Econ 223 TEST with some answers (in red or ** - red colour isn't always working on this page ...)

Thursday October 13, 2005

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) any time after 7pm.WHEN YOU LEAVE SIMPLY LEAVE YOUR ANSWER ON THE DESKTOP IN FRONT OF YOU – I WILL COLLECT THE ANSWER BOOKLETS IN ALPHABETICAL ORDER AFTER THE EXAM IS COMPLETED. No electronic calculators or cell-phones or PDAs etc can be used in this exam – leave them at the front of the room before the exam begins. You exam paper will be considered null and void if you are found to have these or other aids in your possession.

Section A (50 marks) 10 multi choice: 5 marks each

Section B (50 marks): 4 short answer/essay questions

Section A: Multi Choice Questions (10 questions, 50 total marks, 5 marks each)

In your answer booklet clearly identify an area for your answers to these multi choice questions.

A.1 Consider the following two-player games, where higher number represent more favorable outcomes.

game 1

				Player 2
				Action A		Action B	Action C
Player 1		Action X		4 , 4		5 , 2	3 , 6
	Action Y		6 , 1		2 , 5		1 , 3

game 2

				Player 2
				Action A		Action B	Action C
Player 1		Action X		1 , 5		6 , 3	3 , 6
	Action Y		5 , 2		4 , 4		2 , 1

When played as a simultaneous game, each of the above has a unique (pure-strategy) Nash equilibrium, in which Player 1 gets a relatively low payoff. Player 1 may thus be interested in trying to improve his payoff by employing a strategic move. (Ignore the possibility that Player 2 might counter with her own strategic move.) Complete the following. Player 1 can help himself by employing a simple unconditional commitment to choose a particular action in _____. Choose an answer from the list below and enter it in your answer booklet

(a) Game 1 only

**(b)    Game 2 only

(c) both Game 1 and Game 2

(d) neither Game 1 nor Game 2

A 2 The table below represents a 2 player simultaneous game with payoffs in dollars indicated in standard fashion:

	Column
Row	L2	M	R
T	3,5	4,7	12,4
L1	5,4	2,2	6,3
H	2,2	3,1	10,3
B	4,7	5,6	8,5

The first two steps in a process of iterated elimination of dominated alternatives would first eliminate strategy _____and second would eliminate strategy__H then R ___??(Choose an answer from the list of strategies T,L1,H,B,L2,M,R to fill in the blanks in this sentence, writing in the form {first “X” then “Y”} or write “no strategy can be eliminated” and enter your answer in your answer booklet)

A 3 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	14,9	4, Z
B	6, Z	12,6

Suppose that Column Player is able to make a strategic move to commit herself to either action X or action Y before Row Player moves. Column Player ‘s commitment is a observable by Row player and irreversible, and she is confident that Row Player will react in a rational manner according to the payoffs expressed in the table. The payoff Z can take on one of three possible values 4, 8, 10. Complete the following sentence: In this situation, Column Player would choose to commit to action X if Z = _____. (Choose an answer from the list below to fill in the blank and enter it in your answer book)

(a) 8 only

(b) 10 only

(c) 4 only

(c) either 4 or 8

(d) either 8 or 10

(**e) either 4, 8 or 10 (w/o a strategic move there are two pure strategy NE when Z=4 (A,X) (B,Y) , one when Z=8 (A,x) and a mixed strategy NE when Z=10; by committing to action X column can guarantee herself at least as high payoffs in all these circumstances) ; alternatively u can think of this question as asking just whether and when she would commit to X or to Y and given row's best responses (A toX, B to Y) the level of Z is irrelevant....

A 4 Consider a population in which each of the members repeatedly works together in pairs for a short time period, sharing equally in the output. Each player has two strategies, assertive or submissive. Each player must choose their strategy in the stage game simultaneously , but they have payoffs in the repeated game and can learn about what strategies “work” as presumed in evolutionary game theory. When two submissive types meet they produce 6, in total, ie 3 each. When two assertive types meet they produce 10 in total ie 5 each. But when a submissive meets an assertive they can only produce 2 in total, or 1 each. Given these payoffs,_____ (Choose an answer from the list below to fill in the blank and enter it in your answer book)

(a) all players adopting the assertive strategy is the unique evolutionary stable monomorphic equilibrium

(b) all players adopting the submissive strategy is the unique evolutionary stable monomorphic equilibrium

