ANSWERS and GRADING GUIDE

to Econ 223 TEST Thursday October 14, 2004

Instructions:

You have up to three hours to complete this exam. The start time is 6:30, the finish time is 9:30. You may turn in your exam paper and leave the examination room (quietly please) after 7pm.

Be sure to clearly print your name and student ID and sign the multi-choice answer sheet. Multi-answer sheets without names or student ID's will not be marked. Tuck this answer sheet inside the Blue answer booklet assigned to you before you leave. WHEN YOU LEAVE SIMPLY LEAVE YOUR BLUE ANSWER BOOKLET (with multi-choice answer sheet inside) ON THE DESKTOP IN FRONT OF YOU - I WILL COLLECT THE ANSWER BOOKLETS IN ALPHABETICAL ORDER AFTER THE EXAM IS COMPLETED.

You can click on the hyperlink to go to the question(s)/answers that interest you

here is the overall grade distribution to date - ie without accounting for agerotats etc:the mean is about 57

here is the distribution of webct add-on grades ; while only about 60 participated, that's still a good number, and some of the discussion interaction was terrific: thank you

Section A (55 marks) 11 multi choice 5 marks each

A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11

Section B (45 marks)

Question B1 5 marks identifying strategies

Question B2 20 marks achieving credible commitments

Question B3 10 marks labour management game

Question B5 10 marks game theory and love

A Multi Choice Questions (11 questions, 55 total marks, 5 marks each)

here is the multi-choice grade distribution for the final :

On your multi-choice your answer sheet clearly indicate only one response for each of these multi choice questions

A 1 Consider 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 on your multi-choice answer sheet)

(a) no pure strategy Nash equilibria but at least one mixed strategy Nash equilibrium

(b) 5 pure strategy Nash equilibria

(d) 3 pure strategy Nash equilibria

(e) 2 pure strategy Nash equilibria

(f) 1 pure strategy Nash equilibrium

A 2 The following table describes a two-player game. There are two cells in the following payoff matrix where a payoff is unknown to us, but that payoff, indicated by the same value Z in both cases, will equal 4, 8, or 10.

	X	Y
A	4,9	14, Z
B	6, Z	12,6

Suppose that Row Player is able to commit herself to either action A or action B before Column Player moves. Row Player ‘s commitment is a observable by Column player and irreversible, and she is confident that Column Player will react in a rational manner according to the payoffs expressed in the table. Complete the following sentence: In this situation, Row Player would choose to commit to action A if Z = _____. (Choose an answer from the list below to fill in the blank and enter it on your multi-choice answer sheet)

(a) 8 only

(b) 10 only

(d) either 8 or 10

see game tree below (you dont really need to use the game tree to do this); if Z is 4, column will choose X against A and Y against B leaving row with payoffs 4 from A and 12 from B - obviously row doesn't want A. if Z is 8, column will choose X against A and X against B leaving row with payoffs 4 from A and 6 from B - obviously row doesn't want A. FInally if Z is 10, column will choose Y against A and X against B leaving row with payoffs 14 from A and 6 from B - obviously row does want A and would choose to commit to A under those circumstances

A 3 Consider a population in which each of the members of the species may be either aggressive (and are called Hawks) or passive (and are called Doves). Every period, pairs of the species are (randomly) matched together, they interact, and each receives a payoff. Use the following information to complete the sentence given below: When a dove meets a dove, the dove's payoff is +2. When a dove meets a hawk, the dove's payoff is 0. When a hawk meets a hawk, the hawk's payoff is -1 (negative 1). When a hawk meets a dove, the hawk's payoff is +6.

Given these payoffs, _____. (Choose an answer from the list below to fill in the blank and enter it on your multi-choice answer sheet)

(a) all players adopting the Hawk strategy is evolutionary stable in a monomorphic equilibrium

(b) all players adopting the Dove strategy is evolutionary stable in a monomorphic equilibrium

(d) neither Hawk nor Dove is an evolutionary stable strategy and there is no stable polymorphic or monomorphic equilibrium

Answer is c; reasoning follows the ideas in the graph below: you need to sketch the expected (probability weighted average) payoffs to each strategy, H or D, to figure this out; to the right of p* Dove is a better strategy on average against so many hawks....implying the proportion of the population acting like hawks will decrease; to the left of p* hawk is a better strategy on average so proportion of population using it will rise.

