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
The remainder of this paper is devoted to proving the following theorem.
Theorem 1. For the class of public goods economies specified by E, the above feasible and continuous mechanism doubly implements the LCSE correspondence in Nash and
strong Nash equilibria on E.
Proof. The proof of Theorem 1 consists of the following three propositions which show the equivalence among Nash allocations, strong Nash allocations, and LCSE allocations.
Proposition 1 below proves that every Nash allocation is an LCSE allocation. Proposition 2 below proves that every LCSE allocation is a Nash allocation. Proposition
3 below proves that every Nash equilibrium allocation is a strong Nash equilibrium allocation.
To show these propositions, we first prove the following lemmas.
Lemma 2. Suppose x m,Ym P x , y for i [ N. Then agent i can choose a very
i i
i
large g such that X m,Ym P x , y.
i i
i i
¯
Proof. If agent i declares a large enough g , then gm becomes very small since
i n
¯ gmg 1 and thus almost nullifies the effect of other agents in g
o g x m 1
i i 51 i i
n
¯ CYm
o w . Thus, X m 5 gmg x m can arbitrarily approach x m as agent i
i 51 i
i i i
i
wishes. From the hypothesis that x m,Ym P x , y and continuity of preferences, we
i i
i
have X m,Ym P x , y if agent i chooses a very large g . h
i i
i i
Lemma 3. If Xm ,Ym [ N e, then X m [
R for all i [ N.
M,h i
11
Proof. Suppose, by way of contradiction, that X m 5 0 for some i [ N. Let
i
1 ]]]]]]]]]
x 5 w .
i i
1 1 ia 2 a
i 1 ib 2 b i
i i 11
i i 11
G . Tian Mathematical Social Sciences 40 2000 175 –189
183
Then x ,0 P X m ,Ym by indispensability of the private good and w . 0. Thus
i i
i i
n
if agent i chooses y 5 2 o
y , and keeps other components of the message
i j ±i
j
unchanged, then
0 [ Bm ,m .
Thus x m ,m
,Ym ,m 5 x ,0
so that
i 2i
i i
2i i
2i i
x m ,m ,Ym ,m
P X m ,Ym . Then, by Lemma 2, X m ,m ,Ym ,m
i i
2i i
2i i
i i
i 2i
i 2i
P X m ,Ym if agent i chooses a very large g . This contradicts the hypothesis that
i i
i
Xm ,Ym [ N e and thus we must have X m [
R for all i [ N.
h
M,h i
11
Lemma 4. If m is a Nash equilibrium, then a 5 a 5 ? ? ? 5 a
and b 5 b 5
1 2
n 1
2
. . . 5 b . Therefore o
a m 5 0 and o
b m 5 1.
n i [N
i i [N
i
Proof. Suppose, by way of contradiction, that a ± a and or b ± b
for some
i i 11
i i 11
i [ N. Then 1
]]]]]]]]] a m ? Ym 1 b mCYm
w , w .
i i
i i
1 1 ia 2 a
i 1 ib 2 b i
i i 11
i i 11
Let x 5 w 2 a m ? Ym 2 b m CYm .
Then x . X m ,
and thus
i i
i i
i i
x ,Ym P X m ,Ym by monotonicity of preferences. Thus, if agent i chooses
i i
i
a 5 a , b 5 b
, and keeps other components of the message unchanged, we have
i i 11
i i 11
Ym [ Bm ,m and x m ,m
5 x . Hence, x m ,m ,Ym ,m
5 x ,Ym so
i 2i
i i
2i i
i i
2i i
2i i
that x m ,m
,Ym ,m P X m ,Ym .