(c**) both submissive and assertive strategies are evolutionary stable monomorphic equilibria (not relevant 2006)

(d) neither submissive nor assertive strategies are evolutionary stable strategy configurations but there is a stable polymorphic equilibria consisting of an appropriate mix of the two types

(e) neither submissive nor assertive strategies are evolutionary stable strategy configurations but there is an unstable polymorphic equilibria consisting of an appropriate mix of the two types

A 5 Suppose that both players A and B in a 2x2 simultaneous game (that will be repeated multiple times) are using mixed strategies. “Using mixed strategies” means that the players are _____. Choose an answer from the list below to fill in the blank and enter it in your answer book

(a) mixing the order of play; sometimes one player moves first, and sometimes the other moves first

(b) mixing the side of the game that each person takes; sometimes one person takes the part of Player A, and sometimes that same person takes the part of Player B

(c) Both of the above are true.

(d**)   None of the above are true.

A 6 When a player has a strictly 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 in your answer book)

(a**) the payoff he would have earned had he used a different strategy

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

(c) his opponent’s payoff.

(d) Both a and c are correct.

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

A 7 Jude Law (JL) and Sienna Miller (SM) are two movie stars with a relationship problem after JL admitted having an affair with another woman (the nanny to his children from a previous marriage). JL has apologized sincerely to SM and hast stated publicly that he wishes to be reconciled with her. He has also been trying to signal to her that he will be a faithful partner in the future by spending lavish amounts of his personal time, and money on cell phone calls, flowers, trips, presents, etc for her. Naturally SM is uncertain about all of this, asking herself a string of questions. Does it mean anything to signal or not? How is all this signaling behaviour related to the underlying important, but currently unobservable to her, issue of whether JL will be faithful-in-the-future (a “good type” G) or unfaithful-in-the-future (a “bad type” B)? And what is she to do? Accept (A) or reject (R) him? What would she do if he wasn’t signaling ?.

Sienna really has no interest in the signals but has definite preferences over the combinations of her actions and JL’s type: SL’s most preferred outcome is to accept a good type and her worst is to accept a bad type…and in between but she prefers to rejecting a bad type to rejecting a good type . But she can’t observe JL’s type –she can only see whether JL signals (S) or not (NS). JL’s most favored outcome is to be a good type with SM accepting him, but his second best is to be a bad type with SM accepting him. The worst for him is being a good type, rejected. JL knows his type – and knows that SL knows he knows this and she doesn’t….etc.

Each square in the following table represents a combination of critical factors SM and JL need to consider in order to figure out what is reasonable to believe.

Complete the following. The separating equilibrium in which a good type JL signals and a bad type JL doesn’t is found in square(s) _____. (Choose an answer from below to fill in the blank and enter it in your answer book)

(a) I

(b) II

(c) III

(d)IV

(e**) I and III ( a separating equilibria requires a high enough cost of dissembling in order to give incentives for the types to separate out themselves and do different things, in which case the relevant inferences are 100% and 0%....so the prior probabilities or chances of good and bad types alone are no longer relevant)

(f) II and IV

(g) insufficient information to tell

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 ___d and h both __. (Choose your answer from the list a,b,c,d,e,f,g,h that labels each of the cells in the payoff matrixand enter it into your answer booklet)

A9 Consider a two-player repeated game between two rival mobile coffee shops located in one of New Zealand’s National Parks. The shops are run by students from specialized vans that are open 7 days a week – but only during the 4 summer months. National park rules insist that each shop post its prices simultaneously and independently in advance for a whole month, but prices can be changed on the first of each month. Each shop can set either a high or a low price for a standard cup of coffee. If they both set a high price, each receives profits of 6 thousand dollars per month. If one sets a low price while the other sets a high price, the low price firm earns profits of 8 thousand dollars per month while the high price firm earns 2 thousand dollars per month. If they both set a low price, each receives profits of 5 thousand dollars per month. Suppose that the payoffs for each player in the repeated interaction are the sum of their own monetary payoffs in each period (with no time discounting). The Nash equilibrium prediction of the payoffs for the players in this game are:

(a) 24 thousand for each firm

(b) 32 thousand for one firm, 8 thousand for the other

(**c) 20 thousand for each firm(this is a finitely repeated prisoner's dilemma...so cooperation unravels using rollback and each stage game has players playing their stage game dominant stratgeies - cooperation unravels....)

