Multiple choice Section A
A1 ( 3 marks) The game tree below represents a stylized 3 player (A, B, C) sequential game (no payoffs are shown at the terminal nodes).
A appears in 3 "situations: ine the first situation, A.1, she has 2 moves, in A.2 3 moves, and in A.3 3 moves so 2x3x3=18 is the total number of her strategies
Count the number of strategies for player A and write your answer on your multi-choice answer sheet for this question in the three blank spaces provided, selecting from the following list : 2, 3, 4, 5, 6, 8, 9, 12, 16, 18, 24, "none of these"
A2 ( 3 marks) Suppose the game tree for question A1 above is modified with an information set connecting the two nodes A.2 and A.3 for player A. How many strategies are there for player A in this new game ? write your answer on your multi-choice answer sheet for this question in the three blank spaces provided, selecting from the following list : 2, 3, 4, 5, 6, 8, 9, 12, 16, 18, 24, "none of these"
A appears in 2 "situations: ine tthe first situation, A.1, she has 2 moves, in then in the information set surrounding A.2 and A.3, where she has 3 moves; so 2x3=6 strategies. If counting strategies with information sets is not your forte, you might have a look at this short clip on uctv looking at a similar question from 2006.
A3 ( 6 marks) In many Western democracies there is now a separation of powers between the monetary authorities (the Reserve Bank) and politicians in Parliament. In general politicians in Parliament like to spend, running a budget deficit rather than a budget balance, and they like low interest rates rather than high interest rates. Politically, best of all (for them) is where they get to spend up big in the 3 years they are in office (run big deficits) and not have to pay much when they borrow -low interest rates, while worst (for them) is where they have to reign in spending and run a budget balance while voters and businesses that have to borrow are saddled with high interest rates. The Reserve Bank managers differ somewhat, but are not diametrically opposed to those of the politicians. Exact details of each player's preferences are provided in the simultaneous game payoff matrix below. Preference ranks are in letter grades with the conventional ordering (A is better than B is better than C is better than D).
Choose one of the following answers that correctly fills in the blanks : “The simultaneous game illustrated here _IS NOT___a prisoners’ dilemma because __none of these reasons___.”(Choose an answer from below and enter it on your multi-choice answer sheet) Do the best response analysis to find Red has a dominant strategy choosing to run a Deficit while Blue does not have a dominant strategy - prefers to have interest rates low if Parliament balances the budget but prefers interest rates high if parliament wants a deficit. Since only one player has a dominant strategy this can't be a prisoer's dilemma - if you check the reasoning below...in each case the explanation for why it is not a prisoner's dilemma is incorrect.
- (a) is; it has all the characteristics of a prisoners’ dilemma
- (b) is not; because neither player 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 players a mutually beneficial outcome
- (e) is not; it is a battle of the sexes game with each player prefer different equilibria
- (f) is; because Balanced Budgets and low interest rates are better for both than Deficits and High interest rates
- (g) is not, but for none of the above reasons.
A4 (6 marks) Consider the following quote from the a researcher investigating why wearing a bicycle helmet became mandatory in NZ two decades ago. “Teenagers are not likely wear bicycle helmets by individual choice for several reasons - but the most important one appears to be vanity. Teens fear the ridicule of other teens. If everyone else wears a helmet they're tempted to be "cool" by not wearing one. But if no one else wears a helmet they see themselves as a "sucker" by wearing one . Yet they seem to realize that they might all be better off wearing helmets than not, as indicated by the following, typical survey response: “It’s dumb not to wear a helmet, for safety reason.So if the school (or the police) made us all wear them we'd definitely all be better off than if we all didn't wear them. But you look like a nerd if you do! I don’t—because the other kids don’t. I know that’s silly, but most teenagers feel the same way. ” Viewing the wear helmet–don’t wear helmet choice as though it were a 2 player simultaneous game, it most closely resembles:On your multi choice answer sheet for this question write down the identifying letters that are your answer to this question The words "temptation" and "sucker" should have given the show away - this is the language used in the general idea of a prisoner's dilemma. Whether others wear helmets or whether others don't wear helmets a teens best response is not to wear helmets. ie not wearing helmets is a dominant strategy, for everyone, so is the equilibrium prediction. BUT there is a recognitio that if everyone wore helmets they'd all be better off than if no one wore helmets (which is the equilibrium strategy prediction) Presto! the prisoner's dilemma! What you have to check is that the TWO features of a prisoner's dilemma are present: (1) all players have dominant strategies and (2) there is some other feasible combination of strategies in the game where everyone could be better off - ie opportubties for commonality, pareto improvement,
- (a) a game of chicken
- (b) a pure coordination game
- (c) the textbook's tennis shot game (down-the-line or crosscourt)
- (d) a battle of the sexes game
- (e) a stag hunt game
- (f) a centipede game
- (g) a prisoner's dilemma game
- (h) a trust game.
