The technological e¤ect in cost sharing problems
Christian Trudeauy
(preliminary version)
May 6, 2007
Abstract
Cost sharing problems usually assume that all agents have access to
a commonly owned technology and that gains from cooperation are a
matter of scale. However, in many real-life problems agents have private
technologies and at least part of the gain from cooperation come from the
fact they when a coalition grows, the technology that they have access to
improves. A simple model is built, where economies of scale are eliminated
in order to study this e¤ect. We use as the key axiom the property that if
an agent never improves the technology of the coalition he joins, he should
not get any part of this gain. With simple properties of linearity and
symmetry, this axiom characterizes a well-de…ned set of rules. Adding a
property of monotonicity, on technology or the addition of technologically
negligible agents, or a property on the upper limit of individual allocations,
we obtain a unique rule, derived from the familiar Shapley value.
Keywords: Cost sharing, technology, dummy property, Shapley value.
1
Introduction
In cost sharing problems, agents cooperate and combine their demands, to which
is associated a cost. The central planner’s main task is then to divide that cost
among the participating agents.
Cost sharing problems usually have in common the following characteristics:
all agents have access to a commonly owned technology, which associates to any
level of demand a cost. The gains (or losses) associated to cooperation come
from the returns to scale exhibited by the technology (or returns to scope in the
case of heterogeneous goods). However, we argue that gains from cooperation
can come from another important source, through technological cooperation.
Suppose that agents have privately owned technologies of production. When
two agents cooperate, they not only put their demands together, but also their
Many thanks to Yves Sprumont for much discussion and support. I also thank Amadou
Boly, Paul Samuel Njiki Njiki and Adile Tamguicht for their comments and suggestions.
y Département de Sciences Économiques, Université de Montréal, C.P. 6128, succursale
centre-ville, Montréal, Québec, Canada. H3C 3J7, christian.trudeau@umontreal.ca
1
technologies. A possibly complex process, unobserved by the planner, aggregates these technologies to form a new e¢ cient technology for the coalition. In
particular, technologies can be complements, in which case the technology improves for both agents. The planner observes this technology, and the associated
cost.
Consider the following two-player example:
Example 1 Agents demand a homogeneous good. Demands by agents 1 and 2
are respectively 16 and 9 units. Costs for agent 1 alone is 240, for agent 2 alone
it is 180. Suppose that when they cooperate, the total cost is 300.
Case
p 1: Technology is publicly owned, and such that the cost for a coalition
S is 60 x(S); where x(S) is the total quantity demanded by the coalition.
Case 2: Technology is privately owned, and o¤ ers constant returns to scale.
The technology of agent 1 is such that his average (and marginal) cost is 15,
while for agent 2 it is 20. Their technologies are complements and when they
cooperate, average cost falls to 12.
Both cases generate the same coalitional costs, but for distinct reasons. In
case 1, all the gains come from economies of scale. In case 2, there are no
economies of scale, but the gains come from the improvement in technology.
How to allocate the gain generated by this technological e¤ect is not trivial. In
particular, we will probably not want to share it like a gain coming from returns
to scale.
In practice, technology plays an important part in many situations of cooperation. Research joint ventures are the obvious examples, where cooperation
can improve e¢ ciency of the research even without increasing investments (La
Manna, 2006). One can also think of countries cooperating in a space agency,
or di¤erent departments of a company not only putting together their requests
for a project, but also some of the workers and tools they have at their disposal
to complete the task.
It is also important to note that technology is meant in the broadest sense.
For example, in a network problem where the agents consist of the nodes in the
network, and where cooperating agents can only use links between their nodes,
a broader coalition has access to a larger set of possible links. This, in turn, is
a factor that generates gains from cooperation and that has nothing to do with
demands. In problems where agents have to be connected to a single source to
get a good, and where cost on each link depends on the volume of the ‡ow on it,
this e¤ect is su¢ cient to generate stability of the underlying game when there
is decreasing returns to scale on each link (Trudeau, 2006).
The assumption that agents have private technologies has been used in the
output-sharing literature, where agents put together their inputs and where
we must allocate the output among the participants. Leroux (2006) studies a
problem where agents have di¤erent technologies that they combine to enhance
productivity, where he looks at group strategyproof mechanisms.
When the technology-improving process is well-known and depends on some
inputs, and if the problem is more production-oriented, the output-sharing
2
model is more natural. However, the cost sharing model can be used on a
larger set of problems and can accommodate cases where little is known of how
technology improves. For example, problems where some otherwise identical
agents are exogenously able to cooperate e¢ ciently with some and unable with
others can be modelled using the proposed framework.
To better separate the so-called technology and scale e¤ects, a simple model
with constant returns to scale is used, so as to focus on the e¤ects of technology.
To help introduce the key axiom used, consider this second example:
Example 2 Demands are the same as in Example 1 and the privately owned
technologies still give an average cost of 15 to agent 1 and 20 to agent 2. When
they cooperate, however, the average cost is 15. If we apply the Shapley value
to the resulting stand-alone game, it gives allocations of 217.5 to agent 1 and
157.5 to agent 2.
The cooperation in example 2 generates a gain of 45, and with the Shapley
value both agents get an equal share of this gain. However, in this case, agent 2
does not contribute anything technologically. When both agents cooperate, the
e¢ cient way to produce is simply to discard agent 2’s technology. In this case, it
is hard to argue that both agents should get an equal share of this technological
gain.
In fact, if agents are responsible for their technologies, having possibly invested in them, we will want to reward agents with good technologies. In example 2, we can argue that no part of the technological gain should be allocated
to agent 2. More generally, if an agent can be added to any (non-empty) coalition without ever improving the technology of that coalition, then he should
be allocated no part of the technological gains and should pay his stand-alone
cost. This will be the central axiom used in the paper, that we will call the
Technological dummy property.
This property is a close relative of the familiar Dummy property in the classic
cost sharing literature, that says that if an agent can be added to any coalition
at no extra cost, than this agent should not pay anything. This property was
introduced in the original paper by Shapley (1953) where it was the main equity
postulate. It conveys the simple idea that agents should not pay for costs for
which they are not responsible. They are therefore fully responsible for the costs
of their demands. The property has been adapted to models with continuous
demands. In that context, it implies that if there is an agent such that adding
one unit of his good to any demand pro…le can be done at no extra cost, than
this agent should not pay anything. See Moulin and Sprumont (2006) for a
discussion of the implications of the property and a comparison to properties
yielding partial responsibility.
The technological dummy property di¤ers in the sense that we look at variations of average cost, and not total cost, when an agent joins a coalition. This
coalition also needs to be non-empty, which is not the case in the classic property. We di¤er to section 5 further discussion on this point.