(d) 26 thousand for one firm, 20 thousand for the other

(e) 8 thousand for each firm

(f) None of the above

A 10 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 is shown below the tree may have several Nash equilibria. Some, but not all, of the entries in that payoff matrix have been filled in for you. Select as your answer to this multi-choice question the labelled cell(s) a,b,c,d,e,f,g,or h from that payoff matrix that indicates a Nash Equilibrium for this game that is NOT a sub-game perfect equilibrium, or write “none of the cells in the payoff matrix”. (Enter your answer in your answer book“**none of the cells in the payoff matrix” (B, HH is the only NE and it is SPE) )

Section B Short Questions (50 marks)

Question B1 8 marks

Draw and clearly label the game tree for the asymmetric information game between Jude and Sienna in Multi-choice question A7. Use integers 4,3,2,1 for preference ranks of the players (4 highest, 1 lowest). [you do not have to analyse the game for this question].

How many strategies does each player, JL and SM have in this game? List the strategies for each player.

Question B2 10 marks

Row player is a tax payer with three pure strategies A,B and C when filing her tax returns. Each strategy is a more or less complicated way of trying to avoid paying taxes. Column player is the tax authority with 3 pure strategies X,Y and Z for auditing tax returns, each of which is a more or less sophisticated way of detecting improper tax returns. Each combination of strategies leads to a commonly known probability p of the taxpayer successfully passing the audit and avoiding paying taxes, and the corresponding chance 1-p of failing to pass the audit, as indicated in the table below. Imagine that Row player wants to maximize her probability of passing the audit and that Column player wants to minimize this probability, ie maximize the probability of the taxpayer failing the audit.

Use game theory to predict what strategies the taxpayer and the tax authority will and will not play. Briefly explain your reasoning.

Question B3 20 marks

An high school student has the option of handing in an assignment on time (OT) or late (L). The teacher can either be strict and penalize her (P) or be nice and not penalize (N). If the game were played as a simultaneous game the rankings for the two players are given as in the following table (higher numbers indicate more preferred outcomes, student payoff is given first).

	Teacher
Student		P	N
	L	2 , 1	4 , 2
	OT	1 , 3	3 , 4

Suppose the real game is a one-shot sequential game with the student moving first.

(a) Sketch and label the game-tree for this sequential game and use rollback reasoning to predict the outcome.

(b) Imagine that the teacher tries to change the game by making a strategic move to penalize the student before the student decides. Explain the type of strategic move the teacher can make and the credibility problem(s) facing the teacher. Write a short essay explaining what the teacher might do to solve the credibility problems in this game, and how the student might counter them. A good essay will include a brief statement of all the methods Dixit and Skeath identify to deal with credibility problems as well as their applicability (or not) in this situation.

Question B4 12 marks

Granny Smith has a monopoly on iWod digital watches. Granny can make two types, a low end type without video and a high end type with video. Both types plug easily into a personal computer and allow for storage and replay of digital files, as well as for telling time. A low end type iWod without video costs $2 per unit to produce and a high end type with video costs $4 per unit to produce. Granny has 100 customers in a mixture of two types: K of them are lower income (L) users and the rest, 100-K, are wealthy (W) users. Lower income types are prepared to pay a price up to (and including) $4 for a low end iWod and $6 for a high end iWod. Wealthy types are prepared to pay a price up to (and including) $6 for a low end iWod and $9 for a high end iWod. The net payoff for customers when they buy an iWod is the difference between the most they are willing to pay and whatever price they have to pay, as long as that is not less than zero –otherwise they don’t buy . (Assume customers will buy if the price just equals the maximum they are willing to pay- ie when their net payoff is zero – but they will definitely not buy if their net payoff is negative). Granny Smith managers like to maximize profits from their products, when they know what types their customers are, and expected or profits (probability weighted average profits) when they don’t.

