Double implementation of the Lindahl equilibrium by a continuous mechanism

Economic Des ign

ELSEVIER Economic Design 2 (1996) 311-324

Double implementation of the Lindahl �9 equilibrium by a continuous mechanism

B e z a l e l P e l e g *

University of Bielefeld, Institute of Mathematical Economics, P.O. Box 100131 33501 Bielefeld, Germany

Received 1 August 1995; revised I September 1996

Abstract

We consider a class of economies with public goods that have the following properties: (i) The preferences of the agents are convex, interior, and strictly increasing. (ii) The technology for production of public goods is a closed convex cone that satisfies free disposal and an additional mild assumption. No assumptions are made on continuity, completeness or transitivity of preferences. We provide a continuous and feasible mechanism that implements the Lindahl equilibrium by Nash equilibria, and has the following property: For every economy in our class every Nash equilibrium of the game induced by the mechanism is a strong Nash equilibrium.

JEL classification: J41

Keywords: Lindahl equilibrium; Implementation; Nash and strong equilibrium

1. Introduction

The problem of implementing the Lindahl correspondence by Nash equil ibria has been investigated by economists since the publication of Hurwicz (1979a). However, some earlier work must be mentioned at this point. Samuelson (1954, 1955), conjectured the impossibil i ty of the implementation of the Lindahl solution by (incentive compatible) direct mechanisms. Following the ideas of the funda-

* Tel.: (+49) 521-1065647; fax: (+49) 521-1062997; e-mail: [email protected].

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312 B. Peleg / Economic Design 2 (1996) 311-324

mental paper on implementation theory - Hurwicz (I972) - Ledyard and Roberts (1974) proved Samuelson's conjecture (see also Roberts, 1976). Also, the important works of Groves and Ledyard (1977, 1980), paved the way for a solution to the free rider problem.

With regard to our initial problem we found the following (necessarily partial) list of contributions: Hurwicz (1979a,b,1981, 1986a), Hurwicz et al. (1984), Walker (1981), Tian (1989, 1990, 1991), and Tian and Li (1991). For recent surveys of the theory of implementation see Hurwicz (1986b), Groves and Ledyard (1987), and Moore (1992). We shall compare our work with the recent work of Tian after the following review of our results.

We consider public goods economies with the following properties: (i) The initial bundles (of private goods) of the agents are semipositive, and the total supply (of private goods) is strictly positive (see Assumption (2.1)). (ii) The preferences are strictly monotonic, convex, and interior (see Assumptions (2.2)- (2.5)). (iii) The technology of production of public goods from private goods is a closed convex cone that satisfies free disposal and an additional mild property (see Assumptions (2.6)-(2.8)).

Let now 8 ~ be the class of all public goods economies that have at least one Lindahl equilibrium (see also Remark 5.7). We implement the Lindahl equilibrium on g~ by the following game form: F = (S 1 . . . . . Sn;(g2("), g l ( ' ) , f ( ' ) . . . . . f ( ' ) ) ) . Here S i, the strategy space of i, i = 1 . . . . . n, is the set of all price-allocation pairs. The outcome function g = (g2( ' ) , gl( ' ) , f ( ' ) . . . . . f ( . )) is defined in three stages. In Stage 1 the (strictly positive) price system chosen by i, i = 1 . . . . . n, is transformed by f ( . ) into a (strictly positive) price system that is a system of efficiency prices for the production cone (i.e., it yields a member of the dual cone of the production cone). In Stage 2 the agents are ordered on a directed circle. The efficiency prices of player i, i = 1 . . . . . n, now determine the budget set of the next player on the (directed) circle. This allows us to determine the set of all public goods bundles that may be produced subject to the technological and budgetary restrictions. The final bundle of public goods is the feasible bundle which is closest to the announced aggregate bundle of public goods. This completes the description of Stage 2 (and implicitly defines gl(")).

In Stage 3 the (residual) budget sets (for private goods) of the players are given. The final allocation of private goods is the feasible allocation which is closest to the (announced) aggregate allocation. This defines g2( ' ) and completes the description of our game form.

Our main result is that the foregoing game form provides a double implementation of the Lindahl correspondence in the following sense. (i) F is a Nash-implementation of the Lindahl equilibrium on ~; and (ii) for each public goods economy in 8' every Nash equilibrium of the induced game is a strong Nash equilibrium.

There are two important reasons for preferring double implementation (of the Lindahl equilibrium), over Nash implementations. First, obviously, a double

B. Peleg / Economic Design 2 (1996) 311-324 3 1 3

implementation is immune against deviations by coalitions; therefore, it is more stable than a Nash implementation. Secondly, the Lindahl correspondence is a sub-correspondence of the core; thus, it is coalitionally stable. Hence, it seems natural to require that implementations of the Lindahl correspondence also should respect coalitional stability.

