Axiomatizing the Shapley Value without Linearity

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W O R K I N G P A P E R

AXIOMATIZING THE SHAPELY VALUE WITHOUT LINEARITY

I H.P. Young

July 1982 WP-82-64

I n t e r n a t i o n a l I n s t i t u t e for Applied Systems Analysis

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NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

AXIOMATIZING THE SHAPLEY VALUE WITHOUT LINEARITY

H.P. Young July 1982 WP-82-64

Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

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AXIOMATIZING THE SHAPLEY VALUE WITHOUT LINEARITY

H.P. Young

This paper shows that the Shapley value can be uniquely characterized by a simple monotonicity condition without re- sorting to either the linearity or dummy assumptions originally used by Shapley [ 4 1 .

Let N be a fixed finite set. By a _game in characteristic function form is meant a function v which assigns to every sub- set S of N a real number v(S), called the value of S , such that v(@) = 0, and for all disjoint S and T

A v a l u e is a function '4 : V -+ 1~~ where V is the set of all

N N

games on N. Following Shapley we say that P is syrnrnetr-ic if for every permutation of N we have

where ITV is the game defined by (ITV) (S) = ITS) for all S. The value P is e f f i c i e n t if Tvi(v) ₌v(N) for all v.

N

M o n o t o n i c i t y compares a player's inherent claims in different games on the same set N. One formulation of this property, due

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t o Megiddo [ 3 1 , i s t h a t i f two games v a n d w d i f f e r o n l y i n t h a t v ( N )

-

> w ( N )

,

t h e n w e s h o u l d h a v e 'fi ( v )

-

> 'fi ( w ) f o r a l l i. I n o t h e r w o r d s , i f i n p a s s i n g from w t o v t h e c l a i m s o f a l l p r o p e r c o a l i t i o n s s t a y t h e same w h i l e t h e t o t a l amount t o b e d i s t r i b u t e d i n c r e a s e s , t h e n n o p l a y e r ' s a l l o c a t i o n s h o u l d d e c r e a s e . The

a n a l o g o u s c o n c e p t a l s o a r i s e s i n a p p o r t i o n m e n t [ I ] and b a r -

g a i n i n g t h e o r y [ 2 ] . Megiddo shows by example t h a t t h e n u c l e o l u s i s n o t m o n o t o n i c i n t h i s s e n s e .

The f o l l o w i n g s t r o n g e r f o r m u l a t i o n a l l o w s a c o m p a r i s o n o f p l a y e r s ' a l l o c a t i o n s u n d e r more g e n e r a l c h a n g e s i n t h e s t r u c - t u r e o f t h e game. F o r a n y game v and p l a y e r i ' t h e d e r i v a t i v e o f v w i t h r e s p e c t t o i i s t h e f u n c t i o n v i ( S ) d e f i n e d f o r a l l S L N

s u c h t h a t

i i

The v a l u e 'f i s s t r o n g l y m o n o t o n i c i f whenever v ( S )

2

w

( s )

f o r a l l S t h e n Vi(v)

2

Pi ( w )

.

I n o t h e r w o r d s , i f i n p a s s i n g from w t o v , i t s m a r g i n a l c o n t r i b u t i o n t o e v e r y s u b s e t i n c r e a s e s o r s t a y s t h e same, t h e n i ' s a l l o c a t i o n must n o t d e c r e a s e . I n p a r t i c - u l a r s t r o n g r n o n o t o n i c i t y i m p l i e s m o n o t o n i c i t y i n ~ e g i d d o ' s s e n s e .

I t i s c l e a r t h a t t h e S h a p l e y v a l u e i s s t r o n g l y m o n o t o n i c , s i n c e it may b e w r i t t e n

T h e o r e m . T h e S h a p l e y v a l u e 5 s t h e u n i q u e s y m m e t r i c and e f f i c i e n t v a l u e t h a t i s s t r o n g l y m o n o t o n i c .

P r o o f . F i r s t n o t e t h a t s t r o n g m o n o t o n i c i t y means t h a t f o r a n y two games v , w E V N ,

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Next consider the symmetric game w on N which is identically zero on all coalitions, so that w (S) i = 0 for all iIS. BY SYm- metry 'fi(w) = 'fj (w) for all i # j and by efficiency l'fi(w) = 0

,

hence for all i, 'fi(w) = 0. By (1) it follows that for any game N v on N and any i E N ,

( 2 ) i

v (S) = 0 YS implies Pi(v) = 0

.

That is, dummy players get nothing.

We now exploit the fact noted by Shapley that every game v can be expressed as a sum of p r i m i t i v e games

where

The Shapley value can be expressed Pi(v) =

I

^yi(cRvR)⁼

I

c ~ / ( R ~ .

