Richard P. Rumelt

Steven A. Lippman

Thomas M. Liggett

English

Research Papers in Economics

The Asympotic Shapley Value for a Simple Market Game
Thomas M. Liggett†, Steven A. Lippman, and Richard P. Rumelt
Mathematics Department, UCLA
The UCLA Anderson School of Management
The UCLA Anderson School of Management
110 Westwood Plaza
Box 951481 Los Angeles, CA 90095-1481, USA
Email: tml@math.ucla.edu, slippman@anderson.ucla.edu, drumelt@anderson.ucla.edu
December 24, 2005: #19: shapley7.tex
Abstract
We consider the game in which b buyers seek to purchase 1 unit of an indivisible good
from s sellers, each of whom have k units to sell. The good is worth 0 to each seller and
1 to each buyer. Using results from Brownian motion, we ﬁnd a closed form solution for
the Shapley value as s and b increase without bound. This asymptotic value depends upon
the store size k, the limiting ratio b/ks of buyers to items for sale, and the limiting ratio
[ks− b]/
√
b + s of the excess supply relative to the number of market participants.
Keywords Shapley Value; Core; Many Players
†This work was sponsored in part by NSF Grant DMS-03-01795
11 Introduction
An important issue in economics is the extent to which vigorous competition causes ﬁrm proﬁts
to vanish. Adam Smith explained why increasing the demand for a good would lead to a higher
price, but, unlike Edgeworth, he didn’t envisage situations in which a tiny reduction in demand
would induce the price to collapse and cause the sellers’ surplus to vanish.1 Yet this pathology
is the case in the simple many-player market game examined in this note.
In this note we consider a simple market game with many symmetric buyers and sellers.
In this market game, both the core and the Nash equilibrium in prices (Bertrand competition)
implicitly entail competition so vigorous that they can result in the total surplus on one side
of the market falling to zero when there is a slight increase in the number of players on that
side of the market. We show that the Shapley value leads to a natural continuous interpolation
between the two extreme situations, and is thereby the preferred solution concept.
The market game ﬁnds each of s sellers possessing k units of an indivisible good to sell, and
the good is worth 0 to each seller. There are b buyers each seeking to purchase 1 unit, and
the good is worth 1 to each buyer. Obviously, k,s,a n db are positive integers, and this is the
simplest game that captures trade in a market with many agents. When k>1, each seller is a
“store” with k units for sale.
When buyers exceed the number of units for sale (b>k s ), the core of this game has a
transaction price of 1; the Nash equilibrium in prices (Bertrand competition) also predicts a
price of 1. When the number of units for sale exceed the number of buyers (ks > b), these
equilibrium concepts predict a transaction price of 0. On the other hand, when ks = b,t h e s e
solution concepts admit any price between 0 and 1.
While these results parallel our ideas of perfect competition, they are not reasonable pre-
dictions for the outcome of generalized bargaining situations. For example, when k =1a n ds
is large, say s =1 0 6, it is preposterous to predict that s sellers facing b = s + 1 buyers would
each obtain a price of 1, whilst the addition of two more sellers would drop the price to 0.
The Shapley value smooths the discontinuity for this game in a manner consonant with the
economic landscape: the addition of buyers induces a gradual increase in the equilibrium price.
Traditionally, the Shapley value has been viewed as more “cooperative” than the core,
involving the balancing of probabilities of various coalitons being formed. Recently, there has
been considerable interest in non-cooperative rationales for the Shapley value. Two inﬂuential
1At the exact middle of Chapter VII, “Of the Natural and Market Price of Commodities,” Adam Smith [5]
asserts that “A public mourning raises the price of black cloth ... and augments the proﬁts of the merchants
who possess any considerable quantity of it. ... It sinks the price of coloured silks and cloths, and thereby
reduces the proﬁts of the merchants who have any considerable quantity of them upon hand.”
2examples are Gul’s [1] model of random bilaterial contracting and Hart and Mas-Colell’s [2]
model of sequential bilateral oﬀers.
While the behavior of the Shapley value is clear when there are a small numbers of players,
its behavior is not well understood (or known) as the size of the market grows very large.
Shapley and Shubik [4] analyze this problem when k = 1 and provide asymptotic results for
two cases. In the ﬁrst the ratio of buyers to sellers is ﬁxed with b = αs; in the second the
diﬀerence d between the number of buyers and sellers is a ﬁxed constant d: b = s + d. When
α>1, the Shapley value converges to 1 and coincides with the core as the size of the market
increases. Similarly, the Shapley value converges to 0 when α<1, just like the core. When the
diﬀerence between the number of buyers and sellers is d, the Shapley value converges to 0.5a s
the size of the market increases. In sharp contrast, the core yields equilibrium prices of 1 when
d>0, 0 when d<0, and the equilibrium price is indeterminate when d =0 .
We show that the asymptotic Shapley value for this market game is not restricted to the
values 0, 1/2, and 1; in fact, all values between 0 and 1 are possible. Let M ≡ b + s denote
the size of the market, and set d = ks− b. Using an old result about the Brownian bridge, we
provide a closed form expression for the limit of the Shapley value of this game as a function
of u ≡ limb,s→∞ d/
√
M . As long as the diﬀerence d between units for sale and buyers grows in
proportion to the square root of the size of the market, the asymptotic Shapley value (for each
unit) can take on any intermediate value between 0 and 1.
One way to decide which of several solution concepts is most appropriate for a particular
game is to select the concept which produces the most intuitive or plausible answer. On this
ground, we ﬁnd the Shapley value to be more appropriate than either the core or the Nash
equilibrium for our simple market bargaining game.
2 The Closed Form Solution
We now present a closed form solution which gives the asymptotic Shapley value as s and
b increase to ∞. A particularly interesting aspect of our solution is that for each price p
between 0 and 1, there are growth rates for s and b such that the Shapley value converges to p:
asymptotically, the Shapley value traces out each possible price. This stands in stark contrast
to the other solution concepts which yield prices of 0, 1, or an indeterminate price (or all prices)
between 0 and 1.
To begin set k = 1, and let V (b,s) denote the Shapley value for a seller in this game
(naturally, the value added by all players is v ≡ min(b,s) so that a buyer’s Shapley value is
[v−sV (b,s)]/b). To compute V (b,s), consider those permutations of the b+s players in which
3a given seller is the i+1 st player. There is some number j of buyers amongst the ﬁrst i players
in the permutation whence there are i−j sellers amongst the ﬁrst i players. This seller’s value
added is 1 if the number j of buyers is strictly greater than the number i − j of sellers: the
seller’s value added is 1 if and only if 2j>i . Clearly, the probability that this seller is the i+1st
player is 1/(b+s). The product
b
j
s−1
i−j

