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RANDOM MARGINAL and RANDOM REMOVAL values. E. Calvo Universidad de Valencia. SING 3 III Spain Italy Netherlands Meeting On Game Theory VII Spanish Meeting On Game Theory. (2) Random Removal. (3) Random Marginal. [ N ={ 1,…,n } ]. Start. [ S ={ 1,…,s } ]. Active set. Agreement. Y. - PowerPoint PPT Presentation
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RANDOM MARGINAL and RANDOM REMOVAL values
SING 3III Spain Italy Netherlands Meeting On Game Theory
VII Spanish Meeting On Game Theory
E. CalvoUniversidad de Valencia
RM-RR values SING 3
Bargaining: (1) Hart and Mas-Colell (1996)
Start [ N={1,…,n} ]
Active set [ S={1,…,s} ]i S
,S ix YAgreement
N
Breakdown
1
New active set \S i
H&MCi leaves
[ S={1,…,s} ]
RRi S
i leaves
(2) Random Removal
RM
,S iu AgreementY
Ni leaves
(3) Random Marginal
RM-RR values SING 3
\jS ix
\iS jx
jSa
\1j jS S ia x
iSa
,S ja
,S ia
Sa
Sa
1
RM-RR values SING 3
\jS ix
\iS jx
Sa
Sb
Sx
Consistent values , ,S SSa b x
(also Shapley NTU, and Harsanyi solutions)
RM-RR values SING 3
\jS ix
\iS jx0
Sx
,S id
,S jdSd
Sx
,S ju
,S iu
Su
,ix V S
,jx V S
RM-RR values SING 3
Monotonicity , ,, 0i iSS i xiiu S v d
S S Nx
RM “optimistic” , \
1 1, ,i
S S i x S ii S i S
u u S v xs s
RR “pessimistic” , \
1 10,S S i S i
i S i S
d d xs s
RM-RR values SING 3
Characterization of RM and RR values S S Nx
S-egalilitarian
(c) ,i i i j j jS S S S S Sx u x u i j S
(c) ,i i i j j jS S S S S Sx d x d i j S
, ( ) uniqueness
( ) symmetric , symmetric symmetricS S
S S S
u d V S
V S u d x
s.t.SS S N
iS
i S
(b) max : ( )i i iS Sx c c V S
S-utilitarian
Efficient (a) ( )Sx V S
RM-RR values SING 3
Random Marginal value
Hyperplane games
Consistent valueMaschler and Owen (1989)
,S ju
,S iu
S Su x
\\
1,i i i
S x S jj S i
x S v xs
TU-games , , ( ) ( \ )i ix S v S v v S v S i
\\
1,i i i
S S jj S i
x S v xs
! 1 !
( , )!
i iS
T ST i
s t tx S v
s
Shapley value (1953)
RM-RR values SING 3
Random Removal value TU-games
and ( )i i j j iS S S S S
i S
x d x d x v S
\\
1,i av i
S S jj S i
x S v xs
1
, ,av i
i S
S v S vs
! 1 !( , )
!i avS
T ST i
s t tx S v
s
Solidarity value
Nowak and Radzik (1994)
RM-RR values SING 3
\ \\ \
, ,i i i i i i j j j j j jS S x S S S k S S x S S S k
k S i k S i
x S v x x x S v x x
,i i i j j jS S S S S Sx u x u i j S
1( ,..., )n
ˆ ˆ( , ) ( ) ( , ) ( )( ) ( )
i ji i i k j j j k
i k j kk N k N
v x v xx x x x
“mass”
homogeneity
ˆ( , )( ) i i
i
vx i N
Large market games RM value value allocation (core allocation)
RM-RR values SING 3
Large market games RR value Equal split allocation
,i i i j j jS S S S S Sx d x d i j S
\ \\ \
0 0i i i i i j j j j jS S S S S k S S S S S k
k S i k S i
x x x x x x
1( ,..., )n
( ) ( )( ) 0 ( ) 0
i ji i i k j j j k
k kk N k N
x xx x x x
“mass”
homogeneity
ˆ( , )( ) i i
k
k N
vx i N