We now compare our result with Tian (1991). Both papers assume nontotal and nontransitive preferences. All other assumptions on preferences also are similar. Tian has just one private good, whereas we allow any (finite) number of private goods. Therefore, he can construct a mechanism which is completely feasible and of minimal dimension, while our game form is only feasible. However, we implement simultaneously in Nash and strong Nash equilibria, whereas he consid- ers only Nash implementation.

We conclude this section with a discussion of an additional related reference. An Associate Editor of this journal sent me the paper Corchon and Wilkie (1990, last revision May 1996), which provides a double implementation of the ratio equilibrium correspondence in Nash and strong equilibria. The ratio equilibrium was proposed in Kaneko (1977) as an extension of the Lindahl solution to public goods economies whose production set may not satisfy additivity. The mechanism of Corchon and Wilkie (1990) is feasible and continuous. As pointed out by the referee of this paper, the main difference between the two papers is in the assumptions on the production sets: Corchon and Wilkie assume, as Kaneko, non-increasing returns to scale and separability (of the cost function), whereas we assume non-increasing returns to scale and additivity. Furthermore, Corchon and Wilkie (1990) assume that the preferences are represented by utility functions and that there is only one private good.

2. The model

2.1. Economies with public goods

There are n agents, n > 2, who consume L private goods and K public goods. If (x, y) ~ R L++ x then x ~ R z+ is the vector of private goods, and y ~ R +K denotes the vector of public goods. Each agent is characterized by her initial endowment wg~RL+ and her strict preference relation >-g on her consumption set R L+x + �9

There are no initial endowments of public goods. Public goods can be produced from private goods by a technology YcR~• Thus, an economy is a list E = (w 1 . . . . . wn; >" 1 . . . . . >- ,; Y) with the foregoing specifications. We only shall consider economies that satisfy the following conditions:

" g (i.e., > 0 for j--- 1, ,L). wi=/=O,i=l . . . . . n, andw=~g=lW >>0 wj . . .

(2.1)


>'i is

>'i is

and

and

>- i i s

and

then for e v e r y O < ) t < 1, h ( x , y ) + ( 1 - h ) ( x * , y * ) >-

irreflexive. (2.2)

strictly increasing, that is, if ( x, y ) , ( x * y* ) E R L+r

( x , y ) > (x*, y*) ( i . e . , ( x , y ) > ( x * , y * )

( x , y) * ( x * , y * ) ) , then (x , y) >-i(x*, y*). (2 .3)

convex, that is, if ( x, y ) , ( x* , y* ) ~ ..+RL+~C

( x , y ) > - i ( x * , y * ) ,

i ( x * , y * ) . (2.4)

>" i is interior, that is, if ( x, y) >> 0 and ( x *, y * ) ~ OR L+K,+

then ( x , y) >'i ( x * , y * ) (here a~'c+tc v..+ is the boundary of RL+x).

(2.5)

Assumptions (2.1)-(2.5) are necessary for the implementation of the Lindahl Equilibrium (see, e.g., Tian, 1991, p. 251). The reader also should notice that we do not assume any of the following properties: continuity, completeness, or transitivity of >-i, i ~ N (here and in the sequel, N = {l . . . . . n} is the set of agents).

We now specify our assumptions on the technology Y.

Y is a closed convex cone and 0 ~ Y. (2.6)

Free disposal: If ( r, y) ~ Y, r' < r, and y' < y, y' ~ R+ x, then ( r ' , y ' ) ~ Y.

(2.7)

Strict monotonicity: If ( r, y) ~ Y and r' < r, then 3 y' >> y

such that ( r ' , y ' ) ~ Y. (2.8)

Assumptions (2.6) and (2.7) are standard (see, e.g., Tian, 1989). Only (2.8) is new. However, we find that it is necessary for our proof of the continuity of the outcome function of our game form.

2.2. Lindahl equilibria

Let E = (w j . . . . . wn; >-~ . . . . . > -n ;Y) be an economy. An allocation (x I . . . . . xn; y) ~R"+ L+x is feasible if ( - w + ~,n=lx i , y) ~ Y. A Lindahl equilibrium (LE) is a list (xX., . . , ,x . ,n" y . , p . , , q 1 . . . . . . q,. ) that satisfies:

(xl , . . . . . x~. ; y . ) is a feasible allocation. (2 .9)

p. ~RL+,qi. ~R~+ a n d p . - x i . +qi . . y . = p , . w i f o r e a c h i ~ N . (2.10)

B. Peleg / Economic Design 2 (1996) 311-324 3 1 5

t

(If a, b ~ R ' then a - b = ~ ahbh.) h=l

For every i ~ N and ( x, y) L+ K i ~ R + : I f ( x , y ) > - i ( x , , y . )

then p , "x +q' , . y > p . .x'. +q ' , . y . .

