A Note on Biform Analysis of Monopoly∗
Adam Brandenburger†
Harborne Stuart‡
First Version 08/26/04
Current Version 01/16/05
1
Introduction
Kreps [3, 1990, pp.314-315] asks where monopoly power comes from. He summarizes the standard
textbook analysis of monopoly as follows:
It was assumed that the monopoly could name its price . . . or could make a take-orleave offer to potential consumers. You might wonder just why this is supposed to be
true. Consider, for example, our ‘fortunate’ monopoly, which can make a different takeor-leave offer to every consumer and which knows the characteristics of each consumer.
Suppose it makes this take-or-leave offer to consumer X, who says “no thanks,” and then
comes back with a take-or-leave offer of her own. For example, consumer X might offer
to take n units off the hands of the monopoly, at (being generous) $1 above the monopoly’s
marginal cost for producing the n units. Wouldn’t the monopoly take this deal, if it really
believed it was the best deal it could get? And, if so, how do we determine who, in this
sort of situation, does have the bargaining power? Why did we assume implicitly that the
monopoly has all this power (which we most certainly did when we said that consumers
were price takers)? [Footnote in original: Another way to think of this is to consider
that the monopoly dealing separately with a single consumer is in a position of bilateral
monopoly which, you will recall, is meant to have an indeterminate solution.] Standard
stories, if given at all, get very fuzzy at this point. . . . The handwaving typically
involves something about how there is one monopoly and many consumers, and it can do
without any one of them, but they can’t do without it.
∗ We thank Ken Corts and Elon Kohlberg for comments.
Financial support from Columbia Business School,
Harvard Business School, and the Stern School of Business is gratefully acknowledged. b a m p - 0 1 - 1 6 - 0 5
† Address: Stern School of Business, New York University, 44 West Fourth Street, New York, NY 10012,
[email protected], www.stern.nyu.edu/∼abranden
‡ Address:
Graduate School of Business, Columbia University, 3022 Broadway, New York, NY 10027,
[email protected], www.columbia.edu/∼hws7
In this note, we give an answer to Kreps’ question. To do so, we model monopoly as a hybrid
noncooperative-cooperative game, specifically, a biform game as in our [2, 2004]. In the first,
noncooperative stage, the seller chooses a level of capacity. Each capacity choice defines a secondstage, cooperative game between seller and buyers. We analyze this stage using the core, to capture
the idea of unrestricted bargaining among the players.
Section 2 gives an example illustrating the main idea of the note. Section 3 lays out the general
monopoly model and gives some general results. The cases of unitary and non-unitary demand are
treated in Sections 4 and 5. Earlier work on unitary demand includes Shapley and Shubik [6, 1967]
(who consider identical buyers), our [1, 1992], and Moulin [4, 1995, p.58]. Section 6 compares our
core analysis with an analysis using the Shapley Value.
2
An Example
Dollars
Example 2.1 There is one seller and 12 buyers, indexed by j = 1, . . . , 12. Each buyer wishes to
buy one unit from the seller. Let wj = 14 − j be buyer j’s willingness-to-pay for that unit. The
seller has a $1 unit cost of installing capacity, and then zero unit cost of production.
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
Quantity
Figure 2.1
Suppose the seller chooses a capacity of 6 units. Figure 2.1 illustrates the resulting cooperative
game. The total value created is $57 (the sum of the six highest willingness-to-pay numbers minus
the seller’s cost of capacity). This is also the marginal contribution of the seller, since without the
seller no value is created. Next note that buyers 7 through 12 have zero marginal contribution. Now