A 4 Suppose that members of the military have a choice between two actions: doing their duty or electing to duck. Suppose also that as a result of their training, the payoffs of two members of the military are expressed (larger numbers representing more favorable outcomes) as

The strategic interaction between these soldiers can be best described by saying that it is most like which of the following: (Choose an answer from the list below and enter it on your multi-choice answer sheet)

(a) a battle of the sexes game

(b) a chicken game

(d) a pure coordination game

(e) an entry deterrence game

(f) None of the above THIS IS A PRISONER"S DILEMMA GAME

A 5 When a player has a dominant strategy in a certain game, we can say the following. If he uses that strategy, then (given whatever action his opponent has taken) his payoff at the end of the game is certainly higher than _____ is: (Choose an answer from the list below to fill in the blank and enter it on your multi-choice answer sheet)

(a) the payoff he would have earned had he used a different strategy (remember the statement is assuming the opponent has a fixed strategy)

(b) any other payoff that is possibly available to him in the game

(d) Both a and c are correct.

(e) All of a, b, and c are correct.

A 6 Consider a world in which there are two rival species- A and B. Some proportion of species A is weak; the rest is strong. There is some feature that all strong A's naturally possess; at some cost to himself, a weak A can also dissemble and signal this feature. Each member of A chooses whether to challenge a member of B for valuable territory. Strong As always challenge. If the B player fights back, a strong A wins the fight, and a weak A (whether signaling or not) loses. A B player cannot tell the difference between a strong A and a weak A who is signaling merely from observation. A weak A in this scenario has to make a decision [back down or (signal and challenge)], and a B player that is challenged also has to make a decision [fight or retreat]. Two critical factors influence the behavior of weak As and of Bs-the cost to a weak A of signaling, and the proportion of As who are weak. Depending on these factors, a weak A will always, sometimes, or never signal and challenge, and a B will always, sometimes, or never fight. Each square in the following table represents a combination of those two critical factors.

Complete the following. The pooling equilibrium in which weak As always signal and challenge and Bs always retreat is found in square _____. (Choose an answer from below to fill in the blank and enter it on your multi-choice answer sheet) cont'd on next page….

(a) I

(b) II ANSWER (read ch 9 if you want an explanation)

(d)IV

(e) insufficient information to tell

A 7 The 2 player payoff matrix below represents a simultaneous game that can be solved by the method of iterated elimination of dominated alternatives. The first step in this process would eliminate which strategy: T, H, A, B, L, M, or R? (Choose an answer from the list {T, H, L, B, L, M, or R} and enter it on your multi-choice answer sheet)

A 8 The following payoff matrix describes a 3 person simultaneous game. The three players are R, B, and G. Each player has two moves: {T, B} for R, {L, R} for B, and {Yes, No} for G. Payoffs are in dollars, listed in order from left to right first for R, then B then G.

The Nash Equilibrium prediction(s) for the outcome of this game is/are _____. (Choose your answers -there may possibly be more than one - from the list a,b,c,d,e,f,g,h that labels each of the cells in the payoff matrix)

A 9 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). Assume that this is a zero-sum game and that the possible outcomes are ranked as in the following matrix (where larger positive numbers represent outcomes more favorable for the Allies).

Assume that this game is played sequentially, with the Germans' 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)

(a) the Germans defend Calais; the Allies attack Calais

(b) the Germans defend Calais; the Allies attack Normandy ANSWER [draw a game tree with Germans moving first , allies second -the rollback]

(d) the Germans split their defenses; the Allies attack Normandy

(e) the Germans defend Normandy; the Allies attack Calais

(f) the Germans defend Normandy; the Allies attack Normandy

A 10 Suppose that a certain society consists of only two people, A and B, and that each of these drives their own car to work. Suppose that each person could voluntarily choose to put a pollution control device on his or her car at a cost of $100 per car that has to be paid for out of personal income (each has enough income) , but the improved quality of the air is experienced by both A and B. If one car gets a pollution device installed, A values the improvement in her clean air at $80 and if two cars have pollution devices installed she values the improvement in clean air at $150. B has the same preferences as A about clean air. Each player's net payoff subtracts off their own personal cost of installing pollution devices, if any, from any value of clean air that they receive. Each person in this society has two possible actions-Install or Not install. Assuming this game is a simultaneous, one-time game, how would you describe this game? (Choose an answer from the list below and enter it on your multi-choice answer sheet)

(a) a battle of the sexes game, with each player preferring the equilibrium where the other installs a pollution device