p , . r + y " qi. <_p, . - w + Exi, i = I i = 1

(2.11)

n

+ y, " }7~q i, f o r a l l ( r , y ) ~ Y . i = 1

We remark that, by (2.10), (2.12) is equivalent to

p , . r + y " ~ q i , _< 0 for all ( r , y ) ~ Y. i=l

(2.12)

(2.13)

3. The game form

Let ~ be the class of economies that satisfy (2.1)-(2.8) and that, in addition, have at least one LE. We shall implement the LE correspondence on ~ by a game form F = (S 1 . . . . . Sn; g; ~ ) , where S 1 . . . . . S" are the strategy sets of the agents, g is the outcome function of F , and ~ is the outcome space of g.

We now specify the components of F. S i = R nL • R K • A t • A~f for i ~ N, L _ 0]q ~ RK}. where A + - { p > > O I p ~ R L } , and x _ A+-- (q >>

Thus, if si ~ S i then si = ( xi, yi, pi, q i) where ( xi, y i ) ~ RnL • R x is an allocation (which may not be feasible), p'~ is a strictly positive price vector of private goods, and qi is a complete list of positive personalized prices of public goods. (More precisely, xi is a vector of proposed net trades of i with each j ~ N, and y; is a proposed change in the supply of public goods.) Thus, S i is the set of price-al location pairs for any E ~ 8'.

Our outcome space is 1"2 nL R+Kx(AL+x , x ~ = R + • A+ ) . L e t E = ( w I . . . . . w ; >-1 . . . . ,>-~;Y> ~ . The outcome function g ( . ; E ) : S ~ , where S = Xi=lS, , i is defined in the following way. (Because E is fixed, we will write g( . ; E) = g( . ) . ) Let s -- ( s 1 . . . . . s ") ~ S where s i = ( x i, yi, pi, qi) for i ~ N. We distinguish the following steps in the definition of g(s) . ~

Step 1. For each i ~ N we transform the price system ( p i , q i ) into a price system (/3 i, c~ i) which is a system of efficiency prices for Y, that ~/s, (/3 i, El= 1q j )

Y *, where Y * is the dual cone of Y. First, we recall that

Y* = {z ~RL+K{z" ( r , y) < 0 for all ( r , y) ~ Y}. (3.1)

Because E has an LE, Y* contains a strictly positive vector (see (2.3), (2.6),


and (2.13)). Let Y= Y* n RL+ +K. Then Y has an interior point (fi~). Now consider the set Y c RL+ • R+ K defined by

f = ( p , ql . . ,q~)lp~Rm+,qi~R~C i = 1 , . ,nand p,~_,qi ~ ~ . + ~ . ~ o

\ i = l I

( 3 . 2 )

The reader may easily verify that I 7 is a closed and convex cone which contains an interior point (/~, ~1 . . . . . ~") = (/3, @ where ?/= E~'=I~ i (here ~ =

. . . . .

Now, for each point (a, b) = (a, bi,. , b") in RL+• R "K let f ( (a , b)) be the point closest to (a, b) in the interval [(a, b), (/3, @] N Y. The following claim will be proved in the Appendix.

Lemma 3.1. f is continuous.

Clearly, if (a , b} >> 0 then f ( (a , b)) >> 0, Also, notice that for every (a , b) RL+• x, f ( < a , b > ) : ( a , b ) if and only if <a,b>~Y. Now let </3i,~i>= f ( ( pi, qi)) for every i ~ N. This completes Step 1.

Step 2. In this step we will determine the bundle of public goods y(s) that will be produced when s is played. Using the notations of Step 1 let ( i i, ~i)-~ < ^ i - I ^ i - I P , qi ), i = 2 . . . . . n, and ( ill, ~t) = (/3", ~ ' ) . Define the correspondence /3: S ~ ~ R+ x ( ~ ~ denotes a correspondence, i.e., a set-valued function), by

f l ( s ) = { y ~ R X l i i ' w i > ~ i ' y , i = l . . . . . n, a n d ( - w , y ) ~ Y } . (3.3)

/3(s) is simply the set of vectors of public goods that can be produced subject to the budget constraints imposed by the prices ( i 1, ~l) . . . . . ( i " , ~n). Clearly, the budget set of i is determined by the relevant components of the price system announced by i's auctioneer (that is, i - 1 if i > 1, and n if i = 1), after it was transformed into a system of efficiency prices of Y.

/3(s) r g (0 ~/3(s)). Also, the reader may easily verify that /3(s) is convex and /3(-) is upper hemi-continuous (recall that f is continuous). The continuity of /3(-) follows from Lemma 3.2.

Lemma 3.2. /3(.) is lower hemi-continuous.