consider buyer 6. If he leaves the game, he will be replaced by buyer 7. The total value will then
be $56, so that buyer 6’s marginal contribution is $1. More generally, the marginal contribution
of any buyer j, for j = 1, . . . , 6, is wj − 7. The total of these marginal contributions is the area
above the dotted line in Figure 2.1. Since no player can get more than his marginal contribution in
the core, we can conclude: buyers 7 through 12 each get $0, buyer j, for j = 1, . . . , 6, gets at most
wj − 7, and the seller gets at least the area below the dotted line, namely $36.
2
In the next section, we show that, under mild conditions, the core of a monopoly game is fully
characterized by the above constraints.
Note that the seller is guaranteed a minimum amount of value ($36). Conceptually, this is
because of competition among the buyers. There are 12 buyers but only 6 units available. As
a result, the amount of value each ‘included’ buyer (buyers 1 through 6) can get is limited by the
presence of the ‘just-excluded’ buyer (buyer 7). Undersupply by the seller is what gives it this
‘power’ over the buyers.
Capacity
Seller
Choice
Value
0
0
1
[11, 12]
2
[20, 23]
3
[27, 33]
4
[32, 42]
5
[35, 50]
6
[36, 57]
7
[35, 63]
8
[32, 68]
9
[27, 72]
10
[20, 75]
11
[11, 77]
12
[−12, 78]
Table 2.1
Note also that there is a range of values the seller can get—from $36 to the total value of $57.
Table 2.1 gives the range of values for the seller for different capacity choices. The core models how
competition narrows down the division of value between seller and buyers, but leaves a ‘residual’
bargaining problem. A seller who takes an optimistic view of this bargaining would focus on the
upper endpoint of the intervals shown. (In the extreme case, he’ll install 12 units.) A seller with
a more pessimistic view would focus on the lower endpoints. (In the extreme case, he’ll install 6
units.) Or, the seller might focus on the midpoints of the intervals, on the assumption that the
residual pies will be evenly split. (In this case he’ll install 9 units.)
Finally, consider a seller who (unlike the situation Kreps [3, 1990] describes) can name a price
and stick to it. This seller’s optimization problem is: Choose a price wj (for j = 1, . . . , 12) to
maximize j(wj − 1). (We assume that buyer j (just) buys when the price is wj .) Thus, the seller
will choose j to maximize j(13 − j), i.e., it will choose a capacity of 6 or 7.
3
3
General Formulation
Given a set X, let P(X) denote the power set of X, and |X| the cardinality of X. Recall that
a transferable utility (TU) cooperative game consists of a player set N and a characteristic
function v : P(N ) → R, with v(∅) = 0. For each A ⊆ N , we interpret v(A) as the value that
subset A of the players can create. An allocation is a point x ∈ R|N | , where component xi denotes
P
the value received by player i. The core is the set of allocations satisfying i∈N xi = v(N ) and
P
for all A ⊆ N , i∈A xi ≥ v(A). A game is superadditive if for all A, B ⊆ N with A ∩ B = ∅,
v(A ∪ B) ≥ v(A) + v(B).
We begin with an abstract definition of a monopoly game. Assume the game is superadditive.
Assume also that there is one player (the “monopolist”) without whom no value is created. The core
is trivially nonempty (consider the allocation in which the seller receives the total value). Muto,
Nakayama, Potters, and Tijs [5, 1987] show that with a third condition, condition (3.1) below, the
core is easily characterized. The following result is essentially theirs. (A nonnegativity condition,
rather than superadditivity, is used in [5, 1987].)
Proposition 3.1 Fix a superadditive TU cooperative game where N = {m, 1, ..., b} and v(N \{m}) =
0. Suppose the game satisfies, for all A ⊆ N \{m},
X
j∈A
[v(N ) − v(N \{j})] ≤ v(N ) − v(N \A).
(3.1)
Then the core is the nonempty set of allocations x ∈ Rb+1 given by