Conceptually, the property is even closer to the stronger property, for the
classic model with continuous demands, that if there is an agent such that the
3
extra cost of adding one unit of his good to any demand pro…le is always equal to
a, than this agent should pay a times his demand. In that case, there is never any
gains related to scale when the agent joins a coalition, and the property imposes
that he gets none of it. This property does not give anything new in the model
with publicly-owned technology and continuous demands as it is implied by the
usual properties of Additivity, Non-negativity and the classic Dummy property.
The cost sharing rules satisfying these properties have been characterized by
Friedman (2004) and Haimanko (2000). See Wang (1999) for the case with
indivisible demands and Weber (1988) for the case of the stand-alone game.
Combined with adaptations of classic properties of linearity (on both demands and technologies) and symmetry, the technological dummy property generates a well-de…ned class of cost sharing rules. We then de…ne properties of
monotonicity in the technology, a population monotonicity property when the
added agent does not contribute anything technologically, and the classic standalone property. It turns that any one of these three properties, in combination
with the four mentioned before, is su¢ cient to characterize a unique rule.
This rule is such that for all agents i; we de…ne a game that allocates to
each coalition that does not contain i the demand of agent i times the change in
average cost when the coalition joins agent i: The Shapley value is then applied
on this game. An agent is allocated his stand-alone cost to which we add the
sum of his Shapley values on the di¤erent games.
The paper is organized as follows: Section 2 de…nes the framework, the
proposed rule and the central axioms used. Section 3 presents the main characterization results. In Section 4, we extend the framework to variable population
problems and introduce a new characterization of the proposed rule. Section
5 discusses a possible weakening of the key axiom of Technological Dummy
and extensions to more general problems. Independence of axioms is proven in
appendix.
2
2.1
Notations and de…nitions
The model
Let N = f1; :::; ng be the (…xed) set of agents. Let N = fS j ? 6= S N g be
the set of non-empty subsets of N; i.e., the set of possible, non-empty coalitions.
Agents have demands for a single, common good. Let x 2 Rn+ be the demand
vector. Demands are supposed inelastic. If S 2 N and x 2 Rn+ ; write x(S) :=
P
x(S)
i2S xi and x(S) := jSj :
We use the simple assumption that there are constant returns to scale, that is
that at a given technology, marginal cost is constant. Let = c 2 RN
T ) c(S)
+ jS
c 2 represents the average (or marginal) cost for each coalition, with the assumption that this cost cannot increase when we add agents to a coalition.
A problem is a vector (c; x) 2
Rn+ ; that is the cost and demand vectors.
The cost for a coalition S is simply de…ned as c(S)x(S):
4
c(T ) :
P
A rule is a map y :
Rn+ ! Rn s.t.
i2N yi (c; x) = c(N )x(N ). Note
that since c(S)x(S) can decrease when we add agents to coalition S, we do not
impose that yi 0: There is ample justi…cation in this setting to subsidize an
agent if he allows other agents to signi…cantly improve on their technologies.
2.2
A proposed rule
Fix a problem (c; x): For every i 2 N; de…ne a function v i ( ; c) as follow:
v i (S; c) = c(fS [ ig)
c(fig) for each S
i
N n fig
The function v ( ; c) is a TU-game on the player set N n fig : Given the problem (c; x); it gives, for each coalition, the change in the marginal cost when a
coalition joins agent i: Notice that since c(S)
c(T ) for all S
T
N; we
have that v i (S; c)
v i (T; c) for any S
T
N n fig and v i (S; c)
0 for all
S N n fig, all i 2 N and all c 2 :
P
The allocation for the problem (c; x) is then 'i (c; x) = c(fig)xi + j2N nfig xj Shi (v j ( ; c));
where Sh(v j ( ; c)) is the Shapley value on the game v j ( ; c):
Thus, the rule separates the cost of each player. The cost of giving agent i
is demand is c(N )xi ; and agent i is allocated c(fig)xi . Players in N n fig must
be allocated (c(N ) c(fig)) xi : To do so, we use the Shapley value on v i :
2.3
Properties
The following properties are central to the results in the next section.
The …rst two properties are along the lines of the classic Additivity property.
In our context, particularly without the assumption of non-negativity of the
cost-shares, we need the stronger property that the rule be linear in both the
demand and the cost vectors. Because of the assumption of constant returns to
scale, the Demand Linearity property is not particularly strong.
Demand linearity: For x1 ; x2 2 Rn+ , c 2 and 1 ; 2 2 R+ ; y(c; 1 x1 +
2
1
2
2 x ) = 1 y(c; x ) + 2 y(c; x ):
Technological linearity: For c1 ; c2 2 , 1 ; 2 2 R, such that 1 c1 +
2
1
2
1
2
n
2 c 2 ; y( 1 c + 2 c ; x) = 1 y(c ; x) + 2 y(c ; x) for any x 2 R+ :
We also use a weak form of symmetry, where if two agents are completely
identical, both in their demands and how they in‡uence the technology, we
impose that they are allocated the same amount.
Equal shares for equivalent agents: For any problem (c; x) and any
i; j 2 N; if c(S [ fig) = c(S [ fjg) for all S
N n fi; jg and xi = xj ; then
yi (c; x) = yj (c; x):
Finally, to introduce the key axiom discussed in the introduction, we need
the following de…nition:
De…nition For any c 2 ; if c(S [ fig) = c(S) for all ? 6= S
N n fig ; we
say that i is a technological dummy in c.
Technological dummy property: For any problem (c; x); if i 2 N is a
technological dummy in c, then yi (c; x) = c(fig)xi :
5
3
Results
First, remark that on any problem with only two agents, the Technological
Dummy property and Technological Linearity are su¢ cient to characterize ':
To see this, remark that any problem c 2 can be written as the sum of two
technological vectors c1 and c2 such that one agent is a technological dummy in
c1 and the other agent is a technological dummy in c2 : There are many ways to do
this, one simple way being to de…ne c1 such that c1 (f1g) = c(f1g) c(f1; 2g) and
c1 (f2g) = c1 (f1; 2g) = 0: We then de…ne c2 = c c1 : The Technological Dummy
property, combined with the budget-balance condition, de…nes a unique rule on
c1 and c2 : By Technological Linearity (in fact a weaker property of additivity is
su¢ cient), we have also de…ned a unique rule on c:
On problems with larger sets of agents, the budget-balance condition together with the Technological Dummy property will not yield a unique rule, so
more properties will be needed.
Before moving on to the main results, we need to introduce some notations.