We shall prove Lemma 3.2 in the Appendix. Now, the production of public goods is given by the function g~: S ~ R +K where

gL(s) = argmin{[l y -- ~ = , yqIIy ~ f l(s)}. (3.4)

By the Maximum Theorem gt is upper hemi-continuous. Because g~ is single- valued, it is actually continuous.

B. Peleg / Econornic Design 2 (1996) 311-324 3 1 7

Step 3. In this step we shall determine the bundles of private goods that will be consumed when s is played. For i ~ N let

a i ( s ) --'I ( X E RL[ ffi X "~ ~i . g l ( S) ~_~ ffi . w i ) , ( 3 . 5 )

Bi(s) is i ' s budget set for private goods under the assumptions that s is played and g l ( s ) is the bundle of public goods which is produced. Bi(s) is non-empty (0 ~ Bi(s), closed, and convex. Because gl and f are continuous Bi(.) is upper hemi-continuous. Denote B(s)= " i _ >( i f 1 n (s) and

C( s) = {x~R"+Ll(--w + E i= l s t , gl( s) ) ~ Y} tqB(s). (3.6)

Then 0 ~ C(s) and C(s) is closed and convex. Also, C(- ) is upper hemi-continuous because g l is continuous and B(s ) is upper hemi-continuous. The following claim will be proved in the Appendix.

Lemma 3.3. C(.) is lower hemi-continuous.

Now let 2h = )2~=Lx~, h = 1 . . . . . n (recall that s i= (x i, yi, pi, qi)where X i =

i . . , x / ) for i = 1 . . . . . n). We define the allocation of private-goods (when s X �9 ~

is played), by

gz(s) = argmin {llx - -~lll x ~ C ( s ) } . (3.7)

Again, by the Maximum Theorem g z ( ' ) is continuous on S. This completes Step 3.

Now we may define the outcome function g: S ~ 12 by

g ( s ) = ( g 2 ( s ) ; g ~(s) ; f ( ( p~, q ' } ) . . . . . f ( ( pn, q n ) ) ) . (3.8)

By the foregoing analysis g ( . ) is continuous. This concludes our definition of the game form.

Remark 3.4. In the foregoing construction it is possible to replace the cycle (1 . . . . . n) by any cyclic permutation of N.

4. Double implementation of the Lindahl correspondence

Let E = (w ~ . . . . . wn; >-~ . . . . . > -n ;Y> ~ " . For i > 2 let t ( i ) = i - 1 and let t(1) = n. With each i ~ N we associate a strict preference relation >- * ; on S by the following rule. Let s = ( s 1 . . . . . s n) where s i = ( x i, y~,pi, q~}, i ~ N , and


. g ( x i , i s , = ( s l . . . . . . s . ) where s . = , y . , p ' , , q ' . ) , i ~ N , be points in S. Then S > " i S , if:

(i) (g~(s) , g , (s ) ) >- g(g~(s, ), g~(s, )), or (ii) [l( pi, qg) _ (ptr q'~g))ll < I1( pi., qi. ) _ (pt.(i), q~%ll

(here qJ = (q{ . . . . . q~) and q{ = (q{ l . . . . . q { , ) for j ~ N, and I1" II is the Eu- clideaff norm). Now we may ~tefine the (ordinal) game

G = ( S ' , . . . , S " ; g ( . ; E); O; ~- ? . . . . . >- ~, ) (4.1)

which naturally corresponds to E.

Theorem 4.1. Let ( x I . . . . . x"; y; p; q I . . . . . q") be an LE of E. Define s i= ( x i , yi, pi, q i) ~ S i, i ~ N , by

i-I x i = ( O . . . . . O, xi, O . . . . . 0) , y i = y / n , p i = p , a n d q j = q j , j = 1 . . . . . n.

Then s = ( s I . . . . . s ~) is a Nash equilibrium of G (see (4.1)), and g ( s ) = ( x I . . . . . x"; y; p, q . . . . . p, q) where q = (q l . . . . . q , ) .

Proof First we notice that p >> 0 and qJ >> 0, j ~ N, because >- g is strictly increasing for i ~ N (see (2.3)). Hence s g ~ S ~ for all i ~ N. The computation of g ( s ) is straightforward. Thus, it only remains to show that s is a Nash equilibrium. Let i ~ N. Without loss of generality i = n. The budget set of n is

B " ( p , q ) = {(x ' , y') ~ R L + + X l p ' x ' + q " "y' < p " wn}.

Because B"( - ) is determined by n - 1, n cannot unilaterally change her budget set. Also, because ( x " , y ) is maximal with respect to >- n in B"(p , q), n has no profitable deviation from ( x n, y" ) . Finally, by the definition of >--,~ n cannot increase her payoff by deviating for the LE prices ( p , ql . . . . . qn). Q.E.D.