m
x +
b
X
xj = v(N ),
(3.2)
j=1
v(N ) −
b
X
j=1
[v(N ) − v(N \{j})] ≤ xm ≤ v(N ),
0 ≤ xj ≤ v(N ) − v(N \{j}) for j = 1, ..., b.
(3.3)
(3.4)
A monopoly with downward-sloping demand and non-decreasing unit costs satisfies condition
(3.1). With decreasing unit costs, the condition fails. The condition also fails if there is a buyer
‘network effect,’ e.g., if value is created only by a set consisting of all the buyers (and the seller).
This said, the condition is only sufficient for nonemptiness, not necessary.
P
Proof. For any A ⊆ N , define x(A) = i∈A xi . As noted above, the core is clearly nonempty.
Consider any allocation satisfying (3.2)-(3.4). The conditions are clearly necessary for the core, so
4
we show they are sufficient. It is enough to consider an A 3 m. Inequality (3.4) implies
x(N \A) ≤
X
j∈N \A
[v(N ) − v(N \{j})]
≤ v(N ) − v(A),
where the second line uses condition (3.1).
x(N ) − x(N \A) then gives
Equation (3.2) says x(N ) = v(N ).
Writing x(A) =
x(A) ≥ v(N ) − [v(N ) − v(A)]
= v(A),
as required.
We now define a two-stage, noncooperative-cooperative monopoly game. Formally, this is an
example of a biform game (our [2, 2004]). The reader should consult that paper for a general
definition, discussion, properties, and other examples of the biform model.
Definition 3.1 A biform monopoly game is a collection (N, S, V, α), where N = {m, 1, . . . , b},
S = {0, 1, . . . , D} for some D ≥ b, 0 ≤ α ≤ 1, and V is a map from S to the set of maps from P(N )
to R, with V (s)(∅) = 0 for every s ∈ S, and
(a) for every s ∈ S and A ⊆ N \{m}, V (s)(A) = 0 and
X
j∈A
[V (s)(N ) − V (s)(N \{j})] ≤ V (s)(N ) − V (s)(N \A),
(b) for every 0 ≤ s ≤ D − 1, V (s + 1)(N ) > V (s)(N ).
Here, N is the set of players, where m is the seller and 1, . . . , b are the buyers, S is the strategy set
of player m, giving different capacity choices, V associates with each strategy choice a cooperative
game satisfying certain conditions, and α is the confidence index of player m. (Formally, to fit with
Definition 5.1 in [2, 2004], we have to specify (singleton) strategy sets and confidence indices for
players 1, . . . , b. We suppress these here.)
The seller uses the confidence index α to form a weighted average of the greatest and least
P
amounts of value he can receive in the core. Writing g(s) = V (s)(N )− bj=1 [V (s)(N ) − V (s)(N \{j})]
and using Proposition 3.1, the seller will choose s ∈ S to maximize π(s) = αV (s)(N ) + (1 − α)g(s).
Let B(α) denote the set of optimal strategic choices for the seller. The following proposition
characterizes this set as a function of α.
Proposition 3.2 The set B(α) satisfies:
(i) B(0) = argmaxs∈S g(s),
5
(ii) B(1) = {D},
(iii) for 0 ≤ α < 1, max B(α) ≤ min B(α + ε) whenever 0 < ε ≤ 1 − α.
Proof. Part (i) is immediate from Proposition 3.1, and part (ii) is immediate from Proposition 3.1
and condition (b) of Definition 3.1. For part (iii), fix some ε such that 0 < ε ≤ 1 − α, and note that
(α + ε)V (s)(N ) + (1 − α − ε)g(s) = δV (s)(N ) + (1 − δ)π(s)
where δ = ε/(1 − α). Now consider any s0 ∈ B(α) and s00 ∈ B(α + ε), and suppose, contra
hypothesis, that s0 > s00 . Then by condition (b) of Definition 3.1 again, V (s0 )(N ) > V (s00 )(N ), so
that
δV (s0 )(N ) + (1 − δ)π(s0 ) > δV (s00 )(N ) + (1 − δ)π(s0 ).
But π(s0 ) ≥ π(s00 ) since s0 ∈ B(α), so that
δV (s00 )(N ) + (1 − δ)π(s0 ) ≥ δV (s00 )(N ) + (1 − δ)π(s00 ),
from which
δV (s0 )(N ) + (1 − δ)π(s0 ) > δV (s00 )(N ) + (1 − δ)π(s00 ),
contradicting s00 ∈ B(α + ε).
4
Unitary Demand
Let wj , for j = 1, . . . , b, be numbers satisfying w1 ≥ · · · ≥ wb > k + c, where each wj is buyer j’s
willingness-to-pay for a unit of product, k ≥ 0 is the seller’s constant unit cost of capacity, and c ≥ 0
is the seller’s constant unit cost of production. Let D = b, so that S = {0, 1, . . . , b}. For each
s ∈ S, the characteristic function is given by:
where


