prelim1 solns - 1(a(10 points Give the extensive...

Unformatted text preview: 1. (a) (10 points) Give the extensive form (decision tree) of the following perfect information 2—person game. Be sure to mark all branches and outcomes. At the root, player I chooses L or R. I’s choice of R at the root leads to a terminal node with outcome it. PS choice of L at the root leads to a decision node for player II (player 11’s only decision node) with three choices: l, m, 7". Player II’s choice of 1 leads to a terminal node with outcome 1). Player H’s choice of m leads to a decision node for Player I with two choices L, R, leading to terminal nodes with outcomes to and 3;, respectively. Player II’s choice of 7‘ leads to a decision node for Player I with two choices L, R, leading to terminal nodes with outcomes y and 2, respectively. SUBMIT THIS PART WHEN YOU HAVE FIISI-IED 1(a). ONCE YOU HAVE SUBMITTED IT TO A PROCTOR, YOU MAY OPEN YOUR. PART II EXAM BOOK AND BEGIN WORK ON IT. 1. Answer the following questions for the perfect information 2—person game drawn below. (Part 1(a) was to draw the decision tree _ if you are in this part of the LV) 1 MM“ ‘9“wa (c) (7 points) Now change Player I‘s preferences as follows: v>1u>—Im~gz>-w>-Iy Player H’s preferences are unchanged from part (b): y >11 “LU >11 ”U: h’I "U >11 33 NH Z Analyze this version of the game by backward induction. Assuming that both playerjs are rational (in the game—theoretic sense), which strategies will they select? What outcome results? 1' MMQK LEE. 2 n: was. )1 N'beljkhg M" 9“"wa Va. (d) (4 points total) This game is a very simplified version (from Straffin) of a strategic confrontation that occurred in 1962. Player I was Nikita Khr- uschev, Premier of the Soviet Union. Player II was John Kennedy, Presi» dent of the United States. Choice L at the root represented placing inter- mediate range nuclear missiles in Cuba. Choice R at the root represented not placing the missiles in Cuba. Kennedy’s choices Lm and 7", respec— tively, represented the following choices of a response to the placement of the Soviet missiles: (1) do nothing, (m) blockade Cuba, (1:) strike Cuba. The choices for the Soviet Union in response to a US action (m) were: (L) remove the missiles, (R) “escalate the conflict”. The choices for the Soviet Union in response to a US action (r) were: (L) remove the missiles, (R) “escalate the conflict”. Based on the geopolitics of that period, it is presumed that “escalate the conflic ” would lead to nuclear war. i. (2 points) Assume the preferences are as in part (b). Our assumption of perfect information implies that the two players know each others preferences. Also assume that the players are rational (in the game theoretic sense). Describe in a few words how this game ends. ii. (2 points) Repeat the previous part using the preferences from part (c). 2. (35 points total) Consider a zero—sum two-person game where player I’s payoffs are given by the 5 X 3 matrix P ow A; 3 —2 10 "'2... —1 10 5 "l 5—10 3 ”la —2 1610 «at (a) (4 points) Compute a maximin pure strategy for player I (make sure to show all steps). What is player I‘s security level in pure strategies? ”Mi, '2,» “Mr-15$} 43a mmflffi 3M \DEM» Afiflgfiw Sta l. (b) (4 points) Compute a minimax pure strategy for player 11 (make sure to show all steps. What 1s player 11’ 3 security level 1n pure strategies? MM (3 Mic-33 “fli‘g MWU Java/eel W ’3)“: Wfimgw Mg g (c) (6 points) Write down a linear programming (LP) formulation of the prob— lem of finding the value v 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 '0 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. mm: U” Ark! V’ f 3?