University of Canterbury Economics Department

You have 3 hours for this exam. It is a closed book exam -no notes or books in written or electronicform are to be used. Please answer all questions in the exam booklet provided. Please make yourwriting legible - I won't mark what I can't read. There are six questions totalling 100 marks :Question 1 20 basic types of games of strategic interactionQuestion 2 26 credibilityQuestion 3 12 voting gameQuestion 4 12 evolutionary gameQuestion 5 10 ice creams at the beachQuestion 6 20 5 multi choice at 4 marks each)

Question 1 (20 marks)Dixit and Skeath (DS), the author's of our textbook, classify games into a number of various "puretypes" by asking a set of questions. Identify these types and briefly explain the keyconcepts/distinctions used to interpret and understand each type.

Question 2 (26 marks)You are walking down Manchester Street late one night, alone, when a mugger wearing a black maskand holding a big knife jumps out of the shadows and says :”your money or your life”. You get tomove first in this game, either giving the mugger your wallet (G) or not (NG). The mugger stabs (S)you or not (NS), and gets to observe your move before making his move. Your best outcome is that youdon’t have to give up your wallet and you don’t get stabbed; giving up your wallet and getting stabbedis your worst outcome; and you’d prefer giving up your wallet and not being stabbed to not giving upyour wallet and being stabbed. The mugger would rather not stab you whether or not you give up yourwallet, but he also wants the wallet. His worst scenario is when you don’t give up your wallet and hestabs you and his best outcome is that you just hand over your wallet and he doesn’t stab you. As forthe other outcomes that are neither best nor worst for him, he does prefer getting the wallet andstabbing you to not getting the wallet and not stabbing you.2A (8 marks) Draw the game tree for this strategic interaction , using preferences ranks as payoffs (4is best, 1 is worst) and use rollback reasoning to analyse the game and predict the outcome.2B (15 marks) What type of strategic move can the mugger make to change the expected outcome inhis favour? Explain the credibility problems the mugger faces in his strategic move and how he mightsolve them? (A good answer will briefly list all the methods Dixit and Skeath identify and check whichmight be relevant for the mugger)2C (3 marks) What might you do, credibly, to counter any strategic move by the mugger?

Question 3 (12 marks)

The Higher Salaries Commission (HSC) determines salaries for politicians and other high rankingpublic sector officials in NZ. Imagine the HSC has 3 members, two from the North island - A fromAuckland, W from Wellington - and one from the South Island - C from Canterbury. A, W and C areall politicians themselves. Generally each of them prefers a higher salary to a lower salary other thingsequal. But in the world of politics "other things" aren't usually equal. Each HSC member well knowsthat he/she may face a political backlash when the press or television media reports the committee's

voting behaviour. If an HSC member is "seen" by her electorate to be voting to increase her own salaryher political career will be put at risk.

HSC members meet together around a small table, face to face. A minute keeper is present to keepdetailed minutes of the meeting, including a record of how each member of the committee votes.These minutes are traditionally made available publicly by posting them on committee's Internet webpage after a meeting. Suppose that after discussing the pros and cons of salary increases, the recenthistory of salary changes, inflation, and other related issues the HSC comes down to two options: eithera 5% salary increase backdated to Jan 1, 2001, or no salary increase at all this year. The actual methodfor making a decision by the committee is roll-call voting. The chairperson moves a formal motion"That the HSC recommend a salary increase" and each member can vote F="For" or A="Against" athis/her turn. Traditionally voting proceeds in order from the "top" to the "bottom" of NZ: first A thenW then C. Later voters can see how earlier voters voted before casting their votes. A simple majority isrequired for the motion to pass - otherwise it fails.

Suppose the preference rankings of each HSC member are as listed in the following table (highernumbers indicate higher ranked outcomes):

The salary raise motion passes but one's own vote is A=Against 4The salary raise motion fails and one's own vote is A=Against 3The salary raise motion passes but one's own vote is F=For 2The salary raise motion fails and one's own vote is F=For 1

3a How many strategies does each player have?3b Construct and label a game tree for this problem and find the rollback equilibrium3c Explain how you would analyse this voting situation if the committee's voting procedure

changed from roll call voting to secret ballot voting. Here, each member records her vote on apiece of paper (a ballot) in a manner not observable to others, places it in a sealed envelope, andpasses his/her envelope to the minute keeper. The minute keeper leaves the room and under thewatchful eye of an independent scrutiniser counts the votes. She reports back only whether themotion passed or failed, and only the result is publicly reported - i.e. no detail is provided on theextent of the winning margin or on how individual committee members voted.

Question 4 (12 marks)Consider a two-player game between Childlife and Backyard products, each of which produce and sellwooden swing sets for children. Each player can set either a high or a low price for a standard two-swing, one-slide set. If they both set a high price, each receives profits of 64 thousand dollars per year.If one sets a low price while the other sets a high price, the low price firm earns profits of 72 thousanddollars per year while the high price firm earns 20 thousand dollars per year. If they both set a lowprice, each receives profits of 57 thousand dollars per year.