Complete information games

B4.1 Draw and label a game tree for the interaction between Granny Smith and a lower income customer with Granny Smith moving first, setting a price p (in integer levels 3,4,5,6,7,8,9, 10} and the lower income customer moving second (buy b, or not n) . Use your game tree and game theory to predict what price Granny Smith will charge lower income customers, and what payoffs Granny Smith and the lower income customer will receive. Granny can make Hi or Low end (2 choices) for $4 and $2 and can sell to a low income type for $6 and $ 4 respectively for the hi end and low end models, netting $2 from either model (6-4 from hi end, 4-2 from low end) for each of those choices charge a price observes, so she is indifferent about whether low income types buy hi or low end ; in either case the customer recives net payoff of zero - ie customer gets the iwod but at a price which is equal to the absolute max she was willing to pay ;

B4.2 Without drawing the game tree, use game theory to predict what price Granny Smith will charge wealthy customers in a game she plays with a known wealthy customer, and what payoffs Granny Smith and the wealthy customer will receive. Again, Granny can make Hi or Low end (2 choices) for $4 and $2 and can sell to a low income type for $9 and $ 6 respectively for the hi end and low end models, netting $5 from hi end models (9-4 from hi end) and 4 from low end models ( 6-2 from low end); so she clearly wants to sel hi end models to hi income consumers; as before in either case the customer recives net payoff of zero - ie customer gets the iwod but at a price which is equal to the absolute max she was willing to pay ;

Incomplete information game

B4.3 Suppose Granny Smith cannot observe the “type” of her customers so she cannot charge different prices to different types of customers. But Granny can charge different prices for the two different types of products, a price of x for the low-end iWod version, a price of y for the high end iWod version. Write down the incentive compatibility and participation constraints for this problem and use them to explain whether Granny can find prices x and y for the two iWod versions to create a separating equilibrium in the market.

consider low income consumers

incentive compatibility : if a low income customers buys low end products at price x, she gets a net gain of 4-x,ie her value minus the price she pays ; buying a hi end product nets her 6-y, ie her value minus the price she pays ; if we want her to find it prefereable to buy the low end prodcut we must have 4-x>6-y [ you can also say this another way : the price difference y-x must exceed the extra amount the low income customer is willing to pay: y-x>6-4 , y-x>2 ]; to make sure she does purchase the low end we need to ensure she doesn't get a negative gain, ie 4-X≥0, which is the participation constraint for a low income customer ; in words you can't charge more than 4 for a low end product and the price difference has got to be at least 2;

consider hi income consumers

incentive compatibility : if a hi income customers buys hi end products at price y, she gets a net gain of 9-y; buying a lo end product at price x nets her 6-x; if we want her to find it prefereable to buy the hi end prodcut we must have 9-y >6-x [ you can also say this another way - the price difference y-x must be less than the extra amount the hi income customer is willing to pay: y-x<9-6 , y-x<3, otherwise she'll get a larger net payoff buting the lo end product]; to make sure she does purchase the hi end we need to ensure she doesn't get a negative gain, ie 9-y≥0, which is the participation constraint for the hi income customer

put these ideas together: the highest price u can charge to the low income customer is X=4; the biggest price difference you can get away with is 3 so that means the hi end product can be priced at 7 [ think about what happens if you try to price the high end at 9 with a price differenc of 3 or 6 to the low end product...who will buy it??]

so X=4 and Y=7 are the prices that will (just) separate out the market and you make 4-2=2 and 7-4=3 profit on low end and hi end products respectively - and you sell to all 100 ;

total profits will be 2*K +3*(100-k) ; per unit profits will be a weighted avaerage between 2 and 3 depending on the proportion K/100 of low income types

Will Granny Smith find it more profitable to charge different prices for different products or to charge one price? THIS IS TOUGHER

suppose she charges 1 price - only hi end products will be bought (the customers can tell the difference so at any given price, assuming they do buy, they will get more benefit from the more valuable product ), so the seller really has to decide between a price of 6, the highest price she can charge and still have low income customers buy , where everybody buys, hi and lo income, yielding profits of 6-4 = 2 per unit or for 100 units , 2*100=200 ; but this clearly isn't as good for the seller as the separating scheme (do u see why - there her profit per unit is between 3 and 2 and she also sells 100 units ); what about charging $9 per unit, the maximum the hi income types will pay - of course at this price the lo-income types won't buy anything, so you sell less units , 100-K units, but get more per unit ($5=9-4). is 5*(100-K) more or less than 2*K +3*(100-K). the profits from the separating scheme? Clearly this will depend on how big K is. With enough hi income consumers (K small) this may be the better strategy.

How much less profit does Granny make in this asymmetric information situation compared to the situation where she can identify low-income and wealthy types (say via use of a community services ID card)? put the above answers together.... 2*K +3*(100-K) vs 2*K+4*(100-K) , total profits in the case where the seller can observe the customer's type.