A5 (6 marks) Before the Allied invasion of France in 1944, the Germans had to decide where to place their defenses. They had three choices: they could concentrate their defenses at Calais (GC), concentrate at Normandy (GN), or split their defenses between both locations (GS). The Allies had two choices: they could attack at Calais (AC) or at Normandy (AN). The following payoff matrix assumes this is a zero-sum, single shot, simultaneous game played with the indicated numerical payoffs (where larger positive numbers represent outcomes more favorable and negative numbers indicate losses - so less negative is better).
Assume that this game is played sequentially, with the Allies having the first move. In the rollback equilibrium outcome of this sequential game, (Choose an answer from the list below and enter it on your multi-choice answer sheet) Basically you can solve this game by constructing the game tree for the sequential version of this game, then analyse it using rollback. Here is the relevant game tree. Check the uctv video clip for a more detailed explanation. Note - for the sake of completeness, not for this exam question: If you solve the game as a simultaneous game you won't find an equilibrium in pure strategies (but there is one in mixed strategeis -) while if the Germans move first the Nash and rollback equilibrium is (AN GC ) with payoffs -> (4,-4) .
- (a) the Germans defend Calais; the Allies attack Calais
- (b) the Germans defend Normandy; the Allies attack Calais
- (c) the Germans split their defenses; the Allies attack Calais
- (d) the Germans defend Calais; the Allies attack Normandy
- (e) the Germans split their defenses; the Allies attack Normandy
- (f) the Germans defend Normandy; the Allies attack Normandy
A6 ( 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 which columns? On your multi choice answer sheet for this question write down the identifying letters of the rows and/or columns that are your answer to this question. At first cut , starting with the rows, no rows dominate or are dominared by any other rows (see uctv video clip for an explanation). But column W is dominated by Column Z. After eliminating W, B is dominated by both C and D...so can be eliminated.
A7 (6 marks) Millions of people in LDCs have died prematurely of AIDs and its' opportunistic infections in spite of the fact that there are combinations of modern drugs which, if taken regularly, can extend both the quality and length of life for HIV+ infected people. Technically the drugs are "available" to infected people in LDCs - but only at the same high prices that they are available in wealthier western countries. But how can someone in an LDC with 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 LDCs if those countries would impose and enforce restrictions on the resale of those drugs back in wealthy countries like America or Europe. But if BigPharma did actually price AIDS drug low enough to make them affordable to the LDCs, people in the LDCs 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?). This question asks you to use basic game theory to analyze the strategic interaction between the multinational pharmaceutical companies (BigPharma) and the LDCs.
Imagine that Big Pharma and the LDC's are engaged in a one-shot simultaneous game, as in the left hand side of the diagram below. Big Pharma has only two prices they can offer to LDCs, PH, price high , or PL, price low, and that LDCs 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 matrix below (higher numbers indicate more preferred outcomes for that player):
The Nash equilibrium prediction for the payoffs in the simultaneous game is_____(choose one of cells a,b,c,d) and if the game is changed so that the LDC's move first and Big Pharma second the rollback path of play is____(choose one of cells e,f,g,h ) : (write your answer on your multi-choice answer sheet in the answer space provided for this question by choosing exactly 2 letters from the possibilities a,b,c,d,e,f,g,h, ) Answer: a, h
A8 Bargaining (6 marks) Recall the Stop-Go alternating offer game from class, where two players take turns making offers about how to divide up a fixed, but shrinking, pie. In class we had the pie shrinking from 10 to 6 to 4 to 0 chocolate bars, where the shrinking occurs after one player refuses to accept a proposal made by the other, until eventually there was no pie to share out. Imagine that Apple Farmers (AF) and the pickers that work for them (W) are playing a similar alternating offer bargaining game at the start of the harvest season (ie basically the same rules about who can offer and counter offer, and when, with farmers starting) . The apple harvest looks profitable this year, but the crop goes bad fast: the available pie shrinks slowly at first, but then very quickly, from $200K to 160K to $40K to $0 (imagine a week between each bargaining interval). Assume offers and counter offers are made in units of $1K , that each player values only their own $ incomes , and that either party has to be offered a strictly positive amount to induce them to accept as compared to what they can get by rejecting and possibly taking the bargaining to the next round. 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 the space provided on your multi-choice answer sheet. "F" Here one goes to the last round, where a simple ultimatum game is being played. There is $40K to be split. The farmers can extract the most for themselves by taking $39K leaving $1K for the workers. The workers will accept rather than reject and have nothing (note - our assumption is that $1K is better than $0, even though relatively, relative to the farmers take, it is VERY small). Now, at round 2 , when workers make a proposal for dividing $160K , both players know what to expect if the Farmers reject - they can (be expected to) make $39K at the last stage - so the workers must offer them $40K, retaining $12K0 for themselves (this is the best they can do for themselves - if they try to hold out for more than $12K0 for themselves, less than $40K for the farmers, the farmers will reject and take it to the next round. So now we have worked out what to expect at rounds 3 and at rounds 2. So in the opening round, the farmers can think of $79K for themselves and $121K for the workers, out of the total $200K. Why not more for themselves? Because the outside opportunity for the workers is (expected to be) $120K by rejecting any smaller offer and taking it to the next round....in which case farmers can only expect to make $40K...wich is worse for them. Again...we are assuming no "spite" , no concern for equity - these sorts of considerations may become more relevant in a repeated game, or a different game (different assumptions about payoffs, timing, etc ).
