G . Tian Mathematical Social Sciences 40 2000 175 –189
179
with purely private goods or with externalities . Thus, the equilibrium concept can also
be viewed as optimality-guaranteeing equilibrium concepts .
Remark 2. An LCSE allocation does not always result in individually rational allocations
. Wilkie 1990 provided such a counter-example. Therefore, an LCSE allocation may not be in the core
. However, every interior LCSE allocation belongs to the core of the economy
cf. Weber and Wiesmeth, 1991.
n 21
Given profit share vector u [ D , an allocation x , y is a u-Lindahl equilibrium
allocation for an economy e if it is feasible and there are personalized price vectors,
K
q [ R , one for each i, such that: 1 y
maximizes profits q ? y 2 C y; 2
i 1
˚ x 1 q ? y w 1 u [q ? y 2 C y ] for all i [ N; 3 for all i [ N, there does not
i i
i i
˚ exist x , y such that x , y P x , y and x 1 q ? y w 1 u [q ? y 2 C y ]; 4
i i
i i
i i
i i
n
o q 5 q . Denote by Le;u the set of all such allocations.
i 51 i
Mas-Colell and Silvestre 1989 showed that in the convex technology case LCSE allocations are in one-to-one correspondence with Lindahl equilibrium allocations. The
correspondence is established by varying the profit share parameters which characterize Lindahl equilibrium allocations, i.e. LCSEe 5
Le;u . Thus, the existence of
n 21
u [D
a LCSE is guaranteed under the same conditions which guarantee the existence of Lindahl equilibria cf. Foley, 1970; Milleron, 1972; Roberts, 1974. Note that in the
constant returns case, a Linear Cost Share Equilibrium allocation reduces to a Lindahl equilibrium allocation.
Remark 3. Even though the indispensability condition is not necessary for the existence of LCSE
, this assumption cannot be dispensed with for feasible implementation. Tian 1988 showed that the Lindahl correspondence violates Maskin’s 1997 monotonicity
condition without this assumption and thus cannot be Nash-implemented by a feasible mechanism
. Since a Linear Cost Share Equilibrium allocation reduces to a Lindahl equilibrium allocation for constant returns economies
, they are also necessary con- ditions for the feasible and continuous implementation of Linear Cost Share Equilibrium
allocations .
3. Mechanism
In the following we will present a feasible and continuous mechanism which doubly implements the LCSE correspondence in Nash and strong Nash equilibrium.
Let M denote the ith agent’s message domain. Its elements are written as m and
i i
n
called messages. Let M 5
P
M denote the message space. The message spaces of agents
i i 51
are defined as follows. For each i [ N, his her message domain is of the form
nK n 21
K
˚
M 5 0,w ] 3 V 3
D 3 R 3 R
2
i i
1 11
nK nk
n
where V 5 a , . . . ,a [ R :o
a 5 0 . A generic element of M is m 5
h j
1 n
j 51 j
i i
180 G
. Tian Mathematical Social Sciences 40 2000 175 –189
w ,a , . . . ,a , b , . . . ,b , y ,g whose components have the following interpretations.
i i 1
in i 1
in i
i
The component w denotes a profession of agent i’s endowment, the inequality
i
˚ 0 , w w means that the agent cannot overstate his own endowment; on the other
i i
hand, the endowment can be understated, but the claimed endowment w must be
i 4
positive which is necessary to guarantee the feasibility even at disequilibrium points. The intuition here is straightforward: if a mechanism allows agents to overstate their
endowments, then it allows for infeasible outcomes — it will sometimes attempt to allocate more than is possible, given the true aggregate endowment. The component
a
; a , . . . ,a is the side compensation vector profile proposed by individual i. The
i i 1
in
component b ; b , . . . ,b is the direct cost share profile proposed by individual i.
i i 1
in
The component y denotes the proposed level of tax measured in public goods that
i
agent i is willing to contribute a negative y means the agent wants to receive
i
compensation from society. The component g is a shrinking index of agent i used to
i
shrink the private good consumption of other agents. Define the side compensations for consumption of public goods for the ith agent by
a m 5 a 3
i i 11,i
where n 1 1 is to be read as 1. Note that even though a m is only a function of the
i
a-component, a , announced by agent i 1 1 for agent i, we can still write it as a
i 11
function of m without loss of generality. Define the direct cost share for consumption of public goods for the ith agent by
b m 5 b 4
i i 11,i
K
Define a feasible correspondence B: M → →
R by
1 n
K
Bm 5 hy [ R :C y
O
w , and w 2 a m ? y 2 b mC y 0, ;i [ N j
5
1 i
i i
i i 51
which is clearly non-empty, compact, and convex [by the convexity of C ? ] for all m [ M. Notice that 0 [ Bm for all m [ M. We will show the following lemma in
Appendix A.