|’\?1+€F3'1?w+i?5 V é x159, vaa'lDPg +1594 “$5 \r a lot *5?» ”i3 “Will’s 432M at. [I fiA‘:’L ©A172)§A?zalfif (d) (5 points) It has been suggested by a usually reliable tipster that 13 : (0 0, i, 0, $30 is a good mixed strategy for player I. Assuming player I uses =(0 0,14, , 0%) give the best lower bound you can on the expected payoff to player I. ’- I‘ ’P‘ 0J3,“ IS:- {LC‘W ESIIIMIIIIIIIIII M? ‘ III)?” I155 a) A 7 w a 2i 1— M a a lea-m “’1’ , f (e) (6 points) Still assuming player I uses 35 = (0,0, i0, g), and assuming that player II uses a mixed strategy vector q, give the expected payofi to player I in terms of the components of g. If the tipster’s suggestion of p = (0 0,1,0, 3) is in fact optimal for I, What does that imply about the optimal choice of q for II — be as precise as possible. 9:1" x! La: ‘5’“ ’1‘ hII {graffiti I} 53, IIIIII :III II c) ji uIIM'zgh 3374352. A WISE-)6 I; , and 4* ps: ”I I IIIA'fI fig 33% ”is I“ 3" how-d” semi") I (f) (6 points) gnu-Q) fl Evaluate the mixed strategy q“ = (0.8,0.2,0) for player 11, and state as specifically as possible What it reveals about the game. (in; F “iii: CFA.N\ fitter-JEJLQ‘ Wash '3: \A (LA (Ans MK (33- '— IQ‘E‘"; ”if"? Mat :2 JV} AL W a}. e in met ‘3‘ (g) (4 points) Suppose now that the entry 12.53 in row 5 and column 3 of A is changed from 2 to 0. Explain very carefully Why 13 = (0,0, %, 0, g) is no longer as attractive to player I. Wih’l, (15%;: C3 Aga W0 ”3* Z Lair *3 . A A: ’fi‘ ? ”‘16 A23 ..:_-. “a. '2; NW {prllffi‘fi—‘k issgg’ig) , Em l C, (W; e w l :3 w (m eat-ail ,uzm 4—— ;1 a, ”rm {was ~\~-t.u it»: swath. «w "it; ) 4 ”P A A L. :gam . I4 [I 3. (8 points) State von Neumann’s Minimax Theorem. You must include a com— plete description of both the hypothesis and conclusion, and you must define all symbols. 4. (12 points total) Suppose that G is a two-person game with a finite set U of outcomes, player I has pure strategies {31, - - - , Sm}, and player II has pure strategies {t1,- - - ,tn}. (Do not assume that G is a matrix game.) (a) (3 points) Give the definition of strictly competitive in the context of game G. 1‘3 WWW m 11. m. e» MW db 11.1., cm GUM U. mt». Luz; 115 5:. us) a“. 11,511... gluyéflw finite, (b) (4 points) Give the definition of Nash. equilibrium pair ofpure strategies in the context of game G in terms of the players preferences. iA-xhwfiqbiik MfifiemWViwatfi) WEEK"? “19594“ Tl,“ m. Aei fr; :1 (a f) :i a. “and. g WLM W Tam/1.1g ’HN— OLJICDM 'cr'aw (A f\ ,L/g £% EXACJVV‘S‘ fimVV£ASA7E 3%.).5 ‘3le m Page? ET- MUM/a QMCewL'g’T'DV‘ \ (c) (5 points) In this part assume that G' 1s strictly competitive and has mul— tiple Nash equilibrium pairs. Show why it must be that both players a1e indifferent between the outcomes that arise from the different Nash equi— libr ium pairs. Omaha \“ln Wwya imwv 6] 05nd Mumt m ism-t. (A ‘19:: Til (a ' #33 ~33 “ u " A " k ‘ Ba INK e1: {wwvh'aai'fw \ we 1 0a: “w tail. A 6 i gm agmmma a?” ._. --;r Pfirwwwm. l e 53 «LEA Mi Q Eat ‘2’?) Md wag9§t§l’q‘.\/W4«m C... :35 1:) Ekla\ gfi) CA» ijgigfl \ffi,‘ EEFSQJ “1"“? CLNIéN‘lC-Ngl? ¢ng3mguclfi§ifig E 10 . (This BONUS problem, worth 5 Oskars, is a continuation of problem 4 part (c)) Now assume that G is not strictly competitive but still has multiple Nash equi— librium pairs. Is it still the case that both players are indifferent between the outcomes that arise from difierent Nash equilibrium pairs? If so, indicate how to extend your derivation from the previous part. If not, explain what fails in your derivation from the previous part. Ylo , W Mb“ ”it? 14:3 CTZ‘rumt-EE-vwh { W cestcsé-slf; ANthi t“ that“? HVNASLW Er , M44. LJ [I ...
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