if m ∈
/ A,
 0
V (s)(A) =
−ks
if A = {m},

P

−ks + R
χ
(j)(w
−
c)
otherwise,
j
j=1 A


r

 X
χA (j) ≤ min{s, |A| − 1} ,
R = max r :


(4.1)
j=1
and χA (j) is the characteristic function of A (i.e. χA (j) = 0 or 1 according as j ∈ A or j ∈
/ A).
m
1
b
b+1
Let (x , x , . . . , x ) ∈ R
be an allocation. Then Proposition 3.1 yields:
6
Corollary 4.1
(i) Suppose that s = 0. Then in the core, xm = 0 and xj = 0 for every j = 1, . . . , b.
(ii) Suppose that 0 < s < b. Then in the core,
xm +
b
X
xj =
j=1
s
X
[wj − (k + c)],
j=1
s[ws+1 − (k + c)] ≤ xm ≤
j
s
X
[wj − (k + c)],
j=1
0 ≤ x ≤ (wj − ws+1 ) for j = 1, . . . , s,
xj = 0 for j = s + 1, . . . , b.
(iii) Suppose that s = b. Then in the core,
xm +
b
X
j=1
xj
=
b
X
[wj − (k + c)],
j=1
−kb ≤ xm ≤
j
b
X
[wj − (k + c)],
j=1
0 ≤ x ≤ (wj − c) for j = 1, . . . , b.
Part (i) records the obvious fact that if the seller installs no capacity, no value can be created
and so no player can capture any value. Part (ii) covers the case in which the seller installs
a positive amount of capacity, but sufficiently little for there to be at least one excluded buyer
(buyers j = s + 1, . . . , b). Each included buyer (buyers j = 1, . . . , s) cannot capture more than
his willingness-to-pay (wj ) minus the willingness-to-pay of the just-excluded buyer (ws+1 ). Each
excluded buyer captures no value. The seller is guaranteed to receive at least the difference between
the willingness-to-pay of the just-excluded buyer (ws+1 ) and total cost (k + c), times the number
of units of capacity installed (s). Part (iii) describes what happens if the seller installs capacity
sufficient to serve all buyers. Now, the seller is simply involved in a series of bilateral bargaining
problems, and the outcome is completely indeterminate. Specifically, the seller and buyer j (for
j = 1, . . . , b) may divide the value (wj − c) in any way between them.
Proof. First note that for 1 ≤ s ≤ b,
V (s)(N ) = −ks +
7
s
X
(wj − c).
j=1
For 1 ≤ s ≤ b − 1 and j = 1, . . . , b,
V (s)(N ) − V (s)(N \{j}) = −
(
=
R
X
j=1
χS (j)(wj − c)
wj − ws+1
0
if 1 ≤ j ≤ s,
otherwise.
For s = b and j = 1, . . . , b,
V (s)(N ) − V (s)(N \{j}) = wj − c.
The proof then follows from Proposition 3.1, provided inequality (3.1) holds. To show this, consider
P
any 1 ≤ s < b. (The case s = 0 is trivial.) Consider any A ⊆ N \{m}. Let q = sj=1 χN \A (j), so
P
P
that sj=1 χA (j) = s − q. Note that since R
j=1 χN\A (j) ≤ s,
R
X
j=s+1
χN \A (j) ≤ s − q.
Then
V (s)(N ) − V (s)(N \A) =
s
R
X
X
(wj − c) −
χN\A (j)(wj − c)
j=1
=
j=1
s
X
j=1
≥
χA (j)(wj − c) −
s
X
j=1
R
X
j=s+1
χN \A (j)(wj − c)
χA (j)(wj − c) − (s − q)(ws+1 − c)
=
s
X
j=1
χA (j)(wj − ws+1 ) =
X
j∈A
[V (s)(N ) − V (s)(N \{j})] .
The argument for s = b is similar.
The following is immediate from Proposition 3.1 and Corollary 4.1.
Corollary 4.2 The set B(α) satisfies:
(i) B(0) = argmax0≤s≤b−1 s[ws+1 − (k + c)],
(ii) B(1) = {b}.
Suppose the seller has a confidence index α = 0, i.e., the most pessimistic or cautious view of the
residual bargaining with buyers. In this case, the seller will maximize the minimum amount of value
he can capture (maximin). Corollary 4.2(ii) says this means choosing a capacity level s to maximize
8
s[ws+1 −(k +c)]. Note that this is very similar to the maximization problem of a price-setting seller.
(Refer back to the last paragraph of Section 2.) Moulin [4, 1995, p.58] makes a similar observation,
calling the minimum outcome for the seller in the core the “competitive equilibrium profit.” Now
suppose the seller has a confidence index α = 1, i.e., the most optimistic or aggressive view of the
residual bargaining with buyers. If so, the seller will maximize the maximum amount of value he
can capture (maximax). Corollary 4.2(ii) says this means installing sufficient capacity to supply all
the buyers. For intermediate cases, Proposition 3.2(iii) says that a seller with a higher confidence
index will install at least as much capacity as a seller with a lower confidence index.
5
Non-Unitary Demand
Let wi , for i = 1, . . . , D, where D > 1, be positive numbers satisfying w1 ≥ · · · ≥ wD . Let Ψj , for
P
j = 1, . . . , b, be a collection of functions from {1, ..., D} to {0, 1} satisfying bj=1 Ψj (i) = 1 for all
i = 1, . . . , D.
As with unitary demand, the values w1 , . . . , wD determine the points on the demand curve, where
each point is demanded by some buyer. Now, though, we need functions Ψj (i) associating the points
with buyers. We set Ψj (i) = 1 if point i on the demand curve is associated with buyer j. It follows
P
that buyer j wants to buy D
i=1 Ψj (i) units of product. For a given buyer j, let i1 be the first i such
that Ψj (i) = 1, let i2 be the second i such that Ψj (i) = 1, etc. Then buyer j has willingness-to-pay
wi1 for one unit, willingness-to-pay wi1 +wi2 for two units, etc. In the definition of the characteristic
function below, it is implicitly assumed that buyers have non-increasing willingness-to-pay numbers,
i.e. wi1 ≥ wi2 ≥ · · · .
The seller’s strategy set is S = {0, 1, . . . , D}. As before, let k ≥ 0 be the constant unit cost of
capacity, let c ≥ 0 be the constant unit cost of production, and assume that wD > k + c. For each
s ∈ S, the characteristic function is given by:
where