/
s+b−1
i

of binomial coeﬃcients is the probability that
the ﬁrst i players in a permutation has j buyers and i − j sellers given that the i +1 st player
is a seller. Consequently, V (b,s) can be written as
V (b,s)=
1
b + s
s+b−1 
i=0

2j>i
b
j
s−1
i−j

s+b−1
i
 .
More generally, when each seller is a store of integer size k with k ≥ 1, a similar argument
to the one above reveals that the expression for V (s,b), the Shapley value for each seller (who
has k units for sale), is
V (b,s)=
1
b + s
s+b−1 
i=0

(k+1)j>ik
min{k,j − (i − j)k}
b
j
s−1
i−j

s+b−1
i
 .
Theorem:L e tb,s →∞so that b/ks → α.
(a) If α<1, then V (b,s) → 0.
(b) If α>1, then V (b,s) → k.
(c) If α = 1, suppose that
ks− b
√
b + s
→ u.
Then
V (b,s) →
k2
√
2π
 ∞
0
x2
u2 + kx2e−x2/2dx if u ≥ 0,
and
V (b,s) → k −
k2
√
2π
 ∞
0
x2
u2 + kx2e−x2/2dx if u ≤ 0.
Proof : We ﬁrst write V in terms of a simple random walk. Take 0 <p<1, let X1,X 2,... be
independent identically distributed Bernoulli random variables with
P(Xi =1 )=p, P(Xi =0 )=q =1− p,
and let Sm = X1 + ···+ Xm be the corresponding partial sums. Setting N = s + b − 1, we see
that
P(Sb = j | SN = i)=
P(Sb = j)P(Ss−1 = i − j)
P(SN = i)
=
b
j

pjqb−js−1
i−j

pi−jqs−1−i+j
N
i

piqN−i =
b
j
s−1
i−j

s+b−1
i
 .
4Therefore,
V (b,s)=
1
b + s
s+b−1 
i=0

(k+1)j>ik
min{k,j − (i − j)k}P(Sb = j | SN = i)
=
1
N +1
N 
i=0
E

min{k,(k +1 ) Sb − ik};(k +1 ) Sb >i k| SN = i

,
where E(X;A) ≡ E(X · 1A) for a random variable X and a set A.
The process {YN(t):0≤ t ≤ 1} that appears in standard invariance principles is deﬁned
by ([x] ≡ integer part of x):
YN(t)=
S[Nt] − [Nt]p
√
N
, 0 ≤ t ≤ 1.
Deﬁne the random variable Zb,i by Zb,i =( k +1 ) Sb − ik and the set AN,i by AN,i = {SN = i}.
Writing Zb,i and AN,i in terms of YN,w eh a v eZb,i =( k +1 )
√
NY N( b
N)+( k +1 ) bp − ik and
AN,i = {YN(1) =
i−Np √
N }, so that
V (b,s)=
1
N +1
N 
i=0
E

