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Unformatted text preview: 1. Find the Nash bargaining model NTU solution to the bimatrix game —1,2 3,3
“’13) m {(9,5)} (5,3)] with threat point (1, 2). 2 2. Calculate the minimax strategies and the security levels for the two players in
the bimatrix game A: [ I“:
2 '3 (AB) = [ 1 {R
M. [ (1:1) (4,2) (2, 4%)]
(1: "—2) (33 _1) (“12 2) 3 3. Use the Shapley—Shubik power index to solve the 4—player weighted majority
game with Weights wl : 5, mg 2 10,103 = 25, 1.04 = 40. towing?) CU 3/ LT} $54 Lr} QB
UNI} {(3 [/3 {~13 ‘H N H02! 1
(am) :5 i} {(‘3 ch! CV) _. “Lifer: 4, 4 ﬂ! L/ I
.. I I L. w
M") 1;le Mn N V]
‘ i ‘ / N  3 e} .‘l, a???“ ﬂL [P f 11 4. Find the TU solution to the game with bimatrix and state What side payment should be made. 6"" ‘22 “1 O “2] ‘1'“ i: g CVQﬁJWW‘! WWI Cir/g I. / P... §+Rrb 5 5. Find all the Winning moves in the game of Nim with three piles of 17, 21 and 30
chips. 6 6. Prove that if both piayers use the same strategy in a symmetric game with
matrix A, then the expected payoff is 0. .‘ H Y“
Swag/(f 516:4,”wa i/M Eve—“(54¢ i" “A r 7 7. Given the 3~person game (N, v) in coalitional form with characteristic function
MED) : 0,?)(1) 2 2,1:(2) : 1,v(3) = 1,v(12) = 4,1)(13) = 5,v(23) = 2,v(N) = 7, for
what values of a: and y is the vector (113,11, 2) an imputation in the core? 8 8. Prove that if
a b
A — [a a]
has no saddle point and a 2 b, then (p, 1 — p) Where c—d
aﬂb+cwd p I
is an optimal strategy for Player I. 1 a? + (i W “4,9 w} (fa—b) <+(C~(;L))P ‘7 C “42
a __________ f? "r [th A) «(,0 Pwioééa [afﬁﬁfa 4% _‘ . _. K If a C i IL
29 S C W19» “0 3 bk (lath “ﬂipvﬂﬁf‘ I’m JAM g C 9‘ A 3wperson game in strategic form is given by the matrices (0,0,1) (—1,1,0) (4,1,4) (2,0, #1)
(230)—1) (132:0):l [ (0:190) (1:0: *1) where the left matrix is the payoffs if Player I chooses 1 and the right matrix is the
payoffs if Player I chooses 2. The rows correspond to choices of 1 and 2 for Player
11 and the columns choices of 1 and 2 for Player III. Calculate the following values
of the characteristic function of the coalitional form of this game: (a) V(N) (b) V(ll)
(c) V(I II). Cm MN) *1" HMOTK 11.7117
(t7 ‘ u 11 a: 2:2 c
we
17‘ I O l I r“ 2 o 9 I O F
Cd} 1 4
H C) C)
if a; 2K 5 \MIEWTQ
‘1 10 10. Prove that the characteristic function ’U of a game (Nw) in coalitional form
can be written as
“(T) 3 Z CS(U)wS(T)
SCN where ws is the special characteristic function deﬁned by w5(T) : 1 if S Q T and
ms (T) : 0 otherwise and 03(1)) is a constant for each S g N. La? ngho ML WW¢ gm W. 4442“ CTN) : 1/(7') “' C56”)
71? 2‘ CSLV)LOS(77 “" $4 59w " Cg'w]?
563'?
' )
c Cam “1* 2" CSW
’ 5 7:7” ...
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 Spring '10
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