if m ∈
/ A,

 0
V (s)(A) =
−ks
if A = {m},

Pms (A) P

−ks + i=1
Ψ
(i)(w
−
c)
otherwise,
i
j∈A\{m} j

r
 X
ms (A) = max r :

X
i=1 j∈A\{m}
Ψj (i) ≤ min{s,
D
X
i=1 j∈A\{m}
Given s > 0, define the set Ps by
Ps =
(
j ∈ N \{m} :
s
X
i=1
X


Ψj (i)} .

)
Ψj (i) > 0 ,
/ Ps , and ms (N \{j}) > s if j ∈ Ps . Proposition 3.1 then yields:
and note that ms (N \{j}) = s if j ∈
9
Corollary 5.1
(i) Suppose that s = 0. Then in the core, xm = 0 and xj = 0 for every j = 1, . . . , b.
(ii) Suppose that 0 < s < D. Then in the core,
xm +
b
X
xj =
j=1
−ks +
X
j∈Ps
0 ≤ xj ≤
s
X
i=1
ms (N \{j})
X
i=s+1
X
s
X
[wi − (k + c)],
i=1
l∈N \{j,m}
Ψl (i)(wi − c) ≤ xm ≤
ms (N\{j})
Ψj (i)(wi − c) −
X
i=s+1
X
l∈N \{j,m}
s
X
[wi − (k + c)],
i=1
Ψl (i)(wi − c) for j ∈ Ps ,
(5.1)
xj = 0 for j ∈
/ Ps .
(iii) Suppose that s = D. Then in the core,
xm +
b
X
xj =
j=1
i=1
−kD ≤ xm ≤
0 ≤ xj ≤
D
X
i=1
D
X
[wi − (k + c)],
D
X
[wi − (k + c)],
i=1
Ψj (i)(wi − c) for j = 1, . . . , b.
Proof. First note that for 1 ≤ s ≤ D,
V (s)(N ) = −ks +
s
X
(wj − c).
j=1
For 1 ≤ s ≤ D − 1, the marginal contribution of a buyer j ∈ Ps is then

ms (N \{j})
s
X
X
[wi − (k + c)] − −ks +
V (s)(N ) − V (s)(N \{j}) =
i=1
=
s
X
i=1
ms (N \{j})
(wi −c)−
X
i=1
X
l∈N \{j,m}
i=1
Ψl (i)(wi −c) =
s
X
i=1
10
X
l∈N \{j,m}
ms (N\{j})
Ψj (i)(wi −c)−
X
i=s+1

Ψl (i)(wi − c)
X
l∈N \{j,m}
Ψl (i)(wi −c),
and the marginal contribution of a buyer j ∈
/ Pu is

s
s
X
X
[wi − (k + c)] − −ks +
V (s)(N ) − V (s)(N \{j}) =
i=1
=
i=1 l∈N \{j,m}
s
X
i=1
X
(wi − c) −
s
X

Ψl (i)(wi − c)
X
i=1 l∈N\{j,m}
Ψl (i)(wi − c) = 0.
For s = D, first note that

r
 X
mD (N \{j}) = max r :