Notation 1 For each T 2 N nN; de…ne
T (S)
=
T
2
by
0 if S [ T = N
1 otherwise
Moreover, de…ne N 2 by N (S) = 1 for all S 2 N :
Let 0 = f T j T 2 N g. By de…nition1 , 0
: Note that T is such that
all i 2 T are technological dummies in T ; while agents in N nT are not.
We are now ready to introduce the family of rules obtained by adding properties of Demand Linearity and Equal Shares for Equivalent Agents.
Lemma 1 If a rule y satis…es the Technological Dummy property, Technological
Linearity, Demand Linearity and Equal Shares for Equivalent Agents then there
exists a list of n 2 real numbers = 1 ; :::; n 2 such that for all T 2 N ,
x 2 Rn+ and i 2 N; yi ( T ; x) is equal to
)
(
xi
if i 2 T
(1)
yi ( T ; x) =
jT j
(xi x(N n(T [ fig))) nx(TjT)j if i 2
=T
n 1
with the convention that x(?) = 0 and
unique rule on
Rn+ :
= 0: Each value of
determines a
Proof. The proof is in two steps. First, we will show that 0 is a base of RN : In
the second step, we will determine cost allocations on problems ( T ; x) 2 0 Rn+
using the properties of Technological Dummy, Demand Linearity and Equal
Shares for Equivalent Agents. Technological Linearity then allows us to compute
cost allocations on any (c; x) 2
Rn+ :
Step 1: Show that 0 is a base of RN
1 Throughout
this paper,
denotes strict inclusion, i.e. ( :
6
We show that 0 is a generating set of RN : Since j 0 j = jN j ; 0 is a base
of RN :
For all S 2 N ; de…ne S 2 RN
+ such that, for all S 2 N ; S (T ) = 1 if
S T and 0 otherwise. The set = f S gS2N is often used in the cost sharing
literature, and it is well-known that it is a generating set. We will show that
every game in is a linear combination of games in 0 :
First, for all S 2 N nN; S = N
N nS , as, by de…nition of N and N nS ;
(T
)
=
1
if
S
T
and
0
otherwise.
S
De…ne
X
( 1)jSj+1 S
N = N
S2N nN
We check that for T 2 N nN;
N (T )
=
=
N (T )
N (T )
1
X
= 0 while
X
( 1)jSj+1
1
X
1
X
S (T )
( 1)jSj+1
( 1)jSj+1 +
S2N nN
=
= 1.
S2N nN
S2N nN
S[T 6=N
=
N (N )
X
( 1)jSj+1
S2N nN
S[T =N
( 1)jSj+1 +
S N
S6=?
X
jT j+jRj+1
( 1)n
R T
= 1 + ( 1)1 + ( 1)n
= 0
1
( 1)n
1
Since S (N ) = 0 for all S 2 N nN and N (N ) = 1; N (N ) = 1:
As claimed, every game in can be written as a linear combination of games
in 0 :
Since 0 is a generating set and j 0 j = jN j ; every c 2 can be written as a
unique linear combination of the elements of 0 :
Step 2: Show that for all i 2 N and x 2 Rn+ shares on the base problems
( T ; x) 2 0 Rn+ can be written as in (1).
This will be done in three substeps. First, we …nd the allocation yi ( T ; x)
when i 2 T; which, with budget balance, allows us to determine all cost allocations when jT j n 1: Next, we …nd the allocation yi ( T ; x) when i 2
= T and
show that it depends on a parameter T : Finally, we show that jSj = jT j )
T
= S:
Step 2.1: Show that yi ( T ; x) = xi for all i 2 T; T 2 N ; x 2 Rn+ :
For any T 2 N ; we have that T (S [ fig) = T (S) for every ? 6= S N n fig
and every i 2 T; that is each i 2 T is a technological dummy in T . By the
Technological Dummy Property, yi ( T ; x) = xi for all i 2 T and all T 2 N :
Combined with budget balance, it implies that for any i 2 N , yi ( N nfig ; x) =
x(N n fig); since N nfig (N ) = 0:
7
We have thus de…ned unique allocations on ( N nfig ; x) and ( N ; x):
Step 2.2: Show that, for every T 2 N such that jT j < n 1; we have
yi (
T ; x)
=
T
(xi
x(N n(T [ fig)))
x(T )
if i 2
=T
n jT j
with T 2 R:
Fix T 2 N such that jT j < n 1 and i; j 2 N nT until the end of step 2.2.
Since T (N ) = 0; Step 2.1 and budget balance imply that
X
yk ( T ; x) = x(T )
(a)
k2N nT
For any S 2 N ; de…ne 1S 2 Rn+ as follows: 1Sk = 1 if k 2 S and 1Sk = 0
otherwise: In particular, 1k is such that agent k demands 1 P
while everybody
k
else has demand 0: Any vector x can be rewritten as x =
k2N xk 1 : Let
l
T
kl := yk ( T ; 1 ) for k; l 2 N be the cost allocation of agent k in the problem
( T ; 1l ): By Step 2.1 we have,
if k 2 T;
T
kl
= 0 for all l 6= k and
T
kk
=1
(b)
Then, we have
yi (
T ; x)
= yi (
=
X
T;
X
xk 1k )
k2N
xk yi (
k
T;1 )
(By Demand Linearity)
k2N
=
X
T
ik xk
k2N
From (a) and (b)
X
T
kl
=
1 for l 2 T
(2)
T
kl
=
0 for l 2 N nT
(3)
k2N nT
X
k2N nT
Since i; j 2 N nT; we have that T (S [ fig) = T (S [ fjg) for all S 2
N n fi; jg : If xi = xj ; then by Equal Shares of Equivalent Agents, yi ( T ; x) =
yj ( T ; x):
Consider the demand vector 1k , for k 2 T: Since 1ki = 1kj = 0, by Equal
Shares for Equivalent Agents we must have that yi ( T ; 1k ) = Tik = Tjk =
yj ( T ; 1k ): Combined with (2), we have that Tik = n 1jT j for all k 2 T:
Case 1: jT j = n 2: Consider the demand vector 1fi;jg : By Equal Shares
for Equivalent Agents, yi ( T ; 1fi;jg ) = Tii + Tij = Tji + Tjj = yj ( T ; 1fi;jg ):
8
T
T
T
ij =
jj and
ji =
fi;jg
T
) and
jj = yj ( T ; 1
x(T )
T
;
x)
=
(x
x
)
i
j
T
2 :
yi (
T
Therefore, yi ( T ; 1fi;jg ) = Tii
jj =
T
T
T
= jj : De…ne
= jj : For any x 2 Rn+ ;
T
ii :
By (3),
T
ii +
T
ii
Case 2: jT j < n 2: Consider the demand vector 1k for k 2 N n (T [ fi; jg) :
Since 1ki = 1kj = 0, by Equal Shares for Equivalent Agents we must have that
yi ( T ; 1k ) = Tik = Tjk = yj ( T ; 1k ): Combined with (3), we have that
T
ik
=
T
kk
n
jT j
(4)
1
for all k 2 N nT:
Consider the demand vector 1fi;jg : By Equal Shares for Equivalent Agents,
yi ( T ; 1fi;jg ) = Tii + Tij = Tji + Tjj = yj ( T ; 1fi;jg ): By (4), it implies that
T
ii
T
jj
T
ii
n jT j 1 =
De…ne T = Tjj :
n jT j 1
+
T
jj :
After simpli…cation, we get that
T
ii
=
T
jj :
As claimed, we have that for every T 2 N such that jT j < n 1 and every
k2
=T
1
0
X xl
X
x
l
T @
A
(5)
yk ( T ; x) =
xk
n jT j 1
n jT j
l2T
l2N n(T [fkg)
T
=
(xk
x(N n(T [ fkg)))
x(T )
n jT j
Step 2.3: Show that for any S; T 2 N such that jSj = jT j ; then S = T
Fix i; j 2 N , S N n fi; jg and x 2 Rn+ : De…ne = S[fig + S[fjg : Then,
for R 2 N
8
6 N and R [ S [ fjg =
6 N
< 2 if R [ S [ fig =
1 if R [ S [ fkg = N and R [ S [ flg =
6 N for fk; lg = fi; jg
(R) =
:
0 if R [ S = N
Clearly, 2 : We have that (R [ fig) = (R [ fjg) for all R 2 N n fi; jg
and by Equal Shares for Equivalent Agents, if xi = xj ; then yi ( ; x) = yj ( ; x):
By Technological Linearity, y( ; x) = y( S[fig ; x) + y( S[fjg ; x):
Then, by (5),
yi ( ; x) = xi +
S[fig
(xj
S[fjg
(xi
x(N n(S [ fi; jg)))
x(N n(S [ fi; jg)))
x(S [ fjg)
=
n jSj 1
x(S [ fig)
+ xj = yj ( ; x)
n jSj 1
which directly implies that S[fig = S[fjg : This in turn implies that S = T
if jSj = jT j :
Summing up the results of steps 2.1, 2.2 and 2.3, we get that y = y : With
the result of Step 1, it implies that de…nes a unique rule on
Rn+ :
9
Rules in this set di¤er only by how they allocate costs between non-technological
dummies in the basic problems, which results in di¤erent values for : In particular, ' corresponds to = (1; :::; 1):
To arrive to a characterization of ', we need to introduce new properties.
The …rst one is the classic stand-alone property that requires that every agent
should not be allocated more than his stand-alone cost.
Stand alone property: For any i 2 N; c 2 ; x 2 Rn+ ; yi (c; x) c(fig)xi
We also introduce a second property, a monotonicity property, that imposes
that starting from a cost vector c 2 ; if the average cost decreases for a coalition
S and stays the same for all other coalitions; then all agents in S should not see
their cost share increase following this change.
Technological Monotonicity: For any i 2 N; if c; c0 2 are such that
there is a S
N n fig with c(S [ fig) > c0 (S [ fig) and c(T ) = c0 (T ) for all
T 6= S [ fig ; then yi (c; x) yi (c0 ; x) for any x 2 Rn+ :
With these properties, we are ready for the main characterization results.
Theorem 1 A rule y satis…es the Technological Dummy property, Technological
Linearity, Demand Linearity, Equal Shares for Equivalent Agents and Stand
alone property if and only if y = ':
Proof. See the appendix for the proof that ' satis…es the …ve properties. We
show that they imply a unique rule.
Fix i 2 N: Fix T N n fig, with T 6= ?:
By Lemma 1; we have yi ( T ; 1i ) = jT j . By de…nition of T ; T (fig) = 1:
By the Stand-alone property, we must have
jT j
1
(6)
De…ne cT = fig + N nfig + T [fig
T : By de…nition of the elements of
0 ; we have, for any S 2 N
8
= S and S [ T 6= N n fig
< 2 if i 2
1 if [N n fig * S and i 2 S] or [N n fig * S and S [ T = N n fig]
cT (S) =
:
0 if N n fig S
(7)
Therefore, cT (S [ fjg) cT (S) for all S 2 N and all j 2 N , which imply
cT 2 :
By Technological Linearity, yi (cT ; x) = yi ( fig ; x)+yi ( N nfig ; x)+yi ( T [fig ; x)
yi ( T ; x): By Lemma 1; we have
yi (cT ; 1i )
= 1+0+1
jT j
= 2
jT j
By (7), cT (fig) = 1: By the Stand-alone property, we must have
yi (cT ; 1i )
jT j
2
cT (fig)
1
1
jT j
10
With (6), we conclude that jT j = 1:
We thus have that k = 1 for all k 2 1; 2; :::; n 2: By Lemma 1, we have
de…ned a unique rule on
Rn+ and we must have that y = ':
Theorem 2 A rule y satis…es the Technological Dummy property, Technological
Linearity, Demand Linearity, Equal Shares for Equivalent Agents and Technological Monotonicity if and only if y = ':
Proof. See the appendix for the proof that ' satis…es the …ve properties. We
show that they imply a unique rule.
Fix i 2 N: Fix T N n fig, with T 6= ?:
By de…nition of T and N ; we have for all S N n fig ; T (S) = N (S) = 1:
We can then construct a sequence c0 ; :::; ck such that c0 = T and ck = N with
the property that for each l 2 [1; ::; k] ; cl 2 and there exists a S 2 N such that
i 2 S, cl 1 (S) > cl (S) and cl 1 (R) = cl (R) for every R 2 N nS: To do so, let =
( 1 ; :::; 2jT j ) be an ordering of the elements of the set fS 2 N j ? 6= S N nT g.
Let l 2 N be the element in position l: We select an ordering such that for
l < m; j l j j m j : Since for all ? 6= S
N nT; T (S) = 0; we construct the
jT j
sequence c0 ; :::; c2 as follows: at each step l = 1; :::; 2jT j ; let cl be such that
cl ( l ) = cl 1 ( l ) + 1 = 1 and cl (R) = cl 1 (R) for every R 2 N n l :
Successive applications of Technological Monotonicity imply that yi ( T ; x)
yi ( N ; x): By Lemma 1; we have yi ( T ; 1i ) = jT j and yi ( N ; 1i ) = 1: Therefore,
jT j
We claim that jT j 1
To prove this claim, de…ne
For R 2 N ; we have that
i
(R)
=
=
If R = fig ; fS
i
=
1
N nfig
N nfig (R)
+
N nfig (R)
+
(8)
+
X
P
S N nfig (
( 1)jSj+n
1)jSj+n
S[fig (R)
S N nfig
X
( 1)jSj+n
S N nfig
R[S[fig6=N
N n fig j R [ S [ fig = N g = ?. Therefore,
X
i
(fig) =
( 1)jSj+n
N nfig (fig) +
S N nfig
=
0+
n
X2
( 1)k+n
k=0
2n 1
=
( 1)
= 1
11
(n 1)!
k!(n k 1)!
