prelim1 solns 2011

prelim1 solns 2011 - a” Liv-33$ a glee“{1w\eee...

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Unformatted text preview: a”??? Liv-33$ a glee“ \ {1w \eee , ”W“ "men Max:333 1. (10 points total) In the context of zero—sum , two-person, extensive form games, give the relevant definition in each part below. (a) subgame lecfl 6%sz +Lk cgagmm magnet/36 ex clem‘eiww emerge 3g, “Mi all) eeccwerfl‘ ’W* VMWW%Q{3LNM& fifth; "If“ {fix/“flan: 44%ch fliez‘egewee rise ll Eweéll eff WE fagfifl % c £5.17,er Magi" mg wen e 306231491 titer we ”fl “5:"? 75m . in C} 5% @” Yemen-An” \f" , w (6) perfect information “ ggewfi A? " a may“ Wk (31/ch1994 loom. 1,113 Lime e 5L“: await gig“ o ELQ {rm C M" “fig W? 2. State Zermelo’s Theorem for win~lose games. You must include all assumptions in the hypothesis-of‘-Zermelt¥s¢PheoremYnoebbreviations. , it 3% am. 7 #5,?“ en We “peak- ALE“ 44‘" F‘“ ’ a} / g (:3 fill—“Wee {4'31 ””33 M £3; BL if? \L r [E K a fipw’i; New »« lay (gem-ills mm‘cfimk _) firm e Plibfifl’etu 6. {lo pomts total ) b‘or analyzmg nmte extensive—form games we typically rely on backward induction. The point of this question is to have you explain, carefully but briefly, the elementary step of backward induction - you do not need to explain the entire process, and you should NOT appeal to any prior knowledge of backward induction. Given a finite extensive form game G with It _>_ 2 decision nodes, explain how to reduce it to an equivalent game with k — 1 decision nodes. (Your approach must be general — it will not suffice to analyze a specific example.) How is the assumption of finiteness used in your argument? 7 we“ I .3 ' ‘ c; 9163_ @43sz (Emu its so baa J at to; _WH n We y pm, a o {,«f My Ame N 0 l1, ”$1; an a l m P (WM ‘ ;ythNh=-llfit"‘sr; "Qua M3345}?! a (Q) "a": am I rtiVfiQ-Q Wax éyf‘kgolg cl 5*? a}; W p " an C ‘ =‘H V‘iL W L} 121 i; t: a qét Lgéa‘ma b mt. T 1% 1L E E F ) {l f l aawmac’ “in a 5L Lei L ".73: 1“ x- . #Ugmaa 5%” Mara; ed A 3% amp all {:3 C32. ”.45.!th g f \¢\§J~flt CL «EM 3 % @W‘lmw‘ {J UK AF) 7%.!) {1353” El QM “g “ F¥§Mlxifl¥vbrvfifiwm WM} Al l» , xi 3 1/! 4. Consider a zero—sum two—person game Where player I’s payoiilgs are given. by the Heme 4 X 3 matrix —4 1 —1 0 —1 —2 A _ —3 0 1 l —2 —4 Based on a quick glance at the payofl matrix1 you might anticipate that the value of this game is negative. (a)- C’Grfip‘ute a maximin pure strategy for player I. he}. as, l m {Quads "‘ if fan It)?" VV‘YWw 1.3 ‘3 m .2” 2; a, “2’. ’51 3 I73; , ,3 I 1kg Lil am if C. Limefiitfi J fit” {T is: 2- I £11: :5: "2 ins”? 2 “Vb-”#2" ‘I\ ‘F‘A-l'rs 8 b r E “is iT‘th 5,133) (b) Write down a linear programming (LP) formulation of the problem of finding the value '0 of the game in mixed strategies and a. maximin mixed strategy for player I, a mixed strategy for player I that guarantees player I an expected outcome of at least 1} no matter what strategy player II selects. (The LP formulation should be written out in full detail — no summation signs or matrix notation.) DO NOT ATTEMPT TO SOLVE THE LP. Relate the optimal values of the variables in the LP, to the value of the game and to the strategy for player I. NMtHinah 7 k | ) }.. )Yiléfl Ad: e/ f “Lila “3% “i l W L a w ’2 ’3 t C34 5?: lift” i i r; #3 f i ' .r r. 2. ’3“; "l l itawé’l {by Dzé¢'\\\ 3‘5 )_,§jb{-P§?l a ,E p "(a Ag": f [‘1‘ a, max- “1%; it?!" \ J 2 . ve ! ‘ y. L U534” Gm «am L? as we gt 3,! o! x l a}; like: fitter-wt cwoc if?) ) if} ’M C» MQIQ- u fiifififix “in“ I a [,1 1 —4 1—1 1'73 0—1—2 / -3 o 1 1—2 —4 (c) It has been suggested by a usually reliable tipster that p: (O,- 3, 31-,OJi1as good mixed strategy for player I. Assuming player I uses 19— — (0, g, g, 0), give the best lower bound you can on the expected payofi to player 1. film; ”'i ”QM?” I” W131; affix "I ”i id}: 1“) C} 9 