Prelim 1 solutions - Name: N etID: Section: OR 4350 Prelim...

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Unformatted text preview: Name: N etID: Section: OR 4350 Prelim 1 Spring 2010 Be sure to give complete answers for each question. In particular, explain your reasoning for each answer. A correct answer with incomplete (or nonexistent) reasoning will receive only partial credit. (Think about what you would have to say to convince a classmate that your answer is really correct.) Part I: True (T) or False (30 points) (No reasoning is needed for T/ / l 1. In each 2-person perfect information finite game without chance moves, in which each outcome is either a Win or a Loss for Player I (and the Oppositive outcome for Player II), one of the two players must have a winning strategy. f. I 2. Every strictly competitive perfect information finite 2-person game without chance moves has a value. ‘1‘" l 3. Every strictly competitive perfect information finite 2—person game has a subgame—perfect Nash equilibrium. (J 4. Every finite 2-person game with perfect information but no chance moves in which there are only two outcomes is strictly competitive. F 5. In a 2—person strictly competitive perfect information finite game without chance moves, suppose that there is a strategy—stealing argument to prove that if Player II has a strategy that assures the outcome L (which is a win for her), then Player I also has a strategy that assures the outcome W (that is, a win for him). Then Player I has a winning strategy. ,0-1 ll 6. Consider a 2- erson strictly competitive perfect information ame P g without chance moves for which the game tree (extensive form) is not finite. Then it is not possible for Player I to have a winning strategy. i 7. Consider a 2—person strictly competitive perfect information finite game with only two possible outcomes (W or L). Let the rank of a node in a game tree be the length of the longest path to a leaf (or terminal node). Thus, terminal nodes (corresponding to outcomes) have rank 0. Suppose that each node of rank 1 is a chance node, with two branches each of equal probability, and the two branches lead to different outcomes (i.e., one is labeled W and the other is labeled L). Then the value of the game is 1/2. E 8. Consider a 2-person strictly competitive perfect information finite game without chance moves, where the outcomes are either W, D, or L (with the standard preferences); then the value of the game is D. I 9. If a strictly competitive finite 2-person game with perfect informa- tion has a unique Nash equilibrium (i.e., there is exactly one Nash equilib— rium), then it must have a unique subgameperfect Nash equilibrium. 2 f I 10. If a strictly competitive finite 2—person game with perfect infor- mation has a unique subgame—perfect Nash equilibrium, then it must have a unique Nash equilibrium. 11. (5 points) For one of the previous 10 for which you answered True, give an explanation (or cite the appropriate Theorem from class) to explain why the statement is true. . ' - L‘ ?’ (mlfiéA rdl’jfll'yfj ‘Lz‘vfer/er agjnu‘miLHeHCfi/EQC “balk 0‘? (“"9 i 311/1 Wtme 2.5le 2 5A? = 3: gem ' a - £4 a gut Mach or mi, k w flu. am J”— 910 Web a ({fi, (,1, LLLCLWCAr hush) I mi}, 01‘? Vihk {(44 Has (€60 (‘FW 6?» (J‘ELEj-fixkfl unit) 2 UL ‘Hu. Max L"! :6; 0% (‘FJ‘L G mach) i frsfliémnq y Vilma, a“? ramk k (0 {Jame—3 f“) I}; Hfimal’ Val/mi 66:6}. 12. (5 points) For one of the 10 true—false statements for which you answered False, give a simple counterexample to the statement. g. CWJMQA-A tum What? +1”; J'tralfixj .C-{wcaalafi 1 {mm [:3 Value, ‘ 1pm {£614 ’é: i; 221"". Part II: (30 points) A popular schoolyard game is the following “grid game”. The board con- sists of a 6 by 6 grid, like the one below, and the two players take turns, selecting an edge of the grid and making it bold. When the fourth side of a square is made bold, that square has been captured by the player who chose that fourth side. The player that captures the most squares wins, and it is a draw if both players capture the same number of squares. (Unlike the 1. (20 points) Explain why one of the two players must have a winning strategy. (Focus on an extremely simple, yet complete, explanation.) glint;- ‘3 an Oceal. 