between two partners (who are friends) in a business . There are 4 weeks holiday in total and the problem is to determine who is going to take how many weeks of holiday time. The possible divisions are 4,0 or 3,1 etc as indicated by the pairs of numbers (x, y ) at the terminal nodes in the one-shot sequential game tree below, x the # (number) of weeks of holiday for R (Red) and y the # of weeks of holiday for B (Blue). Red, the senior partner, moves first and proposes a division of the total , (x, y ) ,where x+y=4, and Blue moves second either accepting (A) or rejecting (R) Red's proposal. If Blue rejects the proposal neither of them take any holidays at all. If the players were only self interested then these numbers (x, y ) also indicate the players' payoffs. But suppose R and B are not purely self interested - rather, each is altruistic, selflessly concerned about the amount of holiday time going to their partner: Blue only cares about the # of weeks of holiday going to Red (more for Red is better for Blue ) and Red only cares about the # of weeks of holiday going to Blue (more for Blue is better for Red ). In case of ties in outcomes for the other player an altruist will take account of what their own holiday time is and prefer more for himself/herself if that is possible, but never if that choice leads him/her to have all 4 weeks of holiday and the other player none. Work out the rollback equilibrium payoffs for this game played and Write the letter from the multi options list for your answer on your multi choice answer sheet.

note the pruning at node B.2e. Blue won't accept 4 for himself given his options at node B.2e because red has no holidays no matter what he does , and Blue would prefer he also have none if red has none rather than him have all of the holiday time and Red none. At node B.2d though Blue accepts the proposal rather than rejects it becasue he prefers 1 week of holiday for Red as compared to no holiday time for Red.

Part B 3 Questions

Question B1: Dixit and Skeath Questions defining various types of games (18 marks) (write answers in answer book)

Dixit and Skeath (DS) classify games into a number of different "pure types" by asking and answering several 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.)

B2: Order of play and Bullying (15 marks) (write answers in answer book)

Note - during the test I made a public announcement and changed the payoffs in the game tree becasue a student pointed out an inconsistency in the description in the text of the question " Blue on the other hand likes it best when he acts like a bully and Red is nice , passively accepting the bullying, while his worst is being nice and having Red fight back." and the payoffs as indicated. I made a mistake in an on the spot judgement. I should have said ignore that text description and stick with the payooffs as in the game tree. Please accept my apologies for the mistake and the confusion it must have casued many of you. I'll mark this question VERY carefully to make sure I don't disadvantage anyone...and if it's pretty obvios you are disadvantaged i'll just throw the question out as far as determining your grade. My mistake here, not yours. The problem is that the original question DOES lead to a change in behaviour whereas the modified one doesn't - and I wanted you to explain the changed behaviour. So I need to take that into account when i grade your answer. If you feel my grading of this question is unfair to you in ANY way when you get your paper back please come and see me about it.

So let's answer the original question first.

Bullying in schools is a problem. Imagine two players Blue and Red. Blue can act like a bully (B) or be nice (N). Similarly Red can fight back (F) or be nice (N). Red's most preferred combination is where everybody is nice and his worst is where he gets bullied and has to fight back. Blue on the other hand likes it best when he acts like a bully and Red is nice , passively accepting the bullying, while his worst is being nice and having Red fight back. In between preference ranks are indicated as in the following game tree for a one-shot (ie not repeated) bullying game where Blue moves first. Higher numbers reflect higher preference ranks, and payoffs are in the conventional order (Blue first, then Red).. 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

B2.a - (5 marks) Analyse and predict the outcome of the game using rollback reasoning.

Applying rollback to the above game tree: Blue strategy is B to be a bully and Red's strategy (complete specification of evrything he will do in the circumstances he finds himself in) in the rollback equilibrium will be N, to Not fight back at R.1 and N, to not fight at R.2. The rollback path of play is for Blue to bully and for Red to not fight back, and the rollback payoffs are 4 for Blue (his best) and 2 for Red (his next to worst).

B2.b - (10 marks) Then change (only) the order of play in the game so that Red moves first, analyse the new game using rollback reasoning. Explain briefly the key strategic idea here that works to change the equilibrium outcome of the second game for Red . Applying rollback to the changed order of play game (see below -note I have also changed the order of the payoffs) game tree: red's strategy is to fight and Blue's strategy (complete specification of everything he will do in the circumstances he finds himself in) in the rollback equilibrium will be N, to Not bully if he sees Red committed to fighting and to bully if he sees Red committed to not fighting. The rollback path of play is for Red to commit himself to fighting and for Blue to not bully.

The difference between the two games is that when there is flexibility on the part of Red to respond to a behaviour that Blue commits himself to (the original game) Blue can rationally and intelligently expect Red to not fight back. The bully doesn't determine Red's choice, but by committing himslef to bullying behaviour (ie being unable to back off if Red does fight) he knows and can reliably infer from Red's preferences that Red will not fight. In the second game these expectations or beliefs are changed becasue now it is Red who can commit to a behaviour, then Blue has the flexibility to respond. Why would Red commit to behaviour - fighting - which he doesn't want to do? Because he can the rationally and intelligently expect the Bully to back off becaue he knows the bully's preferences are - don't bully if the other guy fights. The Bully's belief's change about what Red will (and can) do in this game - notice that once the bully's choice is made Red can't back out - there is no future in this one shot game where that is "allowed" as part of the rules of the game. For fun you might try to ad in a 3rd stage where Red gets a chance to "reconsider" his earlier decision IF the bully does decides to bully him...- that "opportunity" to reconsider later, that flexibility, undoes the new expectations of the bully and will lead to Red deciding NOT to commit to fighting!