Lemma 1. B ? is continuous on M.
n K
˜ Let y 5
o y . Define the outcome function for public goods Y: M
→ R by
i 51 i
1
˜ Ym 5
hy:min iy 2 yij
6
y [Bm
˜ which is the closest point to y. Then Ym is single-valued and continuous on M.
Define a preliminary private good consumption x : M →
R by
i 1
w
i
]]]]]]]] x m 5
2 a m ? Ym 2 b mCYm 7
i i
i
1 1 ia 2 a
i 1 ib 2 b i
i i 11
i i 11
4
When goods are physical goods, this requirement can be guaranteed by asking agents to exhibit their reported endowments to the designer.
G . Tian Mathematical Social Sciences 40 2000 175 –189
181
for each i [ N. Define a shrinking index correspondence A: M
→ → R by
1 n
n
Am 5 hg [ R :gg 1, ;i [ N, and g
O
g x m 1 CYm
O
w j
8
1 i
i i i
i 51 i 51
which is clearly a continuous correspondence with non-empty, compact and convex values.
¯ ¯
¯ Let gm be the largest element of A, i.e. gm [ Am, gm g for all g [ Am.
¯ Thus, g ? is continuous on M.
Finally, define the outcome function for private good consumption Xm: M →
R by
1
¯ X m 5 gmg x m
9
i i i
which is agent i’s consumption resulting from the strategic configuration m. It may be ¯
¯ remarked that, because gmg 1 and gmg
→ 1 as g
→ `, X m x m and
i i
i i
i
X m →
x m as g →
`.
i i
i
Thus the outcome function is continuous and also feasible on M since, by the
n 1K
construction of Bm, Xm,Ym [ R
and
1 n
n n
˚
O
X m 1 CYm
O
w
O
w 10
i i
i i 51
i 51 i 51
for all m [ M. Note that the last inequality comes from the assumption that agents cannot overstate their endowments.
n 1K
Denote h:M →
R the outcome function, or more explicitly, h m 5 X m,Ym.
1 i
i
Then the mechanism consists of kM,hl which is defined on E.
Remark 4. Note that the mechanism constructed above is a destruction mechanism
. That is
, the unreported endowments are destroyed but not consumed. One can also construct a withholding mechanism by using the techniques similar to those in Tian
1993. A message m 5 m , . . . ,m [ M is said to be a Nash equilibrium of the
1 n
mechanism kM,hl for an economy e if, for each i [ N and m [ M , it is not true that
i i
h m ,m P h m
11
i i
2i i
i
where m ,m 5 m , . . . ,m
,m ,m , . . . ,m . hm is then called a Nash
equilib-
i 2i
1 i 21
i i 11
n
rium allocation of the mechanism for the economy e. Denote by N
e the set of all
M,h
such Nash equilibrium allocations. The mechanism
kM,hl is said to Nash-implement Linear Cost Share Equilibrium allocations LCSE on E, if, for all e [ E, N
e 5 LCSEe.
M,h
A message m 5 m , . . . ,m [ M is said to be a strong Nash equilibrium of the
1 n
mechanism kM,hl for an economy e if there does not exist any coalition S and
m [
P
M such that for all i [ S,
S i
i [S
h m ,m P h m
12
i S
2S i
i
182 G
. Tian Mathematical Social Sciences 40 2000 175 –189
hm is then called a strong Nash equilibrium allocation of the mechanism for the
economy e. Denote by SN e the set of all such strong Nash equilibrium allocations.
M,h
The mechanism kM,hl is said to doubly implement the Linear Cost Share Equilibrium
allocations on E, if, for all e [ E, SN e 5 N
e 5 LCSEe.
M,h M,h
Remark 5. Note that our mechanism works not only for three or more agents, but also for a two-agent world. While most mechanisms which implement market-type social
choice correspondences such as Walrasian, Lindahl, Ratio, or LCSE allocations in the existing literature need to distinguish the case of two agents from that of three or more
agents, this paper gives a unified mechanism which is independent of the number of agents.
4. Double implementation