S[fig :
If R 2 N n fig ;
i
(R)
=
=
=
=
N nfig (R)
+
N nfig (R)
+
N nfig (R)
X
( 1)jSj+n
X
( 1)jSj+n
X
S N nfig
( 1)jSj+n
S N nfig
R[S[fig=N
P
S N nfig
2n 1
( 1)
X
( 1)jP j
jRj+2n 1
R
2n 1
+ ( 1)
N nfig (R)
Therefore, i (R) = 0 if i 2 R and R 2 N n fig ; 1 otherwise and i 2 :
By de…nition of i and T ; we have, for all S
N n fig ; that i (S [ fig)
i
(S) = T (S): By the same argument as earlier, successive
T (S [ fig) and
applications of Technological Monotonicity yield
yi ( i ; x)
yi (
T ; x)
(9)
P
By Technological Linearity, yi ( i ; x) = yi ( N nfig ; x)+ S N nfig ( 1)jSj yi ( S[fig ; x):
By Lemma 1 and our selected x; yi ( N nfig ; x) = 0 and yi ( S[fig ; x) = 1 for all
S N n fig : Then,
X
yi ( i ; x) =
( 1)jSj = 1
(10)
S N nfig
By Lemma 1 and our selected x; yi ( T ; x) = jT j :
Together with (9) and (10), this implies that jT j 1:
jT j
With (8) ; we conclude that
= 1.
Therefore, for all k 2 1; 2; :::; n 2; k = 1: By Lemma 1, we have de…ned a
unique rule on
Rn+ and we must have that y = ':
4
Variable population problems
We now look at problems with variable population. The model is adapted to
this framework as follows:
Let N = f1; :::; ng be the (…nite) set of admissible agents. For any T 2 N ;
jT j
let T = fS j ? 6= S T g be the set of non-empty subsets of T . Let x 2 R+ be
T
T
the demand vector. Let n = c 2 R+ j R S 2 T ) c(R) o c(S) :
Our domain is D = (T; c; x) j T 2 N ; c 2
T
jT j
; x 2 R+
and a rule is a
P
jSj
jT j
map y : D ! [S2N R s.t. y(T; c; x) 2 R and i2T yi (T; c; x) = c(T )x(T ).
To de…ne ' in this setting, …x a problem (T; c; x): For every i 2 T; de…ne a
function vTi ( ; c) as follow:
vTi (S; c) = c(fS [ ig)
c(fig) for each S
vTi (
T n fig
The function
; c) is a TU-game on the player set T n fig : The allocation
P
for the problem (T; c; x) is then 'i (T; c; x) = c(fig)xi + j2T nfig xj Shi (vTj ( ; c)):
12
Notation 2 For S 2 T ; and x 2 RjT j ; de…ne xs such that xsi = xi for all i 2 S:
For each T 2 N and S 2 N nN; de…ne TS 2 T by
T
S (R)
Moreover, de…ne
T
T
2
T
by
=
T
T (S)
0 if R [ S = T
1 otherwise
= 1 for all S 2 T : Let
T
0
=
T
S
jS2T .
Note that all axioms de…ned in the previous section have direct extensions
on variable population problems. The result of Lemma 1 can also be extended
to this framework.
Lemma 2 If a rule y satis…es the Technological Dummy property, Technological
Linearity, Demand Linearity and Equal Shares for Equivalent Agents then there
exists, for each T 2 N ; such that jT j 2; a list of jT j 2 real numbers (T ) =
jT j
1
(T ); :::; jT j 2 (T ) such that for all S 2 T , x 2 R+ and i 2 T; yi (T; TS ; x)
is equal to
(
xi
if i 2 S
T
yi (T; S ; x) =
x(S)
jSj
=S
(T ) (xi x(T n(S [ fig))) jT j jSj if i 2
with the convention that x(?) = 0 and jT j 1 (T ) = 0: Each value of
f (T ) j T 2 N ; jT j > 2g determines a unique rule on D:
=
Proof. Follows directly from lemma 1.
In variable population problems, a property on the addition of technological
dummies can be de…ned. More speci…cally, the following property imposes that
when we add a new agent to a problem, if that agent is technologically dummy
and all other costs and demands stay the same, the cost shares of any of the
original agents should not increase.
T-dummy monotonicity: For any T 2 N , if we have, for j 2 N nT; c 2 T
and c0 2 T [fjg ; that c0 (S) = c(S) and c0 (S) = c0 (S [ fjg) for all S 2 T ; then
jT j+1
:
yi (T [ fjg ; c0 ; x) yi (T; c; xT ) for all i 2 T and x 2 R+
Adding the T-dummy Monotonicity property to the four central axioms
again allows us to characterize ':
Theorem 3 A rule y satis…es the Technological Dummy property, Technological
Linearity, Demand Linearity, Equal Shares for Equivalent Agents and T-dummy
Monotonicity if and only if y = ':
Proof. See the appendix for the proof that ' satis…es the …ve properties. We
show that they imply a unique rule.
