RANDOM MARGINAL and RANDOM REMOVAL values SING 3 III Spain Italy Netherlands Meeting On Game Theory...
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Transcript of RANDOM MARGINAL and RANDOM REMOVAL values SING 3 III Spain Italy Netherlands Meeting On Game Theory...
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