Now as for the modifed question the game tree analysis is here: call the GAME MOD !

and when you change the order of play the analysis is here: GAME MOD 2

The Red player still acts nice and the Blue player still bullies, so it hasn't changed the outcome of the game. Beliefs and expectations change from one game to the other but not in such a way that makes any difference. For example in the original Game Mod 1 the bully makes an expectation about what red will do if he bullies and if he doesn't, namely not fight back in both cases, while in Game Mod 2 he believes that REd will NOT fight back becasue he expects Red to believe that Blue will be a bully no matter what Red does. SO technically we have different beliefs from one game to the other. But in the modified game mod 2 Red committing to fight back doesn't change the bully's behaviour at all.

Question B3: Alternating Offer Bargaining (11 marks) (write your answer in your answer book)

Shareholders (S) and Managers (M) in a small publishing business are playing an alternating offer bargaining game about managerial compensation payments in the form of revenue sharing options. However, the longer the bargaining goes on, the smaller the overall revenues there are to share out. revenue opportunities are $100K this month, $75K next month, $20K in month 3 , and will be zero after that. Due to circumstances beyond anyone’s control S and M can only meet monthly to make offers and ratify agreements . The bargaining rules are that Shareholders S make an offer of a division of revenues in month 1. Managers M either accept or reject that offer. If they accept , the revenues available at that time are split as agreed. If the offer is rejected they both have to wait until month 2 at which time M can make a counter offer to S. At that time (month 2) S can either accept and split the smaller amount of money as agreed, or reject , wait until month 3 then come back with a counter-counter-offer to M. This last offer is either accepted by M, and the money available at that time shared out as agreed, or it is rejected , and they each take away nothing at all - ie revenue opportunities have completely dried up. Assume each party is interested only in their own monetary payoff. So shareholders are only interested in the share of the revenues that they receive and managers are interested in their share of the revenues as well but unless they receive at least $10k they won't put in the effort necessary to actually take advantage of whatever revenue opportunities are available. Offers and counter offers are made in units of at least $1K (ie no fractions) and assume as we did in class for "ties" that either party has to be offered something positive to not carry through to the next round in case a current offer would only just get them what they could expect to make taking the bargaining to the next round.

B3.a (6 marks) Use game theory to predict the rollback equilibrium strategies, outcomes and payoffs in this game. Briefly explain your reasoning.

There is no need to draw the game tree - think of the alternating offer bargaining game in class. Start form the end. The total rvenues available are$20k. Managers have to get at least 10k to be willing to work. If they re offerred exactly 10 k in revenues then their net gain is zero - which means by assumption they will take the game to its final stage and no one receives anything. So Shareholders have to offer them $11k, proposing to keep $9k for themselves. Now go to the second round. There are $75K in revenues. The shareholders have to be offerred at least 10k - a 9k offer will give them what they could expect in the next round, leaving 10k only for the managers in the 3rd round. But a 10k offer to the shareholders will make them better off than carrying bargaining to the 3rd round AND Managers will get 65 k in revenue or a net 55k in benefit after subtracting off their 10k effort costs. 65 vs 11k gross for Managers? They will offer 10k to the shareholders and expect them to accept this and keep 65k in revenues for themselves. Now back up to the initial round where there is $100k on the table to share. The shareholders expect that if they offer 65K or less the managers will reject - and take bargaining to the next round, expecting to get 65k in revenues there. So they need to offer $66k and keep $34 k for themsleves, and they would expect Managers to accept this. So the predicted path of play is for the shareholders to offer 66k to the managers keeping 33k themsleves and for the managers to accept that. The strategies are that IF bargaining reaches thesecond round mangers will keep 65k in revenues for themsleves offeringn 10k to the shareholders, and expect them to accept this; if the third round is reached shareholders will offer managers 11k keeping 9k for themselves and expect managers to accept this.

B3.b (5 marks) Would the shareholders agree to a change in the bargaining rules that gave "more initiative" to the managers. To be specific, would they agree to a change that gave the managers the initiative to propose divisions of the revenue at the first and last rounds (months) in this bargaining game, with the shareholders having only the option to propose an intiative at round 2? Explain briefly.

Analyse te new game. At stage 3 with 20k in revenues to divide up, shareholders will accept anyhting more than 0, so $1k will do, and managers can keep 19k, with a net payoff of 9k after subtracting off their effort costs of $10k. At stage 2 shareholders can keep managerial revenue shares down to $20k ($1k more than the $19k they can expect in the last round) and keep 55K out of the 75k total for themsleves. Then at round 1 the managers have to offer the shareholders 56k in order to just make them willing to acept and not take bargaining to the next round where shareholders could expect to get 55k . Managers in the first round would then only receive 45k in the first round. SO shareholders could be expected to support this change of rules (and mangers oppose it) - even though in the last round - if they ever reached there - they would be much more vulnerable by doing so.