By Theorem 4.1 every LE of E is an outcome of a Nash equilibrium of G. Theorem 4.2 proves the converse result, that is, the outcome of a Nash equilibrium of G i s a n L E o f E.

Theorem 4.2. I f s = ( s I . . . . . s n) is a Nash equilibrium of G, where S i =

(X i, yi, pi, qi) , i ~ N, then the following conditions are satisfied: (i) There-exist p . ~ AL+ and q{ ~ A K +, j = 1 . . . . . n, such that ( p i , ql . . . . . q i )

= ( p . , ql. . . . . . q", ) for all i ~ N . (ii) ( g2(s); gl(s) ; f ( ( p , , q~. . . . . . q7 ) ) ) is an LE of E.

Proof First we prove the following claim.

Claim 1. gi(s) >> 0 for i = 1,2.

B. Peleg / Economic Design 2 (1996) 311-324 319

Proof of Claim 1. Let g2(s) = (21 . . . . . 2") and g l ( s )=~ . Assume, on the contrary, that there exists i ~ N such that ( 2 ; , ~ ) ~ OR e+*: Choose 2>> 0 and + -

9 >> 0 sufficiently small such that (i) fii .2 i + ~1 i .-9 < i . wi; (ii) ~J .~ <f i J . w i, j ~ N\{i}; and (iii) ( - w + 2 i, ~ ~ Y). (i) and (ii) are possible because ~ - w ~ > 0 for all j ~ N. Also, if - w + 2 i < 0 then there is (a sufficiently small) -9 >> 0 such that ( - w + 2 i, -9) ~ Y (see (2.8)). Now consider the following deviation of i:

i - ! g i = ( 2 i , y ' , p ' , q ' ) where ~ i+F_ . j . i x J= (O . . . . . O ,x ,O . . . . O)

and ~ i + ~ y j =~. j4, i

Clearly, after the deviation i gets the bundle (2 i, -9). By (2.5) (2 i, -9) >- i(2 i, ~). Because s is a Nash equilibrium of G, the desired contradiction has been obtained. Q.E.D.

We proceed with the proof of our second claim.

Claim 2. For each i ~ N (g~(s), gl(s)) is maximal in her budget set

B*(s) = ( (x , y) - - ~ L + K , - ~ . X + 71 i" " ~ ~ + I P Y <~ fi ' " wi]

Proof of Claim 2. Denote, again, g l ( s ) = p and g2(s)= (21 . . . . . 2"). Assume, on the contrary, that there exist i ~ N and (U, -9)~ Bi(s) such that (2 i, -9)>i (2 i, p). For 1 >_ h > 0 denote (x[, yz) = / ( U , ~) + (1 - I )(2 ~, p). Then (x[, y,)

Bi(s) and (xia, ya) >- i(2 i, ~). Also, by Claim 1, 2J >> 0 for j = 1 . . . . . n. Hence, for h > 0 sufficiently small ( - w + x~, Ya) ~ Y (see (2.8)). Furthermore, 2 j >> 0,j

N, also implies g/J - Yx < PJ " wj for ,~ > 0 small enough. Consider now the deviation gi = ( 2i, ~i, pi, qi) of i where

i - 1

2 ~+ ]~xJ (0 . . . . . O, _ = x a , 0 . . . . . 0 ) jq=i

and }i + y,.j, iy j = ya (here A > 0 is sufficiently small). After the deviation i gets i i the bundle (x A, y~). As (x~, YA) >- i( 2~, -~) and s is a Nash equilibrium of G, the

desired contradiction has been obtained. Q.E.D.

Claim 2 has the following corollaries:

Corollary 1. For each i ~ N

fii. g~( s ) + 77 i. gt( s) = fii. w i. (4.2)

Corollary 1 follows from the strict monotonicity of >- ~ (see (2.3)).

Corollary 2. There exis tp , ~ At+ and q~, ~ AK+, i = 1 . . . . . n, such that ( p~, q~) = ( p , , q ' . . . . . . q~.) f o ra l l i ~ N .


Proof o f Corollary 2. Assume, on the contrary, that there exists i ~ N such that (p i , q i ) ~ (pt~i),qt<i)). Let j = t - l ( i ) . For 1 > A > 0 let ( p ~ , q ~ ) = A(ptU), qt(i)) "1- (1 - A)(p i, qi). Because gJ2(s) >> 0 and f ( . ) is continuous, for 1 > A > 0 sufficiently - ^i small q;~;. g~(s) < ~ . w j where f ( ( p], q~)) = (/3~, ~ ) . Denote, again, ~ = gl(s) and 2 = (~1 . . . . . 2 " ) = g2(s). Then gi = (~i , ~i, p~, q~), where

i - I ~ i .,1_ E X h = ( 0 . . . . . 0 , x i , 0 . . . . . 0 ) ,

h~,i

~iq_ ~_~h~,iyh =~ , and 1 > h > 0 is sufficiently small, is a deviation of i which yields him a consumption bundle (2i, ~) = (g~(s), gl(s)). Moreover, I1( pA, q~) - ( pt(i), q,(O)l [ < i[(p~, qi) _ ( p,(i), q,(O)[i. Thus s[g i = ( s 1 . . . . . si-1, gi, si+-l, s " ) >-/'s, 'which is the desired contradfction. Q.E.D.

