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