We use the assumption that y(T; ; ) = '(T; ; ) for all T 2 N such that
jT j = k; with 1 < k < n and show that it implies that y(T 0 ; ; ) = '(T 0 ; ; ) for
all T 0 2 N such that jT 0 j = k + 1:
jT j+1
Fix T 2 N nN; i 2 T , ? 6= S
T n fig and j 2 N nT: Fix x 2 R+
such
that xi = 1 and xk = 0 for all k 6= i:
13
Step 1: Show that k (T [ fjg) = 1 for 1 < k jT j 1
T [fjg
By de…nition, TS and S[fjg are such that j is a technological dummy in
T [fjg
S[fjg
T [fjg
T
S (R) = S[fjg (R) for all R 2 T : By T-dummy Monotonicity, we
T [fjg
yi (T; TS ; xT ) yi (T [ fjg ; S[fjg ; x): By Lemma 2 and our selected
jSj
jSj+1
and
have that
x; this amounts to
'(T; ; ); we have
jSj
(T )
(T [ fjg): Since, by assumption, y(T; ; ) =
(T ) = 1: Therefore,
jSj+1
(T [ fjg)
1
(11)
T
De…ne cS = Tfig + TT nfig + TS[fig
S:
T
By de…nition of the elements of 0 ; we have that, for any R 2 T
8
= R and R [ S 6= T n fig
< 2 if i 2
1 if fT n fig * R and i 2 Rg or fT n fig * R and R [ S = T n figg
cS (R) =
:
0 if T n fig R
and thus cS 2 T :
T [fjg
T [fjg
T [fjg
T [fjg
De…ne cSj = fi;jg + T [fjgnfig + S[fi;jg
S[fjg :
Sj
Since c is built using the same technological vectors as cS ; to which we
have added j as a technological dummy, cS (R) = cSj (R) for all R 2 T and
j is a technological dummy in cSj : By Technological Linearity, yi (T; cS ; xT ) =
yi (T; Tfig ; xT ) + yi (T; TT nfig ; xT ) + yi (T; TS[fig ; xT ) yi (T; TS ; xT ): By Lemma
jSj
2 and our selected x; yi (T; cS ; xT ) = 1 + 0 + 1
(T ): Since, by assumption,
y(T; ; ) = '(T; ; ); we have jSj (T ) = 1 and yi (T; cS ; xT ) = 1: In the same
jSj+1
(T [ fjg): By T-dummy Monotonicity,
manner, yi (T [ fjg ; cSj ; x) = 2
yi (T; cS ; xT ) yi (T [ fjg ; cSj ; x); which gives
jSj+1
(T [ fjg)
1
Together with (11), this gives that k (T [ fjg) = 1 for 1 < k
Step 2: ShowPthat 1 (T [ fjg) = 1
De…ne c? = R2T nT ( 1)jRj+1 TR : For all P 2 T nT; we have
c? (P )
=
X
( 1)jRj+1
T
R (P )
=
R2T nT
=
X
R2T nT
=
X
( 1)jRj+1
( 1)
X
1
( 1)jRj+1
R2T nT
P [R6=N
R2T nT
P [R=N
M
R T
R6=?
=
X
( 1)jRj+1
X
jT j
( 1)jRj+1
( 1)jT j
jP j+jM j+1
P
( 1)jT j+1 + ( 1)jT j+1 = 1
while
c? (T ) = 0: Clearly, c? 2 T : By Technological Linearity, yi (T; c? ; xT ) =
P
jRj+1
yi (T; TR ; xT ): By Lemma 2, the assumption that y(T; ; ) =
R2T nT ( 1)
14
'(T; ; ) and our selected x; yi (T; TR ; xT ) = 1 for all R 2 T such that R 6= T n fig
and yi (T; TT nfig ; xT ) = 0: Therefore,
X
yi (T; c? ; xT ) =
( 1)jRj+1 ( 1)jT j
R2T nT
=
jT j 1
X
( 1)k+1
k=1
=
=
( 1)
jT j!
k!(t k)!
( 1)jT j+1
( 1)jT j
( 1)jT j
1
T [fjg
T [fjg
; c? (R) = fjg (R) for all R 2 T and j is
fjg
T [fjg
in fjg : By T-dummy Monotonicity, yi (T; c? ; xT )
By de…nition of
logical dummy
fjg ;
T [fjg
; x);
fjg
which implies that
1
(T [ fjg)
1
a technoyi (T [
(12)
To show that 1 (T [ fjg)
1; we need to de…ne two more technological
vectors. De…ne cT nfig = 2 Tfig + TT nfig c? : By de…nitions of c? and the
elements of T0 ; for any R 2 T
8
= R and R 6= T n fig
< 2 if i 2
1 if i 2 R and R 6= T
cT nfig (R) =
:
0 if T n fig R
and cT nfig 2 T :
De…ne cj = 2
such that
in
T [fjg
fi;jg :
T [fjg
T [fjg
fi;jg + T [fjgnfig
T [fjg
T
fi;jg (R) = fig (R) for
j
j
By de…nition,
T
fig
and
T [fjg
fi;jg
are
all R 2 T and j is a technological dummy
The same is true for the pair
seen that it is true for c? and
T nfig
T [fjg
:
fjg
T [fjg
:
fjg
T [fjg
T
T nfig
and
T [fjg
T [fjgnfig :
We have also
Therefore, we have that it also holds for
c
and c : Moreover, c 2
: By T-dummy Monotonicity, we must have
yi (T; cT nfig ; xT ) yi (T [ fjg ; cj ; x):
Once again using Technological Linearity, the assumption that y(T; ; ) =
'(T; ; ) and our selected x; we get that yi (T; cT nfig ; xT ) = 2 1 + 0 1 = 1 and
1
(T [ fjg):
yi (T [ fjg ; cj ; x) = 2 + 0
We obtain
1
(T [ fjg) 1
Together with (12), this gives that 1 (T [ fjg) = 1.
Step 3: Show that y(T; ; ) = '(T; ; ) for any T 2 N
We have shown that y(T; ; ) = '(T; ; ) for all T 2 N such that jT j = k;
with 1 < k < n implies that y(T 0 ; ; ) = '(T 0 ; ; ) for all T 0 2 N such that
jT 0 j = k + 1:
As previously discussed, we have that y(T; ; ) = '(T; ; ) for all T 2 N such
that jT j = 2: Thus, y(T; ; ) = '(T; ; ) for any T 2 N and y = ':
15
5
Discussions and extensions
One can weaken the Technological Dummy property by slightly modifying the
framework. Suppose that there exists a freely available but poor technology
that gives an average cost of c: The (average) cost vectors are now de…ned on
all subsets of N: In particular, c(?) = c: We can now de…ne a weaker technological dummy property by saying that if c(S [ fig) = c(S) for all S N n fig ;
then yi (c; x) = cxi : On this framework, even the weaker Technological Dummy
property leaves little latitude. The new degree of freedom consists in how we
allocate costs when all agents have average costs of 0; but where c > 0:2 It
seems extremely natural to impose in those cases that all agents pay 0; which
is prescribed by the rule ':
However using this weaker Technological Dummy property on this framework, Theorem 1, that uses the Stand-alone property, still holds, but Theorem
2, that uses Technological Monotonicity, does not. In the case of Theorem 3,
using T-dummy Monotonicity, it follows only if the set of admissible agents is
in…nite.
The model presented here is obviously simpli…ed, as it imposes constant
returns to scale, but shows the importance of the so-called technological e¤ect
and how the concept of the Technological Dummy property restricts the set of
potential rules. On more general models, we would like to be able to separate
the technological and the scale e¤ects, and use appropriate methods on both.
