Final exam PS 30 December 2009

Transcription

1 Fial exam PS 30 December 2009 Name: UID: TA ad sectio umber: This is a closed book exam. The oly thig you ca take ito this exam is yourself ad writig istrumets. Everythig you write should be your ow work. Cases of academic dishoesty will be referred to the Dea of Studets office, which has the power to susped ad expel studets. Partial credit will be give: math mistakes will ot jeopardize your grade. There are eight parts i this exam. Each part is weighted equally (12 poits for each part). Please show all steps of your work ad explai what you are doig at each step. Correct aswers aloe are worth othig without a clear ad correct explaatio of where the aswers come from. Clarity ad legibility are factors i the grade. If you have a questio, raise your had ad hold up the umber of figers which correspods to the part you have questios about (if you have a questio o Part 2, hold up two figers). If the TA resposible for a give questio is ot i the room at the time, work o other parts of the exam ad hold the questio util that TA rotates to your exam locatio. Whe the ed of the exam is aouced, please stop workig immediately. People who cotiue workig after the ed of the exam is aouced will have their grades pealized by 25 percet. If you eed to leave the room to use the bathroom durig the exam, please write your ame dow o the bathroom log before you leave. A perso caot leave the room more tha oce durig the exam (a perso who leaves for a secod time will be cosidered to have completed his or her exam). Please tur i your exam to oe of the TAs. Whe you had i your exam, please write your ame dow o the log. Please write all aswers o this exam if you write o the reverse side of pages, please idicate this clearly. Please tur off ad put away all cell phoes ad other electroic gadgets. Please put away all otes ad close all bags. Before you had i your exam, make sure you flip through the exam ad at least look at all questios sometimes two pages get stuck to each other ad you ca miss a etire sectio of the exam. Good luck! total

2 Part 1. Cosider the followig two-perso game. 2a 2b 2c 2d 1a 8, 5 1, 0 2, -1 0, 4 1b 4, 8 3, 6-1, 9 1, 5 1c 7, -1 5, 2 3, 0 2, 3 1d 0, 7 6, 1 7, -2 1, 2 a. Iteratively elimiate strogly ad weakly domiated strategies. Elimiate as may as possible. Show the order of deletio. (4 poits). b. Fid all pure strategy Nash equilibria. (4 poits) c. Fid all mixed strategy Nash equilibria. Please write your aswer out i words (writig p=1/7, q=5/8 is ot sufficiet). (4 poits)

3 Part 2. Cosider a world with two states who are players 1 ad 2. Player 2 has a oil field i its territory that player 1 wats. Player 1 starts off with three choices: it ca do othig (N) ad allow player 2 to keep the oilfield, it ca threate (T) player 2 with force, or it ca lauch a surprise attack (S) at player 2. If player 1 chooses to do othig, the game eds. If player 1 threates, player 2 ca either capitulate (C) ad give up the oil field (i which case the game eds), or reject (R) player 1 s threat ad prepare for war. If player 2 rejects player 1 s threat, the player 1 ca follow up by either attackig (A) player 2, which starts a war, or backig dow (B) from its threat ad allowig player 2 to keep the oil field. Both of these actios ed the game. If player 1 istead lauches a surprise attack, player 2 ca either give up (G) or try to defed (D) itself. Player 1 ideally wats player 2 to capitulate ad give up the oil field without a fight, but gettig player 2 to give up the oil field after a surprise attack is a secod best optio. If player 2 will ot capitulate or give up the oil field, player 1 prefers to attack player 2. However, if he does attack player 2, player 1 would prefer to lauch a surprise attack rather tha attack after player 2 has had a chace to prepare for war. Player 1 s worst optio is to back dow from the use of force after makig a threat, ad oly slightly better tha this is to lose the oil field by doig othig. Player 2 ideally wats to keep the oil field without a fight. It ca do this if player 1 either does othig or backs dow, but player 2 would prefer that player 1 back dow after makig a threat because it makes player 2 look strog i the eyes of the iteratioal commuity. If player 2 caot keep the oil field without a fight, it prefers to fight for the oil field. However, if player 2 must fight, it prefers to fight player 1 after it has had a chace to reject player 1 s threat ad prepare for war, istead of defedig itself from a surprise attack. If player 2 does ot fight for the oil field, the it either capitulates to player 1 s threats or gives up after a surprise attack from player 1. However, give a choice betwee the two, it would prefer to capitulate to threats aloe, sice this avoids ay coflict takig place o player 2 s territory. a. Write out this game i extesive form. For actios, use the choices i the game idicated i brackets (i.e. N, T, S, A, ad B for player 1, ad C, R, G, D for player 2). For each player payoffs, you ca use the payoff values {1, 2, 3, 4, 5, 6}. You will ed up usig all 6 values for both players. (4 poits)