Then, by
Lemma 2,
i i
2i i
2i i
i
X m ,m ,Ym ,m
P X m ,Ym if agent i chooses a very large g . This
i i
2i i
2i i
i i
contradicts the hypothesis that Xm ,Ym [ N e. Thus we must have a 5 a 5
M,h 1
2
. . . 5 a and b 5 b 5 . . . 5 b , and therefore
o a m 5 0 and
o b m 5
n 1
2 n
i [N i
i [N i
1. h
˚ Lemma 5. If m
is a Nash equilibrium, then w 5 w and consequently X m 1
i i
i
˚ a m ? Ym 1 b m CYm 5 w for all i [ N.
i i
i
˚ Proof. Suppose, by way of contradiction, that w ± w for some i [ N. Then X m 1
i i
i
˚ ˚
a m ? Ym 1 b m CYm w , w . Let
x 5 w 2 a m ? Ym 2
i i
i i
i i
i
b m CYm . Then we have x . X m and thus x ,Ym P X m ,Ym by
i i
i i
i i
n
˚ monotonicity of preferences. Thus if agent i chooses w 5 w, y 5 Ym 2
o y , and
i i
j ±i j
keeps other components of the message unchanged, then x m ,m ,Ym ,m
5
i i
2i i
2i
x ,Ym so that x m ,m ,Ym ,m
P X m ,Ym . Therefore, by Lemma 2,
i i
i 2i
i 2i
i i
X m ,m ,Ym ,m
P X m ,Ym if agent i chooses a very large g . This
i i
2i i
2i i
i i
˚ contradicts the hypothesis that Xm ,Ym [ N
e. So we must have w 5 w .
M,h i
i
Thus, by
the definition
of X m,
we have
X m 5 w 2 a m ? Ym 2
i i
i i
˚ b m CYm 5 w 2 a m ? Ym 2 b m CYm
and therefore
X m 1
i i
i i
i
˚ a m ? Ym 1 b m CYm 5 w for all i [ N.
h
i i
i
¯ Lemma 6. If Xm ,Ym [ N
e, then gm g 5 1 for all i [ N and thus
M,h i
Xm 5 xm . ¯
Proof. This is a consequence of Lemma 5. Suppose gm g , 1 for some i [ N. Then
i
184 G
. Tian Mathematical Social Sciences 40 2000 175 –189
¯ X m 5 gm g x m , x m , and therefore X m 1 q m ? Ym , x m 1
i i
i i
i i
i
˚ q m ? Ym w . But this is impossible by Lemma 5.
h
i i
Proposition 1. If the mechanism defined above has a Nash equilibrium m , then the
Nash allocation Xm ,Ym is a Linear Cost Share Equilibrium allocation with a m , . . . ,a m and b m , . . . ,b m as the parameters of the linear cost
1 n
1 n
share system, i.e. N e 7 LCSEe for all e [ E.
M,h
Proof. Let m be a Nash equilibrium. We need to prove that Xm ,Ym is an LCSE
allocation with a m , . . . ,a m and b m , . . . ,b m as the parameters of the
1 n
1 n
n
linear cost share system. Note that the mechanism is feasible, o
a m 5 0, and
i 51 i
n
˚
o b m 5 1 as well as X m 1 a m ? Ym 1 b m CYm 5 w for all i [ N
i 51 i
i i
i i
by Lemmas 4 and 5. So we only need to show that each individual is maximizing
11K
his her preferences. Suppose, by way of contradiction, that there is some x , y [ R
i 1
˚ such that x , y P X m ,Ym and x 1 a m ? y 1 b m C y w . Let:
i i
i i
i i
i
x 5 lx 1 1 2 lX m
li i
i
y 5 ly 1 1 2 lYm
l
Then by the convexity of preferences we have x , y P X m ,Ym for any
li l
i i
11K
˚
0 , l , 1. Also x , y [ R
and x 1 a m ? y 1 b m C y w by convexity
li l
1 li
i l
i l
i
of the cost function and non-negativity of b m . Now suppose that player i chooses
i n
y 5 y 2 o
y , and keeps w , a , b , and g unchanged. Since w 2 a m ?
