Name:+
MA 103-001
Exam2
(Showwork for all problems.Problems
with littleor no workwill receivelittle or no credit.)
l.
(20pts.)For eachof the following, answertrue or false.
a. TheDivider-Chooser
methodalwaysproduces
anenvy-free
division. O , F
b. In theLone-Divider
methodit is possible
thatthereareno C-pieces, T /
e
thatis,no chosenpieces.
Theweighted
votingsystems
(D,
...,wx]and
F
[Qiw1,w2,w3,w4,
\-/
cwl,cw2,cw3,cw4,
...,
cwN]
have
always
the
same
Shapely-Shubik
[cq:
powerdistribution.
d. In a weightedvotingsystemwith 4 playersthereare20 possible
sequential
coalitions.
e. In a fairdivisiongamewith 7 playerstheLast-Diminisher
method
will have7 rounds.
I
r
t2
rud:(6€
,/o
,,o
r{:
2. (20pts.)Consider
thefollowingweighted
votingsystem[16:9, 8, 7].
a. Write down all sequentialcoalitionsand in eachcoalition identif, the pivotal player.
(?,,%Pr)
(K,be,)
(Pr,\Pr)
(?^,?u,D
j
\%rL,?r)
(P,,?r,Q)
b. Find the Shapely-Shubik power distributionfor this weightedvoting system.
35, r Ll
3 52- I
55v-l
r?
4
L1
vt
b
Y L' *€3= '
J--
b
(20 pts.)Supposethat P1and P2jointly purchasea $9 half turkey-half roastbeef foot-long sandwich. The plan is to
divide the sandwichfairly using the divider-choosermethod. Supposefurther that P1prefersturkey twice as
much as roastbeef and that Pzlikes turkey and roastbeefequally.
a. Describeexplicitly how P1cutsthe sandwichinto two shares,s1and s2.
1"4 x
fue-eb pk{{zr{derpb lt4 va{ur4 {-p&e,t
-lren 2x c\er,des
{to- va|,uo* 'lht fu*J tl6O
-{'t^cr-lwu" 2x tK ,' 3t $q + x'b
lp| ,q danoL {a
e+S
4
{d"
' IdU"t%L
piece biw bh}
't",t""{
"* (ta)=k'mJ9 - fu
,0..8',n.'$"
*,."s.::1a'*${t
ttt X &nab {d' valt-- d RE 'b Pu " I1ftn
\i +{^u- vo.[tra- 4 d^.tey as r^lelt. -flrus
Lx=$q
6r x ${,so I
\to,\ul- pg _tl,r4_e.S i,t
K
ds^"t- fl^r- PzU
U
lnt
lwtJ
gt.ce ? ct^-t.
=gA(rt) =sb
co'"Rb
qra?e
tt^L
-t{^,,"
,xb
W3
\
[2,/\
/
(2 5 p t s ) Su pp o se th a tP 1 ,P za n d P r ar etodivideahalfstr awb.,o- n' ,"5.$H
P.g
S*b"31 t,g sts fitz
VanitlaI trz
tra $ z.t
howP1makeshis initialdivisionif thepiecehe is to cut is to beentirelystrawberry.
a. Describe
A^
I BOo
g) .-{l 6- =) x = l5o'
6,
\'
Specifuthe piecefrom a) which P2takes,andthendescribehow P2would subdivideit into threepiecessL
32, Ofld 53.
eXrlatl1Y, "{fu1'r\cs
fl\n,g fl,P, Yalusshawhrrl
,Jvl
{\e
qts <qB
n*ror'tlrt
it'
*rawW-r1
\str gtecr {
{\u-
V4'*>
e
vo't'*Ldat
3\eca7
3o*#t*U
, Ya"ito' an-nc\
$2 t ,
?' roit\ c.-t
K V {,.L *YL d€ a' uon'\\t" Pt*
lA
-ll"€$ #Qa )=$-l a \- 791 -tlru,ue
?z yvLew-%
.f,ru 10" cwls r+ vani\[^,UW aMrWit {f vnn,bDo
Jtrnl(15pts) A cakevaluedat $30 is dividedamong4 players(P;,P2,P3,Pa)
usingthe last-diminishermethod. P1
makestheinitialcutandtakesclaimof thecurrentC-piece.Thefollowingtablegivesthevalueof thecurrentCpie ce in the e yesofaplayeratthetimeit is t h a t p |a y e r' s t u rn t o p |a y . @
e6'6a$'i'25
*ff'flS;oh*e
ls8m
C - ,yr.* rn^st b' r^-aa-*tdb fu*&r('
a Determine
-*n",:-!r:,,:::,,",*'
I
D
JL
b. Determinewhich player receivesa fair shareat the end of round 1.
P
'2-.
c. Let R denotethg.piecethat remainsafter round L To eachof the remainingplayersR hasvalue greater
thanor equatb$z$I\
=&fu.Wso =h-zro
1+ot r l{ov sh,are