Work as yet to be done on this subject, but one intuitive way to separate the
two e¤ects would be to estimate the scale e¤ect by evaluating costs at a constant
technology, and to estimate technological e¤ect on the function de…ned by the
di¤erence between the real cost function and the one used to isolate the scale
e¤ect. A classic cost sharing method could be used to determine the shares on
the scale gains, while ' or other methods built with the technological e¤ect in
mind could be used to determine the shares on the technological gains.
References
[1] FRIEDMAN, E. (2004), "Paths and consistency in additive cost sharing",
International Journal of Game Theory, Vol. 32, No.4, pp. 501-518.
[2] HAIMANKO, O. (2000), "Partially Symmetric Values", Mathematics of
Operations Research, Vol. 25, No. 4, pp. 573-590.
[3] LA MANNA, M.M.A. (2006), The Hidden Surplus From Research Joint
Ventures: An Application Of Systems Reliability Theory, CRIEFF Discus2 Formally,
let N 0 = N [ f?g and
0
=
n
0
c 2 RN
+ jS
T ) c(S)
o
c(T ) : De…ne, for
? 6= T
N; 0T such that 0T (S) = 0 if S [ T = N and 1 otherwise. De…ne 0N such that
0 (S) = 1 for all S and 0 (S) = 0 for all S 2 N and 0 (?) = 1: De…ne 0 =
0
?
?
0
N
S S2N 0 :
0 is a base of 0 ; and on all 0 such that T 6= ?; the weak and the strong Technological
0
T
Dummy property imply the same thing, and results are the same as in previous sections. It
remains 0? ; for which the weak Technological Dummy property does not say anything.
16
sion Papers 0610, Centre for Research into Industry, Enterprise, Finance
and the Firm.
[4] LEROUX, J. (2006), "Cooperative production under diminishing marginal
returns: interpreting …xed-path methods", forthcoming in Social Choice
and Welfare.
[5] MOULIN, H. (2002), "Axiomatic Cost and Surplus-Sharing", in Handbook
of Social Choice and Welfare, by K. Arrow, A. Sen and K, Suzumura,
editors, North-Holland.
[6] MOULIN H. and Y. SPRUMONT (2006), "Responsibility and CrossSubsidization in Cost Sharing", Games and Economic Behaviour, Vol. 55
No. 1, pp. 152-188.
[7] SHAPLEY L.S. (1953), "A Value for n-person Games". In Contributions to
the Theory of Games, volume II, by H.W. Kuhn and A.W. Tucker, editors.
Annals of Mathematical Studies v. 28, pp. 307-317. Princeton University
Press.
[8] TRUDEAU C. (2006), "On the Core of Single-Source Uncapacitated Network Flow Problems", mimeo, Département de Sciences Économiques, Université de Montréal.
[9] WANG Y. (1999), "The additivity and dummy axioms in the discrete cost
sharing model", Economic Letters, Vol. 64, pp. 187-192.
[10] WEBER R. (1988), "Probabilistic values for games", In The Shapley Value,
by A. Roth, editor, Cambridge University Press.
17
6
Appendix
Properties of '
6.1
Lemma 3 In …xed-population problems, ' is a budget balanced rule that satis…es the Technological Dummy property, Technological Linearity, Demand Additivity, Equal Shares for Equivalent Agents, Technological Monotonicity, and the
Stand-alone property.
Proof. Fix i; j 2 N .
Budget balance:
X
'k (c; x)
=
k2N
X
c (fkg) xk
k2N
=
X
X
xl Shk (v l ( ; c))
k2N l2N nfkg
c (flg) xl
l2N
=
X
X
X
xl Shk (v l ( ; c))
l2N k2N nflg
X
c (flg) xl
l2N
= c(N )
X
X
xl (c(flg)
c(N ))
l2N
xl = c(N )x(N )
l2N
Technological Dummy: Suppose that c is such that c(S [ fig) = c(S) for
all ? 6= S
N n fig : For any k 2 N n fig and any S 2 N n fi; kg : v k (S [
fig ; c) = c(S [ fi; kg) c(fkg) = c(S [ fkg) c(S) = v k (S; c): We also have
that v k (fig ; c) = c(fi; kg) c(fkg) = 0: Thus, all marginal contributions of
agent i are equal to zero and Shi (v k ( ; c)) = 0 for all k 2 N n fig : Therefore,
'i (c; x) = c(fig)xi :
Technological Linearity: For c1 ; c2 2
and 1 ; 2 2 R such that 1 c1 +
2
N n fjg
2 c 2 ; we have that, for any j 2 N; and any ? 6= S
v j (S;
1
1c
+
2
2c )
=
=
1
2
1 c (S [ fjg) + 2 c (S [
j
1
j
2
1 v (S; c ) + 2 v (S; c )
1
1 c (S)
fjg)
2
2 c (S)
Therefore,
'i (
1
1c
+
2
2 c ; x)
=
1
1 c (fig)
+
2
2 c (fig)
X
+
xj Shi (v j ( ;
1
1c
+
2
2 c ))
j2N nfig
=
=
1
1 c (fig)
1
1 'i (c
+
; x) +
2
2 c (fig)
2
2 'i (c
+
X
xj
1 Shi (v
j
(S; c1 )) +
j2N nfig
; x)
Demand Additivity. Follows directly from the fact that demands enter ' in
a linear fashion.
Equal Shares for Equivalent Agents: Suppose that we have c such that
c(S [ fig) = c(S [ fjg) for all S N n fi; jg and x such that xi = xj : Clearly,
18
2 Shi (v
j
(S; c2 ))
the …rst terms in 'i (c; x) and 'j (c; x) are equal. For any S
k 2 N n fi; jg
v k (S [ fig ; c) = c(S [ fi; kg)
c (fkg) = c(S [ fj; kg)
N n fi; jg and any
c (fkg) = v k (S [ fjg ; c)
Therefore, Shi (v k ( ; c)) = Shj (v k ( ; c) for any k 2 N n fi; jg : For any S
N n fi; jg ; we have
v i (S [ fjg ; c)
v j (S [ fig ; c)
v i (S; c)
=
=
v j (S; c) =
=
c(S [ fi; jg)
c(S [ fi; jg)
c(S [ fi; jg)
c(S [ fi; jg)
c (fig) c(S [ fig) + c (fig)
c(S [ fig)
c (fjg) c(S [ fjg) + c (fjg)
c(S [ fjg)
We thus have that v i (S [ fjg ; c) v i (S; c) = v j (S [ fig ; c) v j (S; c) for
any S
N n fi; jg ; which implies that Shj (v i ( ; c)) = Shi (v j ( ; c)): Therefore,
'i (c; x) = 'j (c; x):
Technological Monotonicity: Fix x 2 Rn+ and S 2 N such that i 2 S. Take
0
c; c 2 such that c(S)
c0 (S) and c(T ) = c0 (T ) for all T 2 N nS: For any
k
k 2 N n fig ; v (Sn fkg ; c) = c(S) c(fkg)
c0 (S) c(fkg) = v k (Sn fkg ; c0 );
k
k
0
while v (T; c) = v (T; c ) for all T 6= Sn fkg : By the properties of the Shapley
value, Shi (v k ( ; c)) Shi (v k ( ; c0 )). Then, 'i (c; x) 'i (c0 ; x):
Stand-alone property: Follows directly from the fact that Shi (v j ; c) 0 for
all i; j 2 N and all c 2 ; and from the fact that x 2 Rn+
Lemma 4 In variable population problems, ' is a budget balanced rule that
satis…es the Technological Dummy property, Technological Linearity, Demand
Additivity, Equal Shares for Equivalent Agents, Technological Monotonicity, the
Stand-alone property and T-dummy Monotonicity.