4 b. Write this game i strategic form. (4 poits) c. Fid all pure strategy Nash equilibria of this game. (2 poits) d. Fid all Subgame Perfect Nash Equilibria of this game. (2 poits)

5 Part 3. Say that there are five people V, W, X, Y, ad Z, who choose amog four cadidates a, b, c, ad d. Their prefereces are show i the table below, where the most preferred is listed first ad the least-preferred is listed last. For istace, perso V likes a best ad d worst. V W X Y Z a c b c a b d c d b c b a b c d a d a d a. Is there a Codorcet wier? If so, who? (1 poit) b. Is there a ageda i which they decide o c? If there is, show oe. If ot, explai why ot. (1 poit) c. Who is the Borda cout wier? (1 poit)

6 (Part 3 cotiued) Now suppose there are three groups i society with preferece orders as show below (the best o top ad worst at the bottom). Group X has 5 people, group Y has y people, where y is assumed to be odd, ad group Z has 3 people. X Y Z a c a b b c c a b d. What is the smallest value of y for which c is a Codorcet wier? (3 poits) e. Suppose this society uses the Borda cout system. List all the possible values of y for which a is the Borda cout wier. Remember that y is assumed to be odd. (3 poits)

7 f. Depedig o the value of y, ca b be a Borda cout wier? If so, fid the smallest value of y for which b is a Borda cout wier. Remember that y is assumed to be odd. If ot, explai why ot. (3 poits)

8 Part 4. Say that there are three city coucilpeople, Trudy, Ursula, ad Victor, who make decisios by majority rule. They choose amog four alteratives, c, d, e, ad f. Their prefereces are give by the followig table. Trudy Ursula Victor Best c d e d e f e f c Worst f c d For example, Trudy likes c best, d secod-best, e third-best, ad f worst. a. What is the top cycle? (3 poits) b. Is it possible to write a ageda i which f wis? If so, write oe dow. If ot, explai why ot. (3 poits)

9 (Part 4 cotiued) Now say that you are a iterest group. By cotributig to a coucilperso s campaig, you make that coucilperso s prefereces reverse completely. For example, if you cotribute to Victor, he ow likes d best, c secod-best, f third-best, ad e worst. You ca cotribute to just oe coucilperso, two coucilpersos, or all three, but to save cash you would rather cotribute to as few of them as possible. c. By makig cotributios, is it possible to chage coucilpersos prefereces so that c is chose o matter what the ageda is? If so, which coucilperso(s) should you cotribute to, give that you wat to cotribute to as few as possible? If ot, explai why ot. (3 poits) d. By makig cotributios, is it possible to chage coucilpersos prefereces so that f is chose o matter what the ageda is? If so, which coucilperso(s) should you cotribute to, give that you wat to cotribute to as few as possible? If ot, explai why ot. (3 poits)

10 Part 5. Sister (S) ad Brother (B) are sittig aroud the fireplace ejoyig their holiday whe their mother serves them a plate of m cookies. Sice they just fiished takig a game theory class, they decide to make a game out of it. They take turs eatig the cookies. Each perso ca eat 1, 2, or 3 cookies. Whoever eats the last cookie wis. However, if you choose the same umber that your oppoet chose last time, the the game is immediately over ad you lose. This is true eve if you eat the last cookie. For example, say they start with four cookies (m=4). If Sister eats 1 cookie, the if Brother eats 1 cookie he immediately loses because he chose the same umber (1) as Sister. If Sister eats 1 cookie, the Brother ca wi by eatig 3 cookies. If Sister eats 2 cookies, the if Brother eats 2 cookies, he loses because he chose the same umber as Sister, eve though he ate the last cookie. Here each perso gets a payoff of 1 from wiig, ad a payoff of 0 from losig. For coveiece, write payoffs as (Sister, Brother). Sister always gets to go first. a. Say m=5, i other words they start with 5 cookies. Write dow this game as a extesive form game. (4 poits) b. Write dow a subgame perfect Nash equilibrium (SPNE) of this game by writig arrows i the tree you wrote above (i.e. you do t have to write dow the SPNE i words). Please make your arrows ice ad clear. If there is more tha oe SPNE, just write dow oe of them; you do t have to write dow all of them. (4 poits)