Proof o f Theorem 4.2. It only remains to prove profit maximization in the production of public goods. Denote, again, g l ( s ) = ~ and g2 ( s )= (~1 . . . . . 2n). By the definition of g ( . ) ( - w + E~= i xi, Y) ~ Y. Furthermore, by Corollaries 1 and 2 ~, . ( E T = t 2 i - w ) + ~ . E7=l~, = 0 where f ( ( p , , q l , . . . . . q , ~ ) ) = ( /3 , , ~l . . . . . ~,~ ). Also, by the definition of f ( . ) , ( /3 , , ET= ~c?i,) ~ P c y* (see

n --i Section 3). Hence, the production plan ( - w + F~i= i x , ~) is profit maximizing. Q.E.D.

The double implementation result is proved by the following theorem.

Theorem 4.3. Every Nash equilibrium of the game G (see (4.1)) is a strong Nash equilibrium.

The proof of Theorem 4.3 is exactly the same as the proof of Theorem 4.1 in Peleg (1996). Nevertheless, for the sake of completeness, we outline the proof. Let s = ( s I . . . . . s n) be a Nash equilibrium of G and let s i = ( x i, yi, pi, qi) , i ~ N. By Theorem 4.2 there exist p . ~ At+ and q~ ~ A + K, j = 1 . . . . . n , - such that ( p i , q~ . . . . . q i ) = ( p , , ql, . . . . . q~ . ) f o r all i ~ N , and (g2(s) ; g l ( s ) ; f ( ( p . , q l . . . . . . q" .))) is an LE of E. Let g2(s) = (21 . . . . . 2n), g l ( s ) = y, and f ( ( p , , q l , , . . . . . . . , q , ~ ) ) = ( ~ , ~-1 , q ~ ) As (21 . . . . . ~ , ; p; fi; ?/i . . . . . ~") is an LE (21 . . . . . Yc n, ~) is Pareto-optimal. Thus N has no improvement upon s. Now let T c N, T v~ Q,N. Choose i ~ T such that t(i) q~ T. Then T cannot change the budget set of i because it is determined by t(i). Furthermore, (2 i, p) is maximal with respect to >- i in i 's budget set. Hence, T cannot (strictly) improve the position of i.


5. Concluding remarks

We provide a game form F with the following two properties: (i) F implements by Nash equilibria the correspondence of Lindahl equilibria (see Theorems 4.1 and 4.2). (ii) For every admissible economy every Nash equilibrium of the game induced by F is a strong Nash equilibrium (see Theorem 4.3). (i) and (ii) imply that F is a double implementation of the Lindahl correspondence. We now make the following remarks on Sections 2, 3, and 4.

Remark 5.1. Our assumptions on the initial bundles and preferences are similar to those of Tian (1991). Both papers consider nontotal-nontransitive preferences that are convex and interior (see (2.5)). The only difference is that we consider strictly increasing preferences, whereas Tian assumes only local nonsatiation of preferences.

Remark 5.2. Our assumptions on the technology that produces public goods are standard except (2.8). However, (2.8) is usually satisfied in case that there is only one private good (see, again, Tian, 1991). In our opinion it is a reasonable assumption that allows us to obtain strong results.

Remark 5.3. As far as we know only Suh (1993) investigates double implementation (by Nash and strong Nash equilibria), of the Lindahl correspondence. However, the outcome function of his mechanism is not continuous.

Remark 5.4. Our game form is feasible but not balanced (that is, out of equilibrium the budget inequality of an agent may be satisfied as a strict inequality). We do no know whether there exists a double implementation (in our sense) of the Lindahl correspondence by a single-valued, continuous, feasible, and balanced mechanism.

Remark 5.5. Some useful ideas of Peleg (1996) are used in this paper. However, this paper is very far from being a corollary of the foregoing paper. The implementation problem for the Lindahl correspondence is much more difficult than its special case of implementing the Walras correspondence (compare, e.g., Tian (1992) and Tian and Li (1991)).

Remark 5.6. The strategy space of an agent according to our mechanism is the set of all price-allocation pairs (for the case of public goods). Although this space is finite-dimensional and 'natural', it is not minimal. We could not find a reference for a minimal implementation of the Lindahl correspondence when rain(L, K) > 2 (we recall that L is the number of private goods and K is the number of public goods).