Proof. Budget balanced, the Technological Dummy property, Technological
Linearity, Demand Additivity, Equal Shares for Equivalent Agents, Technological Monotonicity and the Stand-alone property follow from Lemma 3. It remains
to look at T-dummy Monotonicity.
jT j+1
and c 2 T and c0 2 T [fjg such
Fix T 2 N nN , i 2 T; j 2 N nT; x 2 R+
0
0
0
that c (S) = c(S) and c (S) = c (S [ fjg) for all S 2 T :
For any k 2 T and ? 6= S T n fkg ; vTk [fjg (S; c0 ) = c0 (S [ fkg) c0 (fkg) =
c(S [fkg) c(fkg) = vTk (S; c) and vTk [fjg (S [fjg ; c0 ) = c0 (S [fj; kg) c0 (fkg) =
c0 (S [ fkg) c0 (fkg) = vTk [fjg (S; c): In addition, vTk [fjg (fjg ; c0 ) = c0 (fj; kg)
c0 (fkg) = 0: By the properties of the Shapley value, Shj (vTk [fjg ( ; c0 )) = 0
and Shi (vTk [fjg ( ; c0 )) = Shi (vTk ( ; c)): Since Shi (vTj [fjg ( ; c0 ))
'i (T [ fjg ; c0 ; x) 'i (T; c; xT ):
6.2
0; we have
Independence of axioms
De…ne C(S; c; x) = c(S)x(S), the function that assigns the total cost to each
coalition. De…ne y 1 (c; x) = Sh(C( ; c; x)) as the Shapley value on the standalone game C: One can verify that y 1 satis…es Budget-balance, Technological
19
Linearity, Demand Additivity, Equal Shares for Equivalent Agents, Technological Monotonicity and the Stand-alone property. By example 2, it fails the
Technological Dummy property. The adaptation to variable-population problems, y 1 (T; c; x); satis…es T-dummy monotonicity.
P
h
i
P
=S N nfig c(S[fig) c(S)
c(N
)x(N
)
c(fjg)x
For all i 2 N; de…ne yi2 (c; x) = c(fig)xi + P ?6P
j
j2N
j2N
?6=S N nfjg c(S[fjg) c(S)
P
P
if j2N ?6=S N nfjg c(S [ fjg) c(S) 6= 0 and yi2 (c; x) = c(fig)xi otherwise.
One can verify that y 1 satis…es Budget-balance, the Technological Dummy property, Demand Additivity, Equal Shares for Equivalent Agents, Technological
Monotonicity and the Stand-alone property, but fails Technological Linearity.
The adaptation to variable-population problems, y 2 (T; c; x); satis…es T-dummy
monotonicity.
Let D(c) = fi 2 N j i is a technological dummy in cg : For all i 2 N; de…ne
y 3 such that, for all i 2 N
8
c(fig)xi if i 2 D(c)h
>
>
i
<
P
xi
3
c(N
)x(N
)
c(N
)x
+
c(fjg)x
if i 2
= D(c) and x(N nD(c)) 6= 0
i
j
j2D(c)
yi (c; x) =
x(N nD(c))h
i
>
P
>
1
: c(N )xi +
if i 2
= D(c) and x(N nD(c)) = 0
j2D(c) c(fjg)xj
n jD(c)j c(N )x(N )
One can verify that y 3 satis…es Budget-balance, the Technological Dummy
property, Technological Linearity, Equal Shares for Equivalent Agents, Technological Monotonicity and the Stand-alone property, but fails Demand Additivity.
The adaptation to variable-population problems, y 3 (T; c; x); satis…es T-dummy
monotonicity.
For all j 2 N and all T 2 N ; de…ne v j;T ( ; c) as follows:
v i;T (S; c) = c(fS [ ig)
c(fig) for each S
T n fig
Fix S 2 N and de…ne y 4 such that:
(
P
c(fig)xi + j2Snfig xj Shi (v j;S ( ; c)) if i 2 S
4
P
yi (c; x) =
c(fig)xi + j2N n(S[fig) xj Shi (v j;N nS ( ; c)) if i 2 N nS
One can verify that y 4 satis…es Budget-balance, the Technological Dummy
property, Technological Linearity, Demand Additivity, Technological Monotonicity and the Stand-alone property, but fails Equal Shares for Equivalent Agents.
The adaptation to variable-population problems, y 4 (T; c; x); satis…es T-dummy
monotonicity.
P
n
For a problem (c; x); de…ne c = S2N ( 1)jSj+1 c(S): Then, de…ne c0 2 R2
such that c0 (?) = c and c0 (S) = c(S) for all S 2 N : Then, for all i 2 N;
yi5 = cxi + Shi (c0 )x(N ):
One can verify that y 5 satis…es Budget-balance, the Technological Dummy
property, Technological Linearity, Demand Additivity and Equal Shares for
Equivalent Agents, but fails Technological Monotonicity and the Stand-alone
property. It corresponds to the case where jT j = nn jTjTj j 1 : The adaptation to
variable-population problems, y 5 (T; c; x); fails T-dummy monotonicity.
20
The following table, where "+" signi…es that the property is satis…ed, and
"-" that it is not, summarizes the results
Technological Dummy property
Technological Linearity
Demand Linearity
Equal Shares for Equivalent Agents
Stand-alone property
Technological Monotonicity
T-dummy Monotonicity
21
'
+
+
+
+
+
+
+
y1
+
+
+
+
+
+
y2
+
+
+
+
+
+
y3
+
+
+
+
+
+
y4
+
+
+
+
+
+
y5
+
+
+
+
-