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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 FIISIIED 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 twoperson 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,» “Mr15$} 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 Mic33 “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 leam “’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" howd” semi") I (f) (6 points) gnuQ)
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\ ﬁtterJEJLQ‘ 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 eatail ,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 twoperson 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. iAxhwﬁqbiik 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 ismt. (A ‘19:: Til (a ' #33 ~33 “ u "
A " k ‘ Ba INK e1: {wwvh'aai'fw
\ we 1 0a: “w tail.
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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‘lCNgl? ¢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‘rumtEEvwh { W cestcséslf; ANthi t“ that“? HVNASLW Er , M44. LJ [I ...
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 Spring '08
 SHMOYS

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