Remark 5. 7. The assumption that each economy in our domain has at least one LE may be relaxed in the following way. The (weaker) assumption that the technology has at least one efficient input-output pair is enough (it is wel l -known that a (finite-dimensional) convex and closed set that has an efficient point has a supporting hyperplane with a strictly positive normal (see, e.g., Peleg (1972)).

Appendix A. Proofs of the Lemmata of Section 3

Proof of Lemma 3.1. Let (a(t) , b( t ) ) > ~ bZ+ • R "r t = 1, 2, ( a ( t ) , b ( t ) ) ~ ( a , b ) , and let z ( t ) = f ( ( a ( t ) , b ( t ) ) ) . Notice that the sequence (z(1) , z(2) . . . . ) is bounded. Assume now, on the contrary, that there is a subsequence z( t k) ~ z * and z * 4: f ( ( a, b ) ). By definition

z ( t k ) = a ( t k ) ( a ( t k ) , b ( t k ) ) + ( l - - a ( t k ) ) ( f i , q ) , k = 1 , 2 . . . . .

for some a ( t k) ~ [0, 1]. Without loss of generality, a( t k) ~ a. Also, we may assume that (a(t~), b( tk)) ff Y, k = 1, 2 . . . . . Thus, z* = a(a , b ) + (1 - a ) ( / 5 , q ) and z* r Because I ~ is closed, z* ~ I ~. Hence, z* ~ I n t ( Y ) . Tht~s, z(t k) ~ Int ( l ~) for k sufficiently large. However, because (a(t~), b(tk)) f~ Y, we must have z( t k) ~ O I ~, k = 1,2 . . . . . Therefore, the desired contradiction has been obtained. Q.E.D.

Proof of Lemma 3.2. Let s ~ S, y ~ / 3 ( s ) , and s ( t ) --*s, where s = ( s I . . . . . s " ) , s i = ( x i, yi, pi, qi) , i = 1 . . . . . n, si(t) = (x i ( t ) , yi(t) , pi(t), qi( t )) , t = 1,2 . . . . .

i = 1 . . . . . n, a n d " s ( t ) = ( s l ( t ) . . . . . s"( t)) . Because 0 ~/3(s( t-)) , t = 1,2 . . . . . we may assume y4~ 0. Let y(t) be the closest point to y in the interval [0, y] A f l(s(t)) , y ( t ) is well defined because 0 ~ / 3 ( s ( t ) ) . We claim that y( t ) ~ y. Assume, on the contrary, that there exists a subsequence y(tk), k = 1,2 . . . . . such that y( t k) ~ Yo and y r Y0. Then, there exists k 0 such that for every k > k 0 there exists i = i (k) such that ~i(tk). y( t k) = ~i( tk) . w i. (Notice that ( - w, y ' ) ~ Y for every y ' ~ [ 0 , y].) Thus, there is l < i < n such that ~ i . y 0 = f i i . w e > ~ i . y . Because ~i >> 0, y > 0, and Y0 ~ [0, y), the inequality 7/i �9 Y0 > ~ i . y is impossi- ble. Q.E.D.

Proof of Lemma 3.3. Let s ~ S, s = ( s I . . . . . s " ) , where s i = ( x i, yi, pi , q i ) , i = l . . . . . n and let z ~ C(s). Furthermore, let s( t) ~ s , where ~( t ) = ( s l ( t ) . . . . . s"( t ) ) and-s /O) =~(xi(t), yi(t), pi(t), qi( t)) , i = 1 . . . . . n, t = 1,2 . . . . .

We need to consider only the case z > 0 (0 ~ C(s'(t)), t = 1,2 . . . . ). Now let z( t) be the point in [0, z] N C(s(t)) which is closest to z. We claim that z ( t ) ~ z . Indeed, assume on the contrary that there exists a subsequence z ( t k) ~ z . and z , < z . Let ~ = ( _ z + z , ) / 2 . Then z < z . Denote N+={i l z i>O} . Then N + v a Q and for each i ~ N+ ~i . zi, + 7t i " gz(s) < ~i . w i because ,~i >> 0. Because z ( t k)


~ z . and g l and f are con t inuous , there is k* such tha t for k > k* ~i(tk)" zi(tk ) + ~i(tk)" gl (s( tk)) < ~i(tk)" W i for i ~ N+. M o r e o v e r , b y (2.8) , the re exis ts y

R+ r , y >> g l ( s ) , such that ( - w + E7=1~ i, y ) ~ Y. W e m a y a s s u m e w i t h o u t loss o f

genera l i ty t ha t y >> gl(s(tk)) and z( t k) < ~ for k > k * . Hence , ( - w + n i t ~]i~ ~ Z ( k ) , Y) ~ Y" NOW, the fo rego ing inequal i t i es con t r ad ic t the de f in i t ion o f

Z(tk), k > k *. Q.E.D.