(b) a chicken game, with each player preferring the equilibrium where the other installs a pollution device and if they don't coordinate there is a bad outcome for both

(c) an assurance game, where they both prefer that two cars have pollution devices installed to an equilibrium where only one car has a pollution device installed

(d) a pure coordination game where they both realize it is too expensive to install pollution devices on both cars but they prefer equilibrium, where either one or the other has a pollution device installed, to a non-equilibrium

ANSWER (e) a prisoner's dilemma game with an equilibrium where neither installs a pollution device but both would be better off if both did pay for and install pollution devices. See theGraph below - the payoffs reveal it is a prsioner's dilemma game

(f) None of the above

A 11 The following game tree diagram describes a 2 player sequential game between player R and player B, each having two moves. The payoff matrix associated with the analysis of this sequential game (shown below the tree) has several Nash equilibria. Select as your answer to this multi-choice question the labelled cell a,b,c,d,e,f,g,or h from that payoff matrix that indicates the Nash Equilibrium for this game that is NOT a sub-game perfect equilibrium ANSWER f.

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Section B Short Questions (45 marks)

Question B1 5 marks

1. The game tree below represents a stylised 3 player (A, B, C) sequential game (no payoffs are shown at the crosshatched terminal nodes). How many strategies does each player, A, B and C, have in this game? List the strategies for each player.

A appears in 2 situations a.1 and a.2 and has 2x3=6 strategies ut,um,ub,dt,dm,db

B appears in 2 situations b.1 and b.2 and has 2x2=4 startegies uu,ud,du,dd

C appears in 2 situations, the first is the information set {c.1,c.2,c.3} which you treat as one big node with 2 actions possible, then at c.4 where she has 3 actios: 3x2=6 uu,um,ud,du,dm,dd

Grade advice: you can see from the grade hstorgram that 1,3 and 5 were the most frequent marks. The 3's mostly came becasue one of the sets of strategies was incorrectly described, usually for player A: just becasue A choosing "u" at her first node means the game doesn't ever arrive at her second node a.2 doesn't mean that a strategy will ignore what she can possibly do at a.2: stargeies have a lot of built in redunacny "what-if" type reasoning...; rememebr even if this information in A's stratgey doesn't seem useful for player A, it may be for other players: other players may want to reason about what A will do when and if A ever did get to a.2

Question B2 20 marks

In a scene from the movie Manhattan Murder Mystery, Woody Allen and Diane Keaton are at an ice hockey game in Madison Square Garden. She is obviously not enjoying herself, but he tells her: "Remember our deal. You stay here with me for the entire hockey game, and next week I'll come to the opera with you and stay until the end." Later in the movie, we see them coming out of the Metropolitan Opera House while inside the music is still playing. Keaton is visibly upset and says, emotionally: "What about our deal? I stayed to the end of the hockey game, and you were supposed to stay till the end of the opera." Allen answers: "You know I can't listen to too much Wagner. At the end of the first act, I already felt the urge to invade Poland."

Write a short essay commenting on this scene from a strategic perspective. A good essay will include an explanation of methods that may be used to improve the credibility of strategic moves generally, and in this specific setting between Keaton and Allen. Organise your essay and write no more than 3 double spaced pages. Be clear and concise.

Grade explanation

here are the grade distributions for this question: in general a full 10 point list with a 1-2 sentence explanation of each point + a short description of why credibility was a problem was worth 15; then extra marks were added according to how well you applied these ideas to the promises or threats or committments that might solve some of the credibility issues in this game; the modal mark was 20 with 15 the next most frequent- most of you did a good jobe listening to what i was asking you to do, reading the material, organising what you were going to say, and doing it. Thank you!

Question B3 10 marks

The labor union and the management of a company are submitting a wage dispute to an arbitration board. The board is to consist of one representative of each side and a chairperson. The game is between the union and the management, and the strategy for each is to choose its own representative on the board. Under the guidance of the arbitrator and the rules of arbitration there will be negotiations and eventually an outcome in the form of a wage increase, measured in dollars per week. Each side has four possible candidates, with different degrees of commitment to their own side and different degrees of aggression/skill in pursuing their side's case. We will label each side's candidates as unbiased (U), biased (B), soft spoken (S), and hard line (H). We will also label the sides as labor (L) and management (M); thus labor's choosing a hard-line nominee will be strategy LH, management's choosing an unbiased nominee MU, and so on. The payoff matrix for this simultaneous, zero-sum game is known to both players and is as follows in Table 1. You will see that generally the union wants large wage increases and the management wants small ones, but note that a hard-line or nominee or one biased in favor of one's own side is not always an advantage: for example, that type of nominee may alienate the chair-person, making her less inclined to seriously consider that representative's argument.