1% 0‘? Sidmfj 4i (straw L‘q 394363?le g3 %€4MJDJJ 6?... I male {.3 fit M. Sam Lo/o QLKWHU. Woven“) 1: Cam. WW Talia} wflwfléfl" CMmM—a MTgfzaj‘ flag 15‘) 2. (10) Give a “strategy stealing argument” that allows you to prove which of the two players has a Winning strategy. (This really has 3 subparts: state a claim about What strategy stealing means here; prove the claim; Show how this implies that a particular player has a Winning strategy. Credit will be given for the 3 subparts independently — 2 points, 5 points, and 3 points, respectively.) .' , '9‘; ( g-lr‘flfig‘fi g'lifillilg 3 F-Plcifym. I Can MIMIC, L J [3 “‘j ‘li’u ywaim m ant] Muriel éflp'lLW’kf‘ (Jun—LL jib-‘3), filmy; =g £319 seem/Lu; bum 175 gal if; afffmanex bum (03 1" . . 6 m This l-Mf’lvifl Haj—2A I [/qu amxIWELTx-tjjjéi Miami! ' ' l Meme/3 M‘s . karma) @4561 P L Q _ WWI ‘ M i [w {Amie/i Z 4 0‘? Slow”; H 9 a at can; 42% - In tub " S 5, . up (4%Awe allasmwj (Mirage: “ilk-e. C%+M[{EJ 2. “0 (WA-‘5” I“ t (0,0) all WW was no (gal (See Marta, I 6:? :chlr-d Ec‘lafl Lu: Anj £?o{_a,a_,u CcvalLur—écg, W. {Miran {Maw (3L, “4] M'l” MW. (at m «a Jen! at emf a 70‘: +0 3:; Emil.) ll (a 3 #5 C Mi ‘3 .L , lit-Mi £# Cd‘FiVM‘l “I g’f’ ‘ 63’? um M .fll’t ' ‘HM silo/t ftmclfi .filfiqfrwl J; at W l w IR {lull :5ch I 'Gicgjwj am mm mm leL/‘IL “Pl M‘ mg‘hrm. MA. 10’ win? I await q 11le I 7% MW: m' 1%. film: flu, Cir—mafia Wm , #11 “HQ, Mr Mu) MM NM: (“mt W67 W hhcé‘m" gi‘m” 4 B r WICL-flflft: I/l M I , If 6M, in» ME“ “‘1’ at M“ W “W, w mm ‘ 1,7 (mlrmlmtm W}! w | l : Part III: (40 points) Consider the following variant of a game called Gale’s Roulette. There are three wheels, and player I picks one of the wheels, starts it spinning, and while it is still Spinning, player II picks one of the remaining wheels and starts it spinning too. Wheel 1 has the numbers 2, 4, 6, and 9; wheel 2 has the numbers 1, 5, 6, and 8; wheel 3 has the numbers 3, 4, 5, and 7. Each of the four outcomes on each wheel is equally likely to be chosen as the result of spinning that wheel. The player whose wheel is spun to reveal the higher number wins. If both wheels end up on the same number, then the players must choose the same wheels and both spin again, until it is not a tie. 1. (5 points) Draw the extensive form of this game. Since this is quite large, you need not draw all of it, but you must cempletely show the initial moves of both players, as well as the complete range of possible outcomes when player I chooses wheel 1 and player H chooses wheel 2. 2. (5) If player I chooses Wheel 1 and player II chooses Wheel 2, show that the probability 39 that player I Wins satisfies p = 1/2 + (1/ 16);). 1:"L juqu’u 62“. [M waflf :15 OWL" .- _ 7 . F : Frfié. I OWN wt“? j Jflbi + Frog ,1, (MW: du/ 1 Jinn : a? wmmq» minim”; A“ I (Ma-(mm? “Jam ‘ F? I r /F—/’//— Mir/1f [in {Mimi 6%le H: owl-(W (N wmvlmlmg i .L : 2. 4H: P fl7rzg/lg 3. (5) Give the extensive form for an equivalent game in which there are no moves of chance, and each outcome is a lottery for Which you must indicate the precise probability with which player I wins. Pap‘cai‘l' wilexlm/ul will 2 41,1 I 6 E: F: A at a jig—fl szpfip :71::#_ I ii i Chum“ CLLIU'K '1.— 41‘ a. I t Ji 73’ l 3 7’ 3 s i .5. J I” IS [3th [7‘1]: 4. ’5 1C IS r'U {- é~* 7'2? gf‘g/‘F ?\7_l-u [S ,u 4. (5) Compute the value of this game, Btlal 65%}.3 4:46 17,—(2/1Mito ‘1 5&5 _Jl’/L4;Lom 619W. ":33 Value =3 67:5 (ii-MAE» “:91" “’1 QMW 3 5. (10) Give the jiirirfal (or strategic) form of this game. The following table might b ful (though you need not use all rows/ columns). Ii i ll 3 I so as 9% —.._. -—_.._' 6. (10) Indicate in the table above all Nash equilibria and at least one subgame—perfeot Nash equilibrium. Does there exist a Nash equilibrium that is not subgame—perfect, and if so, how do you know that it is a Nash equilibrium? ~ who» ’z Chi-{aw gab/i Q €N~F, Hmi Garrang (a Cdcdcri‘wvh $4 .. _ hm Anj Seatian {wiri 6:3“? Nvt~ be max M C é Mivl M W~ a ' (JD (ti/cad) i (5 3i i Iaiz'xfg 71L?” In W i Z} i k W moi” MM! T 2 m I l. L, 13(25 {Mi ‘Lw {\MJ |V\ ‘4 run.) h 1; Col/Ltmm‘ ‘( mag {gm ’3‘ M WU“ 1"1 “M l” W; Qua/qu’h 50 flickch-é] at M '10: 9‘1 (ibbqu‘CL‘LCE‘L'l [FD iii“ Dim oi {MQ Vii Ami mi P‘W 5" iiw’wr‘ M mimb a £2le Mi’iw’“ ...
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Prelim 1 solutions - Name: N etID: Section: OR 4350 Prelim...

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