if? "the? {593% ‘1 I \‘l 3: W “11 ates/1,1111) (d) Still assuming player I uses p = (0,— 3, 3,0), and assuming that player II uses a mixed strategy vector (1, give the expected payoff to player I in terms of the components of g. If the tipster’ s suggestion of p— — (0,— 3, 3,0) is in fact optimal for I, what does that imply about the optimal choice of q for II n be as precise as possible. g” __ __ ________ 1— 111511 1» ~ u. a- W l C‘a‘l a 3135‘: "a ‘" A] .1 E4 ddddd 1 w“ 1 a ll E 5 5 j [/2 ( flirafi-LJ: IF’E‘ t? {5) l? (e) g. (:2 i=3: —4 1 —1 w 3.": 0 —1 —2 m. ‘ Evaluate the mixed strategy q = (0.5,0, 0.5] for player II, and state as specifically as possible What it reveals about the game. 1-9 iffrlr‘xém :37 Wk)! fig 5: we\ 5. (33 points total) Now consider the extensive-form two—person game in the figure below. The ordered pair on each terminal node is (payoff to I, payoff to II) at that node. In . \— (a) Explain in just a few words how you can tell that this game is NOT strictly competitive. } MN Cc. 4131.! .1? Ci: _ a; ‘1”! wi" #01 ”11.1111 4’11 H.113 1.111111 .1 i L». 9111.11.11. «K111? sides... 121: if $11,119». T. (b) Someone just suggested that he thinks the decision tree is almost correct — the only thing that IS wrong, he says, is that nodes 5, c and d should be in the same information set. He is wrong; how can you be certain of that? It” 1-4-4 Mil-Ida. .1 _ (hots is isss “211er 111.1 as flaw-111111111» if? C, L 19% 1511311111 L i F3 - \{fiw‘c If} liai-‘awln A}: 9:31 Proceed to analyze the game as drawn — the decision tree is correct.I J 9.1" 7 (c) (10 points) Use backward induction to analyze the game and determine a subgame perfect Nash Equilibrium. Show in the decision tree yourcom— plete analysis determining the value of each subgame to both players. (d) (16 points) Write out the strategic form of this game and use it to find all Nash equilibria — make sure to shortr all work. Indicate which Nash equilibria are subgame perfect and which are not. 9 6. OPTIONAL BONUS QUESTION (Adapted from Dixit and Sheath.) Two pro- posals, A and B, are under consideration by the U.S. federal government. The Congressional leadership is deciding how to proceed with them, if at all, during ' the current session. There are four possible outcomes: (1.51) A becomes law but B does not; (ug) 13 becomes law but A does not; (1L3) both A and B become law; (1&4) neither becomes law. The Congress (player I) can decide how to package the proposals and then the President (player II) has a choice of vetoing or sign— ing whatever bills the Congress passes. A bill becomes lav;r only if the Congress passes it and the President signs it. (Congress does not have enough votes on these matters to override a veto.) Congress’s preferences are ul >0 U3 >0 U4 >0 ”2 While the President’s preferences are 1132 F13 U3 >13 U4 >13 1151. Player I Will start at the root of the game by selecting from one of four choices: do nothing, pass A only, pass B only, pass both A and B as separate bills. (a) Draw the decision tree for this game and use backward induction to analyze the game and determine all subgame perfect Nash Equilibria. Shovs.r in the decision tree your complete analysis determining the value of each subgame to both players. (b) Both the Congressional leadership and the White House have game—theoretically savvy interns and both already know how this game will play out (to the“ 5%; subgame perfect Nash equilibrium you just found). However, before play has started, someone suggests a change to the game: add a branch at the root which corresponds to packaging A and B as a single bill. Modify the decision tree and repeat your analysis. “pm ' ii ‘1 i i , \ (va’iibcfiv‘ae QM? Ls: PC3335“ Alf-3“ r gggL‘ h. if“ if; rm; {{3} sarefi) I V lg ...
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