min{k,Zb,i};Zb,i > 0 | AN,i

.
Suppose [ks − b]/
√
b + s → u with u ∈ [−∞,∞]a n db/(b + s) → t. Because Zb,i is
integer-valued, we have
E[min{k,Zb,i};Zb,i > 0 | AN,i]=E[Zb,i ;1≤ Zb,i <k| AN,i]+kP(Zb,i ≥ k | AN,i).
By [3, Theorem 4], (YN( b
N) | AN,i)=( YN( b
N) | YN(1) = (i − Np)/
√
N ) converges weakly to
Y (t)=

p(1 − p)B0(t), where B0(t) is the standard Brownian bridge. As a consequence, the
ﬁrst term is seen to go to 0 as N →∞because YN( b
N) | AN,i is approximately normal so that
the probability that Zb,i =( k +1 )
√
NY N( b
N)+( k +1 ) bp − ik lies in a bounded interval goes
to zero.
To ﬁnd the second term, set i =[ ( b + s)p]. Again, using [3, Theorem 4], we have
P(Zb,i ≥ k | AN,i)=P(YN(
b
N
) ≥
k(i +1 )− (k +1 ) bp
(k +1 )
√
N
| AN,i) → P(Y (t) ≥
up
k +1
). (1)
Using (1) and the bounded convergence theorem, we have
V (b,s) → k
 1
0
P

p(1 − p)B0(t) >
up
k +1
	
dp = k
 1
0
P

B0(t) >
u
k +1


p
1 − p
	
dp. (2)
5If t<k / (k + 1), then u =+ ∞, so the right of (2) is zero. If t>k / (k +1 ),u = −∞,s ot h e
right of (2) is 1.
If t = k/(k+1), then u can take any value. We show the result for the case u ≥ 0 (the case
u ≤ 0 is almost identical). Because B0(k/(k+1))isN(0,k/(k+1)2), B0(k/(k+1))/[
√
k/(k+1)]
is a standard normal random variable, which we denote by Z. Deﬁne the function g(x,p)t o
be 1 if x ≥ u
√
p/

k(1 − p) and 0 otherwise. Then utilizing (2) and interchanging the order of
integration, we obtain
V (b,s) → k
 1
0
P

Z>
k +1
√
k
·
u
k +1
√
p
√
1 − p
	
dp = k
1
√
2π
 1
0
[
 ∞
0
e−x2/2g(x,p)dx]dp
= k
1
√
2π
 ∞
0
[
 1
0
g(x,p)dp]e−x2/2dx = k
1
√
2π
 ∞
0
kx2
u2 + kx2 e−x2/2dx.
The equilibrium market price for each unit of the item is V (b,s)/k. In a large market, the
closed form solution reveals that when 0 <u<∞, limb,s→∞V (b,s)/k is strictly increasing in
k, with limit 1/2: when there is an excess supply (ks > b), larger stores induce an increase in
the equilibrium market price. [The opposite is true when −∞ <u<0.]
References
[1] Gul, Faruk: “Bargaining foundations of Shapley value,” Econometrica 57, 81-95 (1991).
[2] Hart, Sergiu and Andreu Mas-Colell: “Bargaining and value,” Econometrica 64, 357-380
(1996).
[3] Liggett, Thomas M.: “An invariance principle for conditioned sums of independent random
variables,” J. Math. Mech 18, 559-570 (1968).
[4] Shapley, Lloyd S. and Martin Shubik: “Pure Competition, Coalitional Power, and Fair
Division.” International Economic Review 10, 337-362 (1969).
[5] Smith, Adam: The Wealth of Nations. Norman S. Berg, Dunwoody, Georgia, 1776 (1976).
6

An invariance principle for conditioned sums of independent random variables,”

Bargaining and value,”

Faruk: “Bargaining foundations of Shapley value,”

The Wealth of Nations. Norman S.

The Asympotic Shapley Value for a Simple Market Game

http://www.dklevine.com/archive/refs4122247000000001011.pdf

The Asympotic Shapley Value for a Simple Market Game

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