i l
j ±i j
i i
i i
j j
Ym 2 b m CYm . 0 for all j [ N by noting the fact that Xm . 0, by the
j
continuity of the cost function and outcome functions, we have w 2 a m ,m ? y 2
j j
i 2i
l
b m ,m C y . 0 for all j [ N as l is sufficiently small. Hence y [ Bm ,m
and
j i
2i l
l i
2i
therefore Ym ,m 5 y
as well as, by Lemma 6 and the convexity of C ? ,
i 2i
l
˚ ˚
X m ,m 5 x m ,m 5 w 2 a m ? Ym ,m 2 b m CYm ,m 5 w 2
i 2i
i i
2i i
i i
2i i
i 2i
i i
˚ ˚
a m ? y 2 b m C y l[w 2 a m ? y 2 b m C y] 1 1 2 l[w 2 a m ?
i l
i l
i i
i i
i
Ym 2 b m CYm ] 5
lx 1 1 2 lX m 5
x . From
x , y P
i i
i il
il l
i
X m ,Ym , we have:
i
X m ,m ,Ym ,m P X m ,Ym
i 2i
i 2i
i i
i
This contradicts the hypothesis that Xm ,Ym [ N e.
h
M,h
Proposition 2. If x , y is a LCSE allocation with a , . . . ,a and b , . . . ,b as the
1 n
1 n
parameters of the linear cost share system, then there is a Nash equilibrium m such
that X m 5 x , a m 5 a , and b m 5 b , for all i [ N, Ym 5 y , i.e.
i i
i i
i i
LCSEe 7 N e for all e [ E.
M,h n
Proof. We first note that x [ R
by the assumption that the private good is
11
indispensable. We need to show that there is a message m such that x , y is a Nash
˚ equilibrium allocation. Let a 5 a , . . . ,a , b 5 b , . . . ,b , w 5 w , y 5 y n,
i 1
n i
1 n
i i
i
G . Tian Mathematical Social Sciences 40 2000 175 –189
185
and g 5 1 for all i [ N. Then, a m 5 a , b m 5 b , Ym 5 y , and X m 5
i i
i i
i i
x , for all i [ N. Notice that agent i cannot change a m and b m by changing m .
i i
i i
Then, a m ,m ,b m ,m
5 a m ,b m for all m [ M . Also, Xm ,m ,
i i
2i i
i 2i
i i
i i
i 2i
11K
˚
Ym ,m [
R and
X m ,m 1 a m ? Ym ,m
1 b m CYm ,m w
i 2i
1 i
i 2i
i i
2i i
i 2i
i
for all i [ N and m [ M . Therefore, we know that it is not true that:
i i
X m ,m ,Ym ,m
P X m ,Ym
i i
2i i
2i i
i
for otherwise it contradicts the fact that X m ,Ym is a LCSE allocation. h
i
Proposition 3. Every Nash equilibrium m of the mechanism defined above is a strong
Nash equilibrium, that is N e 7 SN
e for all e [ E.
M,h M,h
Proof. Let m be a Nash equilibrium. By Proposition 1, we know that Xm ,Ym is
a Linear
Cost Share
Equilibrium allocation
with a m , . . . ,a m
and
1 n
b m , . . . ,b m as the parameters of the linear cost share system. Then
1 n
Xm ,Ym is Pareto optimal and thus the coalition N cannot be improved upon by any m [ M. Now for any coalition S with 5 ± S ± N, choose i [ S such that i 1 1 [
⁄ S.
Then no strategy played by S can change the budget set of i since a m and b m are
i i
determined by a and b
, respectively. Furthermore, because Xm ,Ym [
i 11,i i 11,i
LCSEe, it is P -maximal in the budget set of i, and thus S cannot improve upon
i
Xm ,Ym . h
Since every strong Nash equilibrium is clearly a Nash equilibrium, by combining Propositions 1–3, we know that N
e 5 LCSEe for all e [ E and thus the proof of
M,h
Theorem 1 is completed. h
5. Concluding remarks