(a) Verify that this game has a prisoners’ dilemma structure by looking at the ranking of payoffsassociated with the different strategy combinations (both cooperate, both defect, one cheats,etc.). What are the Nash equilibrium strategies and payoffs in the simultaneous-play game if theplayers meet and make price decisions only once?

(b) If the two firms decide to play this game for a fixed number of periods, say for four years, usegame theory to predict each firm’s total profits at the end of the game? (Don’t discount.)Explain how you arrived at your answer.

(c) Suppose that the two firms play this game repeatedly forever. Let each of them use a grim

strategy in which they both price high unless one of them cheats, after which they price low forthe rest of the game. What is the one-time gain from cheating? How much do you lose, in eachfuture period, after you cheat once? For what values of discount factor d or interest rate r d=1/(1 + r)) would this strategy be able to sustain cooperation between the two firms? (You maysimply apply a formula or explain your reasoning here; you do not have to carry out all of thecalculations.)

(d) Suppose that the firms play this game repeatedly year after year, neither’s expecting any changeto their interaction. If the world were to end after four years, without either’s having anticipatedthis event, what would each firm’s total profits (not discounted) be at the end of the game?Compare your answer here with the answer in part b. Explain any differences or similarities.

Question 5 (10 marks)Along a stretch of a beach, there are 500 children in five clusters of 100 each. Label the clusters A, B,C, D, and E in that order. Two ice-cream vendors are deciding simultaneously where to locate. Theymust choose the exact location of one of the clusters. If there is a vendor in a cluster, all 100 children inthat cluster will buy an ice cream. For clusters without a vendor, 50 of the 100 children are willing towalk to a vendor who is one cluster away, only 20 are willing to walk to a vendor two clusters away,and none are willing to walk the distance of three or more clusters. The ice cream melts quickly, so thewalkers cannot buy for the non-walkers. If the two vendors choose the same cluster, each will get a50% share of the total demand for ice cream. If they choose different clusters, then those children(locals or walkers) for whom one vendor is closer than the other will go to the closer one, and those forwhom the two are equidistant will split 50% each. Each vendor seeks to maximize his or her sales.(a) Construct the five-by-five payoff matrix for their location game; (eg If both vendors choose tolocate at A, each sells 85 units; If the first vendor chooses B and the second chooses C, the first sells150 and the second sells 170.)(b) Eliminate dominated strategies as far as possible and in the remaining matrix, identify all pure-strategy Nash Equilibria.(c) If the game is altered to one with sequential moves, where the first vendor chooses his location firstand then the second vendor follows, what are the locations and the sales that result in the subgame-perfect equilibrium?

Multi Choice Questions 6 (5 questions, 20 total marks , 4 marks each)In your answer book clearly indicate only one response for each of these multi choice questions

6a Each of two movie studios has to pick the week during which it will begin showing (or open) itspotential summer blockbuster movie, Harriot and the Potter for Studio A, and Bored of the Rings forStudio B. There are three possible weeks during which the movies could open. Suppose that the gamebetween the studios can be modeled as a simultaneous game and that the following matrix shows the

Complete the following. In this game between the studios, there is (are) _____ (pure-strategy) Nashequilibrium (equilibria).(a) 0(b) 1(c) 2(d) 3

6b The following table describes a two-layer game. There is one payoff that is unknown to us, butthat payoff (Z) will equal 5, 7, or 9.

X YA 8,8 Z,6B 8,8 6,10

Complete the following. Suppose that Player 2 is able to commit itself to either action X or action Ybefore Player 1 moves. Player 2’s commitment is public and irreversible, and Player 2 is confident thatPlayer 1 will react in a rational manner. In this situation, Player 2 would choose to commit to action Yis Z = _____.(a) 5 only(b) 9 only(c) either 5 or 7(d) either 7 or 9

6c. 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. A strategy is calledevolutionary stable if, when the entire population uses that strategy, no other strategy can successfully“invade” the population. Use the following information to complete the sentence given below: When adove meets a dove, the dove’s payoff is 4.When a dove meets a hawk, the dove’s payoff is 2.When a hawk meets a hawk, the hawk’s payoff is 1.When a hawk meets a dove, the hawk’s payoff is 6.Given these payoffs, _____.(a) Hawk is an evolutionary stable strategy(b) Dove is an evolutionary stable strategy(c) both Hawk and Dove are evolutionary stable strategies(d) neither Hawk nor Dove is an evolutionary stable strategy

6d Suppose that members of the military have a choice between two actions: doing their duty orelecting to duck. Suppose also that as a result of their training, the payoffs of two members of themilitary 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 likewhich of the following:

(a) a prisoners’ dilemma(b) a chicken(c) an assurance(d) None of the above is correct.

6e When a player has a dominant strategy in a certain game, we can say the following. If he usesthat strategy, then (given whatever action his opponent has taken) his payoff at the end of the game iscertainly higher than _____ is.(a) his opponent’s payoff.(b) any other payoff that is possibly available to him in the game(c) the payoff he would have earned had he used a different strategy(d) Both a and c are correct.(e) All of a, b, and c are correct.