- a) the Apple Farmers will will hold out till week 3 and exploit the workers, dividing the $40K to keep $39K for themselves and offer the bare minimum $1K for the workers, which the workers will grudgingly accept.
- b) The Apple Farmers will offer the workers $161K on round 1 , slightly more than the entire harvest available at round 2, which the workers will accept;
- c) The Apple Farmers will keep $198K for themselves in the first week and offer the worker’s $1K more than the minimum $1K they will work for in the last round, and the workers will accept, grudgingly.
- d) Same as above (c) except that workers reject this initial 198/2 split and hold out till week 2 where they propose a $80K, $80K split of the $160K proceeds , and the Apple Farmer's will accept rather than run the risk of either not getting an agreement before the season runs out or of having to divide the smaller $40K crop. [Note: typos corrected "live" in the exam]
- e) The Apple Farmers will offer to split the first week harvest evenly $100, $100 each and the workers will accept rather than delay and run the risk of get a lower share of a smaller pie later.
- f) The Apple Farmers will offer the workers $121K on round 1 , the workers will not refuse.
A8 Giving Birth (note - the question numbering here went astray...so the two A8 questions were given identifying labels) (6 marks) Giving birth can be very painful. The intensity of the pain is highly variable, but about 70% of women in childbirth choose to have some form of anesthesia, ie pain relief, before or during the birthing process (narcotics taken orally or by injection, or an injection into their spine that blocks pain signals from the uterus). Yet, as Nobel prize winning economist Thomas schelling observes. "An increasingly familiar occurrence for obstetricians is being asked by patients to withhold anesthesia during deliver of a baby. The physician often proposes that a facemask be put beside the patient who may inhale nitrous oxide as she needs it. But some determined patients ask that no such opportunity be provided: if gas is available they will use it, and they want not to be able to."The strategic situation being described here is :(Choose one answer from the list below and write the answer in the space provided on your multi-choice answer sheet) This is an example of a simple sequential trust game played between a player and their own future self - similar to the smoking addiction game discussed in the text pp 51-54 and in class.
- a) a prisoners’ dilemma game.
- b) a centipede game with a growing pie, the intensity of the mother's pain and the health of the baby, the"pie" at stake
- c) a battle of the sexes game
- d) an assurance game
- e) a trust game
- f) the female equivalent of the stag hunt game.
A9 (6 marks) The minimum effort game where the players receive a $2 salary , has a payoff table like the following one:
- a) a prisoners’ dilemma game.
- b) a stag hunt game.
- c) a battle of the sexes game
- d) an assurance game
- e) a trust game
- f) a threat game
This game is strategically similar to which of the following stylized 2 player games (choose ONE option from the list and write it on your multi choice answer sheet. )
2x2 Stag Hunt - basically an assurance game with the poorer equilibrium - 0 effort from both - being the "safe" option. Classic bureaucracy!! Have a look at the accompanying graph that blocks out a number of the options to reveal the stag hunt pattern . I gave half marks for an assurance game answer.
A10 (6 marks) Consider the situation of two
groups of firms that produce digital entertainment products: regular DVDs and
new higher definition/higher capacity optical disks. Think Sony/Philips for
BluRay style disks and Toshiba/Hitachi for HD-DVD style disks. (You don't need
to know the technical details of these systems to appreciate the strategic
interaction between the two groups of firms). The most important long run concern
for these two firms is that they coordinate their product designs and agree
on one set of technical standards (either both pick BluRay or both pick HD-DVD)
so their products are compatible with new generation of DVD type players. Such
coordination will leave both firms much better off than they would be with
no coordination.(so in the accompanying graph we want
the diagonals to have positively ranked payoffs compared to the off diagonals
where actions are not coordinated) As a secondary concern, each firm
has its own favorite product design that it would most like to see both firms
adopt. (so
in the accompanying graph we want the diagonals to have positively ranked payoffsbut
with some conflict of interest...) The game between these two firms
(in which each chooses a product design simultaneously and independently )
can be best described by saying that is it most like a Battle
of sexes (both agree that equilibria and coordination are better than not,
but there is disagreement about which is preferred) :
(Choose
one answer from the list below and write the answer in the space provided on
your multi-choice answer sheet)
(a) a prisoners’ dilemma game.
(b) a centipede game with a growing pie to share out
(c) a battle of the sexes game.
(d) an assurance game.
(e) a minimum effort coordination game
(f) a voluntary contributions (VCM) game