11 c. Now let m, the startig umber of cookies, go from 1 to 20, as show i the table below. For each value of m, fid out which perso wis the game i a SPNE ad write it i the table below. Remember that Sister (S) always goes first. For example, whe m=1, there is oly oe cookie at the start, ad Sister obviously wis by takig oe cookie right at the start. So the table etry whe m=1 is already filled i for you as a example. It is crucial to explai your reasoig here; simply fillig out the table is ot sufficiet without a explaatio of where your aswers come from. (4 poits) m Who wis? S

12 Part 6. Mike ad Sarah are competig i a electio to be their party's omiee for presidet. The oly issue i the electio is how may troops to sed to a peacekeepig missio i aother coutry. Their optios (i army divisios) are to sed 1, 2, 3, 4 or 5. Assume that 32% of the populatio wats to sed 1 divisio, 18% wats to sed 2 divisios, 4% wats to sed 3 divisios, 8% wats to sed 4 divisios, ad 38% wats to sed 5 divisios. Mike ad Sarah each choose to take a positio o this issue: either 1, 2, 3, 4, or 5 divisios. Oce they choose their positios, each voter votes for the cadidate whose positio is closest to their ow. If the two cadidates are equally far away from a voter's positio, the half of the voters at that positio vote for oe cadidate, ad half vote for the other cadidate. For example, if Mike takes positio 1 ad Sarah takes positio 3, the their positios are equally far away from the 18% of voters at positio 2. Thus half of these voters (9%) vote for Mike ad half (9%) vote for Sarah. Each cadidate's payoff is the percetage of votes he or she receives. a. Model this as a strategic form game ad fid all pure strategy Nash equilibria. (6 poits)

13 b. Now there is a third player, Rush. Rush has three possible actios: he ca either edorse Mike, edorse Sarah, or ot make a edorsemet. If he chooses to edorse a cadidate, the the voters act the same as before except for oe chage. Now if two cadidates are equally far away from a voter's positio, the all voters vote for the cadidate whom Rush edorses. For example, if Mike takes positio 1 ad Sarah takes positio 3, the their positios are equally far away from the 18% of voters at positio 2. If Rush edorses Mike, all of these voters (18%) vote for Mike ad oe (0%) vote for Sarah. Of course, voters at positio 1 still vote for Mike ad voters at positios 3, 4, ad 5 still vote for Sarah. Rush's edorsemet affects oly those voters who would otherwise be idifferet betwee Mike ad Sarah. As before, Mike ad Sarah's payoffs are the percetage of votes they get. Rush's payoffs are as follows. If he edorses a cadidate, he gets a payoff of 1 if he edorses a cadidate who gets 50% of the vote or more; otherwise he gets a payoff of 0. If he does ot edorse a cadidate, the he gets a payoff of 1 if Mike ad Sarah tie ad get the same percetage of votes (because he looks like a statesma) but he gets a payoff of 0 if there is a clear wier. Fid all pure strategy Nash equilibria of this game. (Hit: oe ca aswer this questio without writig dow the whole game, although of course writig dow the whole game is oe way to do it.) (3 poits)

14 c. Agai, Rush has three possible actios: he ca either edorse Mike, edorse Sarah, or ot make a edorsemet. But ow oly the 38% of voters at positio 5 care about Rush's edorsemet. All other voters do ot care. For example, say Rush edorses Sarah. If Mike takes positio 2 ad Sarah takes positio 2, the their positios are equally far away from the 4% of voters at positio 3. Sice the voters at positio 3 could care less about Rush, half of these voters (2%) vote for Mike ad half (2%) vote for Sarah. But the 38% of voters at positio 5 care about Rush, ad all 38% vote for Sarah. Rush's edorsemet affects oly those voters at positio 5, ad oly whe they would otherwise be idifferet betwee Mike ad Sarah. Mike ad Sarah's payoffs, ad Rush's payoffs, are the same as before. Fid all pure strategy Nash equilibria of this game. (Hit: agai, oe ca aswer this questio without writig dow the whole game, although of course writig dow the whole game is oe way to do it.) (3 poits)

