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