Use game theory to predict what is NOT likely to happen in this game as well as what the outcome of this game is likely to be.

Table 1

You can use iterated elimination of dominated strategies to figure out some strategies that won't be played namely LU and LH for labour as they are dominated by LB. Once these are eliminated, MU and MH can be elimnated for management, becasue they will be dominated...assuming LU and LH aren't expected.

This leaves us wuth a 2x2 game that has no pure strategy Nash equilibrium, so we expect some sort of mixed strategy nash equilibrium.

You do not have to calcluate these mixed strategy NE but you can show how it would be calcluated by sketching the payoff functions for each player against the other player's p and q mix..., findoing best responses (shaded sections) then the equilibrium mixes p* and q*...as illustrated in the graph.

advice on grades for B3:

5-6/10 if you used iterative dominance and recognised there was no Nash Equilibri in pure stratgeies but didn't make any attempt to discuss mixed strategies; if you then proceeded with minimax that was typically worth 8/10; unfortunately most people (see the grade for 5-6 below) simply didn't consider the possibility of mixed strategies - it was only a small part of the exam, 4 points, but i was hoping..... :-)

2-3 if you used best response methods to figure out there was nop Nash equilibrium...but weren't using dominated alternatives reasoning

grade distribution for B3

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Question B4 10 marks

This passage of text should be familiar to you. It comes from Chapter 1 of our text.

One upon a time in New York City there lived a man and a woman who had separate rent-controlled apartments, but their relationship had reached the point at which they were using only one of them. The woman suggested to the man that they give up the other apartment. The man, an economist, explained to her the fundamental principle of his subject: it is always better to be flexible and have more choice available. The probability of their splitting up might be small but, given even a small risk, it would be useful to retain the second low-rent apartment. The woman took this very badly and promptly ended the relationship.

Dixit and Skeath (DS) comment:

"Words are cheap anyone can say, "I love you". If the man had put his property where his mouth was and had given up (sold) his rent-controlled apartment that would have been concrete evidence of his love. The fact that he refused to do so constituted hard evidence of the opposite and the woman did right to end the relationship".

Imagine you are a therapist for the economist, visibly upset at being dumped by his lady friend. You advise your grieving client to look at the situation through the eyes of a game theorist using the ideas of perfect Bayesian equilibrium, separating/pooling equilibria. At first your client looks at you through his tear stained eyes as if you're the crazy one and in need of a therapist. But you ease his pain by suggesting to him that once he figures this out for himself he should be able to go back and talk to his girlfriend/partner, explain to her that the (strategic) situation is a bit more complicated than she was imagining and ask her to reconsider her understanding of this game they are playing. Write a short, clear essay about what the economist could say to his girlfriend, credibly, as a game theorist to try to change the outcome of their romantic interaction. Organise your essay, be clear and concise, and write no more than 3 double spaced pages.

Grades for B4

YOU CAN SEE FROM THE GRADE DISTRIBUTION THAT RESULTS ON THIS QUESTION WEREN'T SO GOOD

Some of you wrote some very insightful short essays, but most didn't really have much of a clue as to how to adress this question. I had suggested in class that a concept (not calcuation) type essay on precisely this text would be how i would examine the asymmetric info material...but either the message didn't get htough or i didn't spend enough time getting it through (although the issue did come up in the extra help sessions...before the exam). Perhaps start with definitions of PBE (beliefs, inferences, and best responses) , pooling equilib, separating equilib? emphasiaze beliefs and inferences (by bothplayers), best response thinking to beliefs and stratgeies, the roles of beliefs about types, strategies conditional on type, and costs of dissembling in figuring out when pooling and separating equilibria might be likely, and then incorporate the idea that multiple equilibria can exist and that there might be real coordination problems (just as in any coordination rpoblem like chicekn or battle of sexes or assurance where one player is thinking one equilibrium and the other is thinking another...when deciding on their stratgeies..)) when players chose strategies from different nash equilibria; at the end of the day one has to ask: if actions speak louder than words then what do these actions, embedded in strategies and beliefs, have to say??

END OF TEST