X
i=1 l∈N \{j,m}
Ψl (i) ≤ min{D,

r
 X
= max r :

D
X
X
i=1 l∈N \{j,m}
X
i=1 l∈N \{j,m}


Ψl (i)}

Ψl (i) ≤
D
X
X
The marginal contribution of buyer j is then

D
D
X
X
V (s)(N ) − V (s)(N \{j}) =
[wi − (k + c)] − −kD +
i=1
=
i=1
=
D
X
i=1
X
i=1 l∈N \{j,m}
D
D
X
X
(wi − c) −
X
i=1 l∈N \{j,m}
Ψl (i)
i=1 l∈N \{j,m}



= D.

Ψl (i)(wi − c)
Ψl (i)(wi − c)
Ψj (i)(wi − c).
It remains to show that inequality (3.1) holds. Consider any 1 ≤ s < D. (The case s = 0 is
trivial.) For any A ⊆ N \{m},
ms (N \A)
s
X
X
V (s)(N ) − V (s)(N \A) =
(wi − c) −
i=1
=
s X
X
i=1 j∈A
≥
i=1
X
j∈N \(A∪{m})
ms (N\A)
Ψj (i)(wi − c) −
s
XX
j∈A i=1
X
i=s+1
Ψj (i)(wi − c) −
11
Ψj (i)(wi − c)
X
j∈N \(A∪{m})
ms (N \(A∩Ps ))
X
i=s+1
Ψj (i)(wi − c)
X
j∈N \((A∩Ps )∪{m})
Ψj (i)(wi − c)
≥
s
XX
j∈A i=1
Ψj (i)(wi − c) −
=
X
j∈(A∩Ps )
X

ms (N \{j})
j∈(A∩Ps )

