Practice Exam 2019, questions PDF

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Examination Paper SEMESTER 1: UNIT:

Sample Final Examination

EFB337 – Game Theory and Applications

DURATION OF EXAMINATION: WORKING

PERUSAL -

10 MINUTES

2 HOURS

EXAMINATION MATERIAL SUPPLIED BY THE UNIVERSITY Examination Booklets

EXAMINATION MATERIAL SUPPLIED BY THE STUDENT Writing Implements Calculators – Any Type; Programmable computers are to be reset at beginning of exam

INSTRUCTIONS TO STUDENTS Notes may be made on the examination paper during perusal time. However, writing is not permitted on examination booklets, mark sense sheets, drawing paper, graph paper etc. until such time as the Supervisor so authorises. SECTION A – ALL THREE QUESTIONS TO BE ATTEMPTED. SECTION B – THREE OUT OF SIX QUESTIONS TO BE ATTEMPTED.

THIS EXAMINATION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM.

Queensland University of Technology

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Section A Attempt all questions in this section.

(25 marks) QUESTION 1 For each of the following games answer these questions: (i) What is the equilibrium if neither player can use any strategic moves? (Also find the mixed strategy equilibrium). (ii) Can one player improve his or her payoff by using a strategic move (commitment, threat, or promise) or a combination of such moves?

If so, which player makes what strategic move? (iii) Give

examples of real world situations that have a similar payoff structure. (a) Joan Rig Left Righ

ht 4, 3

3, 4

t Left

2, 1

1, 2

Ji m

(b) Column Rig Left Righ

ht 3, 3

2, 4

t Left

4, 1

1, 2

Ro w

(10 marks) QUESTION 2 Choose four of the following six terms, define each one you choose and briefly describe the significance of each one to game theory, and its economic applications: (i) Winner’s curse (ii) Salami tactics (iii)

Brinkmanship

(iv)Signaling

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(v) Separation and Pooling (vi)Screening (10 marks) QUESTION 3 Suppose there is a game called “the dollar auction game.” This game involves auctioning a dollar bill to two individuals. The rules of the game are as follows: Bidding starts at $.05 and increases in 5 cent increments. A bidder can drop out of the auction at any time by raising a white card that says “Surrender.” When this happens, the dollar bill goes to the bidder who did not drop out and the price is the amount of his last bid. The loser, however, must also pay the auctioneer the amount of his last bid. For example, assume that player 2 bids $.80, and player 1 bids $.85. Then player 2 drops out, player 1 wins the dollar for $.85 cents, and player 2 must pay $.80 to the auctioneer. Is there a Nash equilibrium for this game? If not, why not ? (5 marks)

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Section B

Answer three of the following six questions. All questions have equal weight. (75 marks) QUESTION 4 Bargaining negotiations attempt to divide surplus (excess value) that is available to parties if an agreement can be reached. Bargaining can be analysed as a cooperative game in which parties find and implement a solution jointly or as a non-cooperative game in which parties choose strategies separately. Use a simple non-cooperative, alternating offers model/ numerical example to discuss the rollback equilibrium in the following cases: (a). Surplus value decays with refusals. (b). A delay in agreement is costly to players because they are impatient, and the players involved have different rates of impatience. In this case show that the equilibrium offer shares the surplus roughly in inverse proportion to the parties’ rate of impatience. Provide real world examples to illustrate each of these situations. (25 marks) QUESTION 5 (a) Explain one of the following statements: (vii)

Sellers can elicit the true valuations from bidders using a second-

price sealed-bid auction. (viii)

If bidders are risk neutral and have independent valuation

estimates, all auction types will yield the same outcome. (b) You are in the market for used cars and see an ad for a model you like. The owner has not set a price but invites potential buyers to make offers. Your pre purchase inspection gives you only a very rough idea of the value of the car; you think it is equally likely to be anywhere in the range of $2000 to $6000. The current knows the exact value, and will accept your offer if it exceeds that value. If your offer is accepted and you get the car, then you will find out the truth. But you have some special repair skills, and know that once you own the car, you will be able to work on it and increase its value by one fourth (25%) of whatever it is worth when it comes into your hands. What is your expected profit if you offer $4000? Should you make such an offer? What is the highest offer you can make without losing money on the deal?

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(25 marks)

QUESTION 6 Do one of the following, i.e either (a) or (b): (a)

Consider a group of 50 residents attending a town meeting in Alphaville. They must choose one of three proposals for dealing with town garbage. Proposal 1 asks the town to provide garbage collection as one of its services; Proposal 2 asks the town to hire a private garbage collector to provide collection services; and Proposal 3 calls for residents to be responsible for their own garbage. There are three types of voters. The first type prefers Proposal 1 to Proposal 2, and Proposal 2 to Proposal 3; there are 20 of these voters. The second type prefers Proposal 2 to Proposal 3 and Proposal 3 to Proposal 1; there are 15 of these voters. The third type prefers Proposal 3 to Proposal 1 and Proposal 1 to Proposal 2; there are 15 of them. (i) Under the plurality voting system, which proposal wins? (ii) Suppose voting proceeds with use of a Borda count in which voters list the proposals, in order of preference, on their ballots. The proposal listed first gets 3 points; the proposal listed second gets 2 points; and the proposal listed last gets 1 point. In this situation, with no strategic voting, how many points does each proposal gain? Which proposal wins? (iii)

What strategy can the second and third types of voters use to alter the outcome of the Borda-count vote in part (b) to one that both types prefer? If they use this strategy, how many points does each proposal get, and which wins?