~#R=N r . - R: i€R

A game u is s g n m e t r i c if for all i#j there is a rermutation.~taking i to j such that u(nS) = u(S) for all S. Letting cr = max c

R: 1 R I =r ^R

-

- c a n d u =

1

C I ~ I V ~ , v c a n b e for 1 < r < n, cR = ClR1

R I

@#LY.

rewritten in the form

where

cR 2

0 YR and u is symmetric. Define the i n d e x I of v to be the minimum number of terms with E R > 0 in some expression for v of form (4). The theorem is proved by induction on I.

If I = 0, v = u is symmetric so 'Qi(v) = V.(V) for all i # ^j, I

whence by efficiency 'fi(v) = v(~)/n, which is the Shapley value.

If I = 1, v = u

-

² v for some R ~ N . For i g R , v (S) i = R R

ui(s) for all S, hence by monotonicity

Pi

(v) = u (N) /n. By symmetry

Pi

(v) = P . (v) for all i, j E R; combined with efficiency this says that 3

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u ( N ) / n - e R / I R I f o r i E R

u ( N ) /n f o r i @ R

which i s t h e S h a p l e y v a l u e f o r v .

Assume now t h a t v ( v ) i s t h e S h a p l e y v a l u e whenever t h e i n d e x o f v i s a t most I . I n p a r t i c u l a r t h i s means t h a t i f v = u

-

l a

^v ^{t h e n}

k=l Rk Rk

L e t v have i n d e x 1+1 w i t h e x p r e s s i o n

1+1

L e t R = n Rk and suppose i e R . D e f i n e t h e game k= 1

and n o t e t h a t w i s s u p e r a d d i t i v e . The i n d e x o f w i s a t most I and w i ( s ) = v i ( s ) f o r a l l S f s o u s i n g i n d u c t i o n i t f o l l o w s t h a t

t h e l a t t e r s i n c e P i ( C R vR ) = 0 whenever i

4 R

by ( 2 ) . But t h i s

k k k '

i s j u s t t h e S h a p l e y v a l u e f o r i.

I t r e m a i n s t o show t h a t Vi(v) i s t h e S h a p l e y . v a l u e when

1+1 1+1 ^-

i E R = n

~ i ,

i . e . , t h a t

vi

^{( v )} ⁼

^{p i}

^{( u )}

- 1 ^s .

S i n c e v i s

k= 1 k=l Rk

1+1 symmetric on R it s u f f i c e s t o show t h a t

1

^{( ' f i}^{( u )}

-

^lpi ^{( v )}⁾ ⁼I RI

1

^S

.

R k=l Rk

T h i s f o l l o w s by o b s e r v i n g t h a t

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1

^(pi(u)

-

^(Pi(v))=

,I ,I

^yi(eRkvRk¹

N-R k=l i€N-R

the latter since

R C R ~

for all k.

Thus by efficiency

Observe that the proof only requires the assumption that a player's value depends just on the vector of his marginal contri- butions. This condition is also implicit in Shapley's axiom

i i

scheme, which requires that dummies get nothing: if v ( S ) = w (S) for all S then i is a dummy in (v-w) so 'fi(v-w) ₌ 0: combined with linearity it follows that 'Pi(v) ₌ Vi(w). On the other hand, the condition that vi (S) = w i ( S ) implies

'Pi

(v) = Pi(u) is much weaker than the dummy and linearity axioms, indeed seems only slightly stronger than the dummy axiom itself. What we have shown is that by taking full advantage of efficiency and symmetry we can use it to deduce linearity and effectively characterize the Shapley value.

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REFERENCES

[ I ] M.L. B a l i n s k i a n d H.P. Young, F a i r R e p r e s e n t a t i o n . New Haven, Conn: Yale U n i v e r s i t y P r e s s ( 1 9 8 2 ) .

[ 2 ] E . K a l a i a n d M . S m o r o d i n s k y , " O t h e r s o l u t i o n s t o N a s h ' s b a r g a i n i n g p r o b l e m " , E c o n o m e t r i c a , - 43 ( 1 9 7 5 )

,

51 3-51 8 . [ 3 ] N . Megiddo, "On t h e n o n m o n o t o n i c i t y o f t h e b a r g a i n i n g s e t ,

t h e k e r n e l , a n d t h e n u c l e o l u s o f a game", SIAM J o u r . AppZ.

M a t h . , - 27 ( 1 9 7 4 ) , 355-358.

[ 4 ] L . S . S h a p l e y , "A v a l u e f o r n - p e r s o n g a m e s " , I n H . W . Kuhn a n d A.W. T u c k e r ( E d s . ) , C o n t r i b u t i o n s t o t h e T h e o r y of Games, 11 ( A n n a l s o f Math. S t u d i e s 2 8 ) , P r i n c e t o n , N . J . : P r i n c e t o n U n i v e r s i t y P r e s s ( 1 9 5 3 )

,

303-306.