15 Part 7. Say there are two me A ad B ad two wome X ad Y. Each perso wats to match up with a member of the opposite sex. Each perso would rather be matched with someoe tha ot have a parter at all. Each perso raks his or her potetial parters from best to worst. a. (2 poits) Say that their prefereces are give by the followig tables. A B Best X Y Worst Y X Best Worst X Y A B I other words, ma A likes woma X best ad woma Y worst ad ma B likes woma Y best ad woma X least. Woma Y likes ma A best ad ma B least. Note that woma X's prefereces are left blak. I other words, X could either like A best ad B worst or he could like B best ad A worst. Is it possible to fill i woma X's prefereces so that there exists exactly oe stable match? If so, fill i her prefereces i the table above ad show that there exists exactly oe stable match. If ot, explai why ot. b. (2 poits) Say that their prefereces are agai give by the followig tables. A B Best X Y Worst Y X Best Worst X Y A B Is it possible to fill i woma X's prefereces so that there exist exactly two stable matches? If so, fill i her prefereces i the table above ad show that there exist exactly two stable matches. If ot, explai why ot.

16 c. (4 poits) Now say there are three me A, B, ad C, ad three wome X, Y, ad Z. Their prefereces are give by the followig tables. A B C Best X Y Z Y Z X Worst Z X Y X Y Z Best C A A B Worst B C Note that woma X's prefereces are left blak. Is it possible to fill i woma X's prefereces so that there exist exactly three stable matches? If so, fill i her prefereces i the table above ad show that there exist exactly three stable matches. If ot, explai why ot.

17 d. (4 poits) Agai, say that their prefereces are give by the followig tables. A B C Best X Y Z Y Z X Worst Z X Y X Y Z Best C A A B Worst B C Note that woma X's prefereces are left blak. Is it possible to fill i woma X's prefereces so that there exists exactly oe stable match? If so, fill i her prefereces i the table above ad show that there exists exactly oe stable match. If ot, explai why ot.

18 Part 8. A revolutioary group is tryig to start a isurgecy to topple the govermet. They eed to gather eough soldiers to costitute a serious threat. However, they face two hurdles: first they eed to covice people to participate ad secod they eed to keep them i the raks oce they ve joied. Sacrificig yourself for the good of others is oble but havig someoe else die for your coutry is easier o the wardrobe. Thus, eve if you support the revolutio, there is a great temptatio to say, Why do they eed me? I ca drop out ad free ride o the isurgets success. These dyamics are depicted below i a threshold model. There are two groups i the populatio, the Sympathizers (S) ad the Moderates (M), ad eve withi these groups there is a good mix of thresholds, as show below. Lower threshold Upper threshold t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8 S 0 9 S 1 2 S 1 2 S 2 2 M 3 3 M 3 4 M 4 6 M 6 6 M 7 9 A perso ca either participate (p) i the isurgecy or ot participate (). As explaied i class, a perso participates if the total umber of other people who participate is greater tha or equal to her lower threshold ad less tha or equal to her upper threshold. a. Say that they start from a situatio (t=0) i which o oe participates. Fill i the table above to show how participatio chages over time. How may people will participate i the isurgecy at t=3? How may people will participate i the isurgecy at t=5? (4 poits)

19 (Part 8 cotiued) The isurgets ca wi if they get 7 or more soldiers. However, they will lose ad be crushed by the govermet if they ever drop below 3 soldiers after t=1. The revolutioary leaders have two strategies they ca follow to try to wi: they ca either resort to coercio to keep soldiers or give out beefits to ecourage recruits. If they resort to coercio they will raise the upper thresholds of the sympathizers by 2 but raise the lower thresholds of the moderates by 2. If they decide to give out beefits they lower the lower thresholds of the sympathizers by 2 ad lower the upper thresholds of the moderates by 2. b. Say that they resort to coercio. Agai, assume that they start from a situatio i which o oe participates. Will they wi or lose if they resort to coercio? You ca use the table below to help work out the problem. (4 poits) Lower threshold S S S S M M M M M Upper threshold t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8

20 c. Now say that they give out beefits. Agai, assume that they start from a situatio i which o oe participates. Will they wi or lose if they give out beefits? You ca use the table below to help work out the problem. (4 poits) Lower threshold S S S S M M M M M Upper threshold t=0 t=1 t=2 t=3 t=4 t=5 t=6 t=7 t=8