References

Corchon, L. and S. Wilkie, 1990, Double implementation of the ratio correspondence by a market mechanism, Economic Design.

Groves, T. and J. Ledyard, 1977, Optimal allocation of public goods: A solution to the 'free rider' problem, Econometrica 45, 783-811.

Groves, T. and J. Ledyard, 1980, The existence of efficient and incentive compatible equilibria with public goods, Econometrica 48, 1487-1506.

Groves, T. and J. Ledyard, 1987, Incentive compatibility since 1972, In: T. Groves, R. Radner and S. Reiter, eds., Information, Incentives, and economic mechanisms: Essays in honor of Leonid Hurwicz (University of Minnesota Press, Minneapolis, MN) 48-111.

Hurwicz, L., 1972, On informationally decentralized systems, In: R. Radner and C.B. McGuire, eds., Decision and organization, Volume in honor of J. Marschak (North-Holland, Amsterdam) 297-336.

Hurwicz, L., 1979a, Outcome functions yielding Walrasian and Lindahl allocations at Nash equilibrium points, Review of Economic Studies 46, 217-225.

Hurwicz, L., 1979b, Balanced outcome functions yielding Walrasian and Lindahl allocations at Nash equilibrium points for two or more agents, In: Jerry R. Green and Jos6 A. Scheinkman, eds., General equilibrium, growth, and trade (Academic Press, New York).

Hurwicz, L., 1981, On incentive problems in the design of non-wasteful resource allocation systems, In: J. Los, ed., Studies in economic theory and practice, Volume in honor of Edward Lipinski (North-Holland, Amsterdam) 93-106.

Hurwicz, L., 1986a, On the implementation of social choice rules m irrational societies, In: W.P. Heller, R.M. Ross and D.A. Starrett, eds., Social choice and public decision making: Essays in Honor of Kenneth J. Arrow, Volume 1 (Cambridge University Press, Cambridge) 75-96.

Hurwicz, L., 1986b, Incentive aspects of decentralization, In: K.J. Arrow and M.D. Intriligator, eds., Handbook of mathematical economics, Vol. 3 (North-Holland, Amsterdam) 1441-1482.

Hurwicz, L., E. Maskin and A. Postlewaite, 1984, Feasible implementation of social choice correspon- dences by Nash equilibria, Mimeo.

Kaneko, M., 1977, The ratio equilibrium and a voting game in a public good economy, Journal of Economic Theory 16, 123-136.

Ledyard, J. and J. Roberts, 1974, On the incentive problem with public goods, Department of Economics, Mimeo. (Northwestern University, Evanston, IL).

Moore, J., 1992, Implementation, contracts, and renegotiation in environments with complete information, In: Jean-Jacques Laffont, ed., Advances in economic theory: Sixth world congress (Cambridge University Press, Cambridge) 182-282.

Peleg, B., 1972, Topological properties of the efficient point set, Proceedings of the American Mathematical Society 35, 531-536.

Peleg, B., 1996, A continuous double implementation of the constrained Walras equilibrium, Economic Design 2, 89-97.

Roberts, J., 1976, The incentives for correct revelation of preferences and the number of consumers, Journal of Public Economics 6, 359-374.


Samuelson, P., 1954, The pure theory of public expenditure, Review of Economics and Statistics 36, 387-389.

Samuelson, P., 1955, Diagrammatic exposition of a theory of public expenditure, Review of Economics and Statistics 37, 350-356.

Suh, S.-C., 1993, Implementation with coalition formation: The case of unknown cooperation possibili- ties, Mimeo.

Tian, G., 1989, Implementation of the Lindahl correspondence by a single-valued, feasible, and continuous mechanism, Review of Economic Studies 56, 613-621.

Tian, G., 1990, Completely feasible and continuous Nash-implementation of the Lindahl correspondence with a message space of minimal dimension, Journal of Economic Theory 51,443-452.

Tian, G., 1991, Implementation of Lindahl allocations with nontotal-nontransitive preferences, Journal of Public Economics 46, 247-259.

Tian, G., 1992, Implementation of the Walrasian correspondence without continuous, convex, and ordered preferences, Social Choice and Welfare 9, 117-130.

Tian, G. and Q. Li, 1991, Completely feasible and continuous implementation of the Lindahl correspondence with any number of goods, Mathematical Social Sciences 21, 67-79.

Walker, M., 1981, A simple incentive compatible scheme for attaining Lindahl allocations, Economet- rica 49, 65-73.

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Double implementation of the Lindahl equilibrium by a continuous mechanism