X
i=s+1
X
l∈N\{j,m}

ms (N \{j})
s
X
X

Ψj (i)(wi − c) −
i=1
i=s+1

Ψl (i)(wi − c)
X
l∈N \{j,m}
=

Ψl (i)(wi − c)
X
j∈A
[V (s)(N ) − V (s)(N \{j})] .
The argument for s = D is similar.
Although Corollary 5.1 is a bit involved, it is conceptually similar to Corollary 4.1. (Both are, of
course, based on Proposition 3.1.) Consider inequality (5.1). The right-hand side is the marginal
contribution of a buyer who has positive marginal contribution. The first term,
s
X
i=1
Ψj (i)(wi − c),
is a buyer’s gross contribution to the total value created. With unitary demand, this term would
be wj . The second term,
ms (N \{j})
X
X
Ψl (i)(wi − c),
i=s+1
l∈N \{j,m}
represents the value of the best substitutes for the buyer in question. If buyer j were not in the game,
then the next highest willingness-to-pay numbers would be found in the range s + 1 to ms (N \{j}),
and they would be associated with buyers other than buyer j. Hence the double summation. In the
case of unitary demand, this term would be ws+1 . Thus, the same general story can be told, with
one change. With unitary demand (and undersupply), excluded buyers provide competition for the
included buyers. With non-unitary demand, it is the excluded units that provide competition. In
particular, a buyer who is able to purchase some of but not all of the units he wants can provide
competition for another included buyer.
6
Shapley Value
The Shapley Value can give very different answers in the monopoly game. We start with a simple
example.
Example 6.1 There are 2 buyers (i.e. b = 2), each with a willingness-to-pay of $1 for one unit
(i.e. w1 = w2 = 1). The cost parameters are k = c = 0.
If s = 1, the core of the resulting game is (xm , x1 , x2 ) = (1, 0, 0). If s = 2, the core of the
resulting game is (xm , x1 , x2 ) = (2 − x − y, x, y), for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. The seller might
12
choose s = 1 or s = 2, depending on his view of the residual bargaining. (If α < 1/2, the seller will
choose s = 1.)
If s = 1, the Shapley Value of the resulting game is (xm , x1 , x2 ) = (2/3, 1/6, 1/6). If s = 2, the
Shapley Value of the resulting game is (xm , x1 , x2 ) = (1, 1/2, 1/2). The seller will always choose
s = 2.
Here is a general result, for the monopoly game of Section 4 (unitary demand). We assume
that the seller chooses capacity s to maximize what he gets under the Shapley Value in the resulting
game.
Proposition 6.1
(i) If wb ≥ kb + (k + c), the seller chooses s = b.
(ii) If w1 ≤
k
b
+ (k + c), the seller chooses s = 0.
Note, in particular, that if k = 0, the seller will always choose to serve the whole market. Under
the Shapley Value, a seller only chooses to undersupply to avoid the cost of capacity. Evidently,
the Shapley Value behaves very differently from the core in this game.
Proof. The Shapley Value gives player i ∈ N the amount
X
ϕi =
T ⊆N,T 3i
(t − 1)!(n − t)!
[v(T ) − v(T \{i})]
n!
where t = |T |. Consider an s ≥ 1. Using equation (4.1), the seller’s allocation can be written as
X
ϕm (s) =
T ⊆N,T 3m
=
(t − 1)!(n − t)!
V (s)(T )
n!
b+1
X
(i − 1)!(n − i)!
n!
i=1
where w =
1
b
X
V (s)(T )
T 3m,|T |=i
R(s)
= −sk +
b+1
X
(i − 1)!(n − i)!
T 3m,|T |=i j=1
= −sk +
s+1
X
(i − 1) (w − c)
b+1
X
(i − 1)!(n − i)!
+
n!
i=s+2
Pb
j=1
i=2
i=2
n!
n
X
X
χT (j)(wj − c)
X
R(s)
X
T 3m,|T |=i j=1
χT (j)(wj − c),
wj . For any T 3 m with b > |T | ≥ s + 2, define ∆(s, T ) by
R(s+1)
∆(s, T ) =
X
j=1
R(s)
χT (j)(wj − c) −
13
X
j=1
χT (j)(wj − c).
Note that ∆(s, T ) ≥ wb − c and ∆(s, T ) ≤ ws+1 − c. We have
ϕm (s + 1) − ϕm (s) = −k +
≥ −k +
= −k +
= −k +
b+1
X
(i − 1)!(n − i)!
n!
i=s+2
b+1
X
(i − 1)!(n − i)!
n!
i=s+2
b+1
X
(wb − c)
n
i=s+2
X
∆(s, T )
T 3m,|T |=i
X
(wb − c)
T 3m,|T |=i
(b − s)
(wb − c).
b+1
Thus, if (b − s) (wb − c) ≥ k (b + 1), we get ϕm (s + 1) ≥ ϕs (s). Setting s = b − 1 gives (wb − c) ≥
k (b + 1), from which wb ≥ kb + (k + c), establishing part (i).
For part (ii), note that
ϕm (s + 1) − ϕm (s) = −k +
≤ −k +
= −k +
= −k +
b+1
X
(i − 1)!(n − i)!
n!
i=s+2
b+1
X
(i − 1)!(n − i)!
n!
i=s+2
b+1
X
(ws+1 − c)
n
i=s+2
X
∆(s, T )
T 3m,|T |=i
X
(ws+1 − c)
T 3m,|T |=i
(b − s)
(ws+1 − c).
b+1
Thus, if (b − s)(ws+1 − c) ≤ k (b + 1), we get ϕm (s + 1) ≤ ϕm (s). In particular, setting s = 0 gives
the sufficient condition b(w1 − c) ≤ k (b + 1), or w1 ≤ kb + (k + c), as required.
14
References
[1] Brandenburger, A., and H. W. Stuart, Jr., “Modelling Business Strategy: A Two-stage Approach,” in H. W. Stuart, Jr., Non-Equilibrium and Non-Procedural Approaches to Game Theory, Ph.D. dissertation, Harvard University, 1992.
[2] Brandenburger, A., and H. W. Stuart, Jr., “Biform Games,” 2004.
www.stern.nyu.edu/∼abranden.
Available at
[3] Kreps, D., A Course in Microeconomic Theory, Princeton University Press, Princeton, 1990.
[4] Moulin, H., Cooperative Microeconomics: A Game-Theoretic Introduction, Princeton University
Press, 1995.
[5] Muto, S., M. Nakayama, J. Potters, and S. Tijs, “On Big Boss Games,” Economic Studies
Quarterly, 39, 1987, 303-321.
[6] Shapley, L., and M. Shubik, “Ownership and the Production Function,” Quarterly Journal of
Economics, 81, 1967, 88-111.
[7] Stuart, Jr., H. W., “Does
www.columbia.edu/∼hws7.
Buyer
Symmetry
15
Matter?”
2004.
Available
at