(b)

Discuss in detail the median voter theorem and its significance in economics. Your discussion should include an example that clearly illustrates the result/implication of the theorem (25 marks)

QUESTION 7 Do one of the following; either (a) or (b) (a) An economy has two types of jobs, Good and Bad, and two types of workers, Qualified and Unqualified. The population consists of 60% Qualified and 40% Unqualified. In a Bad job, either type of worker produces 10 units of output. In a good job a Qualified worker produces 100 and an Unqualified worker produces 0. Companies have numerous job openings of each type and must pay each type of job what they expect the appointee will produce. Also, companies cannot directly observe the workers type before hiring, but Qualified workers can signal their qualification by getting educated. The cost of getting educated to level n for a

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Qualified worker is 6n, and for an Unqualified worker is 12 n. These costs are measured in the same units as output, and n must be an integer. Find the minimum level of n that will achieve separation, and compare the outcome with the situation in which the signal is not available. Also consider an alternative cost structure, where the cost of getting educated is n 2/2 for a Qualified worker and n2 for an unqualified worker. (b)Smith and Co., a producer of hand tools, wishes to hire a researcher to speed the development of the next generation of left-handed screwdrivers. If the researcher works hard, there is an 80% chance that she will make the crucial breakthrough and allow her firm to earn $50000 in revenues and only a 20% chance that she will make no breakthrough and earn the firm no additional revenue. If the researcher shirks in her duties, there is only a 30% chance that she’ll be able to make the breakthrough. (i). Suppose that a typical contract offered by Smith and Co. specifies the wages in the event of a breakthrough and if there is no breakthrough. If the researchers can earn $20,000 in other jobs, what contract should Smith and Co. offer to a researcher? Assume that the hard effort by a researcher is equivalent to a cost of $10,000, and shirking is equivalent to a cost of $0. Also assume that Smith and Co. cannot observe the effort level of the researcher and there is no cost of effort in the other jobs. (ii). If researchers can earn $45,000 in other jobs what is the minimum acceptable level of wages and ? Should Smith and Co. offer such a contract? (iii). What is the maximum amount that researchers can earn in outside jobs such that Smith and Co. would still find it profitable to hire a researcher? (25 marks) QUESTION 8 Discuss in detail the evolutionary perspective of the one-shot, twice-repeated and thrice-repeated Prisoner’s dilemma game. Wherever applicable comment on parallels with the rational-play versions of these games. (25 marks)

QUESTION 9 Rosencrantz and Guildenstern pass the time on their voyage to England by playing the following game. Each makes an initial bet of 8 ducats. Then each separately tosses a fair coin (probability 1/2 each of Heads and Tails). Each sees the outcome of his own toss but not that of the other; Hamlet acts as the impartial referee and observes and records both outcomes to prevent cheating. Then Rosencrantz decides whether to pass or to bet an additional 4 ducats. If he chooses to pass, the two coin tosses are compared. If the outcomes are different, the one who has Heads collects the whole pot. The pot has 16 ducats, of which he himself contributed 8, so his winnings are 8

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ducats. If the outcomes are the same, the pot is split equally and each gets his 8 ducats back. If Rosencrantz chooses to bet, then Guildenstern has to decide whether to concede or match with his own additional 4 ducats. If Guildenstern concedes, then Rosencrantz collects the pot irrespective of the numbers tossed. If Guildenstern matches, then the coin tosses are compared. The procedure is the same as that in the previous paragraph, with Heads the winner, but the pot is now bigger. (a). Show the game in extensive form. (b). When this game is written in the normal form, Rosencrantz has four strategies: pass always (PP), bet always (BB), bet if his own coin comes up Heads and pass if Tails (BP), and pass on Heads and bet on Tails (PB). Similarly, Guildenstern has four strategies: concede always (CC), match always (MM), match on Heads and concede on Tails (MC), and the other way round (CM). Show that the table of payoffs to Rosencrantz is as follows.

ROSENCRANT Z

PP BB BP PB

CC 0 8 2 6

GUILDENSTERN MM MC 0 0 0 1 1 0 -1 1

CM 0 7 3 4

(c). Eliminate dominated strategies as far as possible. Find the mixed-strategy equilibrium in the remaining matrix, and the expected payoff to Rosencrantz in the equilibrium. Explain intuitively, using your knowledge of the theory of signalling and screening why the equilibrium has mixed strategies. (25 marks)

End of Paper...